Index of /~history/Mathematicians

Index of /~history/Mathematicians Name

Last modified

Parent Directory

29-Aug-2001 13:31

-

Abbe.html

10-Dec-2001 16:56

8k

Abel.html

23-Nov-2001 13:51

25k

Abraham.html

23-Nov-2001 13:51

9k

Abraham_Max.html

09-Aug-2001 13:00

9k

Abu'l-Wafa.html

23-Nov-2001 13:51

13k

Abu_Kamil.html

23-Nov-2001 13:51

12k

Ackermann.html

09-Aug-2001 13:00

5k

Adams.html

23-Nov-2001 13:51

9k

Adams_Frank.html

09-Aug-2001 13:00

10k

Adelard.html

10-Jan-2002 15:43

11k

Adler.html

09-Aug-2001 13:00

5k

Adrain.html

23-Nov-2001 13:51

13k

Aepinus.html

23-Nov-2001 13:51

10k

Agnesi.html

23-Nov-2001 13:51

17k

Ahlfors.html

09-Aug-2001 13:00

10k

Ahmed.html

09-Aug-2001 13:00

8k

Ahmes.html

23-Nov-2001 13:51

6k

Aida.html

09-Aug-2001 13:00

9k

Aiken.html

09-Aug-2001 13:00

9k

Airy.html

09-Aug-2001 13:00

21k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (1 of 62) [2/16/2002 10:55:55 PM]

Size

Description

Index of /~history/Mathematicians

Aitken.html

09-Aug-2001 13:00

11k

Ajima.html

09-Aug-2001 13:00

4k

Akhiezer.html

09-Aug-2001 13:00

6k

Al'Battani.html

09-Aug-2001 13:00

1k

Al'Biruni.html

09-Aug-2001 13:00

1k

Al'Haitam.html

09-Aug-2001 13:00

1k

Al'Kashi.html

09-Aug-2001 13:00

1k

Al'Khwarizmi.html

09-Aug-2001 13:00

1k

Al-Baghdadi.html

09-Aug-2001 13:00

12k

Al-Banna.html

09-Aug-2001 13:00

11k

Al-Battani.html

23-Nov-2001 13:51

16k

Al-Biruni.html

23-Nov-2001 13:51

26k

Al-Farisi.html

09-Aug-2001 13:00

13k

Al-Haitam.html

09-Aug-2001 13:00

1k

Al-Haytham.html

28-Nov-2001 16:11

23k

Al-Jawhari.html

09-Aug-2001 13:00

9k

Al-Jayyani.html

09-Aug-2001 13:00

10k

Al-Karaji.html

23-Nov-2001 13:51

17k

Al-Kashi.html

23-Nov-2001 13:51

17k

Al-Khalili.html

09-Aug-2001 13:00

9k

Al-Khazin.html

09-Aug-2001 13:00

13k

Al-Khujandi.html

09-Aug-2001 13:00

10k

Al-Khwarizmi.html

23-Nov-2001 13:51

26k

Al-Kindi.html

09-Aug-2001 13:00

11k

Al-Maghribi.html

09-Aug-2001 13:00

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (2 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Al-Mahani.html

09-Aug-2001 13:00

9k

Al-Nasawi.html

23-Nov-2001 13:51

9k

Al-Nayrizi.html

09-Aug-2001 13:00

8k

Al-Qalasadi.html

09-Aug-2001 13:00

13k

Al-Quhi.html

09-Aug-2001 13:00

14k

Al-Samarqandi.html

09-Aug-2001 13:00

5k

Al-Samawal.html

23-Nov-2001 13:51

17k

Al-Sijzi.html

09-Aug-2001 13:00

9k

Al-Tusi_Nasir.html

23-Nov-2001 13:51

19k

Al-Tusi_Sharaf.html

09-Aug-2001 13:00

12k

Al-Umawi.html

09-Aug-2001 13:00

11k

Al-Uqlidisi.html

23-Nov-2001 13:51

11k

Albanese.html

09-Aug-2001 13:00

6k

Albert.html

09-Aug-2001 13:00

4k

Albert_Abraham.html

23-Nov-2001 13:51

15k

Alberti.html

23-Nov-2001 13:51

5k

Albertus.html

09-Aug-2001 13:00

6k

Alcuin.html

23-Nov-2001 13:51

7k

Aleksandrov.html

23-Nov-2001 13:51

21k

Aleksandrov_Aleksand..> 09-Aug-2001 13:00

12k

Alexander.html

23-Nov-2001 13:51

11k

Alexander_Archie.html

09-Aug-2001 13:00

10k

Ampere.html

23-Nov-2001 13:51

23k

Amringe.html

09-Aug-2001 13:00

6k

Amsler.html

09-Aug-2001 13:00

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (3 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Anaxagoras.html

09-Aug-2001 13:00

14k

Anderson.html

09-Aug-2001 13:00

11k

Andreev.html

09-Aug-2001 13:00

5k

Angeli.html

09-Aug-2001 13:00

7k

Anstice.html

09-Aug-2001 13:00

9k

Anthemius.html

23-Nov-2001 13:51

7k

Antiphon.html

08-Feb-2002 17:10

13k

Apastamba.html

10-Jan-2002 15:43

7k

Apollonius.html

10-Dec-2001 16:56

20k

Appell.html

09-Aug-2001 13:00

14k

Arago.html

28-Aug-2001 16:02

8k

Arbogast.html

09-Aug-2001 13:00

11k

Arbuthnot.html

23-Nov-2001 13:51

7k

Archimedes.html

28-Nov-2001 16:11

30k

Archytas.html

23-Nov-2001 13:51

15k

Arf.html

09-Aug-2001 13:00

14k

Argand.html

23-Nov-2001 13:51

12k

Aristaeus.html

23-Nov-2001 13:51

9k

Aristarchus.html

23-Nov-2001 13:51

16k

Aristotle.html

09-Aug-2001 13:00

31k

Arnauld.html

09-Aug-2001 13:00

6k

Aronhold.html

09-Aug-2001 13:00

7k

Artin.html

10-Jan-2002 15:43

23k

Aryabhata.html

09-Aug-2001 13:00

1k

Aryabhata_I.html

23-Nov-2001 13:51

18k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (4 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Aryabhata_II.html

09-Aug-2001 13:00

6k

Atiyah.html

23-Nov-2001 13:51

13k

Atwood.html

27-Aug-2001 16:15

5k

Auslander.html

08-Oct-2001 14:36

11k

Autolycus.html

23-Nov-2001 13:51

10k

Avicenna.html

23-Nov-2001 13:51

18k

Babbage.html

10-Jan-2002 15:43

25k

Bachet.html

23-Nov-2001 13:51

6k

Bachmann.html

23-Nov-2001 13:51

8k

Backus.html

09-Aug-2001 13:00

7k

Bacon.html

09-Aug-2001 13:00

9k

Baer.html

23-Nov-2001 13:51

10k

Baire.html

09-Aug-2001 13:00

17k

Baker.html

09-Aug-2001 13:00

6k

Baker_Alan.html

23-Nov-2001 13:51

10k

Ball.html

09-Aug-2001 13:00

4k

Balmer.html

10-Jan-2002 15:43

8k

Banach.html

23-Nov-2001 13:51

23k

Banneker.html

09-Aug-2001 13:00

9k

Banu_Musa.html

09-Aug-2001 13:00

13k

Banu_Musa_Ahmad.html

09-Aug-2001 13:00

4k

Banu_Musa_Muhammad.html 09-Aug-2001 13:00

5k

Banu_Musa_al-Hasan.html 09-Aug-2001 13:00

5k

Barbier.html

09-Aug-2001 13:00

8k

Bari.html

09-Aug-2001 13:00

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (5 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Barlow.html

23-Nov-2001 13:51

8k

Barnes.html

09-Aug-2001 13:00

9k

Barocius.html

09-Aug-2001 13:00

5k

Barrow.html

23-Nov-2001 13:51

21k

Bartholin.html

09-Aug-2001 13:00

6k

Batchelor.html

09-Aug-2001 13:00

11k

Bateman.html

10-Jan-2002 15:43

10k

Battaglini.html

09-Aug-2001 13:00

5k

Baudhayana.html

10-Jan-2002 15:43

8k

Bayes.html

23-Nov-2001 13:51

9k

Beaugrand.html

09-Aug-2001 13:00

5k

Bell.html

09-Aug-2001 13:00

7k

Bellavitis.html

19-Sep-2001 10:05

10k

Beltrami.html

23-Nov-2001 13:51

13k

Bendixson.html

09-Aug-2001 13:00

13k

Benedetti.html

09-Aug-2001 13:00

5k

Bergman.html

09-Aug-2001 13:00

7k

Berkeley.html

23-Nov-2001 13:51

7k

Bernays.html

09-Aug-2001 13:00

11k

Bernoulli_Daniel.html

28-Nov-2001 16:11

21k

Bernoulli_Jacob(II)...> 28-Nov-2001 16:11

6k

Bernoulli_Jacob.html

28-Nov-2001 16:11

21k

Bernoulli_Johann(II)..> 28-Nov-2001 16:11

6k

Bernoulli_Johann(III..> 28-Nov-2001 16:11

6k

Bernoulli_Johann.html

28-Nov-2001 16:11

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (6 of 62) [2/16/2002 10:55:55 PM]

24k

Index of /~history/Mathematicians

Bernoulli_Nicolaus(I..> 28-Nov-2001 16:11

9k

Bernoulli_Nicolaus(I..> 28-Nov-2001 16:11

7k

Bernstein_Felix.html

09-Aug-2001 13:00

6k

Bernstein_Sergi.html

23-Nov-2001 13:51

8k

Bers.html

09-Aug-2001 13:00

8k

Bertini.html

19-Sep-2001 10:05

6k

Bertrand.html

23-Nov-2001 13:51

8k

Berwald.html

09-Aug-2001 13:00

7k

Berwick.html

09-Aug-2001 13:00

6k

Besicovitch.html

23-Nov-2001 13:51

17k

Bessel.html

23-Nov-2001 13:51

17k

Betti.html

23-Nov-2001 13:51

14k

Beurling.html

09-Aug-2001 13:00

6k

Bezout.html

23-Nov-2001 13:51

12k

Bhaskara.html

09-Aug-2001 13:00

1k

Bhaskara_I.html

09-Aug-2001 13:00

10k

Bhaskara_II.html

22-Jan-2002 10:38

23k

Bianchi.html

27-Aug-2001 16:15

9k

Bieberbach.html

23-Nov-2001 13:51

11k

Bienayme.html

09-Aug-2001 13:00

8k

Billy.html

09-Aug-2001 13:00

6k

Binet.html

10-Jan-2002 15:43

8k

Bing.html

10-Jan-2002 15:43

14k

Biot.html

10-Jan-2002 15:43

8k

Birkhoff.html

23-Nov-2001 13:51

18k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (7 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Birkhoff_Garrett.html

09-Aug-2001 13:00

13k

Bjerknes_Carl.html

09-Aug-2001 13:00

7k

Bjerknes_Vilhelm.html

09-Aug-2001 13:00

12k

Black.html

09-Aug-2001 13:00

12k

Blaschke.html

28-Aug-2001 16:02

6k

Blichfeldt.html

09-Aug-2001 13:00

9k

Bliss.html

09-Aug-2001 13:00

9k

Bloch.html

09-Aug-2001 13:00

5k

Bobillier.html

09-Aug-2001 13:00

7k

Bocher.html

09-Aug-2001 13:00

10k

Bochner.html

09-Aug-2001 13:00

7k

Boethius.html

23-Nov-2001 13:51

12k

Boggio.html

19-Sep-2001 10:05

7k

Bohl.html

31-Oct-2001 09:23

7k

Bohr_Harald.html

23-Nov-2001 13:51

9k

Bohr_Niels.html

10-Jan-2002 15:43

23k

Boltzmann.html

09-Aug-2001 13:00

14k

Bolyai.html

23-Nov-2001 13:51

8k

Bolyai_Farkas.html

09-Aug-2001 13:00

6k

Bolza.html

09-Aug-2001 13:00

9k

Bolzano.html

23-Nov-2001 13:51

11k

Bombelli.html

10-Jan-2002 15:43

19k

Bombieri.html

10-Jan-2002 15:43

12k

Bonferroni.html

10-Jan-2002 15:43

6k

Bonnet.html

19-Sep-2001 10:05

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (8 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Boole.html

23-Nov-2001 13:51

12k

Boone.html

23-Nov-2001 13:51

7k

Borchardt.html

09-Aug-2001 13:00

6k

Borda.html

19-Sep-2001 10:05

5k

Borel.html

23-Nov-2001 13:51

7k

Borgi.html

09-Aug-2001 13:00

4k

Born.html

09-Aug-2001 13:00

12k

Borsuk.html

23-Nov-2001 13:51

13k

Bortkiewicz.html

09-Aug-2001 13:00

9k

Bortolotti.html

09-Aug-2001 13:00

6k

Bosanquet.html

09-Aug-2001 13:00

7k

Boscovich.html

08-Oct-2001 14:36

6k

Bose.html

10-Dec-2001 16:56

7k

Bossut.html

09-Aug-2001 13:00

5k

Bougainville.html

09-Aug-2001 13:00

11k

Bouguer.html

09-Aug-2001 13:00

9k

Boulliau.html

09-Aug-2001 13:00

6k

Bouquet.html

09-Aug-2001 13:00

7k

Bour.html

19-Sep-2001 10:05

7k

Bourgain.html

23-Nov-2001 13:51

11k

Bourgainville.html

09-Aug-2001 13:00

10k

Boutroux.html

09-Aug-2001 13:00

9k

Bowditch.html

09-Aug-2001 13:00

15k

Bowen.html

09-Aug-2001 13:00

5k

Boyle.html

09-Aug-2001 13:00

22k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (9 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Boys.html

09-Aug-2001 13:00

6k

Bradwardine.html

23-Nov-2001 13:51

11k

Brahe.html

31-Oct-2001 09:23

8k

Brahmadeva.html

09-Aug-2001 13:00

5k

Brahmagupta.html

23-Nov-2001 13:51

16k

Braikenridge.html

09-Aug-2001 13:00

13k

Bramer.html

09-Aug-2001 13:00

5k

Brashman.html

09-Aug-2001 13:00

5k

Brauer.html

09-Aug-2001 13:00

21k

Brauer_Alfred.html

09-Aug-2001 13:00

14k

Brianchon.html

23-Nov-2001 13:51

9k

Briggs.html

23-Nov-2001 13:51

20k

Brill.html

09-Aug-2001 13:00

8k

Brillouin.html

09-Aug-2001 13:00

6k

Bring.html

09-Aug-2001 13:00

6k

Brioschi.html

19-Sep-2001 10:05

6k

Briot.html

09-Aug-2001 13:00

4k

Brisson.html

09-Aug-2001 13:00

7k

Britton.html

09-Aug-2001 13:00

9k

Brocard.html

23-Nov-2001 13:51

5k

Brodetsky.html

09-Aug-2001 13:00

8k

Broglie.html

09-Aug-2001 13:00

16k

Bromwich.html

09-Aug-2001 13:00

8k

Bronowski.html

09-Aug-2001 13:00

16k

Brouncker.html

28-Nov-2001 16:11

8k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (10 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Brouwer.html

23-Nov-2001 13:51

9k

Brown.html

09-Aug-2001 13:00

8k

Browne.html

10-Jan-2002 15:43

10k

Bruno.html

09-Aug-2001 13:00

6k

Bruns.html

09-Aug-2001 13:00

5k

Bryson.html

09-Aug-2001 13:00

9k

Budan_de_Boislaurent..> 09-Aug-2001 13:00

4k

Buffon.html

23-Nov-2001 13:51

7k

Bugaev.html

27-Aug-2001 16:15

9k

Bukreev.html

09-Aug-2001 13:00

6k

Bunyakovsky.html

09-Aug-2001 13:00

10k

Burali-Forti.html

23-Nov-2001 13:51

8k

Burchnall.html

09-Aug-2001 13:00

7k

Burgi.html

23-Nov-2001 13:51

6k

Burkhardt.html

09-Aug-2001 13:00

7k

Burkill.html

09-Aug-2001 13:00

7k

Burnside.html

23-Nov-2001 13:51

11k

Caccioppoli.html

31-Oct-2001 09:23

16k

Cajori.html

09-Aug-2001 13:00

11k

Calderon.html

09-Aug-2001 13:00

12k

Callippus.html

23-Nov-2001 13:51

10k

Campanus.html

10-Jan-2002 15:43

6k

Campbell.html

09-Aug-2001 13:00

10k

Camus.html

09-Aug-2001 13:00

9k

Cannell.html

09-Aug-2001 13:00

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (11 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Cantelli.html

27-Aug-2001 16:15

5k

Cantor.html

23-Nov-2001 13:51

29k

Cantor_Moritz.html

09-Aug-2001 13:00

8k

Caramuel.html

31-Oct-2001 09:23

6k

Caratheodory.html

09-Aug-2001 13:00

7k

Carcavi.html

09-Aug-2001 13:00

7k

Cardan.html

23-Nov-2001 13:51

24k

Carlitz.html

09-Aug-2001 13:00

10k

Carlyle.html

09-Aug-2001 13:00

6k

Carnot.html

23-Nov-2001 13:51

9k

Carnot_Sadi.html

23-Nov-2001 13:51

15k

Carslaw.html

09-Aug-2001 13:00

8k

Cartan.html

23-Nov-2001 13:51

9k

Cartan_Henri.html

27-Aug-2001 16:15

6k

Cartwright.html

09-Aug-2001 13:00

16k

Casorati.html

19-Sep-2001 10:05

4k

Cassels.html

09-Aug-2001 13:00

8k

Cassini.html

23-Nov-2001 13:51

8k

Castel.html

09-Aug-2001 13:00

6k

Castelnuovo.html

23-Nov-2001 13:51

16k

Castigliano.html

19-Sep-2001 10:05

7k

Castillon.html

09-Aug-2001 13:00

6k

Catalan.html

09-Aug-2001 13:00

6k

Cataldi.html

23-Nov-2001 13:51

7k

Cauchy.html

28-Nov-2001 16:11

24k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (12 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Cavalieri.html

28-Nov-2001 16:11

11k

Cayley.html

28-Nov-2001 16:11

15k

Cech.html

28-Aug-2001 16:02

15k

Cesaro.html

23-Nov-2001 13:51

12k

Ceva_Giovanni.html

28-Nov-2001 16:11

7k

Ceva_Tommaso.html

09-Aug-2001 13:00

6k

Ch'in.html

23-Nov-2001 13:51

5k

Chandrasekhar.html

27-Aug-2001 16:15

6k

Chang.html

09-Aug-2001 13:00

9k

Chaplygin.html

09-Aug-2001 13:00

7k

Chapman.html

09-Aug-2001 13:00

10k

Chasles.html

23-Nov-2001 13:51

14k

Chatelet.html

09-Aug-2001 13:00

6k

Chebotaryov.html

23-Nov-2001 13:51

8k

Chebyshev.html

28-Nov-2001 16:11

9k

Chern.html

27-Aug-2001 16:15

9k

Chernikov.html

09-Aug-2001 13:00

12k

Chevalley.html

09-Aug-2001 13:00

8k

Chisholm_Young.html

09-Aug-2001 13:00

8k

Chowla.html

08-Oct-2001 14:36

13k

Christoffel.html

09-Aug-2001 13:00

18k

Chrysippus.html

31-Oct-2001 09:23

11k

Chrystal.html

09-Aug-2001 13:00

24k

Chu.html

23-Nov-2001 13:51

5k

Chuquet.html

23-Nov-2001 13:51

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (13 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Church.html

23-Nov-2001 13:51

6k

Clairaut.html

23-Nov-2001 13:51

21k

Clapeyron.html

09-Aug-2001 13:00

8k

Clarke.html

31-Oct-2001 09:23

7k

Clausen.html

19-Sep-2001 10:05

7k

Clausius.html

09-Aug-2001 13:00

21k

Clavius.html

28-Nov-2001 16:11

8k

Clebsch.html

09-Aug-2001 13:00

10k

Cleomedes.html

09-Aug-2001 13:00

12k

Clifford.html

09-Aug-2001 13:00

11k

Coates.html

08-Oct-2001 14:36

10k

Coble.html

09-Aug-2001 13:00

8k

Cochran.html

09-Aug-2001 13:00

7k

Cocker.html

23-Nov-2001 13:51

8k

Codazzi.html

09-Aug-2001 13:00

4k

Cohen.html

23-Nov-2001 13:51

8k

Cohn.html

09-Aug-2001 13:00

12k

Cole.html

09-Aug-2001 13:00

7k

Collingwood.html

09-Aug-2001 13:00

8k

Collins.html

09-Aug-2001 13:00

10k

Condorcet.html

23-Nov-2001 13:51

10k

Connes.html

23-Nov-2001 13:51

11k

Conon.html

09-Aug-2001 13:00

10k

Conway.html

22-Jan-2002 10:38

17k

Conway_Arthur.html

31-Oct-2001 09:23

11k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (14 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Coolidge.html

27-Aug-2001 16:15

11k

Cooper.html

09-Aug-2001 13:00

8k

Copernicus.html

23-Nov-2001 13:51

9k

Copson.html

09-Aug-2001 13:00

7k

Coriolis.html

23-Nov-2001 13:51

10k

Cosserat.html

09-Aug-2001 13:00

6k

Cotes.html

23-Nov-2001 13:51

7k

Coulomb.html

23-Nov-2001 13:51

17k

Courant.html

09-Aug-2001 13:00

19k

Cournot.html

23-Nov-2001 13:51

9k

Couturat.html

19-Sep-2001 10:05

5k

Cox.html

09-Aug-2001 13:00

5k

Coxeter.html

09-Aug-2001 13:00

10k

Craig.html

09-Aug-2001 13:00

7k

Cramer.html

23-Nov-2001 13:51

18k

Cramer_Harald.html

09-Aug-2001 13:00

13k

Crank.html

09-Aug-2001 13:00

6k

Crelle.html

23-Nov-2001 13:51

12k

Cremona.html

19-Sep-2001 10:05

17k

Crighton.html

09-Aug-2001 13:00

12k

Cunha.html

09-Aug-2001 13:00

9k

Cunningham.html

09-Aug-2001 13:00

7k

Curry.html

09-Aug-2001 13:00

5k

Cusa.html

23-Nov-2001 13:51

7k

D'Alembert.html

23-Nov-2001 13:51

22k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (15 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

D'Ovidio.html

19-Sep-2001 10:05

8k

Dandelin.html

09-Aug-2001 13:00

7k

Danti.html

28-Nov-2001 16:11

6k

Dantzig.html

09-Aug-2001 13:00

8k

Dantzig_George.html

10-Jan-2002 15:43

7k

Darboux.html

19-Sep-2001 10:05

11k

Darwin.html

09-Aug-2001 13:00

6k

Dase.html

09-Aug-2001 13:00

4k

Davenport.html

28-Aug-2001 16:02

11k

Davidov.html

09-Aug-2001 13:00

8k

Davies.html

09-Aug-2001 13:00

6k

De_Beaune.html

23-Nov-2001 13:51

8k

De_Groot.html

09-Aug-2001 13:00

7k

De_L'Hopital.html

19-Sep-2001 10:05

6k

De_Moivre.html

28-Nov-2001 16:11

9k

De_Morgan.html

23-Nov-2001 13:51

11k

De_Prony.html

23-Nov-2001 13:51

11k

De_Rham.html

09-Aug-2001 13:00

10k

De_Witt.html

23-Nov-2001 13:51

8k

Dechales.html

09-Aug-2001 13:00

5k

Dedekind.html

23-Nov-2001 13:51

21k

Dee.html

09-Aug-2001 13:00

6k

Dehn.html

23-Nov-2001 13:51

10k

Delamain.html

09-Aug-2001 13:00

6k

Delambre.html

19-Sep-2001 10:05

6k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (16 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Delaunay.html

23-Nov-2001 13:51

7k

Deligne.html

23-Nov-2001 13:51

8k

Delone.html

09-Aug-2001 13:00

8k

Delsarte.html

09-Aug-2001 13:00

7k

Democritus.html

09-Aug-2001 13:00

17k

Denjoy.html

23-Nov-2001 13:51

12k

Deparcieux.html

09-Aug-2001 13:00

8k

Desargues.html

28-Nov-2001 16:11

11k

Descartes.html

28-Nov-2001 16:11

19k

Dickson.html

23-Nov-2001 13:51

10k

Dickstein.html

09-Aug-2001 13:00

10k

Dieudonne.html

09-Aug-2001 13:00

6k

Digges.html

28-Nov-2001 16:11

7k

Dilworth.html

10-Jan-2002 15:43

13k

Dinghas.html

27-Aug-2001 16:15

12k

Dini.html

28-Aug-2001 16:02

11k

Dinostratus.html

09-Aug-2001 13:00

8k

Diocles.html

23-Nov-2001 13:51

13k

Dionis.html

09-Aug-2001 13:00

8k

Dionysodorus.html

09-Aug-2001 13:00

11k

Diophantus.html

23-Nov-2001 13:51

21k

Dirac.html

09-Aug-2001 13:00

21k

Dirichlet.html

23-Nov-2001 13:51

21k

Dixon.html

09-Aug-2001 13:00

10k

Dixon_Arthur.html

09-Aug-2001 13:00

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (17 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Dodgson.html

08-Oct-2001 14:36

6k

Doeblin.html

08-Oct-2001 14:36

11k

Domninus.html

08-Oct-2001 14:36

8k

Donaldson.html

23-Nov-2001 13:51

11k

Doob.html

27-Aug-2001 16:15

6k

Doppelmayr.html

09-Aug-2001 13:00

6k

Doppler.html

09-Aug-2001 13:00

21k

Douglas.html

23-Nov-2001 13:51

7k

Dowker.html

10-Jan-2002 15:43

11k

Drach.html

09-Aug-2001 13:00

6k

Drinfeld.html

23-Nov-2001 13:51

8k

Du_Bois-Reymond.html

09-Aug-2001 13:00

6k

Du_Val.html

09-Aug-2001 13:00

7k

Dubreil.html

09-Aug-2001 13:00

8k

Dudeney.html

09-Aug-2001 13:00

7k

Duhamel.html

09-Aug-2001 13:00

6k

Duhem.html

09-Aug-2001 13:00

17k

Dupin.html

09-Aug-2001 13:00

9k

Dupre.html

31-Oct-2001 09:23

6k

Durer.html

28-Nov-2001 16:11

24k

Dynkin.html

08-Oct-2001 14:36

11k

Eckert_John.html

23-Nov-2001 13:51

11k

Eckert_Wallace.html

27-Aug-2001 16:15

9k

Eckmann.html

08-Oct-2001 14:36

10k

Eddington.html

10-Dec-2001 16:56

19k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (18 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Edge.html

09-Aug-2001 13:01

14k

Edgeworth.html

23-Nov-2001 13:51

9k

Egorov.html

09-Aug-2001 13:01

7k

Ehrenfest.html

31-Oct-2001 09:23

21k

Ehresmann.html

28-Aug-2001 16:02

9k

Eilenberg.html

23-Nov-2001 13:51

18k

Einstein.html

23-Nov-2001 13:51

21k

Eisenhart.html

09-Aug-2001 13:01

10k

Eisenstein.html

09-Aug-2001 13:01

22k

Elliott.html

09-Aug-2001 13:01

9k

Empedocles.html

09-Aug-2001 13:01

11k

Engel.html

23-Nov-2001 13:51

9k

Enriques.html

28-Aug-2001 16:02

8k

Enskog.html

23-Nov-2001 13:51

7k

Eotvos.html

09-Aug-2001 13:01

5k

Epstein.html

09-Aug-2001 13:01

9k

Eratosthenes.html

23-Nov-2001 13:51

19k

Erdelyi.html

09-Aug-2001 13:01

9k

Erdos.html

28-Nov-2001 16:11

21k

Erlang.html

28-Aug-2001 16:02

7k

Escher.html

10-Dec-2001 16:56

29k

Esclangon.html

19-Sep-2001 10:05

8k

Euclid.html

10-Jan-2002 15:43

30k

Eudemus.html

23-Nov-2001 13:51

13k

Eudoxus.html

23-Nov-2001 13:51

23k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (19 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Euler.html

23-Nov-2001 13:51

44k

Eutocius.html

23-Nov-2001 13:51

12k

Evans.html

09-Aug-2001 13:01

13k

Ezra.html

09-Aug-2001 13:01

9k

Faa_di_Bruno.html

19-Sep-2001 10:05

8k

Faber.html

09-Aug-2001 13:01

6k

Fabri.html

09-Aug-2001 13:01

7k

Fagnano_Giovanni.html

09-Aug-2001 13:01

6k

Fagnano_Giulio.html

23-Nov-2001 13:51

10k

Faltings.html

23-Nov-2001 13:51

8k

Fano.html

19-Sep-2001 10:05

9k

Faraday.html

10-Jan-2002 15:43

18k

Farey.html

09-Aug-2001 13:01

11k

Fatou.html

09-Aug-2001 13:01

9k

Faulhaber.html

09-Aug-2001 13:01

9k

Fefferman.html

23-Nov-2001 13:51

9k

Feigenbaum.html

10-Jan-2002 15:43

18k

Feigl.html

31-Oct-2001 09:23

6k

Fejer.html

23-Nov-2001 13:51

8k

Feller.html

09-Aug-2001 13:01

8k

Fermat.html

28-Nov-2001 16:11

26k

Ferrar.html

23-Nov-2001 13:51

9k

Ferrari.html

23-Nov-2001 13:51

10k

Ferrel.html

09-Aug-2001 13:01

5k

Ferro.html

23-Nov-2001 13:51

14k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (20 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Feuerbach.html

28-Nov-2001 16:11

8k

Feynman.html

09-Aug-2001 13:01

8k

Fibonacci.html

28-Nov-2001 16:11

22k

Fields.html

09-Aug-2001 13:01

6k

Finck.html

09-Aug-2001 13:01

6k

Fincke.html

09-Aug-2001 13:01

6k

Fine.html

09-Aug-2001 13:01

5k

Fine_Henry.html

09-Aug-2001 13:01

7k

Finsler.html

27-Aug-2001 16:15

9k

Fischer.html

09-Aug-2001 13:01

6k

Fisher.html

23-Nov-2001 13:51

7k

Fiske.html

09-Aug-2001 13:01

8k

FitzGerald.html

23-Nov-2001 13:51

24k

Flamsteed.html

09-Aug-2001 13:01

10k

Flugge-Lotz.html

09-Aug-2001 13:01

7k

Fomin.html

09-Aug-2001 13:01

14k

Fontaine_des_Bertins..> 09-Aug-2001 13:01

9k

Fontenelle.html

09-Aug-2001 13:01

6k

Forsyth.html

09-Aug-2001 13:01

9k

Fourier.html

23-Nov-2001 13:51

20k

Fowler.html

09-Aug-2001 13:01

21k

Fox.html

09-Aug-2001 13:01

7k

Fraenkel.html

23-Nov-2001 13:51

7k

Francais_Francois.html

09-Aug-2001 13:01

9k

Francais_Jacques.html

09-Aug-2001 13:01

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (21 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Francesca.html

31-Oct-2001 09:23

10k

Francoeur.html

09-Aug-2001 13:01

6k

Frank.html

09-Aug-2001 13:01

9k

Franklin.html

09-Aug-2001 13:01

6k

Franklin_Benjamin.html

10-Dec-2001 16:56

8k

Frattini.html

09-Aug-2001 13:01

11k

Frechet.html

23-Nov-2001 13:51

7k

Fredholm.html

23-Nov-2001 13:51

7k

Freedman.html

23-Nov-2001 13:51

13k

Frege.html

23-Nov-2001 13:51

7k

Freitag.html

09-Aug-2001 13:01

7k

Frenet.html

09-Aug-2001 13:01

4k

Frenicle_de_Bessy.html

23-Nov-2001 13:51

8k

Frenkel.html

08-Oct-2001 14:36

9k

Fresnel.html

27-Aug-2001 16:15

5k

Freudenthal.html

09-Aug-2001 13:01

13k

Freundlich.html

09-Aug-2001 13:01

15k

Friedmann.html

09-Aug-2001 13:01

22k

Friedrichs.html

09-Aug-2001 13:01

6k

Frisi.html

09-Aug-2001 13:01

8k

Frobenius.html

23-Nov-2001 13:51

23k

Fubini.html

19-Sep-2001 10:05

12k

Fuchs.html

10-Jan-2002 15:43

10k

Fueter.html

09-Aug-2001 13:01

8k

Fuller.html

09-Aug-2001 13:01

5k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (22 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Fuss.html

09-Aug-2001 13:01

6k

Galerkin.html

09-Aug-2001 13:01

7k

Galileo.html

23-Nov-2001 13:51

12k

Gallarati.html

08-Oct-2001 14:36

7k

Galois.html

23-Nov-2001 13:51

21k

Galton.html

23-Nov-2001 13:51

6k

Gassendi.html

09-Aug-2001 13:01

7k

Gauss.html

28-Nov-2001 16:11

26k

Gegenbauer.html

09-Aug-2001 13:01

4k

Geiringer.html

09-Aug-2001 13:01

14k

Geiser.html

09-Aug-2001 13:01

7k

Gelfand.html

09-Aug-2001 13:01

13k

Gelfond.html

23-Nov-2001 13:51

8k

Gellibrand.html

09-Aug-2001 13:01

6k

Geminus.html

23-Nov-2001 13:51

15k

Gemma_Frisius.html

23-Nov-2001 13:51

9k

Genocchi.html

19-Sep-2001 10:05

10k

Gentzen.html

08-Oct-2001 14:36

15k

Gergonne.html

28-Nov-2001 16:11

12k

Germain.html

23-Nov-2001 13:51

14k

Gherard.html

23-Nov-2001 13:51

8k

Ghetaldi.html

09-Aug-2001 13:01

6k

Gibbs.html

23-Nov-2001 13:51

13k

Girard_Albert.html

23-Nov-2001 13:51

7k

Girard_Pierre.html

09-Aug-2001 13:01

5k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (23 of 62) [2/16/2002 10:55:55 PM]

Index of /~history/Mathematicians

Glaisher.html

09-Aug-2001 13:01

10k

Glenie.html

09-Aug-2001 13:01

5k

Gnedenko.html

09-Aug-2001 13:01

18k

Godel.html

22-Jan-2002 10:38

12k

Gohberg.html

09-Aug-2001 13:01

9k

Goldbach.html

23-Nov-2001 13:51

7k

Goldstein.html

09-Aug-2001 13:01

6k

Gompertz.html

23-Nov-2001 13:51

8k

Goodstein.html

27-Aug-2001 16:15

6k

Gopel.html

09-Aug-2001 13:01

6k

Gordan.html

09-Aug-2001 13:01

12k

Gorenstein.html

08-Oct-2001 14:36

14k

Gosset.html

23-Nov-2001 13:51

7k

Goursat.html

23-Nov-2001 13:51

5k

Govindasvami.html

09-Aug-2001 13:01

5k

Graffe.html

09-Aug-2001 13:01

4k

Gram.html

09-Aug-2001 13:01

11k

Grandi.html

23-Nov-2001 13:51

9k

Granville.html

08-Oct-2001 14:36

15k

Grassmann.html

23-Nov-2001 13:51

9k

Grave.html

09-Aug-2001 13:01

8k

Green.html

23-Nov-2001 13:51

23k

Greenhill.html

09-Aug-2001 13:01

9k

Gregory.html

10-Jan-2002 15:43

27k

Gregory_David.html

23-Nov-2001 13:51

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (24 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Gregory_Duncan.html

09-Aug-2001 13:01

6k

Grosseteste.html

09-Aug-2001 13:01

7k

Grossmann.html

09-Aug-2001 13:01

5k

Grothendieck.html

23-Nov-2001 13:51

11k

Grunsky.html

08-Oct-2001 14:36

9k

Guarini.html

27-Aug-2001 16:15

11k

Guccia.html

10-Jan-2002 15:43

7k

Gudermann.html

10-Jan-2002 15:43

8k

Guenther.html

09-Aug-2001 13:01

8k

Guinand.html

09-Aug-2001 13:01

9k

Guldin.html

23-Nov-2001 13:51

5k

Gunter.html

23-Nov-2001 13:51

6k

Haar.html

23-Nov-2001 13:51

7k

Hachette.html

27-Aug-2001 16:15

10k

Hadamard.html

28-Nov-2001 16:11

24k

Hadley.html

23-Nov-2001 13:51

6k

Hahn.html

27-Aug-2001 16:15

6k

Hajek.html

08-Oct-2001 14:36

9k

Hall.html

23-Nov-2001 13:51

7k

Hall_Marshall.html

23-Nov-2001 13:51

8k

Halley.html

23-Nov-2001 13:51

24k

Halmos.html

09-Aug-2001 13:01

14k

Halphen.html

09-Aug-2001 13:01

10k

Halsted.html

09-Aug-2001 13:01

6k

Hamill.html

27-Aug-2001 16:15

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (25 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Hamilton.html

23-Nov-2001 13:51

25k

Hamilton_William.html

09-Aug-2001 13:01

6k

Hamming.html

23-Nov-2001 13:52

9k

Hankel.html

09-Aug-2001 13:01

9k

Hardy.html

10-Jan-2002 15:43

23k

Hardy_Claude.html

09-Aug-2001 13:01

6k

Harish-Chandra.html

28-Aug-2001 16:02

8k

Harriot.html

23-Nov-2001 13:52

28k

Hartley.html

09-Aug-2001 13:01

7k

Hartree.html

09-Aug-2001 13:01

8k

Hasse.html

10-Dec-2001 16:56

15k

Hausdorff.html

28-Nov-2001 16:11

7k

Hawking.html

09-Aug-2001 13:01

13k

Heath.html

09-Aug-2001 13:01

6k

Heaviside.html

09-Aug-2001 13:01

14k

Heawood.html

23-Nov-2001 13:52

9k

Hecht.html

09-Aug-2001 13:01

4k

Hecke.html

09-Aug-2001 13:01

10k

Hedrick.html

09-Aug-2001 13:01

9k

Heegaard.html

23-Nov-2001 13:52

8k

Heilbronn.html

08-Oct-2001 14:36

19k

Heine.html

23-Nov-2001 13:52

6k

Heisenberg.html

09-Aug-2001 13:01

22k

Hellinger.html

09-Aug-2001 13:01

14k

Helly.html

09-Aug-2001 13:01

11k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (26 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Helmholtz.html

09-Aug-2001 13:01

21k

Heng.html

09-Aug-2001 13:01

5k

Henrici.html

09-Aug-2001 13:01

5k

Hensel.html

23-Nov-2001 13:52

7k

Heraclides.html

09-Aug-2001 13:01

11k

Herbrand.html

27-Aug-2001 16:15

7k

Herigone.html

09-Aug-2001 13:01

5k

Hermann.html

09-Aug-2001 13:01

7k

Hermann_of_Reichenau..> 23-Nov-2001 13:52

6k

Hermite.html

23-Nov-2001 13:52

20k

Heron.html

28-Nov-2001 16:11

20k

Herschel.html

23-Nov-2001 13:52

23k

Herschel_Caroline.html

09-Aug-2001 13:01

15k

Herstein.html

09-Aug-2001 13:01

6k

Hesse.html

23-Nov-2001 13:52

6k

Heuraet.html

23-Nov-2001 13:52

6k

Heyting.html

10-Dec-2001 16:56

12k

Higman.html

10-Jan-2002 15:43

15k

Hilbert.html

23-Nov-2001 13:52

21k

Hill.html

09-Aug-2001 13:01

9k

Hille.html

09-Aug-2001 13:01

14k

Hindenburg.html

09-Aug-2001 13:01

8k

Hipparchus.html

23-Nov-2001 13:52

24k

Hippias.html

23-Nov-2001 13:52

12k

Hippocrates.html

23-Nov-2001 13:52

15k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (27 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Hironaka.html

23-Nov-2001 13:52

7k

Hirsch.html

27-Aug-2001 16:15

9k

Hirst.html

09-Aug-2001 13:01

15k

Hniedenko.html

09-Aug-2001 13:01

1k

Hobbes.html

09-Aug-2001 13:01

6k

Hobson.html

09-Aug-2001 13:01

7k

Hodge.html

23-Nov-2001 13:52

7k

Holder.html

23-Nov-2001 13:52

10k

Hollerith.html

09-Aug-2001 13:01

13k

Holmboe.html

27-Aug-2001 16:15

5k

Honda.html

10-Jan-2002 15:43

7k

Hooke.html

23-Nov-2001 13:52

9k

Hopf.html

23-Nov-2001 13:52

17k

Hopf_Eberhard.html

28-Aug-2001 16:02

13k

Hopkins.html

09-Aug-2001 13:01

9k

Hopkinson.html

10-Jan-2002 15:43

6k

Hopper.html

09-Aug-2001 13:01

15k

Hormander.html

23-Nov-2001 13:52

11k

Horner.html

23-Nov-2001 13:52

5k

Houel.html

09-Aug-2001 13:01

8k

Householder.html

27-Aug-2001 16:15

12k

Hsu.html

08-Oct-2001 14:36

11k

Hubble.html

27-Aug-2001 16:15

9k

Hudde.html

23-Nov-2001 13:52

7k

Humbert_Georges.html

09-Aug-2001 13:01

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (28 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Humbert_Pierre.html

09-Aug-2001 13:01

8k

Hunayn.html

09-Aug-2001 13:01

9k

Huntington.html

09-Aug-2001 13:01

9k

Hurewicz.html

23-Nov-2001 13:52

9k

Hurwitz.html

09-Aug-2001 13:01

7k

Hutton.html

09-Aug-2001 13:01

5k

Huygens.html

28-Nov-2001 16:11

25k

Hypatia.html

23-Nov-2001 13:52

12k

Hypsicles.html

23-Nov-2001 13:52

10k

Ibrahim.html

09-Aug-2001 13:01

11k

Ingham.html

27-Aug-2001 16:15

9k

Ito.html

31-Oct-2001 09:23

14k

Ivory.html

27-Aug-2001 16:15

7k

Iwasawa.html

08-Oct-2001 14:36

13k

Iyanaga.html

10-Jan-2002 15:43

11k

Jabir_ibn_Aflah.html

23-Nov-2001 13:52

7k

Jacobi.html

23-Nov-2001 13:52

25k

Jacobson.html

23-Nov-2001 13:52

8k

Jagannatha.html

09-Aug-2001 13:01

6k

James.html

09-Aug-2001 13:01

11k

Janiszewski.html

09-Aug-2001 13:01

16k

Janovskaja.html

09-Aug-2001 13:01

6k

Jarnik.html

09-Aug-2001 13:01

8k

Jeans.html

31-Oct-2001 09:23

20k

Jeffrey.html

09-Aug-2001 13:01

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (29 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Jeffreys.html

09-Aug-2001 13:01

11k

Jensen.html

09-Aug-2001 13:01

8k

Jerrard.html

09-Aug-2001 13:01

6k

Jevons.html

23-Nov-2001 13:52

16k

Joachimsthal.html

27-Aug-2001 16:15

8k

John.html

02-Nov-2001 16:01

10k

Johnson.html

27-Aug-2001 16:15

7k

Johnson_Barry.html

09-Aug-2001 13:01

9k

Jones.html

23-Nov-2001 13:52

6k

Jones_Burton.html

10-Jan-2002 15:43

10k

Jones_Vaughan.html

23-Nov-2001 13:52

11k

Jonquieres.html

09-Aug-2001 13:01

6k

Jordan.html

09-Aug-2001 13:01

20k

Jordanus.html

23-Nov-2001 13:52

5k

Jourdain.html

09-Aug-2001 13:01

4k

Juel.html

09-Aug-2001 13:01

8k

Julia.html

28-Nov-2001 16:11

5k

Jungius.html

09-Aug-2001 13:01

7k

Jyesthadeva.html

09-Aug-2001 13:01

8k

Kac.html

09-Aug-2001 13:01

16k

Kaestner.html

10-Jan-2002 15:43

9k

Kagan.html

27-Aug-2001 16:15

7k

Kakutani.html

31-Oct-2001 09:23

10k

Kalmar.html

27-Aug-2001 16:15

9k

Kaluza.html

08-Feb-2002 17:10

10k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (30 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Kaluznin.html

09-Aug-2001 13:01

18k

Kamalakara.html

09-Aug-2001 13:01

6k

Kantorovich.html

09-Aug-2001 13:01

9k

Kaplansky.html

27-Aug-2001 16:15

12k

Karman.html

09-Aug-2001 13:01

6k

Karp.html

27-Aug-2001 16:15

8k

Katyayana.html

10-Jan-2002 15:43

5k

Keill.html

31-Oct-2001 09:23

5k

Kelland.html

10-Jan-2002 15:43

8k

Kellogg.html

31-Oct-2001 09:23

8k

Kemeny.html

31-Oct-2001 09:23

10k

Kempe.html

23-Nov-2001 13:52

6k

Kendall.html

09-Aug-2001 13:01

9k

Kendall_Maurice.html

09-Aug-2001 13:01

12k

Kepler.html

28-Nov-2001 16:11

32k

Kerekjarto.html

09-Aug-2001 13:01

5k

Keynes.html

23-Nov-2001 13:52

19k

Khayyam.html

28-Nov-2001 16:11

21k

Khinchin.html

09-Aug-2001 13:01

12k

Killing.html

09-Aug-2001 13:01

7k

Kingman.html

10-Jan-2002 15:43

10k

Kirchhoff.html

09-Aug-2001 13:01

7k

Kirkman.html

09-Aug-2001 13:01

13k

Kleene.html

09-Aug-2001 13:01

7k

Klein.html

23-Nov-2001 13:52

21k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (31 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Klein_Oskar.html

31-Oct-2001 09:23

20k

Klingenberg.html

27-Aug-2001 16:15

8k

Kloosterman.html

08-Oct-2001 14:36

9k

Klugel.html

27-Aug-2001 16:15

7k

Kneser.html

09-Aug-2001 13:01

6k

Kneser_Hellmuth.html

09-Aug-2001 13:01

12k

Knopp.html

09-Aug-2001 13:01

5k

Kober.html

09-Aug-2001 13:01

6k

Koch.html

28-Nov-2001 16:11

11k

Kochin.html

09-Aug-2001 13:01

4k

Kodaira.html

08-Oct-2001 14:36

8k

Koebe.html

28-Aug-2001 16:02

6k

Koenigs.html

09-Aug-2001 13:01

5k

Kolmogorov.html

23-Nov-2001 13:52

22k

Kolosov.html

09-Aug-2001 13:01

4k

Konig_Denes.html

09-Aug-2001 13:01

5k

Konig_Julius.html

09-Aug-2001 13:01

7k

Konig_Samuel.html

09-Aug-2001 13:01

11k

Konigsberger.html

09-Aug-2001 13:01

8k

Korteweg.html

19-Sep-2001 10:05

6k

Kotelnikov.html

09-Aug-2001 13:01

7k

Kovalevskaya.html

09-Aug-2001 13:01

15k

Kramer.html

10-Jan-2002 15:43

16k

Kramp.html

09-Aug-2001 13:01

6k

Krawtchouk.html

09-Aug-2001 13:01

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (32 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Krein.html

09-Aug-2001 13:01

14k

Kreisel.html

09-Aug-2001 13:01

7k

Kronecker.html

31-Oct-2001 09:23

22k

Krull.html

23-Nov-2001 13:52

11k

Krylov_Aleksei.html

09-Aug-2001 13:01

7k

Krylov_Nikolai.html

09-Aug-2001 13:01

5k

Kulik.html

27-Aug-2001 16:15

5k

Kumano-Go.html

08-Oct-2001 14:36

8k

Kummer.html

23-Nov-2001 13:52

16k

Kuratowski.html

23-Nov-2001 13:52

21k

Kurosh.html

09-Aug-2001 13:01

13k

Kurschak.html

09-Aug-2001 13:01

5k

Kutta.html

23-Nov-2001 13:52

5k

Kuttner.html

27-Aug-2001 16:15

7k

La_Condamine.html

09-Aug-2001 13:01

7k

La_Faille.html

09-Aug-2001 13:01

5k

La_Hire.html

23-Nov-2001 13:52

8k

La_Roche.html

09-Aug-2001 13:01

5k

Lacroix.html

23-Nov-2001 13:52

6k

Lagny.html

28-Nov-2001 16:11

7k

Lagrange.html

23-Nov-2001 13:52

29k

Laguerre.html

09-Aug-2001 13:01

6k

Lakatos.html

23-Nov-2001 13:52

12k

Lalla.html

10-Jan-2002 15:43

7k

Lamb.html

09-Aug-2001 13:01

11k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (33 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Lambert.html

23-Nov-2001 13:52

9k

Lame.html

23-Nov-2001 13:52

14k

Lamy.html

09-Aug-2001 13:01

5k

Lanczos.html

09-Aug-2001 13:01

11k

Landau.html

28-Nov-2001 16:11

10k

Landau_Lev.html

09-Aug-2001 13:01

6k

Landen.html

09-Aug-2001 13:01

6k

Landsberg.html

09-Aug-2001 13:01

6k

Langlands.html

23-Nov-2001 13:52

11k

Lansberge.html

09-Aug-2001 13:01

7k

Laplace.html

23-Nov-2001 13:52

33k

Larmor.html

31-Oct-2001 09:23

15k

Lasker.html

23-Nov-2001 13:52

7k

Laurent_Hermann.html

09-Aug-2001 13:01

5k

Laurent_Pierre.html

09-Aug-2001 13:01

4k

Lavanha.html

09-Aug-2001 13:01

6k

Lavrentev.html

09-Aug-2001 13:01

7k

Lax.html

09-Aug-2001 13:01

5k

Le_Fevre.html

09-Aug-2001 13:01

8k

Le_Paige.html

09-Aug-2001 13:01

7k

Le_Tenneur.html

09-Aug-2001 13:01

6k

Le_Verrier.html

09-Aug-2001 13:01

8k

Lebesgue.html

23-Nov-2001 13:52

8k

Ledermann.html

09-Aug-2001 13:01

19k

Leech.html

27-Aug-2001 16:15

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (34 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Lefschetz.html

08-Oct-2001 14:36

19k

Legendre.html

28-Nov-2001 16:11

22k

Leger.html

09-Aug-2001 13:01

6k

Leibniz.html

23-Nov-2001 13:52

34k

Lemoine.html

08-Feb-2002 17:10

9k

Leonardo.html

09-Aug-2001 13:01

11k

Leray.html

23-Nov-2001 13:52

10k

Lerch.html

09-Aug-2001 13:01

9k

Leshniewski.html

08-Oct-2001 14:36

13k

Leslie.html

27-Aug-2001 16:15

4k

Leucippus.html

09-Aug-2001 13:01

12k

Levi-Civita.html

23-Nov-2001 13:52

9k

Levi.html

23-Nov-2001 13:52

7k

Levinson.html

10-Jan-2002 15:43

20k

Levy_Hyman.html

09-Aug-2001 13:01

10k

Levy_Paul.html

09-Aug-2001 13:01

10k

Levytsky.html

09-Aug-2001 13:01

5k

Lexell.html

09-Aug-2001 13:01

11k

Lexis.html

23-Nov-2001 13:52

8k

Lhuilier.html

09-Aug-2001 13:01

12k

Libri.html

09-Aug-2001 13:01

6k

Lie.html

23-Nov-2001 13:52

25k

Lifshitz.html

09-Aug-2001 13:01

8k

Lighthill.html

08-Feb-2002 17:10

18k

Lindelof.html

09-Aug-2001 13:01

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (35 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Lindemann.html

10-Jan-2002 15:43

12k

Linnik.html

23-Nov-2001 13:52

7k

Lions.html

23-Nov-2001 13:52

10k

Liouville.html

23-Nov-2001 13:52

22k

Lipschitz.html

09-Aug-2001 13:01

10k

Lissajous.html

19-Sep-2001 10:05

6k

Listing.html

10-Jan-2002 15:43

16k

Littlewood.html

10-Dec-2001 16:56

8k

Littlewood_Dudley.html

09-Aug-2001 13:01

9k

Livsic.html

27-Aug-2001 16:15

10k

Llull.html

09-Aug-2001 13:01

6k

Lobachevsky.html

23-Nov-2001 13:52

21k

Loewner.html

08-Oct-2001 14:36

17k

Loewy.html

27-Aug-2001 16:15

6k

Lopatynsky.html

09-Aug-2001 13:01

6k

Lorentz.html

23-Nov-2001 13:52

11k

Love.html

09-Aug-2001 13:01

5k

Lovelace.html

23-Nov-2001 13:52

6k

Lowenheim.html

31-Oct-2001 09:23

5k

Loyd.html

09-Aug-2001 13:01

5k

Lucas.html

09-Aug-2001 13:01

8k

Lueroth.html

23-Nov-2001 13:52

7k

Lukacs.html

09-Aug-2001 13:01

13k

Lukasiewicz.html

09-Aug-2001 13:01

9k

Luke.html

09-Aug-2001 13:01

12k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (36 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Luzin.html

09-Aug-2001 13:01

19k

Lyapunov.html

23-Nov-2001 13:52

6k

Lyndon.html

09-Aug-2001 13:01

9k

MacCullagh.html

09-Aug-2001 13:01

12k

MacLane.html

23-Nov-2001 13:52

9k

MacMahon.html

28-Aug-2001 16:02

8k

Macaulay.html

23-Nov-2001 13:52

8k

Macdonald.html

09-Aug-2001 13:01

5k

Macintyre.html

27-Aug-2001 16:15

7k

Maclaurin.html

28-Nov-2001 16:11

24k

Madhava.html

23-Nov-2001 13:52

11k

Magnitsky.html

09-Aug-2001 13:01

5k

Magnus.html

23-Nov-2001 13:52

9k

Mahavira.html

23-Nov-2001 13:52

10k

Mahendra_Suri.html

09-Aug-2001 13:01

5k

Mahler.html

09-Aug-2001 13:01

9k

Maior.html

09-Aug-2001 13:01

4k

Malcev.html

09-Aug-2001 13:01

15k

Malebranche.html

09-Aug-2001 13:01

15k

Malfatti.html

09-Aug-2001 13:01

8k

Malus.html

09-Aug-2001 13:01

8k

Manava.html

23-Nov-2001 13:52

6k

Mandelbrot.html

28-Nov-2001 16:11

17k

Mannheim.html

23-Nov-2001 13:52

7k

Mansion.html

09-Aug-2001 13:01

6k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (37 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Mansur.html

09-Aug-2001 13:01

13k

Marchenko.html

27-Aug-2001 16:15

7k

Marcinkiewicz.html

08-Oct-2001 14:36

12k

Marczewski.html

27-Aug-2001 16:15

12k

Margulis.html

23-Nov-2001 13:52

12k

Marinus.html

09-Aug-2001 13:01

8k

Markov.html

23-Nov-2001 13:52

7k

Mascheroni.html

10-Jan-2002 15:43

6k

Maschke.html

09-Aug-2001 13:01

7k

Maseres.html

09-Aug-2001 13:01

5k

Maskelyne.html

09-Aug-2001 13:01

8k

Mason.html

09-Aug-2001 13:01

7k

Mathews.html

09-Aug-2001 13:01

9k

Mathieu_Claude.html

09-Aug-2001 13:01

4k

Mathieu_Emile.html

27-Aug-2001 16:15

9k

Matsushima.html

31-Oct-2001 09:23

13k

Mauchly.html

23-Nov-2001 13:52

13k

Maupertuis.html

09-Aug-2001 13:01

7k

Maurolico.html

23-Nov-2001 13:52

7k

Maxwell.html

22-Jan-2002 10:38

24k

Mayer_Adolph.html

28-Aug-2001 16:02

8k

Mayer_Tobias.html

09-Aug-2001 13:01

7k

Mazur.html

08-Oct-2001 14:36

13k

Mazurkiewicz.html

23-Nov-2001 13:52

14k

McClintock.html

09-Aug-2001 13:01

8k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (38 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

McDuff.html

09-Aug-2001 13:01

13k

McShane.html

09-Aug-2001 13:01

10k

Meissel.html

09-Aug-2001 13:01

6k

Mellin.html

09-Aug-2001 13:01

9k

Menabrea.html

19-Sep-2001 10:05

5k

Menaechmus.html

28-Nov-2001 16:11

14k

Menelaus.html

28-Nov-2001 16:11

13k

Menger.html

09-Aug-2001 13:01

8k

Mengoli.html

23-Nov-2001 13:52

6k

Menshov.html

09-Aug-2001 13:01

13k

Meray.html

23-Nov-2001 13:52

9k

Mercator_Gerardus.html

09-Aug-2001 13:01

7k

Mercator_Nicolaus.html

23-Nov-2001 13:52

6k

Mercer.html

10-Jan-2002 15:43

7k

Merrifield.html

09-Aug-2001 13:01

4k

Merrill.html

09-Aug-2001 13:01

8k

Mersenne.html

23-Nov-2001 13:52

9k

Mertens.html

09-Aug-2001 13:01

5k

Meshchersky.html

09-Aug-2001 13:01

4k

Meyer.html

09-Aug-2001 13:01

5k

Miller.html

09-Aug-2001 13:01

5k

Milne.html

09-Aug-2001 13:01

12k

Milnor.html

23-Nov-2001 13:52

12k

Minding.html

09-Aug-2001 13:01

8k

Mineur.html

27-Aug-2001 16:15

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (39 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Minkowski.html

09-Aug-2001 13:01

7k

Mirsky.html

09-Aug-2001 13:01

8k

Mises.html

23-Nov-2001 13:52

19k

Mittag-Leffler.html

23-Nov-2001 13:52

21k

Mobius.html

23-Nov-2001 13:52

15k

Mohr.html

23-Nov-2001 13:52

5k

Molien.html

09-Aug-2001 13:01

1k

Molin.html

09-Aug-2001 13:01

10k

Monge.html

23-Nov-2001 13:52

25k

Monte.html

09-Aug-2001 13:01

6k

Montel.html

09-Aug-2001 13:01

4k

Montmort.html

09-Aug-2001 13:01

7k

Montucla.html

09-Aug-2001 13:01

7k

Moore_Eliakim.html

08-Oct-2001 14:36

14k

Moore_Jonas.html

09-Aug-2001 13:01

10k

Moore_Robert.html

10-Jan-2002 15:43

16k

Morawetz.html

27-Aug-2001 16:15

12k

Mordell.html

09-Aug-2001 13:01

7k

Mori.html

23-Nov-2001 13:52

10k

Moriarty.html

08-Feb-2002 17:10

5k

Morin.html

09-Aug-2001 13:01

4k

Morin_Jean-Baptiste...> 09-Aug-2001 13:01

8k

Morley.html

09-Aug-2001 13:01

8k

Morse.html

09-Aug-2001 13:01

8k

Mostowski.html

27-Aug-2001 16:15

6k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (40 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Motzkin.html

08-Oct-2001 14:36

11k

Moufang.html

27-Aug-2001 16:15

9k

Mouton.html

09-Aug-2001 13:01

4k

Muir.html

09-Aug-2001 13:01

7k

Mumford.html

23-Nov-2001 13:52

9k

Mydorge.html

23-Nov-2001 13:52

7k

Mytropolshy.html

27-Aug-2001 16:15

7k

Naimark.html

23-Nov-2001 13:52

8k

Napier.html

23-Nov-2001 13:52

20k

Narayana.html

09-Aug-2001 13:01

7k

Nash.html

10-Jan-2002 15:43

20k

Navier.html

23-Nov-2001 13:52

12k

Neile.html

23-Nov-2001 13:52

6k

Nekrasov.html

09-Aug-2001 13:01

8k

Netto.html

09-Aug-2001 13:01

9k

Neuberg.html

09-Aug-2001 13:01

7k

Neugebauer.html

09-Aug-2001 13:01

12k

Neumann_Bernhard.html

31-Oct-2001 09:23

16k

Neumann_Carl.html

09-Aug-2001 13:01

8k

Neumann_Franz.html

09-Aug-2001 13:01

11k

Neumann_Hanna.html

10-Dec-2001 16:56

7k

Nevanlinna.html

09-Aug-2001 13:01

7k

Newcomb.html

09-Aug-2001 13:01

9k

Newman.html

09-Aug-2001 13:01

10k

Newton.html

28-Nov-2001 16:11

34k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (41 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Neyman.html

09-Aug-2001 13:01

7k

Nicolson.html

27-Aug-2001 16:15

7k

Nicomachus.html

23-Nov-2001 13:52

14k

Nicomedes.html

23-Nov-2001 13:52

9k

Nielsen.html

31-Oct-2001 09:23

6k

Nielsen_Jakob.html

31-Oct-2001 09:23

14k

Nightingale.html

08-Feb-2002 17:10

15k

Nilakantha.html

31-Oct-2001 09:23

10k

Niven.html

09-Aug-2001 13:01

6k

Noether_Emmy.html

23-Nov-2001 13:52

13k

Noether_Max.html

09-Aug-2001 13:01

6k

Novikov.html

23-Nov-2001 13:52

10k

Novikov_Sergi.html

23-Nov-2001 13:52

10k

Nunez.html

09-Aug-2001 13:01

7k

Ockham.html

09-Aug-2001 13:01

5k

Oenopides.html

09-Aug-2001 13:01

11k

Ohm.html

27-Aug-2001 16:15

17k

Oka.html

09-Aug-2001 13:01

6k

Olivier.html

09-Aug-2001 13:01

4k

Oresme.html

23-Nov-2001 13:52

6k

Orlicz.html

23-Nov-2001 13:52

21k

Ortega.html

28-Nov-2001 16:11

5k

Osgood.html

09-Aug-2001 13:01

7k

Osipovsky.html

09-Aug-2001 13:01

5k

Ostrogradski.html

23-Nov-2001 13:52

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (42 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Ostrowski.html

09-Aug-2001 13:01

16k

Oughtred.html

23-Nov-2001 13:52

9k

Ozanam.html

09-Aug-2001 13:01

6k

Pacioli.html

23-Nov-2001 13:52

19k

Pade.html

09-Aug-2001 13:01

5k

Padoa.html

09-Aug-2001 13:01

7k

Painleve.html

28-Aug-2001 16:02

6k

Paley.html

09-Aug-2001 13:01

8k

Paman.html

09-Aug-2001 13:01

16k

Panini.html

10-Jan-2002 15:43

11k

Papin.html

09-Aug-2001 13:01

7k

Pappus.html

28-Nov-2001 16:11

22k

Paramesvara.html

10-Jan-2002 15:43

9k

Pars.html

09-Aug-2001 13:01

7k

Parseval.html

09-Aug-2001 13:01

11k

Pascal.html

28-Nov-2001 16:11

20k

Pascal_Etienne.html

09-Aug-2001 13:01

6k

Pasch.html

09-Aug-2001 13:01

5k

Patodi.html

09-Aug-2001 13:01

7k

Pauli.html

10-Jan-2002 15:43

20k

Peacock.html

23-Nov-2001 13:52

8k

Peano.html

28-Nov-2001 16:11

22k

Pearson.html

22-Jan-2002 10:38

6k

Pearson_Egon.html

22-Jan-2002 10:38

14k

Peirce_Benjamin.html

23-Nov-2001 13:52

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (43 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Peirce_Charles.html

09-Aug-2001 13:01

8k

Pell.html

10-Jan-2002 15:43

7k

Penney.html

09-Aug-2001 13:01

8k

Peres.html

31-Oct-2001 09:23

8k

Perron.html

09-Aug-2001 13:01

9k

Perseus.html

09-Aug-2001 13:01

7k

Peter.html

09-Aug-2001 13:01

8k

Petersen.html

22-Jan-2002 10:38

6k

Peterson.html

09-Aug-2001 13:01

6k

Petit.html

09-Aug-2001 13:01

5k

Petrovsky.html

09-Aug-2001 13:01

7k

Petryshyn.html

09-Aug-2001 13:01

7k

Petzval.html

09-Aug-2001 13:01

8k

Peurbach.html

23-Nov-2001 13:52

6k

Pfaff.html

23-Nov-2001 13:52

11k

Pfeiffer.html

09-Aug-2001 13:01

6k

Philon.html

09-Aug-2001 13:01

9k

Picard_Emile.html

10-Jan-2002 15:43

14k

Picard_Jean.html

09-Aug-2001 13:01

6k

Pieri.html

09-Aug-2001 13:01

8k

Pillai.html

09-Aug-2001 13:01

7k

Pincherle.html

09-Aug-2001 13:01

8k

Pitiscus.html

23-Nov-2001 13:52

5k

Plana.html

09-Aug-2001 13:01

10k

Planck.html

23-Nov-2001 13:52

11k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (44 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Plateau.html

09-Aug-2001 13:01

6k

Plato.html

28-Nov-2001 16:11

21k

Playfair.html

09-Aug-2001 13:01

24k

Plessner.html

09-Aug-2001 13:01

9k

Plucker.html

23-Nov-2001 13:52

6k

Poincare.html

23-Nov-2001 13:52

29k

Poinsot.html

10-Jan-2002 15:43

10k

Poisson.html

23-Nov-2001 13:52

9k

Poleni.html

09-Aug-2001 13:01

5k

Polozii.html

09-Aug-2001 13:01

6k

Polya.html

09-Aug-2001 13:01

8k

Poncelet.html

23-Nov-2001 13:52

8k

Pontryagin.html

10-Jan-2002 15:43

16k

Poretsky.html

08-Oct-2001 14:36

6k

Porphyry.html

23-Nov-2001 13:52

10k

Porta.html

09-Aug-2001 13:01

8k

Posidonius.html

09-Aug-2001 13:01

11k

Post.html

10-Jan-2002 15:43

17k

Potapov.html

09-Aug-2001 13:01

8k

Pratt.html

09-Aug-2001 13:01

7k

Pringsheim.html

09-Aug-2001 13:01

4k

Privalov.html

09-Aug-2001 13:01

6k

Privat_de_Molieres.html 09-Aug-2001 13:01

6k

Proclus.html

08-Feb-2002 17:10

16k

Prthudakasvami.html

09-Aug-2001 13:01

5k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (45 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Prufer.html

09-Aug-2001 13:01

5k

Ptolemy.html

23-Nov-2001 13:52

31k

Puiseux.html

31-Oct-2001 09:23

9k

Puissant.html

09-Aug-2001 13:01

4k

Pythagoras.html

28-Nov-2001 16:11

27k

Qadi_Zada.html

09-Aug-2001 13:01

11k

Qalasadi.html

09-Aug-2001 13:01

1k

Quetelet.html

10-Jan-2002 15:43

8k

Quillen.html

10-Jan-2002 15:43

12k

Quine.html

10-Jan-2002 15:43

17k

Rademacher.html

09-Aug-2001 13:01

9k

Rado.html

09-Aug-2001 13:01

6k

Rado_Richard.html

09-Aug-2001 13:01

8k

Radon.html

09-Aug-2001 13:01

6k

Rahn.html

28-Nov-2001 16:11

4k

Rajagopal.html

27-Aug-2001 16:15

6k

Ramanujam.html

31-Oct-2001 09:23

14k

Ramanujan.html

23-Nov-2001 13:52

25k

Ramsden.html

09-Aug-2001 13:01

5k

Ramsey.html

09-Aug-2001 13:01

13k

Ramus.html

09-Aug-2001 13:01

12k

Rankin.html

27-Aug-2001 16:15

15k

Rankine.html

31-Oct-2001 09:23

5k

Raphson.html

28-Nov-2001 16:11

10k

Rasiowa.html

09-Aug-2001 13:01

10k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (46 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Rayleigh.html

09-Aug-2001 13:01

23k

Razmadze.html

09-Aug-2001 13:01

7k

Recorde.html

23-Nov-2001 13:52

6k

Rees.html

09-Aug-2001 13:01

12k

Regiomontanus.html

23-Nov-2001 13:52

7k

Reichenbach.html

09-Aug-2001 13:01

6k

Reidemeister.html

23-Nov-2001 13:52

9k

Reiner.html

08-Oct-2001 14:36

8k

Remak.html

09-Aug-2001 13:01

12k

Remez.html

09-Aug-2001 13:01

6k

Renyi.html

27-Aug-2001 16:15

7k

Rey_Pastor.html

09-Aug-2001 13:01

7k

Reye.html

09-Aug-2001 13:01

7k

Reynaud.html

09-Aug-2001 13:01

7k

Reyneau.html

09-Aug-2001 13:01

6k

Reynolds.html

23-Nov-2001 13:52

10k

Rheticus.html

23-Nov-2001 13:52

14k

Riccati.html

23-Nov-2001 13:52

6k

Riccati_Vincenzo.html

10-Jan-2002 15:43

7k

Ricci-Curbastro.html

23-Nov-2001 13:52

15k

Ricci.html

09-Aug-2001 13:01

6k

Ricci_Matteo.html

09-Aug-2001 13:01

8k

Richard_Jules.html

09-Aug-2001 13:01

5k

Richard_Louis.html

09-Aug-2001 13:01

4k

Richardson.html

23-Nov-2001 13:52

6k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (47 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Richer.html

09-Aug-2001 13:01

7k

Richmond.html

09-Aug-2001 13:01

10k

Riemann.html

23-Nov-2001 13:52

29k

Ries.html

23-Nov-2001 13:52

7k

Riesz.html

23-Nov-2001 13:52

12k

Riesz_Marcel.html

28-Aug-2001 16:02

6k

Ringrose.html

09-Aug-2001 13:01

8k

Roberts.html

09-Aug-2001 13:01

7k

Roberval.html

23-Nov-2001 13:52

8k

Robins.html

09-Aug-2001 13:01

6k

Robinson.html

28-Aug-2001 16:02

15k

Robinson_Julia.html

09-Aug-2001 13:01

8k

Rocard.html

09-Aug-2001 13:01

6k

Rogers.html

27-Aug-2001 16:15

8k

Rohn.html

27-Aug-2001 16:15

5k

Rolle.html

23-Nov-2001 13:52

7k

Roomen.html

23-Nov-2001 13:52

8k

Rosanes.html

09-Aug-2001 13:01

6k

Rosenhain.html

09-Aug-2001 13:01

5k

Rota.html

27-Aug-2001 16:15

15k

Roth.html

27-Aug-2001 16:15

5k

Roth_Klaus.html

27-Aug-2001 16:15

12k

Routh.html

09-Aug-2001 13:01

6k

Rudio.html

09-Aug-2001 13:01

7k

Rudolff.html

23-Nov-2001 13:52

6k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (48 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Ruffini.html

23-Nov-2001 13:52

21k

Runge.html

23-Nov-2001 13:52

7k

Russell.html

23-Nov-2001 13:52

17k

Russell_Scott.html

31-Oct-2001 09:23

9k

Rutherford.html

23-Nov-2001 13:52

6k

Rydberg.html

19-Sep-2001 10:05

7k

SGravesande.html

09-Aug-2001 13:02

5k

Saccheri.html

23-Nov-2001 13:52

7k

Sacrobosco.html

23-Nov-2001 13:52

6k

Saint-Venant.html

09-Aug-2001 13:01

8k

Saint-Vincent.html

09-Aug-2001 13:01

7k

Saks.html

09-Aug-2001 13:01

12k

Salem.html

10-Jan-2002 15:43

14k

Salmon.html

09-Aug-2001 13:01

9k

Samoilemko.html

09-Aug-2001 13:01

6k

Samoilenko.html

31-Oct-2001 09:23

6k

Sang.html

09-Aug-2001 13:01

6k

Sankara.html

10-Jan-2002 15:43

8k

Sasaki.html

31-Oct-2001 09:23

11k

Saurin.html

27-Aug-2001 16:15

6k

Savage.html

09-Aug-2001 13:01

7k

Savart.html

09-Aug-2001 13:01

4k

Savary.html

09-Aug-2001 13:01

4k

Savile.html

09-Aug-2001 13:01

12k

Schatten.html

09-Aug-2001 13:01

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (49 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Schauder.html

09-Aug-2001 13:01

12k

Scheffe.html

09-Aug-2001 13:01

10k

Scheffers.html

09-Aug-2001 13:01

5k

Schickard.html

23-Nov-2001 13:52

6k

Schlafli.html

09-Aug-2001 13:01

6k

Schlomilch.html

09-Aug-2001 13:01

4k

Schmidt.html

09-Aug-2001 13:01

15k

Schoenberg.html

09-Aug-2001 13:01

11k

Schonflies.html

23-Nov-2001 13:52

8k

Schooten.html

23-Nov-2001 13:52

6k

Schottky.html

09-Aug-2001 13:01

7k

Schoute.html

09-Aug-2001 13:01

7k

Schouten.html

09-Aug-2001 13:01

8k

Schreier.html

23-Nov-2001 13:52

11k

Schroder.html

27-Aug-2001 16:15

6k

Schrodinger.html

28-Aug-2001 16:02

24k

Schroeter.html

09-Aug-2001 13:01

6k

Schubert.html

09-Aug-2001 13:01

5k

Schur.html

08-Oct-2001 14:36

18k

Schwartz.html

23-Nov-2001 13:52

10k

Schwarz.html

10-Jan-2002 15:43

14k

Schwarz_Stefan.html

09-Aug-2001 13:01

11k

Schwarzschild.html

09-Aug-2001 13:02

8k

Schwinger.html

10-Jan-2002 15:43

15k

Scott.html

09-Aug-2001 13:02

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (50 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Segner.html

09-Aug-2001 13:02

7k

Segre_Beniamino.html

09-Aug-2001 13:02

8k

Segre_Corrado.html

19-Sep-2001 10:05

9k

Seidel.html

09-Aug-2001 13:02

8k

Seifert.html

10-Jan-2002 15:43

16k

Seki.html

23-Nov-2001 13:52

9k

Selberg.html

28-Nov-2001 16:11

12k

Selten.html

23-Nov-2001 13:52

7k

Semple.html

09-Aug-2001 13:02

6k

Serenus.html

09-Aug-2001 13:02

8k

Serre.html

23-Nov-2001 13:52

11k

Serret.html

09-Aug-2001 13:02

6k

Servois.html

09-Aug-2001 13:02

8k

Severi.html

19-Sep-2001 10:05

10k

Shanks.html

23-Nov-2001 13:52

6k

Shannon.html

23-Nov-2001 13:52

12k

Sharkovsky.html

09-Aug-2001 13:02

6k

Shatunovsky.html

09-Aug-2001 13:02

6k

Shen.html

23-Nov-2001 13:52

6k

Shewhart.html

09-Aug-2001 13:02

4k

Shields.html

09-Aug-2001 13:02

6k

Shnirelman.html

27-Aug-2001 16:15

6k

Shoda.html

10-Jan-2002 15:43

12k

Shtokalo.html

27-Aug-2001 16:15

7k

Siacci.html

19-Sep-2001 10:05

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (51 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Siegel.html

23-Nov-2001 13:52

22k

Sierpinski.html

23-Nov-2001 13:52

19k

Siguenza.html

09-Aug-2001 13:02

7k

Simplicius.html

09-Aug-2001 13:02

12k

Simpson.html

28-Nov-2001 16:11

7k

Simson.html

28-Nov-2001 16:11

17k

Sinan.html

09-Aug-2001 13:02

9k

Sintsov.html

09-Aug-2001 13:02

6k

Sitter.html

09-Aug-2001 13:02

8k

Skolem.html

09-Aug-2001 13:02

5k

Slaught.html

09-Aug-2001 13:02

7k

Sleszynski.html

09-Aug-2001 13:02

6k

Slutsky.html

09-Aug-2001 13:02

6k

Sluze.html

23-Nov-2001 13:52

7k

Smale.html

23-Nov-2001 13:52

12k

Smirnov.html

09-Aug-2001 13:02

8k

Smith.html

09-Aug-2001 13:02

8k

Sneddon.html

10-Jan-2002 15:43

14k

Snell.html

23-Nov-2001 13:52

7k

Snyder.html

09-Aug-2001 13:02

5k

Sobolev.html

23-Nov-2001 13:52

14k

Sokhotsky.html

09-Aug-2001 13:02

7k

Sokolov.html

09-Aug-2001 13:02

6k

Somerville.html

31-Aug-2001 16:25

19k

Sommerfeld.html

09-Aug-2001 13:02

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (52 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Sommerville.html

09-Aug-2001 13:02

8k

Somov.html

23-Nov-2001 13:52

5k

Sonin.html

09-Aug-2001 13:02

5k

Spanier.html

09-Aug-2001 13:02

10k

Spence.html

09-Aug-2001 13:02

5k

Sporus.html

09-Aug-2001 13:02

7k

Spottiswoode.html

09-Aug-2001 13:02

8k

Sridhara.html

23-Nov-2001 13:52

9k

Sripati.html

09-Aug-2001 13:02

8k

Stackel.html

09-Aug-2001 13:02

5k

Stampioen.html

09-Aug-2001 13:02

6k

Steenrod.html

23-Nov-2001 13:52

11k

Stefan_Josef.html

09-Aug-2001 13:02

5k

Stefan_Peter.html

10-Jan-2002 15:43

7k

Steiner.html

28-Nov-2001 16:11

8k

Steinhaus.html

23-Nov-2001 13:52

17k

Steinitz.html

23-Nov-2001 13:52

7k

Steklov.html

09-Aug-2001 13:02

14k

Stepanov.html

09-Aug-2001 13:02

6k

Stevin.html

23-Nov-2001 13:52

7k

Stewart.html

09-Aug-2001 13:02

6k

Stewartson.html

09-Aug-2001 13:02

6k

Stieltjes.html

09-Aug-2001 13:02

16k

Stifel.html

23-Nov-2001 13:52

7k

Stirling.html

28-Nov-2001 16:11

25k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (53 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Stokes.html

23-Nov-2001 13:52

21k

Stolz.html

28-Aug-2001 16:02

5k

Stone.html

09-Aug-2001 13:02

9k

Stott.html

09-Aug-2001 13:02

6k

Struik.html

09-Aug-2001 13:02

12k

Study.html

23-Nov-2001 13:52

8k

Sturm.html

27-Aug-2001 16:15

15k

Sturm_Rudolf.html

09-Aug-2001 13:02

9k

Subbotin.html

27-Aug-2001 16:15

10k

Suetuna.html

08-Oct-2001 14:36

9k

Suter.html

27-Aug-2001 16:15

4k

Suvorov.html

27-Aug-2001 16:15

6k

Swain.html

09-Aug-2001 13:02

7k

Sylow.html

23-Nov-2001 13:52

9k

Sylvester.html

10-Dec-2001 16:56

15k

Synge.html

09-Aug-2001 13:02

12k

Szasz.html

09-Aug-2001 13:02

10k

Szego.html

09-Aug-2001 13:02

8k

Tacquet.html

09-Aug-2001 13:02

5k

Tait.html

10-Jan-2002 15:43

28k

Takagi.html

23-Nov-2001 13:52

15k

Talbot.html

09-Aug-2001 13:02

7k

Taniyama.html

23-Nov-2001 13:52

7k

Tannery_Jules.html

23-Nov-2001 13:52

9k

Tannery_Paul.html

09-Aug-2001 13:02

13k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (54 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Tarry.html

09-Aug-2001 13:02

4k

Tarski.html

23-Nov-2001 13:52

7k

Tartaglia.html

23-Nov-2001 13:52

17k

Tauber.html

09-Aug-2001 13:02

5k

Taurinus.html

09-Aug-2001 13:02

8k

Taussky-Todd.html

09-Aug-2001 13:02

9k

Taylor.html

28-Nov-2001 16:11

19k

Taylor_Geoffrey.html

09-Aug-2001 13:02

9k

Teichmuller.html

09-Aug-2001 13:02

6k

Temple.html

09-Aug-2001 13:02

8k

Tetens.html

09-Aug-2001 13:02

4k

Thabit.html

23-Nov-2001 13:52

20k

Thales.html

28-Nov-2001 16:11

21k

Theaetetus.html

09-Aug-2001 13:02

16k

Theodorus.html

23-Nov-2001 13:52

11k

Theodosius.html

09-Aug-2001 13:02

12k

Theon.html

23-Nov-2001 13:52

12k

Theon_of_Smyrna.html

09-Aug-2001 13:02

11k

Thiele.html

09-Aug-2001 13:02

5k

Thom.html

23-Nov-2001 13:52

12k

Thomae.html

09-Aug-2001 13:02

5k

Thomason.html

09-Aug-2001 13:02

12k

Thompson_D'Arcy.html

09-Aug-2001 13:02

8k

Thompson_John.html

23-Nov-2001 13:52

15k

Thomson.html

10-Dec-2001 16:56

24k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (55 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Thue.html

09-Aug-2001 13:02

8k

Thurston.html

23-Nov-2001 13:52

10k

Thymaridas.html

09-Aug-2001 13:02

6k

Tibbon.html

09-Aug-2001 13:02

6k

Tietze.html

09-Aug-2001 13:02

11k

Tikhonov.html

09-Aug-2001 13:02

12k

Tilly.html

09-Aug-2001 13:02

6k

Tinbergen.html

09-Aug-2001 13:02

9k

Tinseau.html

09-Aug-2001 13:02

5k

Tisserand.html

09-Aug-2001 13:02

9k

Titchmarsh.html

09-Aug-2001 13:02

16k

Todd.html

09-Aug-2001 13:02

11k

Todhunter.html

09-Aug-2001 13:02

12k

Toeplitz.html

09-Aug-2001 13:02

11k

Torricelli.html

28-Nov-2001 16:11

8k

Trail.html

09-Aug-2001 13:02

4k

Tricomi.html

19-Sep-2001 10:05

9k

Troughton.html

09-Aug-2001 13:02

8k

Tschirnhaus.html

23-Nov-2001 13:52

7k

Tsu.html

23-Nov-2001 13:52

6k

Tukey.html

23-Nov-2001 13:52

10k

Tunstall.html

23-Nov-2001 13:52

5k

Turan.html

27-Aug-2001 16:15

6k

Turing.html

23-Nov-2001 13:52

25k

Turnbull.html

09-Aug-2001 13:02

17k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (56 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Turner.html

09-Aug-2001 13:02

5k

Uhlenbeck.html

10-Jan-2002 15:43

16k

Uhlenbeck_Karen.html

09-Aug-2001 13:02

13k

Ulam.html

09-Aug-2001 13:02

15k

Ulugh_Beg.html

23-Nov-2001 13:52

13k

Upton.html

09-Aug-2001 13:02

6k

Urysohn.html

10-Jan-2002 15:43

15k

Vacca.html

27-Aug-2001 16:15

8k

Vailati.html

09-Aug-2001 13:02

8k

Valerio.html

09-Aug-2001 13:02

7k

Vallee_Poussin.html

10-Jan-2002 15:43

17k

Van_Ceulen.html

09-Aug-2001 13:02

6k

Van_Vleck.html

09-Aug-2001 13:02

8k

Van_der_Waerden.html

23-Nov-2001 13:52

11k

Vandermonde.html

10-Jan-2002 15:43

14k

Vandiver.html

31-Aug-2001 16:25

7k

Varahamihira.html

23-Nov-2001 13:52

9k

Varignon.html

09-Aug-2001 13:02

5k

Vashchenko.html

09-Aug-2001 13:02

6k

Veblen.html

09-Aug-2001 13:02

11k

Vega.html

09-Aug-2001 13:02

6k

Venn.html

23-Nov-2001 13:52

8k

Verhulst.html

09-Aug-2001 13:02

7k

Vernier.html

09-Aug-2001 13:02

6k

Veronese.html

19-Sep-2001 10:05

9k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (57 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Vessiot.html

09-Aug-2001 13:02

6k

Viete.html

10-Jan-2002 15:43

23k

Vijayanandi.html

09-Aug-2001 13:02

6k

Vinogradov.html

23-Nov-2001 13:52

11k

Vitali.html

27-Aug-2001 16:15

6k

Viviani.html

23-Nov-2001 13:52

8k

Vlacq.html

09-Aug-2001 13:02

5k

Volterra.html

23-Nov-2001 13:52

9k

Von_Dyck.html

09-Aug-2001 13:02

10k

Von_Neumann.html

23-Nov-2001 13:52

24k

Von_Staudt.html

23-Nov-2001 13:52

6k

Voronoy.html

09-Aug-2001 13:02

6k

Vranceanu.html

09-Aug-2001 13:02

12k

Wald.html

09-Aug-2001 13:02

14k

Walker_Arthur.html

09-Aug-2001 13:02

8k

Walker_John.html

09-Aug-2001 13:02

8k

Wall.html

09-Aug-2001 13:02

9k

Wallace.html

28-Nov-2001 16:11

6k

Wallis.html

23-Nov-2001 13:52

13k

Wang.html

09-Aug-2001 13:02

9k

Wangerin.html

09-Aug-2001 13:02

8k

Wantzel.html

23-Nov-2001 13:52

12k

Waring.html

10-Jan-2002 15:43

15k

Watson.html

09-Aug-2001 13:02

7k

Watson_Henry.html

09-Aug-2001 13:02

8k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (58 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Wazewski.html

09-Aug-2001 13:02

10k

Weatherburn.html

09-Aug-2001 13:02

8k

Weber.html

09-Aug-2001 13:02

6k

Weber_Heinrich.html

23-Nov-2001 13:52

9k

Wedderburn.html

09-Aug-2001 13:02

17k

Weierstrass.html

10-Jan-2002 15:43

25k

Weil.html

23-Nov-2001 13:52

13k

Weingarten.html

09-Aug-2001 13:02

9k

Weinstein.html

09-Aug-2001 13:02

8k

Weisbach.html

27-Aug-2001 16:15

7k

Weldon.html

09-Aug-2001 13:02

8k

Werner.html

09-Aug-2001 13:02

5k

Wessel.html

23-Nov-2001 13:52

14k

West.html

09-Aug-2001 13:02

4k

Weyl.html

23-Nov-2001 13:52

9k

Weyr.html

09-Aug-2001 13:02

5k

Wheeler.html

09-Aug-2001 13:02

9k

Whiston.html

09-Aug-2001 13:02

8k

White.html

09-Aug-2001 13:02

8k

Whitehead.html

08-Feb-2002 17:10

19k

Whitehead_Henry.html

28-Aug-2001 16:02

16k

Whitney.html

09-Aug-2001 13:02

7k

Whittaker.html

09-Aug-2001 13:02

12k

Whittaker_John.html

09-Aug-2001 13:02

8k

Whyburn.html

08-Oct-2001 14:36

12k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (59 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Widman.html

23-Nov-2001 13:52

5k

Wielandt.html

28-Aug-2001 16:02

10k

Wien.html

09-Aug-2001 13:02

8k

Wiener_Christian.html

09-Aug-2001 13:02

5k

Wiener_Norbert.html

23-Nov-2001 13:52

8k

Wigner.html

09-Aug-2001 13:02

21k

Wilczynski.html

09-Aug-2001 13:02

9k

Wiles.html

23-Nov-2001 13:52

14k

Wilkins.html

23-Nov-2001 13:52

8k

Wilkinson.html

09-Aug-2001 13:02

8k

Wilks.html

09-Aug-2001 13:02

8k

Wilson_Alexander.html

09-Aug-2001 13:02

5k

Wilson_Edwin.html

09-Aug-2001 13:02

9k

Wilson_John.html

31-Oct-2001 09:23

6k

Winkler.html

27-Aug-2001 16:15

8k

Wintner.html

10-Jan-2002 15:43

15k

Wirtinger.html

27-Aug-2001 16:15

10k

Wishart.html

27-Aug-2001 16:15

11k

Witt.html

27-Aug-2001 16:15

7k

Witten.html

23-Nov-2001 13:52

13k

Wittgenstein.html

28-Aug-2001 16:02

16k

Wolf.html

31-Oct-2001 09:23

6k

Wolfowitz.html

08-Oct-2001 14:36

10k

Wolstenholme.html

09-Aug-2001 13:02

9k

Woodhouse.html

09-Aug-2001 13:02

6k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (60 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Woodward.html

09-Aug-2001 13:02

6k

Wren.html

23-Nov-2001 13:52

9k

Wronski.html

09-Aug-2001 13:02

7k

Xenocrates.html

09-Aug-2001 13:02

10k

Yang.html

23-Nov-2001 13:52

5k

Yates.html

09-Aug-2001 13:02

7k

Yativrsabha.html

09-Aug-2001 13:02

5k

Yau.html

23-Nov-2001 13:52

11k

Yavanesvara.html

09-Aug-2001 13:02

6k

Yoccoz.html

23-Nov-2001 13:52

10k

Youden.html

27-Aug-2001 16:15

8k

Young.html

27-Aug-2001 16:15

12k

Young_Alfred.html

31-Oct-2001 09:23

14k

Yule.html

27-Aug-2001 16:15

14k

Yunus.html

10-Jan-2002 15:43

12k

Yushkevich.html

09-Aug-2001 13:02

7k

Zarankiewicz.html

09-Aug-2001 13:02

11k

Zaremba.html

09-Aug-2001 13:02

12k

Zariski.html

09-Aug-2001 13:02

17k

Zassenhaus.html

09-Aug-2001 13:02

10k

Zeckendorf.html

09-Aug-2001 13:02

6k

Zeeman.html

27-Aug-2001 16:15

11k

Zelmanov.html

23-Nov-2001 13:52

13k

Zeno_of_Elea.html

10-Jan-2002 15:43

21k

Zeno_of_Sidon.html

09-Aug-2001 13:02

12k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (61 of 62) [2/16/2002 10:55:56 PM]

Index of /~history/Mathematicians

Zenodorus.html

09-Aug-2001 13:02

7k

Zermelo.html

23-Nov-2001 13:52

15k

Zeuthen.html

09-Aug-2001 13:02

11k

Zhukovsky.html

23-Nov-2001 13:52

10k

Zolotarev.html

09-Aug-2001 13:02

9k

Zorn.html

23-Nov-2001 13:52

5k

Zuse.html

27-Aug-2001 16:15

12k

Zygmund.html

09-Aug-2001 13:02

7k

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/ (62 of 62) [2/16/2002 10:55:56 PM]

Abbe

Ernst Abbe Born: 23 Jan 1840 in Eisenach, Grand Duchy of Saxe-Weimar-Eisenach (now in Germany) Died: 14 Jan 1905 in Jena, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernst Abbe's father worked as a spinner who found it extremely difficult to provide for his family. Kendall, writing in [2], describes Ernst's childhood:His childhood was one of privation, his father worked on his feet 16 hours every day with no breaks for meals. Ernst, however, won scholarships and was helped through his studies by his father's employer. Abbe studied at the University of Jena and the University of Göttingen, receiving his doctorate from Göttingen in 1861 with a dissertation on thermodynamics. In 1863 he joined the teaching staff at the University of Jena and he presented the paper Über di Gesetzmässigkeit in der Vertheilung bei Beobachtungsreihen for his teaching qualification. M G Kendall, see [2] or [3], writes:O B Sheynin has recently called attention to a most remarkable paper by Ernst Abbe, presented in 1863, in which Abbe derives not only the c2 distribution, but R L Anderson's (1942) distribution of the serial correlation coefficient. ... The paper is a superbly competent piece of work and perhaps the most remarkable anticipation of later studies of distribution theory that have yet come to light. Abbe was appointed professor of physics and mathematics at Jena in 1870 and, in 1878, he was appointed director of the astronomical observatory at Jena and of the meteorological observatory at Jena. However, Abbe had been approached by Carl Zeiss in 1866 with various optical problems. This turned http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abbe.html (1 of 3) [2/16/2002 10:55:59 PM]

Abbe

his attention towards optics and astronomy. In addition to his university posts, Abbe was made research director of the Zeiss optical works in 1866. In 1868 he invented the apochromatic lens system for the microscope. This important breakthrough eliminates both the primary and secondary colour distortion of microscopes. Other optical advances which Abbe made include a clearer theoretical understanding of limits to magnification and the discovery the Abbe sine condition, as it is called today, which gives conditions on a lens for it to form a sharp image, without the defects of coma and spherical aberration. He also made practical improvements in microscope design including, in 1870, the use of a condenser to give a high-powered even illumination of the field of view. The Carl Zeiss Foundation describes Abbe's work at this time as follows:One year after beginning the manufacture of the Carl Zeiss compound microscope, in 1873, Herr Abbe released a scientific paper describing the mathematics leading to the perfection of this wonderful invention. For the first time in optical design, aberration, diffraction and coma were described and understood. Abbe described the optical process so well that this paper has become the foundation upon which much of our understanding of optical science rests today. As a reward for his efforts Carl Zeiss made Abbe a partner in his burgeoning business in 1876. Becoming wealthy through his optical work and a partnership with Zeiss, Abbe set up and endowed the Carl Zeiss Foundation for research in science and social improvement in 1891. The Carl Zeiss Foundation describes its setting up as follows:This foundation established a new group as the owners of Carl Zeiss. The greater portion of the assets were deeded to the University of Jena, whose Department of Education managed the universities interests. This authority was bound by a set of statutes drawn up by Abbe himself, after studying sociology and law for two years. The balance of the estate was donated to the employees of Carl Zeiss. Abbe introduced industrial relations changes into the Zeiss optical works in 1896 which today sound commonplace but were many years ahead of their time. These included an 8 hour working day, holiday pay, sick pay and pensions. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country Honours awarded to Ernst Abbe (Click a link below for the full list of mathematicians honoured in this way) Lunar features Other Web sites

Crater Abbe 1. West Chester University 2. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abbe.html (2 of 3) [2/16/2002 10:55:59 PM]

Abbe

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR February 1997 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Abbe.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abbe.html (3 of 3) [2/16/2002 10:55:59 PM]

Abel

Niels Henrik Abel Born: 5 Aug 1802 in Frindoe (near Stavanger), Norway Died: 6 April 1829 in Froland, Norway

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Niels Abel's life was dominated by poverty and we begin by putting this in context by looking briefly at the political problems which led to economic problems in Norway. At the end of the 18th century Norway was part of Denmark and the Danish tried to remain neutral through the Napoleonic wars. However a neutrality treaty in 1794 was considered a aggressive act by England and, in 1801, the English fleet destroyed most of the Danish fleet in a battle in the harbour at Copenhagen. Despite this Denmark-Norway avoided wars until 1807 when England feared that the Danish fleet might be used by the French to invade. Using the philosophy that attack is the best form of defence, the English attacked and captured the whole Danish fleet in October 1807. Denmark then joined the alliance against England. The continental powers blockaded England, and as a counter to this England blockaded Norway. The twin blockade was a catastrophe to Norway preventing their timber exports, which had been largely to Britain, and preventing their grain imports from Denmark. An economic crisis in Norway followed with the people suffering hunger and extreme poverty. In 1813 Sweden attacked Denmark from the south and, at the treaty of Kiel in January 1814, Denmark handed over Norway to Sweden. An attempt at independence by Norway a few months later led to Sweden attacking Norway in July 1814. Sweden gained control of Norway, setting up a complete internal self-government for Norway with a government in Christiania (which is called Oslo today). In this difficult time Abel was growing up in Gjerstad in south-east Norway. Abel's father, Soren Georg Abel, had a degree in theology and philology and his father (Niels Abel's grandfather) was a Protestant minister at Gjerstad near Risor. Soren Abel was a Norwegian nationalist who was active politically in the movement to make Norway independent. Soren Abel married Ane

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abel.html (1 of 6) [2/16/2002 10:56:01 PM]

Abel

Marie Simonson, the daughter of a merchant and ship owner, and was appointed as minister at Finnoy. Niels Abel, the second of seven children, was one year old when his grandfather died and his father was appointed to succeed him as the minister at Gjerstad. It was in that town that Abel was brought up, taught by his father in the vicarage until he reached 13 years of age. However, these were the 13 years of economic crisis for Norway described above and Abel's parents would have not been able to feed their family that well. The problems were not entirely political either for [14]:[Abel's] father was probably a drunkard and his mother was accused of having lax morals. Abel's father was, however, important in the politics of Norway and, after Sweden gained control of Norway in 1814, he was involved in writing a new constitution for Norway as a member of the Storting, the Norwegian legislative body. In 1815 Abel and his older brother were sent to the Cathedral School in Christiania. The founding of the University of Christiania had taken away the good teachers from the Cathedral School to staff the University when it opened for teaching in 1813. What had been a good school was in a bad state when Abel arrived. Uninspired by the poor school, he proved a rather ordinary pupil with some talent for mathematics and physics. When a new mathematics teacher Bernt Holmboe joined the school in 1817 things changed markedly for Abel. The previous mathematics teacher had been dismissed for punishing a boy so severely that he had died. Abel began to study university level mathematics texts and, within a year of Holmboe's arrival, Abel was reading the works of Euler, Newton, Lalande and d'Alembert. Holmboe was convinced that Abel had great talent and encouraged him greatly taking him on to study the works of Lagrange and Laplace. However, in 1820 tragedy struck Abel's family when his father died. Abel's father had ended his political career in disgrace by making false charges against his colleagues in the Storting after he was elected to the body again in 1818. His habits of drinking to excess also contributed to his dismissal and the family was therefore in the deepest trouble when he died. There was now no money to allow Abel to complete his school education, nor money to allow him to study at university and, in addition, Abel had the responsibility of supporting his mother and family. Holmboe was able to help Abel gain a scholarship to remain at school and Abel was able to enter the University of Christiania in 1821, ten years after the university was founded. Holmboe had raised money from his colleagues to enable Abel to study at the university and he graduated in 1822. While in his final year at school, however, Abel had begun working on the solution of quintic equations by radicals. He believed that he had solved the quintic in 1821 and submitted a paper to the Danish mathematician Ferdinand Degen, for publication by the Royal Society of Copenhagen. Degen asked Abel to give a numerical example of his method and, while trying to provide an example, Abel discovered the mistake in his paper. Degen had given Abel some important advice that was to set him working on an area of mathematics (see [2]):... whose development would have the greatest consequences for analysis and mechanics. I refer to elliptic integrals. A serious investigator with suitable qualifications for research of this kind would by no means be restricted to the many beautiful properties of these most remarkable functions, but could discover a Strait of Magellan leading into wide expanses of a tremendous analytic ocean. At the University of Christiania Abel found a supporter in the professor of astronomy Christopher Hansteen, who provided both financial support and encouragement. Hansteen's wife began to care for Abel as if he was her own son. In 1823 Abel published papers on functional equations and integrals in a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abel.html (2 of 6) [2/16/2002 10:56:01 PM]

Abel

new scientific journal started up by Hansteen. In Abel's third paper, Solutions of some problems by means of definite integrals he gave the first solution of an integral equation. Abel was given a small grant to visit Degen and other mathematicians in Copenhagen. While there he met Christine Kemp who shortly afterwards became his fiancée. Returning to Christiania, Abel tried to get the University of Christiania to give him a larger grant to enable him to visit the top mathematicians in Germany and France. He did not speak French of German so, partly to save money, he was given funds to remain in Christiania for two years to give him the chance to become fluent in these languages before travelling. Abel began working again on quintic equations and, in 1824, he proved the impossibility of solving the general equation of the fifth degree in radicals. He published the work in French and at his own expense since he wanted an impressive piece of work to take with him when he was on his travels. As Ayoub writes in [6]:He chose a pamphlet as the quickest way to get it into print, and in order to save on the printing costs, he reduced the proof to fit on half a folio sheet [six pages]. By this time Abel seems to have known something of Ruffini's work for he had studied Cauchy's work of 1815 while he was an undergraduate and in this paper there is a reference to Ruffini's work. Abel's 1824 paper begins ([6]):Geometers have occupied themselves a great deal with the general solution of algebraic equations and several among them have sought to prove the impossibility. But, if I am not mistaken, they have not succeeded up to the present. Abel sent this pamphlet to several mathematicians including Gauss, who he intended to visit in Göttingen while on his travels. In August 1825 Abel was given a scholarship from the Norwegian government to allow him to travel abroad and, after taking a month to settle his affairs, he set out for the Continent with four friends, first visiting mathematicians in Norway and Denmark. On reaching Copenhagen, Abel found that Degen had died and he changed his mind about taking Hansteen's advice to go directly to Paris, preferring not to travel alone and stay with his friends who were going to Berlin. As he wrote in a later letter ([7]):Now I am so constituted that I cannot endure solitude. Alone, I am depressed, I get cantankerous, and I have little inclination to work. In Copenhagen Abel was given a letter of introduction to Crelle by one of the mathematicians there. Abel met Crelle in Berlin and the two became firm friends. This proved the most useful part of Abel's whole trip, particularly as Crelle was about to begin publishing a journal devoted to mathematical research. Abel was encouraged by Crelle to write a clearer version of his work on the insolubility of the quintic and this resulted in Recherches sur les fonctions elliptiques which was published in 1827 in the first volume of Crelle's Journal, along with six other papers by Abel. While in Berlin, Abel learnt that the position of professor of mathematics at the University of Christiania, the only university in Norway, had been given to Holmboe. With no prospects of a university post in Norway, Abel began to worry about his future. Crelle's Journal continued to be a source for Abel's papers and Abel began to work to establish mathematical analysis on a rigorous basis. He wrote to Holmboe from Berlin [2]:My eyes have been opened in the most surprising manner. If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum had been

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abel.html (3 of 6) [2/16/2002 10:56:01 PM]

Abel

rigorously determined. In other words, the most important parts of mathematics stand without foundation. It is true that most of it is valid, but that is very surprising. I struggle to find a reason for it, an exceedingly interesting problem. It had been Abel's intention to travel with Crelle to Paris and to visit Gauss in Göttingen on the way. However, news got back to Abel that Gauss was not pleased to receive his work on the insolubility of the quintic, so Abel decided that he would be better not to go to Göttingen. It is uncertain why Gauss took this attitude towards Abel's work since he certainly never read it - the paper was found unopened after Gauss's death. Ayoub gives two possible reasons [6]:... the first possibility is that Gauss had proved the result himself and was willing to let Abel take the credit. ... The other explanation is that he did not attach very much importance to solvability by radicals... The second of these explanations does seem the more likely, especially since Gauss had written in his thesis of 1801 that the algebraic solution of an equation was no better than devising a symbol for the root of the equation and then saying that the equation had a root equal to the symbol. Crelle was detained in Berlin and could not travel with Abel to Paris. Abel therefore did not go directly to Paris, but chose to travel again with his Norwegian friends to northern Italy before crossing the Alps to France. In Paris Abel was disappointed to find there was little interest in his work. He wrote back to Holmboe ([7]):The French are much more reserved with strangers than the Germans. It is extremely difficult to gain their intimacy, and I do not dare to urge my pretensions as far as that; finally every beginner had a great deal of difficulty getting noticed here. I have just finished an extensive treatise on a certain class of transcendental functions to present it to the Institute which will be done next Monday. I showed it to Mr Cauchy, but he scarcely deigned to glance at it. The contents and importance of this treatise by Abel is described in [2]:It dealt with the sum of integrals of a given algebraic function. Abel's theorem states that any such sum can be expressed as a fixed number p of these integrals, with integration arguments that are algebraic functions of the original arguments. The minimal number p is the genus of the algebraic function, and this is the first occurrence of this fundamental quantity. Abel's theorem is a vast generalisation of Euler's relation for elliptic integrals. Two referees, Cauchy and Legendre, were appointed to referee the paper and Abel remained in Paris for a few months [14]:... emaciated, gloomy, weary and constantly worried. He ... could only afford to eat one meal a day. He published some articles, mainly on the results he had already written for Crelle's Journal, then with no money left and his health in a very poor state, he returned to Berlin at the end of 1826. In Berlin, Abel borrowed some money and continued working on elliptic functions. He wrote a paper in which [2]:... he radically transformed the theory of elliptic integrals to the theory of elliptic functions by using their inverse functions ... Crelle tried to persuade Abel to remain in Berlin until he could find an academic post for him and he even offered Abel the editorship of Crelle's Journal. However, Abel wanted to get home and by this time http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abel.html (4 of 6) [2/16/2002 10:56:01 PM]

Abel

he was heavily in debt. He reached Christiania in May 1827 and was awarded a small amount of money by the university although they made sure they had the right to deduct a corresponding amount from any future salary he earned. To make a little more money Abel tutored schoolchildren and his fiancée was employed as a governess to friends of Abel's family in Froland. Hansteen received a major grant to investigate the Earth's magnetic field in Siberia and a replacement was needed to teach for him at the University and also at the Military Academy. Abel was appointed to this post which improved his position a little. In 1828 Abel was shown a paper by Jacobi on transformations of elliptic integrals. Abel quickly showed that Jacobi's results were consequences of his own and added a note to this effect to the second part of his major work on elliptic functions. He had been working again on the algebraic solution of equations, with the aim of solving the problem of which equations were soluble by radicals (the problem which Galois solved a few years later). He put this to one side to compete with Jacobi in the theory of elliptic functions, quickly writing several papers on the topic. Legendre saw the new ideas in the papers which Abel and Jacobi were writing and said ([2]):Through these works you two will be placed in the class of the foremost analysts of our times. Abel continued to pour out high quality mathematics as his health continued to deteriorate. He spent the summer vacation of 1828 with his fiancée in Froland. The masterpiece which he had submitted to the Paris Academy seemed to have been lost and so he wrote the main result down again [3]:The paper was only two brief pages, but of all his many works perhaps the most poignant. He called it only "A theorem": it had no introduction, contained no superfluous remarks, no applications. It was a monument resplendent in its simple lines - the main theorem from his Paris memoir, formulated in few words. Abel travelled by sled to visit his fiancée again in Froland for Christmas 1828. He became seriously ill on the sled journey and despite an improvement which allowed them to enjoy Christmas, he soon became very seriously ill again. Crelle was told and he redoubled his efforts to obtain an appointment for Abel in Berlin. He succeeded and wrote to Abel on the 8 April 1829 to tell him the good news. It was too late, Abel had already died. Ore [3] describes his last few days:... the weakness and cough increased and he could remain out of bed only the few minutes while it was being made. Occasionally he would attempt to work on his mathematics, but he could no longer write. Sometimes he lived in the past, talking about his poverty and about Fru Hansteen's goodness. Always he was kind and patient. ... He endured his worst agony during the night of April 5. Towards morning he became more quiet and in the forenoon, at eleven o'clock, he expired his last sigh. After Abel's death his Paris memoir was found by Cauchy in 1830 after much searching. It was printed in 1841 but rather remarkably vanished again and was not found until 1952 when it turned up in Florence. Also after Abel's death unpublished work on the algebraic solution of equations was found. In fact in a letter Abel had written to Crelle on 18 October 1828 he gave the theorem [13]:If every three roots of an irreducible equation of prime degree are related to one another in such a way that one of them may be expressed rationally in terms of the other two, then the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abel.html (5 of 6) [2/16/2002 10:56:01 PM]

Abel

equation is soluble in radicals. This result is essentially identical to one given by Galois in his famous memoir of 1830. In this same year 1830 the Paris Academy awarded Abel and Jacobi the Grand Prix for their outstanding work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles)

Some Quotations (6)

A Poster of Niels Abel

Mathematicians born in the same country

Some pages from publications

An extract from Abel's On the algebraic resolution of equations (1824)

Cross-references to History Topics

The development of group theory

Other references in MacTutor

Chronology: 1820 to 1830

Honours awarded to Niels Abel (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Abel

Paris street names

Rue Abel (12th Arrondissement)

Other Web sites

1. Eric's treasure troves 2. Interactive Real Analysis 3. Oslo, Norway 4. Norfolk, Va 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Abel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abel.html (6 of 6) [2/16/2002 10:56:01 PM]

Abraham

Abraham bar Hiyya Ha-Nasi Born: 1070 in Barcelona, Spain Died: 1136 in Provence, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Abraham bar Hiyya was a Spanish Jewish mathematician and astronomer. In the Hebrew of his time 'Ha-Nasi' meant 'the leader' but he is also known by the Latin name Savasorda which comes from his 'job description' showing that he held an official position in the administration in Barcelona. Abraham bar Hiyya is famed for his book Hibbur ha-Meshihah ve-ha-Tishboret (Treatise on Measurement and Calculation), translated into Latin by Plato of Tivoli as Liber embadorum in 1145. This book is the earliest Arab algebra written in Europe. It contains the complete solution of the general quadratic and is the first text in Europe to give such a solution. Rather strangely, however, 1145 was also the year that al-Khwarizmi's algebra book was translated by Robert of Chester so Abraham bar Hiyya's work was rapidly joined by a second text giving the complete solution to the general quadratic equation. It is interesting to see the areas of mathematics and the mathematicians with which Abraham was familiar. Of course he knew geometry through the works of Euclid, but he also knew the contributions to geometry from other Greek texts such as Theodosius's Sphaerics in three books, On the Moving Sphere which is a work on the geometry of the sphere by Autolycus, Apollonius's Conics, and the later contributions by Heron of Alexandria and Menelaus of Alexandria. Abraham had also studied some of the important works on algebra by Arab mathematicians, in particular al-Khwarizmi and al-Karaji. Among other texts written by Abraham bar Hiyya was Yesod ha-Tebunah u-Migdal ha-Emunah (The Foundation of Understanding and the Tower of Faith). This work is an encyclopaedia of mathematics, astronomy, optics and music. It is the first encyclopaedia in the Hebrew language. Abraham also wrote a number of texts on astronomy; in particular he wrote on the form of the Earth and the calculation of the paths of the stars on the celestial sphere. His book Tables of the Prince refers to the tables of al-Battani while Abraham's treatise Sefer ha-Ibbur (Book of Intercalation), written in 1122-23, is the first Hebrew work devoted exclusively to a study of the calendar. In the philosophical treatise Hegyon ha-Nefesh ha-Azuva (Meditation of the Sad Soul) Abraham deals with the nature of good and evil and ethics. Megillat ha-Megalleh (Scroll of the Revealer) outlines Abraham's view of history based on astrology. It claims to forecast the messianic future. Perhaps one of the most important features of Abraham bar Hiyya's work is the fact that it appears to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abraham.html (1 of 2) [2/16/2002 10:56:02 PM]

Abraham

have stimulated an interest in Arabic mathematics and, together with the work of Abraham ibn Ezra, marks the beginning of Hebrew scholarly study of mathematics. As the author of [5] writes:The major part of the mathematical 'classics' in Hebrew were translated from Arabic between the second third of the thirteenth century and the first third of the fourteenth century, within the northern littoral of the western Mediterranean. This movement occurred after the original works by Abraham bar Hiyya and Abraham ibn Ezra became available to a wide readership. It is rather difficult to place Abraham bar Hiyya in the development of mathematics since in most respects he did not fit nicely into one culture but spanned several. It may indeed be for just that reason that he is important since he produced a cross-fertilisation of ideas between these cultures. As Levey (the author of [6]) writes in [1], Abraham:... did not definitely belong definitely to one mathematical group. He spent most of his life in Barcelona, an area of both Arab and Christian learning, and was active in translating the masterpieces of Arab science. ... he deplored the lack of knowledge of Arab science and language among the people of Provence. He wrote his own works in Hebrew, but he helped translate ... works into Latin.... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

A Quotation

Mathematicians born in the same country Cross-references to History Topics

Quadratic, cubic and quartic equations

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Abraham.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abraham.html (2 of 2) [2/16/2002 10:56:02 PM]

Abraham_Max

Max Abraham Born: 26 March 1875 in Danzig (now Gdansk), Germany Died: 16 Nov 1922 in Munich, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Max Abraham was born into a Jewish family who had made considerable amounts of money as merchants. He studied at the University of Berlin under Planck, writing his doctoral dissertation in 1897. After this he spent three years at the University of Berlin working as Planck's assistant. In 1900 Abraham was appointed as a Privatdozent at Göttingen. He lectured at Göttingen as a Privatdozent until 1909 which is an unusual length of time for anyone to hold such an unpaid lecturing position. The reason for his failure to obtain a permanent university position during this period was not due to his ability but rather was a result of his personality. Goldberg writes in [1]:... he had no patience with what he considered to be silly or illogical argumentation. Abraham had a penchant for being critical and had no hesitation in publicly chastising his colleagues, regardless of their rank or position. His sharp wit was matched by an equally sharp tongue, and as a result he remained a Privatdozent at Göttingen for nine years. In 1909 Abraham accepted a post at the University of Illinois in the United States. Disliking the small university atmosphere of Illinois, he returned within a few months to Göttingen, going next to Italy at the invitation of Levi-Civita. Abraham was professor of rational mechanics at the the University of Milan until 1914. During this time Abraham and Einstein disagreed strongly about the theory of relativity in a correspondence discussed in [3] and [4]. Einstein also argued about relativity in a correspondence with Levi-Civita and Abraham played a role in this argument too, see for example [4]. Forced to return to Germany at the start of World War I, Abraham worked on the theory of radio transmission. Unable to return to Milan after the War he worked at Stuttgart until 1921, substituting for http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abraham_Max.html (1 of 3) [2/16/2002 10:56:04 PM]

Abraham_Max

the professor of physics at the Technische Hochschule, when he accepted a chair in Aachen. He took ill on the way to Aachen and a brain tumour was diagnosed. He never recovered and [2]:... just as his life was suffering, his end was full of agony. Abraham's work is almost all related to Maxwell's theory and he wrote a text which was the standard work on electrodynamics in Germany for a long time. Goldberg writes in [1]:His consistent use of vectors was a significant factor in the rapid acceptance of vector notation in Germany. But one of the most noteworthy features of the text was that in each new edition Abraham saw fit to include not only the latest experimental work but also the latest theoretical contributions, even if these contributions were in dispute. Furthermore, he had no hesitation, after explicating both sides of a question, in using the book to argue his own point of view. His theory of the electron was developed in 1902, and its case strongly agrued in his text, but in 1904 Lorentz and Einstein produced a different theory. Abraham's study of the structure and nature of the electron led him to the idea of the electromagnetic nature of its mass, and consequently to the dependence of the velocity of electromagnetic waves in a gravitational field. At first his ideas were supported by experiment, particularly work carried out by Wilhelm Kaufmann, but later work was to favour the theory developed by Lorentz and Einstein. Abraham was opposed to relativity all his life. At first he objected both to the postulates on which relativity was based and also to the fact that he felt that the experimental evidence did not support the theory. By 1912 Abraham, who despite his objections was one of those who best understood relativity theory, was prepared to accept that the theory was logically sound. However, he still did not accept that the theory accurately described the physical world. Abraham had been a strong believer in the existence of the aether and that an electron was a perfectly rigid sphere with a charge distributed evenly over its surface. He was not going to give up these beliefs easily particularly since he felt that his views were based on common sense. He hoped that further astronomical data would support the aether theory and show that relativity was not in fact a good description of the real world. As Born and von Laue write in [2]:He loved his absolute aether, his field equations, his rigid electron just as a youth loves his first flame, whose memory no later experience can extinguish. Many would still agree with Abraham that his version of the world was more in line with common sense. However, mathematics and physics over the 20th century has shown that the world we inhabit is at variance with "common sense" when we examine the large scale structure and the small scale structure. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abraham_Max.html (2 of 3) [2/16/2002 10:56:04 PM]

Abraham_Max

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Abraham_Max.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abraham_Max.html (3 of 3) [2/16/2002 10:56:04 PM]

Abu'l-Wafa

Mohammad Abu'l-Wafa Al-Buzjani Born: 10 June 940 in Buzjan (near Jam), Khorasan region (now in Iran) Died: 15 July 998 in Baghdad (now in Iraq) Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Abu'l-Wafa was brought up during the period that a new dynasty was being established which would rule over Iran. The Buyid Islamic dynasty ruled in western Iran and Iraq from 945 to 1055 in the period between the Arab and Turkish conquests. The period began in 945 when Ahmad Buyeh occupied the 'Abbasid capital of Baghdad. The high point of the Buyid dynasty was during the reign of 'Adud ad-Dawlah from 949 to 983. He ruled from Baghdad over all southern Iran and most of what is now Iraq. A great patron of science and the arts, 'Adud ad-Dawlah supported a number of mathematicians and Abu'l-Wafa moved to 'Adud ad-Dawlah's court in Baghdad in 959. Abu'l-Wafa was not the only distinguished scientist at the Caliph's court in Baghdad, for outstanding mathematicians such as al-Quhi and al-Sijzi also worked there. Sharaf ad-Dawlah was 'Adud ad-Dawlah's son and he became Caliph in 983. He continued to support mathematics and astronomy and Abu'l-Wafa and al-Quhi remained at the court in Baghdad working for the new Caliph. Sharaf ad-Dawlah required an observatory to be set up, and it was built in the garden of the palace in Baghdad. The observatory was officially opened in June 988 with a number of famous scientists present such as al-Quhi and Abu'l-Wafa. The instruments in the observatory included a quadrant over 6 metres long and a stone sextant of 18 metres. Abu'l-Wafa is said to have been the first to build a wall quadrant to observe the stars. However, the caliph Sharaf ad-Dawlah died in the following year and the observatory was closed. Like many scientist of his period, Abu'l-Wafa translated and wrote commentaries, which have since been lost, on the works of Euclid, Diophantus and al-Khwarizmi. Some time between 961 and 976 he wrote Kitab fi ma yahtaj ilayh al-kuttab wa'l-ummal min 'ilm al-hisab (Book on what Is necessary from the science of arithmetic for scribes and businessmen). In the introduction to this book Abu'l-Wafa writes that it ([3] or [4]):... comprises all that an experienced or novice, subordinate or chief in arithmetic needs to know, the art of civil servants, the employment of land taxes and all kinds of business needed in administrations, proportions, multiplication, division, measurements, land taxes, distribution, exchange and all other practices used by various categories of men for doing business and which are useful to them in their daily life. It is interesting that during this period there were two types of arithmetic books written, those using http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abu'l-Wafa.html (1 of 4) [2/16/2002 10:56:05 PM]

Abu'l-Wafa

Indian symbols and those of finger-reckoning type. Abu'l-Wafa's text is of this second type with no numerals; all the numbers are written in words and all calculations are performed mentally. Early historians such as Moritz Cantor believed that there were opposing schools of authors, one committed to Indian methods, the other to Greek methods. However, this has since been disproved (see for example [9]), and it is now believed that mathematicians wrote for two differing types of readers. Abu'l-Wafa himself was an expert in the use of Indian numerals but these [1]:... did not find application in business circles and among the population of the Eastern Caliphate for a long time. Hence he wrote his text using finger-reckoning arithmetic since this was the system used for by the business community. The work is in seven parts, each part containing seven chapters Part I: On ratio (fractions are represented as made from the "capital" fractions 1/2, 1/3, 1/4, ... ,1/10). Part II: On multiplication and division (arithmetical operations with integers and fractions). Part III: Mensuration (area of figures, volume of solids and finding distances). Part IV: On taxes (different kinds of taxes and problems of tax calculations). Part V: On exchange and shares (types of crops, and problems relating to their value and exchange). Part VI: Miscellaneous topics (units of money, payment of soldiers, the granting and withholding of permits for ships on the river, merchants on the roads). Part VII: Further business topics. This work is studied in detail in [12] (see also [10]). Of particular interest is the reference to negative numbers in Part II of Abu'l-Wafa's treatise, and this particular aspect is studied in detail in [11] and [12] (see also [1]). This seems to be the only place that negative numbers have been found in medieval Arabic mathematics. Abu'l-Wafa gives a general rule and gives a special case of this where subtraction of 5 from 3 gives a "debt" of 2. He then multiples this by 10 to obtain a "debt" of 20, which when added to (10 3)(10 - 5) = 35 gives the product of 3 and 5, namely 15. Another text written by Abu'l-Wafa for practical use was A book on those geometric constructions which are necessary for a craftsman. This was written much later than his arithmetic text, certainly after 990. The book is in thirteen chapters and it considered the design and testing of drafting instruments, the construction of right angles, approximate angle trisections, constructions of parabolas, regular polygons and methods of inscribing them in and circumscribing them about given circles, inscribing of various polygons in given polygons, the division of figures such as plane polygons, and the division of spherical surfaces into regular spherical polygons. Another interesting aspect of this particular work of Abu'l-Wafa's is that he tries where possible to solve his problems with ruler and compass constructions. When this is not possible he uses approximate methods. However, there are a whole collection of problems which he solves using a ruler and fixed compass, that is one where the angle between the legs of the compass is fixed. It is suggested in [1] that:Interest in these constructions was probably aroused by the fact that in practice they give more exact results than can be obtained by changing the compass opening. Abu'l-Wafa is best known for the first use of the tan function and compiling tables of sines and tangents http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abu'l-Wafa.html (2 of 4) [2/16/2002 10:56:05 PM]

Abu'l-Wafa

at 15' intervals. This work was done as part of an investigation into the orbit of the Moon, written down in Theories of the Moon. He also introduced the sec and cosec and studied the interrelations between the six trigonometric lines associated with an arc. Abu'l-Wafa devised a new method of calculating sine tables. His trigonometric tables are accurate to 8 decimal places (converted to decimal notation) while Ptolemy's were only accurate to 3 places. His other works include Kitab al-Kamil (Complete book), a simplified version of Ptolemy's Almagest. Although there seems to have been little of novel theoretical interest in this work, the observational data in it seem to have been used by many later astronomers.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Longitude and the Académie Royale 2. Arabic mathematics : forgotten brilliance? 3. Arabic numerals 4. The trigonometric functions

Other references in MacTutor

Chronology: 900 to 1100

Honours awarded to Abu'l-Wafa (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Abul-Wafa

Other Web sites

1. Muslim scientists 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abu'l-Wafa.html (3 of 4) [2/16/2002 10:56:05 PM]

Abu'l-Wafa

JOC/EFR November 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Abu'l-Wafa.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abu'l-Wafa.html (4 of 4) [2/16/2002 10:56:05 PM]

Abu_Kamil

Abu Kamil Shuja ibn Aslam ibn Muhammad ibn Shuja Born: about 850 in (possibly) Egypt Died: about 930 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Abu Kamil Shuja is sometimes known as al-Hasib al-Misri, meaning the calculator from Egypt. Very little is known about Abu Kamil's life - perhaps even this is an exaggeration and it would be more honest to say that we have no biographical details at all except that he came from Egypt and we know his dates with a fair degree of certainty. The Fihrist (Index) was a work compiled by the bookseller Ibn an-Nadim around 988. It gives a full account of the Arabic literature which was available in the 10th century and it describes briefly some of the authors of this literature. The Fihrist includes a reference to Abu Kamil and among his works listed there are: (i) Book of fortune, (ii) Book of the key to fortune, (iii) Book on algebra, (vi) Book on surveying and geometry, (v) Book of the adequate, (vi) Book on omens, (vii) Book of the kernel, (viii) Book of the two errors, and (ix) Book on augmentation and diminution. Works by Abu Kamil which have survived, and will be discussed below, include Book on algebra, Book of rare things in the art of calculation, and Book on surveying and geometry. Although we know nothing of Abu Kamil's life we do understand something of the role he plays in the development of algebra. Before al-Khwarizmi we have no information of how algebra developed in Arabic countries, but relatively recent work by a number of historians of mathematics as given a reasonable picture of how the subject developed after al-Khwarizmi. The role of Abu Kamil is important here as he was one of al-Khwarizmi's immediate successors. In fact Abu Kamil himself stresses al-Khwarizmi's role as the "inventor of algebra". He described al-Khwarizmi as (see for example [4] or [5]):... the one who was first to succeed in a book of algebra and who pioneered and invented all the principles in it. Again Abu Kamil wrote:I have established, in my second book, proof of the authority and precedent in algebra of Muhammad ibn Musa al-Khwarizmi, and I have answered that impetuous man Ibn Barza on his attribution to Abd al-Hamid, whom he said was his grandfather. There is certainly no doubt that Abu Kamil considered that he was building on the foundations of algebra as set up by al-Khwarizmi and indeed he forms an important link in the development of algebra between http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abu_Kamil.html (1 of 3) [2/16/2002 10:56:07 PM]

Abu_Kamil

al-Khwarizmi and al-Karaji. There is another reason for Abu Kamil's importance, however, which is that his work was the basis of Fibonacci's books. So not only is Abu Kamil important in the development of Arabic algebra, but, through Fibonacci, he is also of fundamental importance in the introduction of algebra into Europe. The author of [12] presents a list of parallels between Abu Kamil's works on algebra and the works of Fibonacci, and he also discusses the influence of Abu Kamil on two algebra texts of al-Karaji. The Book on algebra by Abu Kamil is in three parts: (i) On the solution of quadratic equations, (ii) On applications of algebra to the regular pentagon and decagon, and (iii) On Diophantine equations and problems of recreational mathematics. The part on the regular pentagon and decagon is studied in detail in [7], while the remainder of the work is described in [10]. The content of the work is the application of algebra to geometrical problems. It is the combination of the geometric methods developed by the Greeks together with the practical methods developed by al-Khwarizmi mixed with Babylonian methods. An important step forward in Abu Kamil's algebra is his ability to work with higher powers of the unknown than x2. These powers are not given in symbols but are written in words, yet the naming of the powers tell us that Abu Kamil had begun to understand what we would write in symbols as xnxm = xn+m. For example he uses the expression "square square root" for x5 (i.e. x2.x2.x), "cube cube" for x6 (i.e. x3.x3), "square square square square" for x8 (i.e. x2.x2.x2.x2). In fact Abu Kamil works easily with the powers up to x8 which appear in the text. The algebra contains 69 problems which include many of the 40 problems considered by al-Khwarizmi, but with a rather different approach to them. The Book on surveying and geometry is studied in detail in [9]. It was written by Abu Kamil, not for mathematicians, but rather for government land surveyors. Because of the people that it was aimed at, the work contains no proofs. Rather it presents a number of rules, some of which are far from easy, each given for the numerical solution of a geometric problem. Each rule is illustrated with a worked numerical example. Mainly the rules are for calculating the area, perimeter, diagonals etc. of figures such as squares, rectangles, and various different types of triangle. Abu Kamil also gives rules to calculate the volume and surface area of various solids such as rectangular parallelepipeds, right circular prisms, square pyramids, and circular cones. The work also deals with circles and here Abu Kamil takes = 22/7. A whole section is devoted to calculating the area of the segment of a circle. The final part of the work gives rules for calculating the side of regular polygons of 3, 4, 5, 6, 8, and 10 sides either inscribed in, or circumscribed about, a circle of given diameter. For the pentagon and decagon the rules which Abu Kamil gives, although without proof in this work, were fully proved in his algebra book. The Book of rare things in the art of calculation is concerned with solutions to indeterminate equations. Sesiano in [11] discusses Abu Kamil's work on indeterminate equations and he argues that his methods are very interesting for three reasons. Firstly Abu Kamil is the first Arabic mathematician who we know solved indeterminate problems of the type found in Diophantus's work. Secondly, as far as we know, Abu Kamil wrote before Diophantus's Arithmetica had been studied in depth by the Arabs. Thirdly, Abu Kamil explains certain methods which are not found in the known books of the Arithmetica. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abu_Kamil.html (2 of 3) [2/16/2002 10:56:07 PM]

Abu_Kamil

Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Other references in MacTutor

Chronology: 900 to 1100

Other Web sites

Karen H Parshall

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Abu_Kamil.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Abu_Kamil.html (3 of 3) [2/16/2002 10:56:07 PM]

Ackermann

Wilhelm Ackermann Born: 29 March 1896 in Schoenebeck (Kr. Altena), Germany Died: 24 Dec 1962 in Luedenscheid, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Ackermann was a mathematical logician who worked with Hilbert in Göttingen. Ackermann received his doctoral degree in 1925 with a thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreihei written under Hilbert, its content was a consistency proof of arithmetic without induction. From 1927 until 1961 he taught as a teacher at the Gymnasien in Burgsteinfeld and in Luedenscheid. He was corresponding member of the Akademie der Wissenschaften in Göttingen, and was honorary professor at the Universität Münster. In 1928, Ackermann observed that A(x, y, z), the z-fold iterated exponentiation of x with y, is an example of a recursive function which is not primitive recursive. A(x, y, z) was simplified to a function P(x, y) of 2 variables by Rosza Peter whose initial condition was simplified by Raphael Robinson, it is the latter which occurs as Ackermann's function in today's textbooks. Also in 1928 there appeared the often reprinted book Grundzuege der Theoretischen Logik by Hilbert and Ackermann. Among Ackermann's later work there are consistency proofs for set theory (1937), full arithmetic (1940), type free logic (1952), further there was a new axiomatization of set theory (1956), and a book Solvable cases of the decision problem (North Holland, 1954). Article by: Walter Felscher, Tuebingen List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ackermann.html (1 of 2) [2/16/2002 10:56:08 PM]

Ackermann

Glossary index

Quotations index

Mathematicians of the day JOC/EFR December 1996

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ackermann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ackermann.html (2 of 2) [2/16/2002 10:56:08 PM]

Adams

John Couch Adams Born: 5 June 1819 in Laneast, Cornwall, England Died: 21 Jan 1892 in Cambridge, Cambridgeshire, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Couch Adams showed great mathematical ability while at school, and he became known for his considerable abilities for accurate numerical calculations. During this period he first became interested in astronomy and, by the age of 16, he had worked out when an annular eclipse of the Sun would be visible in Lidcot. His brother lived at Lidcot, which is near Launceston about 10Km from where John was born. Adams was educated at St John's College, Cambridge. He began his undergraduate course in October 1839 and graduated Senior Wrangler four years later. He is said to have been awarded double the marks of the Second Wrangler which, if true, is an incredible achievement. Also in 1843 he became first Smith's Prizeman and he became a Fellow of Pembroke College. He was to hold this Fellowship, in addition to other posts, until his death. In 1841, while still an undergraduate, he decided to investigate the irregularities of the motion of Uranus...in order to find out whether they may be attributed to the action of an undiscovered planet beyond it . In September 1845 Adams gave accurate information on the position of the new planet to James Challis, director of the Cambridge Observatory. Action was not taken by Cambridge and Urbain Le Verrier's later prediction was published before Adams's. It was Le Verrier's prediction which led to the discovery of Neptune on September 23, 1846 by Galle at the Berlin Observatory. Adams became Regius Professor of Mathematics at St Andrews in 1858. In 1859 he succeeded Peacock as Lowndean Professor of Astronomy and Geometry at Cambridge and held the post for over 32 years. He became director of the Cambridge Observatory in 1861. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adams.html (1 of 3) [2/16/2002 10:56:10 PM]

Adams

Adams's made many other contributions to astronomy, notably his studies of the Leonid meteor shower (1866) where he showed that the orbit of the meteor shower was very similar to that of a comet. He was able to correctly conclude that the meteor shower was associated with the comet. Adams spent much effort on the complex problem of a description of the motion of the Moon, giving one which was more accurate than that of Laplace. He also studied terrestrial magnetism. Adams will be best remembered, however, for his role as the co-discoverer of Neptune. The difference between his role in that discovery and that of Le Verrier is more clearly understood when Adams's character is studied. A fellow undergraduate at Cambridge could hardly remember Adams and described him as:A rather small man, who walked quickly, and wore a faded coat of dark green. Always meticulous, Adams had a reputation for constructing mathematical questions for his students which were admired by all for their beauty (except perhaps the students being examined!). He was a man of great learning studying history, literature, biology and geology. He had a keen interest in politics and he was so affected by the Franco-Prussian war that, according to [6] he could scarcely work or sleep. He never boasted of his achievements and in fact he refused a knighthood which was offered to him in 1847. After the discovery of Neptune, Adams met Le Verrier in Oxford in June 1847. According to [3] He uttered no complaint, he laid no claim to priority, Le Verrier had no warmer admirer. The portrait above was taken when he was in St Andrews by the pioneer photographer John Adamson. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of John Couch Adams Cross-references to History Topics

Mathematicians born in the same country 1. Orbits and gravitation 2. Adams's part in the Mathematical discovery of planets

Honours awarded to John Couch Adams (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1849

Royal Society Copley Medal

Awarded 1848

Lunar features

Crater Adams

Other Web sites

1. St John's College, Cambridge (The discovery of Neptune) 2. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adams.html (2 of 3) [2/16/2002 10:56:10 PM]

Adams

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Adams.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adams.html (3 of 3) [2/16/2002 10:56:10 PM]

Adams_Frank

John Frank Adams Born: 5 Nov 1930 in Woolwich, London, England Died: 7 Jan 1989 in Near Cambridge, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Frank Adams's family was evacuated from London during World War II and he attended school in a number of places. He entered Trinity College, Cambridge in 1949. After taking his degree he started graduate work at Cambridge with Besicovitch on geometric measure theory. He changed to working on algebraic topology with Wylie. Adams became influenced by Henry Whitehead who led the foremost British school of algebraic topology. In fact Adams was appointed to a post at Oxford shortly before completing his doctorate. After a year at Oxford he returned to Cambridge have won a Fellowship with his doctoral thesis on spectral sequences. Adams visited Chicago as a research associate, then he moved to Princeton. Adams said:... I regard the progress of my researches in America as most successful. ... By good luck, moreover, my new methods were sufficiently powerful to answer one of the classical problems of my subject, that proposed by H Hopf in 1935. On his return from the USA he became a College Lecturer at Trinity Hall Cambridge. His work turned towards K-theory, the generalised cohomology theory on vector bundles. After spendind further time in Princeton, Adams took up a post at Manchester as a Reader, being appointed to Newman's chair when he retired. During this time he wrote a series of papers which were highly influential in homotopy theory. In 1964 Adams was elected a Fellow of the Royal Society. In [2] James says:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adams_Frank.html (1 of 3) [2/16/2002 10:56:12 PM]

Adams_Frank

It was in 1965, however, that he suffered the first attack of a psychiatric illness, as a result of which he was on sick leave for some months. It was apparently brought on by the worry caused by his responsibilities as head of department ... In 1970 Adams succeeded Hodge as Lowndean Professor of Astronomy and Geometry at Cambridge. His research continued to be of fundamental importance in homotopy theory of the classifying spaces of topological groups, finite H-spaces and equivariant homotopy theory. He wrote a number of books of major importance, Lectures on Lie groups (1969), algebraic topology: a student's guide (1972), Stable homotopy theory and generalized homology (1974) and Infinite loop spaces (1978). His lectures were well prepared but usually hard. He once received a letter from a second year undergraduate class saying:The class wishes to inform Professor Adams that it has been left behind. He replied:At any rate I have done exterior algebra, even if the second year haven't. Adams received many awards for his work. Among these was the Sylvester Medal of the Royal Society of London which was awarded to him in 1982:... in recognition of his solution of several outstanding problems of algebraic topology and of the methods he invented for this purpose which have proved of prime importance in the theory of that subject. His health continued to cause him problems with another psychiatric illness in 1986. Perhaps his health contributed to his death since he decided to go to London to celebrate the retiral of a friend despite feeling unwell. He was killed in a car crash only a few miles from his home on the return journey. He had apparently always had a reputation as a car driver. According to [2] He drove cars with remarkable skill but in a style that left a lasting impression on his passengers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles)

Some Quotations (2)

Mathematicians born in the same country Honours awarded to Frank Adams (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1964

Royal Society Sylvester Medal

Awarded 1982

LMS Berwick Prize winner

1963

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adams_Frank.html (2 of 3) [2/16/2002 10:56:12 PM]

Adams_Frank

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Adams_Frank.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adams_Frank.html (3 of 3) [2/16/2002 10:56:12 PM]

Adelard

Adelard of Bath Born: 1075 in Bath, England Died: 1160 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Few details of Adelard's life are known with certainty. We do know that he studied in Tours in the Loire Valley in west central France and that he later taught at Laon in the Picardie region of northern France. Laon lies northwest of Reims and northeast of Paris. Adelard may have taught at the theological and exegetical school there which had been founded by Anselm of Laon in about 1100. After leaving Laon, Adelard travelled for about seven years visiting first Salerno southeast of Naples. The medical school at Salerno, considered by many to be the first "modern" European university, was a famous institution at this time, drawing students from all over Europe. From Salerno Adelard travelled to Sicily which at that time was under Norman control but still strongly influenced by Arabic traditions. The Arabs from North Africa had conquered the island in 965 and remained in control for about 100 years but the Normans gained the island in 1060. Adelard next visited Cilicia, an ancient district of southern Anatolia which today is in Turkey. Cilicia was on the north east coast of the Mediterranean Sea and Adelard took the natural coastal route round the east end of the Mediterranean to Syria and then later to Palestine. We know that he returned to Bath and is mentioned in the records of that city for the year 1130. There is no record of Adelard visiting Spain, but many scholars have concluded that he must have visited that country to have had access to the Spanish-Arabic texts which he translated. Certainly Adelard became an expert in the Arabic language which he might have learnt in Spain as did Gherard of Cremona a few years later. However, there is an alternative theory that he learnt Arabic in Sicily. It is quite possible that, if this were the case, then he came in contact with Spanish-Arabic texts in Sicily for several scholars who had lived in Spain were at this time in Sicily. Adelard wrote a number of original works on philosophy. The first work that he is known to have written is a philosophy text written before 1116 and dedicated to William, Bishop of Syracuse. Since Syracuse was one of the most important cities of ancient Sicily, this work is likely to have been written around the time of Adelard's visit to that island. However, since the work is based firmly on Plato's philosophy, without any signs of Arabic influences, it may have been mostly written before Adelard's visits brought him in contact with the learning of the Arabs. It is not as a philosopher that Adelard merits inclusion in this archive. Rather it is because he is [1]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adelard.html (1 of 3) [2/16/2002 10:56:13 PM]

Adelard

... one of the translators who made the first wholesale conversion of Arabo-Greek learning from Arabic into Latin. Adelard made Latin translations of Euclid's Elements from Arabic sources which were for centuries the chief geometry textbooks in the West. In fact there seem to have been three separate versions of Euclid's Elements written by Adelard. Version one is a translation of the whole fifteen books (the 13 original books written by Euclid and the two further books written by Hypsicles). Adelard seems to have taken as his source one of al-Hajjaj's Arabic translations from Greek. The second version of Euclid's Elements by Adelard is quite different. It contains quite different wording of the statements of the propositions to that of version one, while the proofs are often only outlines or indications of how proofs might be constructed. The style of the translation tells experts that Adelard did not produce this from his own version one, but rather that he used some unknown Arabic source different from al-Hajjaj's translations. There is debate as to whether the third version of Euclid's Elements attributed to Adelard is indeed his work. It is a commentary on Euclid's Elements rather than a translation of the original text. We know it was written before 1200 and became quite well known under Adelard's name. Roger Bacon gives quotes from this version in his works. Adelard also translated al-Khwarizmi's tables, wrote on the abacus and on the astrolabe. We should make some further comments on his translation of al-Khwarizmi's tables which became the first Latin astronomical tables of the Arabic type with their Greek influences and Indian symbols. These tables contain, at the end of chapter 4, the date of 26 January 1126 (at least that is what the Arabic date of A.H. 520 Muharram 1 corresponds to). It is hard to see what this date is there for unless it is the date when the chapter was completed, and so it has been taken as the approximate date for Adelard's translation. However, there is a manuscript (written later but a copy of Adelard's translation) which mentions an eclipse of the sun which took place in 1133. It is possible that Adelard's translation took place after 1133 or, equally likely, that the scribe making the later copy added information about a recent eclipse which was not in Adelard's original text. Adelard also wrote arithmetic books, the earliest one of which was written before he studied Arabic arithmetic. It is based on the work of Boethius. A mathematics treatise which is strongly influenced by Arabic ideas has been attributed to Adelard although the attribution is not certain. The work consists of five books, the first three of which are on arithmetic and based on the Indian methods as presented in Arab writings. It has been conjectured that these books are based on an arithmetic book by al-Khwarizmi which is now lost. The remaining two books of the five which compose the treatise cover geometry, which is completely Greek in style, music, and astronomy. The astronomy, like the arithmetic, is Arabic in style. Adelard's Quaestiones naturales consists of 76 scientific discussions based on Arabic science. In this work he promoted the use of experimental data and writes that he [1]:... prefers reason to authority. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adelard.html (2 of 3) [2/16/2002 10:56:13 PM]

Adelard

Click on this link to see a list of the Glossary entries for this page List of References (19 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. How do we know about Greek mathematics? 2. An overview of the history of mathematics

Other references in MacTutor

Chronology: 1100 to 1300

Other Web sites

1. Math Forum 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR November 1999 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Adelard.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adelard.html (3 of 3) [2/16/2002 10:56:13 PM]

Adler

August Adler Born: 1863 in Austria Died: 1923 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

August Adler worked in Vienna. In 1797 Mascheroni had shown that all plane construction problems which could be made with ruler and compass could in fact be made with compasses alone. His theoretical solution involved giving specific constructions, such as bisecting a circular arc, using only a compass. In 1906 Adler applied the theory of inversion to solve Mascheroni's construction problems. Since he was using inversion Adler now had a symmetry between lines and circles which in some sense showed why the constructions needed only compasses. However Adler did not simplify Mascheroni's proof. On the contrary, his new methods were not as elegant, either in simplicity or length, as the original proof by Mascheroni. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adler.html (1 of 2) [2/16/2002 10:56:14 PM]

Adler

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Adler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adler.html (2 of 2) [2/16/2002 10:56:14 PM]

Adrain

Robert Adrain Born: 30 Sept 1775 in Carrickfergus, Ireland Died: 10 Aug 1843 in New Brunswick, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

The names of Robert Adrain's parents are unknown but we do know that his father was a schoolteacher and that he made mathematical instruments. When Adrain was about fifteen years of age both his parents died. Adrain received a good education but this education did not include any mathematics beyond arithmetic. His own curiosity, however, led him to study books on mathematics which explained algebraic notation and so he became essentially a self-taught mathematician. After his parents died, Adrain had to work to support both himself and his four brothers and sisters and it was entirely natural for this well educated young man to earn his living as a teacher. By 1798 Adrain, a young man of 22, had established himself sufficiently well financially to allow him to marry Ann Pollock. They had seven children, one of whom, Garnett Bowditch Adrain, became a Democratic member of Congress. The year 1798 was crucial for Adrain in addition to being the year in which he married. He took part in the Irish rebellion of that year but, in order to understand what this was about, we should give some background to explain the political events which led up to this event that was to result in a complete change in Adrain's life. Irish politics was changed by the American Revolution partly since it led to government troops leaving Ireland, and a Protestant Irish volunteer corps established to defend the country from the French. In 1782 legislative changes gave more power to the Irish Parliament and the position of Roman Catholics was greatly improved. This led to a Protestant backlash and an effect of the French Revolution was to see societies of United Irishmen founded which included both Protestants and middle-class Catholics. The Societies were driven underground and they looked for military support from France. From 1796 to 1798 http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adrain.html (1 of 4) [2/16/2002 10:56:16 PM]

Adrain

France sent several naval expeditions to Ireland to support the Societies but all failed. The United Irishmen provoked a rebellion in May 1798 and Adrain joined the rebels as an officer in their army. The rebellion was unsuccessful in general, but particularly so for Adrain who was shot in the back by one of his own men and badly wounded. After recovering his health Adrain escaped with his wife to the United States where they settled in Princeton, New Jersey. Adrain was appointed as a master at Princeton Academy and remained there until 1800 when the family moved to York in Pennsylvania. In York Adrain became Principal of York County Academy. When the first mathematics journal, the Mathematical Correspondent, began publishing in 1804 under the editorship of George Baron, Adrain became one of its main contributors. One year later, in 1805, he moved again this time to Reading, also in Pennsylvania, where he was appointed Principal of the Academy. After arriving in Reading, Adrain continued to publish in the Mathematical Correspondent and, in 1807, he became editor of the journal. One has to understand that publishing a mathematics journal in the United States at this time was not an easy task since there were only two mathematicians capable of work of international standing in the whole country, namely Adrain and Nathaniel Bowditch. Despite these problems, Adrain decided to try publishing his own mathematics journal after he had edited only one volume of the Mathematical Correspondent and, in 1808, he began editing his journal the Analyst or Mathematical Museum. With so few creative mathematicians in the United States the journal had little chance of success and indeed it ceased publication after only one year. After the journal ceased publication, Adrain was appointed professor of mathematics at Queen's College (now Rutgers University) New Brunswick where he worked from 1809 to 1813. Despite Queen's College trying its best to keep him there, Adrain moved to Columbia College in New York in 1813. He tried to restart his mathematical journal the Analyst in 1814 but only one part appeared. In 1825, while he was still on the staff at Columbia College, Adrain made another attempt at publishing a mathematical journal. Realising that the Analyst had been too high powered for the mathematicians of the United States, he published the Mathematical Diary in 1825. This was a lower level publication which continued under the editorship of James Ryan when Adrain left Columbia College in 1826. Adrain returned to Rutgers College (Queen's College was renamed Rutgers College in 1825 after the philanthropist Henry Rutgers) in 1826. However, leaving his family there, he accepted the post of Professor of Mathematics at the University of Pennsylvania in 1827. In 1828 he was appointed vice-provost of the University and he remained there until 1834. However, he left the University of Pennsylvania having been asked to resign. As Hogan writes in [5] (see also [4]):During his last year [at the University of Pennsylvania] Adrain had serious problems with discipline in his classes. Because the faculty saw no way to aid Adrain and feared that the disturbances would spread to other classes, the university asked for Adrain's resignation. To understand why one of the finest mathematicians in the United States should have discipline problems in his classes we quote again from [5] (Struik in [1] gives essentially the same comments):Although a man of wit and humour, Adrain was often irritable in the classroom. One of his students reported that whenever a student faltered in his recitation (then the principal form of classroom instruction), Adrain would terminate his efforts with a remark such as "If you cannot understand Euclid, dearie, I cannot explain it to you". http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adrain.html (2 of 4) [2/16/2002 10:56:16 PM]

Adrain

Having tended his resignation as requested, Adrain returned to New Brunswick where he earned his living tutoring privately until 1836 when again he went to New York, teaching at the Grammar School attached to Columbia College. In 1840 he retired and returned to New Brunswick where he spent the three years of his retirement before his death. Despite the resignation episode which resulted from Adrain's impatience in the classroom, he had a reputation for being patient and helpful at all other times. It is not uncommon for teachers to change character when they teach a class! Adrain's first papers in the Mathematical Correspondent concerned the steering of a ship and Diophantine algebra (the study of rational solutions to polynomial equations). After publishing further work on Diophantine algebra, he published a paper on the normal law of errors in 1808, one year before Gauss. It was in 1808 that Robert Patterson proposed a surveying problem in the Analyst and, after comments from Bowditch suggesting two procedures, Adrain gave an argument to establish the validity of the normal distribution for the errors, and he then used it to prove the validity of the method of least squares. Taking a number of problems as examples, Adrain showed that one of Bowditch's procedures was equivalent to using the method of least squares. It is unfortunate that despite Adrain's priority over Gauss, it is the latter who has received the credit for this important statistical contribution. See [3] for more details. Other topics which Adrain wrote about include a study of the catenary, and other curves which he called isotomous. In 1818 he published a paper Investigation of the figure of the Earth and of the gravity in different latitudes. In this paper Adrain gave 1/319 as the ellipticity of the Earth, a figure better than that given by Laplace (he gave 1/336), and about halfway between Laplace's figure and the accepted value today of 1/297. Adrain's improvement on Laplace's value was, of course, made because Adrain had been inspired to work on the topic because of the contributions of Laplace. We would know more about Adrain's work today but for an unfortunate incident concerning M J Babb of the University of Pennsylvania. Babb was working on Adrain's manuscripts at the time of his death in 1945 and it appears that both Babb's work and the manuscripts of Adrain on which he was working were inadvertently destroyed after his death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1810 to 1820

Other Web sites

Rutgers (A history of the Department Adrain was in)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adrain.html (3 of 4) [2/16/2002 10:56:16 PM]

Adrain

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Adrain.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Adrain.html (4 of 4) [2/16/2002 10:56:16 PM]

Aepinus

Franz Maria Ulrich Theodosius Aepinus Born: 13 Dec 1724 in Rostock, Mecklenberg-Schwerin (now Germany) Died: 10 Aug 1802 in Dorpat, Russia Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Franz Aepinus was born in Rostock where his father was the Professor of Theology at the University. He came from a famous family of theologians who were originally named Hoeck or Hoch but Franz's great-grandfather had changed the family name to its Greek form. Aepinus studied medicine and mathematics at the universities of Jena and Rostock. He was awarded an MA from Rostock for a dissertation on the paths of falling bodies in 1747. He remained at Rostock, teaching mathematics there until 1755. During this period he undertook research in several different areas of mathematics including algebraic equations, solving partial differential equations, and on negative numbers. Franz was not the only member of the family to be teaching at Rostock during this period, for his elder brother also taught oratory at the University. One of his brother's students, J C Wilcke, took courses given by Franz during 1751-52 and was attracted to a career in mathematics and physics instead of the clerical career that he had been intending to pursue when he entered the university. If Franz had a major effect on Wilcke's career, then the reverse was true a few years later when Wilcke was to suggest problems which led to the most important work of Aepinus's career. In 1755 Aepinus became director of the Observatory in Berlin and he was elected to the Berlin Academy of Sciences. Director of a major observatory may seem a strange appointment given that Aepinus' mathematical interests seemed far removed from astronomy. However, Heilbron writes in [1]:These appointments were apparently merely a device for establishing Aepinus, who had begun to acquire a reputation, in Frederick's capital: he was neither especially interested nor experienced in astronomy, and his closest published approach to the subject during his Berlin sojourn was a mathematical analysis of a micrometer adapted to a quadrant circle. Euler was working at the Berlin Academy of Sciences during the time that Aepinus worked there, and in fact Aepinus lived in Euler's house for the two years that he was in Berlin. Although Aepinus did not make contributions to astronomy while in Berlin, he did his most important work there. Wilcke had moved to Berlin with Aepinus and was writing a dissertation on electricity. Wilcke showed Aepinus the mineral tourmaline, a borosilicate mineral often used as a gem. Tourmaline

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aepinus.html (1 of 3) [2/16/2002 10:56:17 PM]

Aepinus

has piezoelectric properties which means that it can generate electric charge when mechanical stress is applied and can change its shape when voltage is applied to it. Aepinus studied the state of electrical polarisation produced in tourmaline and various other crystals by a change of temperature. The electrical properties of the tourmaline seemed to Aepinus to be similar to those of a magnet and he began to believe that electricity and magnetism were analogous. Aepinus' study of electricity and magnetism led to the publication of his book Tentamen theoriae electricitatis et magnetismi (An Attempt at a Theory of Electricity and Magnetism) in 1759. It was the first work to apply mathematics to the theory of electricity and magnetism and [1]:... is one of the most original and important books in the history of electricity. Before this work was published, however, Aepinus had moved to St Petersburg. In October 1756 he was offered a chair at the Academy in St Petersburg and he requested Frederick to release him from his contract in Berlin so that he could accept the post. Euler supported his request to Frederick and by early 1757 Frederick had agreed that Aepinus could end his duties in Berlin and accept the appointment in St Petersburg. Aepinus was to continue working in St Petersburg until he retired in 1798. Aepinus certainly began his appointment in St Petersburg with the publication of his masterpiece and was held in high esteem by the scientists there [2]:Aepinus studied the relation between conductors and nonconductors, extended Benjamin Franklin's one-fluid theory of electricity, and explained virtually all electric induction in terms of the attraction, repulsion, and flow of electricity in conductors. However, in 1760 he became an instructor for the Corps of Imperial Cadets and this left his little time to devote to his researches at the Academy. Despite continuing to publish on electricity and magnetism he was strongly attacked by Lomonosov who had been annoyed by Aepinus' [1]:... haughtiness towards Russian scientists and quick preferment at court ... Other achievements of Aepinus include improvements to the microscope, and his demonstration of the effects of parallax in the transit of a planet across the Sun's disk (1764). However, except for his masterpiece on electricity and magnetism, his work was no better than competent. As an example of Aepinus' less good work, the authors of [4] relate that in 1763 Aepinus published in Latin in the Commentaries of the St Petersburg Academy a proof of the binomial theorem for real values of the exponent. The proof given by Aepinus, however, appears not to hold. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Other references in MacTutor Other Web sites

Chronology: 1740 to 1760 1. West Chester University 2. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aepinus.html (2 of 3) [2/16/2002 10:56:17 PM]

Aepinus

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Aepinus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aepinus.html (3 of 3) [2/16/2002 10:56:17 PM]

Agnesi

Maria Gaëtana Agnesi Born: 16 May 1718 in Milan, Habsburg Empire (now Italy) Died: 9 Jan 1799 in Milan, Habsburg Empire (now Italy)

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Maria Gaetana Agnesi was the daughter of Pietro Agnesi who came from a wealthy family who had made their money from silk. Pietro Agnesi had twenty-one children with his three wives and Maria was the eldest of the children. As Truesdell writes in [16], Pietro Agnesi:... belonged to a class intermediate between the patricians and the merely rich. Such a bourgeois could have a household fit for a lord, comport himself like a knight, mingle freely with some nobles, occupy himself with the finer things of life, be a patron of men of talent. [Pietro Agnesi] did just that... Some accounts of Maria Agnesi describe her father as being a professor of mathematics at Bologna. It is shown clearly in [12] that this is entirely incorrect, but the error is unfortunately carried forward to [1] and will also be seen in a number of other places. Pietro Agnesi could provide high quality tutors for Maria Agnesi and indeed he did provid her with the best available tutors who were all young men of learning from the Church. She showed remarkable talents and mastered many languages such as Latin, Greek and Hebrew at an early age. At the age of 9 she published a Latin discourse in defence of higher education for women. It was not Agnesi's composition, as has been claimed by some, but rather it was an article written in Italian by one of her tutors which she translated and [16]:... she delivered it from memory to an academic gathering arranged by her father in the garden... In 1738 she published Propositiones Philosophicae a series of essays on philosophy and natural science. The volume contained 191 philosophical theses which Agnesi would defend in disputes with specially http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Agnesi.html (1 of 5) [2/16/2002 10:56:19 PM]

Agnesi

invited audiences of important international and national people who her father would invite to his house. In [4] de Brosses describes one such evening which took place on 16 July 1739:... I was brought into a large fine room, where I found about thirty people from all countries of Europe, arranged in a circle and Mlle Agnesi, all alone with her little sister, seated on a sofa. She is a girl of about twenty years of age, neither ugly nor pretty, with a very simple and very sweet manner. ... Count Belloni, who took me, wanted to make a public show. He began with a fine discourse in Latin to this young girl, that it might be understood by all. She answered him well, after which they entered into a dispute, in the same language, on the origin of fountains and on the causes of the ebb and flow which is seen in some of them, similar to tides at sea. She spoke like an angel on this topic, I have never heard anything so pleasurable. ... She is much attached to the philosophy of Newton, and it is marvellous to see a person of her age so conversant with such abstract subjects. Yet however much I was amazed at her learning, I was perhaps more amazed to hear her speak Latin with such purity, ease and accuracy... It might look as if this were an extremely distasteful affair, with Agnesi's father showing off his daughter's talents like a circus act. To some extent this must be the case, but it is fair to say that shows of this type were relatively common at the time. Certainly, although Agnesi always acted in total obedience to her father's wishes, she was not very happy with the spectacle that he put on. Again we quote [4] where de Brosses wrote:She told me that she was very sorry that the visit had taken the form of a thesis defence, and that she did not like to speak publicly of such things, where for every one that was amused, twenty were bored to death. ... I was much annoyed to hear it said that she wished to enter a convent, and it was not through need, for she is very rich. In [16] Truesdell explains further about her wishing to become a nun:She did ask her father's permission to became a nun. Horrified that his dearest child should desire to leave him, he begged her to change her mind. She agreed to continue living in his house and caring for him on three conditions: that she go to church whenever she wished, that she dress simply and humbly, that she abandon altogether balls, theatres, and profane amusements. Agnesi concentrated her efforts on studying religious books and learning mathematics. Around this time she wrote a commentary on de L'Hôpital's Traite analytique des section coniques but it has never been published. Learning mathematics without proper instruction is an almost impossible task and only a few have ever achieved great things in this way. Agnesi was fortunate, however, in her bid to learn mathematics for a monk, Ramiro Rampinelli, a mathematician who had been a professor at both Rome and Bologna, arrived in Milan and became a frequent visitor to the Agnesi house. With Rampinelli's help Agnesi studied Reyneau's calculus text Analyse démontrée (1708). Agnesi understood the debt she owed to Rampinelli and in the preface to her famous book Instituzioni analitiche ad uso della gioventù italiana she wrote:With all the study, sustained by the strongest inclination towards mathematics, that I forced myself to devote to it on my own, I should have become altogether tangled in the great labyrinth of insuperable difficulty, had not [Rampinelli's] secure guidance and wise direction led me forth from it ...; to him I owe deeply all advances (whatever they might be) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Agnesi.html (2 of 5) [2/16/2002 10:56:19 PM]

Agnesi

that my small talent has sufficed to make. Rampinelli encouraged Agnesi to write a book on differential calculus. She wrote the book in Italian as a teaching text which, according to the preface, attempted to present the material:... endowed with proper clarity and simplicity..., which proceeds with that natural order which provides, perhaps, the best instruction and the greatest light. Agnesi, with her father's money, was able to arrange for the private printing of the book in her own home where she could supervise the whole operation herself. However, she wished to obtain more input from leading mathematicians so, on the 20 July 1745, she wrote to Riccati. It was Rampinelli who suggested that Riccati might offer Agnesi advice and he had clearly contacted Riccati, who had been one of his own teachers, and Riccati had agreed to read the final draft of Agnesi's book and make suggestions. Riccati replied quickly to Agnesi's first letter and promised to pass the text to his two sons, Vincenzo Riccati and Giordano Riccati, so that they could also comment on the work. Once Agnesi received Riccati's comments on the first part of the text she organised printing of that part while later parts were sent to Riccati also for him to comment on. By 1747 Agnesi was sending Riccati later parts of the book and explaining to him that printing of the earlier parts was nearly complete. Riccati wrote to Rampinelli on 1 February 1747, offering Agnesi his some of his earlier work on integration for inclusion in her book. Agnesi included the work with proper acknowledgement to Riccati. In her letters Agnesi tried to get Riccati to reply more quickly giving his notes on her draft since the printer had other work to undertake, and she wrote to Riccati saying that:... if it becomes necessary to suspend the printing again, I do not know when I could start it anew, because even now it has been extremely difficult for me to continue with printing the first part (which soon will be finished). The first volume of Agnesi's famous two volume work Instituzioni analitiche ad uso della gioventù italiana was published in 1748 while Agnesi continued corresponding with Riccati over the material for the second volume which was published the following year. The work was to bring her much fame. A report on it made by a committee of the Académie des Sciences in Paris states:It took much skill and sagacity to reduce, as the author has done, to almost uniform methods these discoveries scattered among the works of modern mathematicians and often presented by methods very different from each other. Order, clarity and precision reign in all parts of this work. ... We regard it as the most complete and best made treatise. Pope Benedict XIV wrote to Agnesi saying that he had studied mathematics when he was young and could see that her work would bring credit to Italy and to the Academy of Bologna. Soon after this he appointed Agnesi to the position of honorary reader at the University of Bologna. Then Agnesi was approached by the president of the Academy of Bologna and three other professors of the Academy and invited to accept the chair of mathematics at the University of Bologna. Indeed, shortly after this, Agnesi received a letter from Pope Benedict XIV written on 26 September 1750:We have had the idea that you should be awarded the well known chair of mathematics, by which it comes of itself that you should not thank us by we you... It is probable that Agnesi neither accepted nor rejected this offer. As Truesdell writes in [16]:In October [Agnesi] received a papal rescript confirming her appointment. She had already

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Agnesi.html (3 of 5) [2/16/2002 10:56:19 PM]

Agnesi

devoted herself to a holy, retired life; while her name remained on the rolls of the university for forty-five years, she never went to Bologna. This does explain the confusion which appears in many accounts as to whether Agnesi ever held a chair of mathematics. Frisi, who was a school friend of one of Agnesi's brothers, visited the Agnesi house after the time that her book was published. He describes in [5] how her father imposed severe constraints on her, and she chose to inhabit rooms of the house away from where the rest of the family lived and there she helped old women who were ill. However [5]:... she immediately, with no apparent difficulty, gave way to her father's wishes ..., taking part in the usual academies in his house with such grace and penetration, propounding or answering questions, problems, and scientific doubts... After the death of her father in 1752, Agnesi devoted herself entirely to charitable work. She [16]:... resumed her study of Catholic doctrine and her costly acts of piety towards the poor and suffering, the hopelessly ill and the demented. First in her late father's house and afterwards in other places she established a hospice for old infirm women. Agnesi spent all her money on this charitable work and she died in total poverty in the poorhouse of which she had been the director. The treatise Instituzioni analitiche ad uso della gioventù italiana contains no original mathematics by Agnesi. Rather the book contains many examples which were carefully selected to illustrate the ideas; one review calls it an:... exposition by examples rather than by theory. The book includes a discussion of the cubic curve now know as the 'witch of Agnesi'. There has been much argument over the reason why the curve is called a 'witch'. The curve was discussed by Fermat and, in 1703, a construction for the curve was given by Grandi. In 1718 Grandi gave it the Latin name 'versoria' which means 'rope that turns a sail' and he so named it because of its shape. Grandi gave the Italian 'versiera' for the Latin 'versoria' and indeed Agnesi quite correctly states in her book that the curve was called 'la versiera'. John Colson, who had translated Newton's De Methodis Serierum et Fluxionum from Latin to English for publication in 1736, translated Agnesi's Instituzioni analitiche ad uso della gioventù italiana into English before 1760 (the year of Colson's death) although his English translation was not published until 1801. Colson mistook 'la versiera' for 'l'aversiera' which means 'the witch' or 'the she-devil'. See [17] for a detailed description of how the curve has become known as the 'Witch of Agnesi'. Article by: J J O'Connor and E F Robertson List of References (17 books/articles) A Poster of Maria Agnesi

Mathematicians born in the same country

Cross-references to Famous Curves

Witch of Agnesi

Other references in MacTutor

Chronology: 1740 to 1760

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Agnesi.html (4 of 5) [2/16/2002 10:56:19 PM]

Agnesi

Honours awarded to Maria Agnesi (Click a link below for the full list of mathematicians honoured in this way) Planetary features

Crater Agnesi on Venus

Other Web sites

1. S B Gray 2. Agnes Scott College 3. The Catholic Encyclopedia 4. University of Alabama 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Agnesi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Agnesi.html (5 of 5) [2/16/2002 10:56:19 PM]

Ahlfors

Lars Valerian Ahlfors Born: 18 April 1907 in Helsingfors, Finland, Russian Empire (now Helsinki, Finland) Died: Oct 1996 in Pittsfield, Massachusetts, USA

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lars Ahlfors' father was professor of mechanical engineering at the Polytechnic Institute in Helsingfors. Tragically his mother died in childbirth when he was born. Ahlfors describes his early years in [2]:As a child I was fascinated by mathematics without understanding what it was about, but I was by no means a child prodigy. As a matter of fact I had no access to mathematical literature except in the highest grades. Having seen many prodigies spoilt by ambitious parents, I can only be thankful to my father for his restraint. the high school curriculum did not include any calculus, but I finally managed to learn some on my own, thanks to clandestine visits to my father's engineering library. Ahlfors entered Helsingfors University in 1924, and there he was taught by Lindelöf and Nevanlinna. He graduated from Helsingfors in 1928. Then Nevanlinna replaced Weyl, who was on leave, in Zurich for the session 1928/29 and Ahlfors went to Zurich with him. In Zurich Nevanlinna lectured on Denjoy's conjecture on the number of asymptotic values of an entire function. Ahlfors modestly writes in [2]:I had the incredible luck of hitting upon a new approach, based on conformal mapping, which with very considerable help from Nevanlinna and Pólya led to a proof of the full conjecture. Ahlfors went to Paris with Nevanlinna for three months before returning to Finland. There he was appointed lecturer in mathematics in Turku. He presented his doctoral thesis in 1930, then in the following two years he made a number of visits to Paris and other European centres.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ahlfors.html (1 of 3) [2/16/2002 10:56:21 PM]

Ahlfors

In 1935, Caratheodory, whom Ahlfors had met in Munich during his travels, recommended him for a post at Harvard in the United States. Ahlfors agreed to a three year trial period. In 1936 he was one of the first two recipients of a Fields Medal at the International Congress in Oslo. In 1938 Ahlfors was offered a chair in mathematics at the University of Helsinki and, being rather homesick, he accepted this rather than remain permanently at Harvard. However a difficult time was approaching with World War II about to begin. The war made severe problems in Finland and the universities were closed. Ahlfors was unfit for military service so, as he states in [2]:Paradoxically I was myself able to do a lot of work during the war, although without the benefit of accessible libraries. Ahlfors' family were evacuated to Sweden during the war and so, when he was offered a chair in Zurich in 1944 it seemed a good chance to reunite his family. He met up with his family in Sweden, where Beurling gave them a great deal of help and friendship, but the war made the trip to Switzerland close to impossible. A flight from Stockholm to Prestwick in Scotland was arranged and, in March 1945, they made the trip. From Glasgow they travelled by train to London, then they made the difficult journey across the Channel, across France via Paris to Switzerland. He writes [2]:I cannot honestly say that I was happy in Zurich. The post-war era was not a good time for a stranger to take root in Switzerland. ... My wife and I did not feel welcome outside the circle of our immediate colleagues. An offer from Harvard in 1946 was therefore gladly accepted and, on this occasion, he remained there, retiring in 1977. His books are of lasting importance. Among them are Complex analysis (1953), Riemann surfaces (with L Sario) (1960). Lectures on quasi-conformal mappings (1966) and Conformal invariants (1973). In addition to the topics covered by these texts, Ahlfors did work of major importance on Kleinian groups. Allow me [EFR] a personal note on Ahlfors' Complex analysis. This was the text recommended to me by Copson who taught me complex analysis and it is indeed a tribute to Ahlfors that Copson, who had himself written a superb book on complex analysis, should recommend Ahlfors' book rather than his own. I found Ahlfors' Complex analysis beautifully written, an example of the very highest quality in mathematical texts, combining clarity with an excitement for the topic. Ahlfors received many honours for his outstanding contributions to mathematics. The award of the first Fields medal, mentioned above, must rank as the most important but another great honour was the award of the Wolf Prize in Mathematics in 1981. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of Lars Ahlfors

Mathematicians born in the same country

Honours awarded to Lars Ahlfors (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ahlfors.html (2 of 3) [2/16/2002 10:56:21 PM]

Ahlfors

Fields' Medal

Awarded 1936

Other Web sites

1. AMS 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ahlfors.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ahlfors.html (3 of 3) [2/16/2002 10:56:21 PM]

Ahmed

Ahmed ibn Yusuf al-Misri Born: 835 in Baghdad (now in Iraq) Died: 912 in Cairo, Egypt Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ahmed ibn Yusuf's father Yusuf ibn Ibrahim was also a mathematician. Yusuf ibn Ibrahim lived in Baghdad but moved to Damascus in about 839. After a little while he moved again, taking his son Ahmed with him, and went to live in Cairo. Although we are far from certain about the date of Ahmed's birth it is believed to have been before the family moved to Damascus. Again it is unclear exactly when the family moved again to Cairo but as Ahmed became known as "al-Misri " meaning "the Egyptian" it is likely that he lived in Cairo from a fairly young age. It is worth saying a word or two about Yusuf ibn Ibrahim, Ahmed's father, since scholars have had some difficulty in deciding which texts are due to the father, which to the son, or perhaps to joint work of the two. Yusuf ibn Ibrahim is known to have been a member of a group of scholars and this must have provided a strong intellectual environment for Ahmed. As well as a text on medicine, Yusuf is known to have written a work on astronomy and produced a collection of astronomical tables. Ahmed was to achieve an important role in Egypt and to understand this we must examine how Egypt achieved relative independence from the Abbasid Caliph. The Caliphs had strengthened their armies in the 9th century with Turkish slaves and began to put their Turkish commanders into positions as governors of certain territories in the Empire. In 868 the Turkish general Babak was put in charge of Egypt and he chose to send his stepson Ahmad ibn Tulun there to take control. Ahmad ibn Tulun soon built up an army under his own control and managed to take control of the finances of the country. Although he never declared complete independence from the Caliph he governed Egypt, and after 878 also Syria which his armies conquered, as an autonomous region. Ahmad ibn Tulun had a large family who formed the administration of Egypt. Ahmed ibn Yusuf was appointed as a private secretary to the family, in particular he was employed by one of Ahmad ibn Tulun's sons. In 884 Ahmad ibn Tulun died but his family continued to rule Egypt until the 905 when the Caliph sent an army to retake Egypt for the Empire. The period had been a fruitful one for Egypt during which agriculture, commerce and industry flourished. More importantly for Ahmed ibn Yusuf, the learning and scholarship of Baghdad was encouraged in Egypt, and he was able to pursue his mathematical researches while working for the Tulunid dynasty. We know of a work by Ahmed on ratio and proportion, a book On similar arcs, a commentary on Ptolemy's Centiloquium and a book about the astrolabe. All these works have survived and historians are confident that they are indeed the work of Ahmed, but several other works which some claim to be due to him are probably by other authors. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ahmed.html (1 of 2) [2/16/2002 10:56:23 PM]

Ahmed

Ahmed's work on ratio and proportion was translated into Latin by Gherard of Cremona. The book is largely a commentary on, and expansion of, Book 5 of Euclid's Elements. It was a carefully constructed work which influenced early European mathematicians such as Fibonacci. However it was not without its defects and Campanus of Novara pointed out a circular argument which occurs in Ahmed's reasoning. The book On similar arcs was also translated into Latin and influenced European mathematicians. In the treatise Ahmed proves that similar arcs of circles can be equal and not equal. The proof, like that on ratio and proportion, is based on Euclid. This time it is Propositions 20 and 21 of Book III of Euclid's Elements which are the main tools used by Ahmed. The complete Arabic text of this treatise is given in [2]. Ahmed ibn Yusuf also gave methods to solve tax problems which appear in Fibonacci's Liber Abaci. He was also quoted by Bradwardine, Jordanus and Pacioli. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ahmed.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ahmed.html (2 of 2) [2/16/2002 10:56:23 PM]

Ahmes

Ahmes Born: about 1680 BC in Egypt Died: about 1620 BC in Egypt Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ahmes is the scribe who wrote the Rhind Papyrus (named after the Scottish Egyptologist Alexander Henry Rhind who went to Thebes for health reasons, became interested in excavating and purchased the papyrus in Egypt in 1858). Ahmes claims not to be the author of the work, being, he claims, only a scribe. He says that the material comes from an earlier work of about 2000 BC. The papyrus is our chief source of information on Egyptian mathematics. The Recto contains division of 2 by the odd numbers 3 to 101 in unit fractions and the numbers 1 to 9 by 10. The Verso has 87 problems on the four operations, solution of equations, progressions, volumes of granaries, the two-thirds rule etc. The Rhind Papyrus, which came to the British Museum in 1863, is sometimes called the 'Ahmes papyrus' in honour of Ahmes. Nothing is known of Ahmes other than his own comments in the papyrus.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ahmes.html (1 of 3) [2/16/2002 10:56:26 PM]

Ahmes

The Rhind papyrus

Article by: J J O'Connor and E F Robertson List of References (4 books/articles)

A Quotation

Mathematicians born in the same country Some pages from publications Cross-references to History Topics

Rhind papyrus 1. Egyptian papyri 2. Squaring the circle

Other references in MacTutor

Chronology: 30000BC to 500BC

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ahmes.html (2 of 3) [2/16/2002 10:56:26 PM]

Ahmes

Other Web sites

1. Simon Fraser University 2. Mathematicians of the African diaspora 3. Kevin Brown

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ahmes.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ahmes.html (3 of 3) [2/16/2002 10:56:26 PM]

Aida

Aida Yasuaki Born: 10 Feb 1747 in Yamagata, Japan Died: 26 Oct 1817 in Edo (now Tokyo), Japan Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aida Yasuaki studied under the mathematician Yasuyuki Okazaki in Yamagata from the age of 15 years. The city of Yamagata in which Aida was born and brought up was (and still is) situated in northern Honshu, Japan nearly 300 km north of present day Tokyo. In 1769, Aida went to Edo, which has been renamed Tokyo. There Aida worked for the shogunate of Tokugawa Ieharu. The shogunate was the government of the shogun, or hereditary military dictator, of Japan and this type of rule lasted from 1192 to 1867. The third shogunate in Japan was established in 1603 ruling from Edo. The shogunate was extremely powerful, controlling the emperor, controlling the religious establishments, administering the lands and forming foreign policy. Aida was employed by the shogunate as a civil engineer working on river control and irrigation systems around Edo. However, this was not the job that Aida aimed for since ever since he was young his aim had been to become the best mathematician in Japan. Also working for the same shogunate at this time was Teirei Kamiya, a mathematician who had been a pupil of Sadasuke Fujita. Aida would have liked to become a pupil of Fujita, for he was one of the leading mathematicians in Japan. Aida saw his friendship with Kamiya as means to be accepted by Fujita and asked Kamiya to arrange for him to be introduced to Fujita. Indeed Kamiya organised the necessary introductions but Aida was not accepted by Fujita. It appears that relations between Fujita and Aida may have been poor even before Kamiya arranged the introduction, although if that were the case it is unclear quite why Aida worked so hard to obtain the introduction. It was the custom of the time for mathematicians to donate tablets inscribed with mathematical problems to religious temples. These tablets represented offerings of scholarship to the gods. Aida had donated some tablets which contained errors and these had been spotted by Fujita. Perhaps Aida was unaware of these errors at the time he sought to become Fujita's pupil. Fujita had published a mathematical work Seiyo sampo in 1781 and in part his high reputation rested on this highly regarded text. Aida now decided to write a work based on the Seiyo sampo yet one which would criticise this work. It is not surprising that relations between Aida and Fujita would deteriorate further when Aida published Kaisei sampo, his critical revision of the Seiyo sampo. The private feud extended to include other mathematicians when Kamiya, who had lost face by arranging the failed introductions, attacked Aida's Kaisei sampo. The argument eventually turned into a public feud between the Seki school of mathematics and the Sijyo school. Ajima was a friend of Fujita, their friendship arising from the fact that both were pupils of the same teacher Nushizumi Yamaji. Naturally Ajima joined the argument on the side of Fujita and since http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aida.html (1 of 2) [2/16/2002 10:56:27 PM]

Aida

Yamaji was a recognised master of the leading Seki school of mathematics, the argument soon involved the whole school. The shogun Tokugawa Ieharu died in 1786 and Tokugawa Ienari became his successor in the following year. Aida lost his post with the arrival of the new shogun and he decided that he would concentrate all his efforts on mathematics [1]:... he would live on his savings and devote himself to the perfection of his studies. He also took pupils, including many from the northeastern provinces; these returned to teach in their native regions, where Aida is still revered as a master of mathematics. Aida compiled Sampo tensi shinan which appeared in 1788. It is a book of geometry problems, developing formulas for ellipses, spheres, circles etc. Aida explained the use of algebraic expressions and the construction of equations. He also worked on number theory and simplified continued fraction methods due to Seki. The remarkable productivity of Aida is commented on in [1] where his contribution is summed up as follows:Aida was hard-working and strong-willed and produced as many as fifty to sixty works a year. Nearly 2000 works survived him, including many on non-mathematical subjects. He was a distinguished teacher of traditional mathematics and a successful populariser of that discipline. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Aida.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aida.html (2 of 2) [2/16/2002 10:56:27 PM]

Aiken

Howard Hathaway Aiken Born: 9 March 1900 in Hoboken, New Jersey, USA Died: 14 March 1973 in St Louis, Missouri, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Howard Aiken studied at the University of Wisconsin, Madison obtaining a doctorate from Harvard in 1939. While he was a graduate student and an instructor in the Department of Physics at Harvard Aiken began to make plans to build a large computer. These plans were made for a very specific purpose, for Aiken's research had led to a system of differential equations which had no exact solution and which could only be solved using numerical techniques. However, the amount of hand calculation involved would have been almost prohibitive, so Aiken's idea was to use an adaptation of the punched card machines which had been developed by Hollerith. Aiken wrote a report on how he envisaged the machine, and in particular how such a machine designed to be used in scientific research would differ from a punched card machine. He listed four main points [2]:... whereas accounting machines handle only positive numbers, scientific machines must be able to handle negative ones as well; that scientific machines must be able to handle such functions as logarithms, sines, cosines and a whole lot of other functions; the computer would be most useful for scientists if, once it was set in motion, it would work through the problem frequently for numerous numerical values without intervention until the calculation was finished; and that the machine should compute lines instead of columns, which is more in keeping with the sequence of mathematical events. The report was sufficient to prompt senior staff at Harvard to contact IBM and an agreement was made that Aiken would build his computer at the IBM laboratories at Endicott, helped by IBM engineers. Working with three engineers, Aiken developed the ASCC computer (Automatic Sequence Controlled

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aiken.html (1 of 3) [2/16/2002 10:56:28 PM]

Aiken

Calculator) which could carry out five operations, addition, subtraction, multiplication, division and reference to previous results. Aiken was much influenced in his ideas by Babbage's writings and he saw the project to build the ASCC computer as completing the task which Babbage had set out on but failed to complete. The ASCC had more in common with Babbage's analytical engine that one might imagine. Although it was powered by electricity, the major components were electromechanical in the form of magnetically operated switches. It weighed 35 tons, had 500 miles of wire and could compute to 23 significant figures. There were 72 storage registers and central units to perform multiplication and division. The gain an idea of the performance of the machine, a single addition to about 6 seconds while a division took about 12 seconds. ASCC was controlled by a sequence of instructions on punched paper tapes. Punched cards were used to enter data and the output from the machine was either on punched cards or by an electric typewriter. Having completed construction of ASCC in 1943 it was decided to move the computer to Harvard University where it began to be used from May 1944. Grace Hopper worked with Aiken from 1944 on the ASCC computer which had been renamed the Harvard Mark I and given by IBM to Harvard University. The computer figured highly in the Bureau of Ordnance's Computation Project at Harvard University, to which Hopper had been assigned, being used by the US navy for gunnery and ballistics calculations. Aiken completed the Harvard Mark II, a completely electronic computer, in 1947. He continued to work at Harvard on this series of machines, working next on the Mark III and finally the Mark IV up to 1952. He not only worked on computer construction, but he also published on electronics and switching theory. In 1964 Aiken received the Harry M Goode Memorial Award, a medal and $2,000 awarded by the Computer Society:For his original contribution to the development of the automatic computer, leading to the first large-scale general purpose automatic digital computer. This was one of many honours which Aiken received for his pioneering work with the development of computers. These awards were from many countries including the United States, France, The Netherlands, Belgium, and Germany. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

A Quotation

Mathematicians born in the same country Other references in MacTutor Other Web sites

A picture of Aiken's Mark I machine 1. Kalamazoo 2. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aiken.html (2 of 3) [2/16/2002 10:56:28 PM]

Aiken

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Aiken.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aiken.html (3 of 3) [2/16/2002 10:56:28 PM]

Airy

George Biddell Airy Born: 27 July 1801 in Alnwick, Northumberland, England Died: 2 Jan 1892 in Greenwich, England

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Airy's father was William Airy while his mother was Ann Biddell. William Airy was from Lincolnshire and Ann was the daughter of a farmer from Suffolk. Originally William had been a farmer too, but he had educated himself and risen to the position of tax inspector. When George was born his parents were living in Northumberland where William was a collector of excise, but in the following year the family moved to Hereford when William was transferred there. George attended Byatt Walker's school in Colchester and at the age of ten he took first place at the end of his primary school career. He had learnt some useful skills at the school such as arithmetic, double-entry book keeping and how to use a slide rule. He had probably learned more, however, from studying his father's books. He wrote in his autobiography that (see [3]):... he was not a favourite with his school mates. Eggen writes [1]:An introverted but not shy child, Airy was, even for the time and especially for his circumstances, a young snob. Nevertheless, he overcame some of the dislike of his schoolmates by his great skill and inventiveness in the construction of peashooters and other such devices. Before Airy left Byatt Walker's school his father had transferred again, this time to Essex. From 1812 Airy spent his summers with his uncle, Arthur Biddell, who had a farm near Ipswich. Clearly Airy was not too happy at home because he asked his uncle if he could live with him rather than with his own family. Things had taken a turn for the worse at home since his father lost his tax collectors job in 1813 and the family were, from that time, living in poverty. Because of the financial circumstances the family http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Airy.html (1 of 6) [2/16/2002 10:56:30 PM]

Airy

seem to have been quite glad that Airy's uncle had almost taken over the role of his father. The fact that Airy spent about half his time with his uncle over the next five years was important for him. Arthur Biddell was a man of learning who had a fine library containing books on chemistry, optics and mechanics which Airy avidly studied, and in addition he had many leading scientists as his friends. Their influence on the young Airy was marked and was a major factor in his seeking an academic career. During these five years, 1814 to 1819, Airy attended Colchester Grammar School where he was [7]:... noted for his memory, repeating in one examination 2394 lines of Latin verse. Airy entered Trinity College, Cambridge in 1819 as a sizar, meaning that he paid a reduced fee but essentially worked as a servant to make good the fee reduction. However it was only because his uncle provided financial support that he was able to undertake university studies at all. To supplement his income Airy took private pupils and this, of course, gave him less time for his own studies. Despite this his performance was outstanding and he graduated as Senior Wrangler (the top First Class student) in 1823 and was a Smith's prizeman. Woodhouse, who had left the Lucasian chair in 1822 to become Plumian Professor of Astronomy, was one of Airy's examiners for the Smith's prize, the other being Thomas Turton who had succeeded Woodhouse to the Lucasian chair. In the following year Airy was awarded a fellowship at Trinity College and began his academic career. We should comment on why Airy did so well in the Tripos examinations, being far ahead of the next best student. The Tripos examinations at that time were less a test of mathematical ability and more a test of the candidates ability to learn vast amounts of material and methods. At this Airy proved exceptionally good, partly because of his excellent memory, but also because of his remarkable organisational abilities. As an undergraduate he kept paper beside him to record every thought he had. Later everything was transferred to the books and diaries which he kept. He maintained this routine throughout his life and this record, almost of his every thought, still exists to provide remarkable evidence of the period [3]:The ruling feature of his character was order. From the time he went up to Cambridge to the end of his life his system of order was strictly maintained. Clerke writes in [7]:He never destroyed a document, but devised an ingenious plan of easy reference to the huge bulk of his papers. In 1824 Airy met Richarda Smith while on a walking holiday. He proposed two days after they first met but her father, Richard Smith, the vicar of a church near Chatsworth, refused to allow the marriage on the grounds that Airy could not support his daughter financially. This made Airy determined to obtain a position with the financial status which would allow him to marry. Only three years after graduating from Cambridge, he was appointed Lucasian Professor of Mathematics at Cambridge. It is rather surprising that the Lucasian Professor only received £99 per year while Airy was already receiving £150 as an assistant tutor. Airy wondered whether he could afford to compete for the chair when he was advised in 1826 that Turton was leaving, but Peacock persuaded him that the status was more important than the money. He became one of three candidates, French and Babbage being the other two. When Babbage stated that he was about to start legal proceedings over the election, French withdrew. Airy triumphed and a rivalry with Babbage which was to last for many years began. In addition to the Lucasian Chair, Airy was appointed a member of the Board of Longitude which gave

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Airy.html (2 of 6) [2/16/2002 10:56:30 PM]

Airy

him another £100 per year provided he attended four meetings. He explained his actions (see [3]):My prospects in the law or other profession might have been good if I could have waited but marriage would have been out of the question and I much preferred a moderate income in no long time. I had now in some measure taken science as my line (but not irrevocably) and I thought it best to work it well for a time at least and wait for accidents. These, of course, are not the words of a man driven by a love of his subject. He certainly still did not have the financial position to allow him to marry Richarda so he tried for other posts. His attempt to secure the vacant post of Astronomer Royal for Ireland failed in 1827. Airy was an examiner for the Smith's Prize and gave lectures while holding the Lucasian Chair. He lectured on light and in these lectures he explained the problem of astigmatism. It was an eye defect which Airy suffered from himself and he had been the first to design glasses to correct it. He had earlier published a paper On a peculiar Defect in the Eye on this problem for which he was the first to provide a practical solution. In 1828 Peacock informed Airy that Woodhouse, the Plumian Professor of Astronomy, had not long to live and advised him to seek this chair. He wrote [3]:I made it known that I was a candidate and nobody thought it worthwhile to oppose me. ... I told everyone that the salary (about £300) was not sufficient and drafted a manifesto to the University for an increase. ... the University had never before been taken by storm in such a manner and there was some commotion about it. I believe that very few people would have taken that step. ... I had no doubt of success. Airy was appointed Plumian Professor of Astronomy at Cambridge and Director of the Cambridge Observatory. The University had complied with his request for the salary to be increased and the £500 per year he received was sufficient to allow him to marry Richarda Smith , which he did on 24 March 1830. He became Astronomer Royal in 1835 moving at that time from Cambridge to Greenwich. There he undertook a reorganisation of the Royal Observatory which was positive in many ways but also had some unfortunate side effects. Since he could not tolerate his staff thinking for themselves no young scientists were trained at the Observatory during his period as Astronomer Royal. However, his considerable engineering ability was put to good use in renovating the instruments at the observatory. He held this post of Astronomer Royal until 1881 when he resigned and lived the rest of his life with his two unmarried daughters in the White House close to Greenwich Park. Airy wrote the text On the Algebraic and Numerical Theory of Errors of Observations and the Combinations of Observations. Although said at the time to be:... unreadable except by those already thoroughly acquainted with the subject, the book was used at Cambridge and influenced Pearson. This text was one of eleven books which Airy published, some of the others being Trigonometry (1825), Gravitation (1834), and Partial differential equations (1866). His remarkable publication record included over 500 papers and reports. This resulted from his extremely hard work and also his highly organised, efficient way of working which enabled him to get through far more work than almost every other scientist. His attitude to mathematics was very much as an applied mathematician who saw no point in the study of the subject in its own right. His son writes in [3]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Airy.html (3 of 6) [2/16/2002 10:56:30 PM]

Airy

His nature was eminently practical, and his dislike of mere theoretical problems and investigations was proportionally great. He was continually at war with some of the Cambridge mathematicians on this subject. Year after year he criticised the Senate House papers and the Smith's Prize papers very severely, and conducted an interesting and acrimonious private correspondence with Professor Cayley on the same subject. Airy's delay, in 1845, of searching for Neptune at the location suggested by Adams prevented Adams obtaining full credit for his work although in many ways he has been unfairly criticised over this episode. Airy did, however, make many major contributions to mathematics and astronomy. He improved the orbital theory of Venus and the Moon, studied interference fringes in optics, made a mathematical study of the rainbow and computed the density of the Earth by swinging a pendulum at the top and bottom of a deep mine. We should note that the value he obtained was too large by a fair amount. Airy was made chairman of the Commission set up to construct Standard Weights and Measures in 1834. He was elected a Fellow of the Royal Society of Edinburgh in 1835, and a Fellow of the Royal Society of London in 1836, having received the Society's Copley Medal in 1831. He gave the Bakerian lecture to the Society entitled On the theoretical explanation of an apparent new polarity of light in 1840. He received the Society's Royal Medal in 1845 for a paper on the Irish tides. The Royal Astronomical Society elected Airy to be their President in 1845. Then, in 1851, Airy was elected President of the British Association, and in 1871 he was elected President of the Royal Society of London holding the post for two years. The Institut de France elected him to membership to fill the position which became vacant on the death of John Herschel in 1872 and in the same year he accepted a knighthood having declined it on three previous occasions on the grounds that he could not afford the fees. Soon after this, in 1874, he organised an expedition to observe the transit of Venus. Outside his professional scientific interests, Airy was a man of broad tastes. He liked poetry, history, theology, antiquities, architecture, engineering, and geology. He even published papers on his other interests including one which tried to identify the exact place where Julius Caesar landed in Britain and the exact place from which he left. In addition he published a number of papers on religious matters. There were certainly sides to his character which made him unpopular with those around him. We have already mentioned how he was a snob at school. In later life he was sarcastic and enforced a rigid discipline on his staff at the Royal Observatory. In his defence we would have to note that he enforced such a rigid discipline on himself that it must have seemed natural to him to expect the same from others. An illustration of Airy's personality is shown from his long running disagreements with Babbage. They had a dispute over the quality of a telescope in 1832, he stated that Babbage's calculating engine was "worthless" ten years later and effectively stopped government funding of the project, and in 1854 he took the side of the narrow gauge for railways while Babbage supported the wide gauge. In all these disputes Airy came out the winner, but it is far from clear that he took the "right" side in the arguments. We should end with a few words on Airy's importance as a scientist. His own words certainly show that he had a realistic view of himself (see for example [1]):... in those parts of astronomy which ... [require] only method and judgement, with very little science in the persons employed, we have done much; while in those which depend exclusively on individual effort we have done little ... our principal progress has been made in the lower branches of astronomy while to the higher branches of science we have not http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Airy.html (4 of 6) [2/16/2002 10:56:30 PM]

Airy

added anything. Eggen writes in [1]:Airy was not a great scientist, but he made great science possible. However, others have a higher opinion of Airy's achievements. Chapman [6] believes that:Airy has been unfairly relegated to the scientific sidelines ... His son summed up Airy's life as follows [3]:The life of Airy was essentially that of a hard-working business man, and differed from that of other hard-working people only in the quality and variety of his work. It was not an exciting life, but it was full of interest ... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) A Poster of George Airy Cross-references to History Topics

Mathematicians born in the same country 1. Orbits and gravitation 2. Mathematical discovery of planets

Cross-references to Famous Curves

Nephroid

Other references in MacTutor

Airy's work in engineering.

Honours awarded to George Airy (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1836

Royal Society Copley Medal

Awarded 1831

Royal Society Royal Medal

Awarded 1845

Royal Society Bakerian lecturer

1840

Fellow of the Royal Society of Edinburgh Lucasian Professor of Mathematics

1826

Lunar features

Crater Airy

Planetary features

Crater Airy on Mars

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Airy.html (5 of 6) [2/16/2002 10:56:30 PM]

Airy

Other Web sites

1. Bob Bruen 2. Eric's treasure troves (Airy Functions) 3. Lassell pages (The discovery of Neptune) 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Airy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Airy.html (6 of 6) [2/16/2002 10:56:30 PM]

Aitken

Alexander Craig Aitken Born: 1 April 1895 in Dunedin, New Zealand Died: 3 Nov 1967 in Edinburgh, Scotland

Click the picture above to see three larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alec Aitken left the Otago Boys' High School in Dunedin in 1913 having won a scholarship to Otago University. He began to study languages and mathematics with the intention of becoming a school teacher but his university career was interrupted by World War I. He enlisted in 1915 and served in Gallipoli, Egypt and France being wounded at the battle of the Somme. His war experiences were to haunt him for the rest of his life. After three months in hospital he was sent back to New Zealand in 1917. The following year he returned to his university studies graduating in 1920 with First Class Honours in French and Latin but only Second Class in mathematics in which he had no proper instruction. Aitken followed his original intention and became a school teacher at his old school Otago Boys' High School. His mathematical genius bubbled under the surface and, encouraged by the new professor of mathematics at Otago University, Aitken came to Scotland in 1923 and studied for a Ph.D. at Edinburgh under Whittaker. Rather remarkably his Ph.D. thesis was considered so outstanding that he was awarded a D.Sc. for it. In 1925 he was appointed to Edinburgh where he spent the rest of his life. After holding lecturing posts in actuarial mathematics, then in statistics, then mathematical economics he became a Reader in statistics in 1936, the year he was elected a Fellow of the Royal Society. Ten years later he was appointed to Whittaker's chair. Aitken had an incredible memory (he knew to 2000 places) and could instantly multiply, divide and take roots of large numbers. He describes his own mental processes in the article [3]. Although some may suggest this has little to do with mathematical ability Aitken himself wrote:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aitken.html (1 of 3) [2/16/2002 10:56:32 PM]

Aitken

Familiarity with numbers acquired by innate faculty sharpened by assiduous practice does give insight into the profounder theorems of algebra and analysis. Aitken's mathematical work was in statistics, numerical analysis and algebra. In numerical analysis he introduced the idea of accelerating the convergence of a numerical method. He also introduced a method of progressive linear interpolation. In algebra he made contributions to the theory of determinants. He also saw clearly how invariant theory came under the theory of groups but wrote that he had never followed through his ideas because of various circumstances of anxiety, or duty, or bad health ... I have observed my talented younger contemporary Dudley Littlewood assault and capture most of this terrain. Aitken wrote several books The theory of canonical matrices (1932) was written jointly with Turnbull. With Rutherford he was editor of a series of the University Mathematical Texts and himself wrote Determinants and matrices (1939) and Statistical Mathematics (1939). In [2], describing his period of recovery from a small operation in 1934 Aitken writes:The nights were bad, in the daytime colleagues and other friends visited me, and I tried to think about abstract things, such as the theory of probability and the theory of groups - and I did begin to see more deeply into these rather abstruse disciplines. Indeed I date a change in my interests and an increase in competence, from these weeks of enforced physical inactivity. Also in [2] Aitken describes the reaction of other mathematicians to his work:... the papers on numerical analysis, statistical mathematics and the theory of the symmetric group continued to write themselves in steady succession, with other small notes on odds and ends. Those that I valued most, the algebraic ones, seemed to attrach hardly any notice, others, which I regarded as mere application of the highly compressed and powerful notation and algebra of matrices to standard problems in statistics or computation found great publicity in America... A colleague at St Andrews was a student in Edinburgh in the early 1960's. He writes:Professor Aitken's first year mathematics lectures were rather unusual. The fifty minutes were composed of forty minutes of clear mathematics, five minutes of jokes and stories and five minutes of 'tricks'. For the latter Professor Aitken would ask for members of the class to give him numbers for which he would then write down the reciprocal, the square root, the cube root or other appropriate expression. From the five minutes of 'stories' one also recalls as part of his lectures on probability a rather stern warning about the evils and foolishness of gambling! In fact Aitken's memory was also a problem for him. For most people memories fade in time which is particularly fortunate for the unpleasant things which happen. However with Aitken memories did not fade and his horrific memories of the battle of the Somme lived with him as real as the day he lived them. He wrote of them in [1] near the end of his life. These memories must have contributed to the ill health he suffered, and eventually led to his death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aitken.html (2 of 3) [2/16/2002 10:56:32 PM]

Aitken

List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Memory, mental arithmetic and mathematics 2. Matrices and determinants

Honours awarded to Alec Aitken (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1936

Fellow of the Royal Society of Edinburgh Honorary Fellow of the Edinburgh Maths Society

Elected 1967 1. New Zealand Edge

Other Web sites

2. David Giles

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Aitken.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aitken.html (3 of 3) [2/16/2002 10:56:32 PM]

Ajima

Chokuyen Naonobu Ajima Born: 1732 in Edo (now Tokyo), Japan Died: 1798 in Shiba, Japan Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Naonobu Ajima studied under Masatada in the Nakanishi school. After this he studied mathematics and astronomy under Yamaji becoming a pupil of the Seki school. He published nothing in his lifetime but his main work Fukyu sampo had a preface written in 1799, one year after Ajima's death, by Kasawa, one of his students. Although the intention was to publish the work then, it did not happen. Ajima's work went towards geometry despite the strong algebraic numerical tradition in the Seki school. He developed methods of computing volumes by double integration and worked on logarithms. In particular he produced log tables which were designed for taking 10th roots and powers of numbers. For this purpose he set the log of the 10th root of 10 to 1. His methods for calculating lengths of arcs of circles were more sophisticated than the Greek methods of Archimedes. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ajima.html (1 of 2) [2/16/2002 10:56:33 PM]

Ajima

JOC/EFR February 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ajima.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ajima.html (2 of 2) [2/16/2002 10:56:33 PM]

Akhiezer

Naum Il'ich Akhiezer Born: 6 March 1901 in Cherikov, Belarus Died: 3 June 1980 in Kharkov, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Naum Akhiezer studied at the Kiev Institute of Public Education, graduating from the in 1924. After graduating he joined the staff there and taught at Kiev until 1933. From 1933 he worked at Kharkov University. There he joined the Kharkov School of function theory and soon became its leading member. He served at President of the Kharkov Mathematical Society and was elected to the Academy of Sciences of the Ukraine. His main work was on function theory and approximation theory, building on the results of Chebyshev, Zolotarev and Markov. His important book Theory of Approximation was awarded the Chebyshev Prize. Petryshyn comments that:His most outstanding work consisted of deep approximation results in the constructive function theory, including the solution of the problem of Zolotarev. In around 1935, Kolmogorov laid the foundation for a new study, namely that of the extremal problem for a class of functons. In 1937 Akhiezer, working with M G Krein, solved the extremal problem for the class of differentiable periodic functions. Akhiezer continued to work on this topic and was later to solve the extremal problem for the class of analytic functions. Akhiezer's later work was on the theory of moments and he did joint work with Sergi Bernstein on completeness of sets of polynomials. Akhiezer wrote 130 papers and 8 books, one of which was the important Theory of Operators in Hilbert

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Akhiezer.html (1 of 2) [2/16/2002 10:56:35 PM]

Akhiezer

Space. He also contributed to the history of mathematics with an important book on Sergi Bernstein and his work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Akhiezer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Akhiezer.html (2 of 2) [2/16/2002 10:56:35 PM]

Al'Battani

Al'Battani This biography is now under Al-Battani. You should be automatically forwarded to the correct page. If your browser leaves you on this page, then press HERE. JOC/EFR September 1999

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al'Battani.html [2/16/2002 10:56:35 PM]

Al-Battani

Abu Allah Mohammad ibn Jabir Al-Battani Born: about 850 in Harran (near Urfa), Mesopotamia (now Turkey) Died: 929 in Qasr al-Jiss, Iraq

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Battani is sometimes known by a latinised version of his name, variants being Albategnius, Albategni or Albatenius. His full name was Abu Allah Mohammad ibn Jabir ibn Sinan al-Raqqi al-Harrani al-Sabi al-Battani. Al-Battani was born in Harran, called Carrhae in earlier times by the Romans, which lies on the Balikh River, 38 km southeast of Urfa. His family had been members of the Sabian sect, a religious sect of star worshippers from Harran. Being worshipers of the stars meant that the Sabians had a strong motivation for the study of astronomy and they produced many outstanding astronomers and mathematicians such as Thabit ibn Qurra. In fact Thabit was also born in Harran and would have still have been living there at the time that al-Battani was born. Al-Battani, unlike Thabit, was not a believer in the Sabian religion, however, for "Abu Allah Mohammad" indicates that he was certainly a Muslim. Although the identification is not absolutely certain, it is probable that al-Battani's father was Jabir ibn Sinan al-Harrani who had a high reputation as an instrument maker in Harran. The name certainly makes the identification fairly certain and the fact that al-Battani himself was skilled in making astronomical instruments is a good indication that he learnt these skills from his father. Al-Battani made his remarkably accurate astronomical observations at Antioch and ar-Raqqah in Syria. The town of ar-Raqqah, where most of al-Battani's observations were made, became prosperous when Harun al-Rashid, who became the fifth Caliph of the Abbasid dynasty on 14 September 786, built several palaces there. The town had been renamed al-Rashid at that time but, by the time al-Battani began observing there, it had reverted to the name of ar-Raqqah. The town was on the Euphrates River just west

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Battani.html (1 of 4) [2/16/2002 10:56:37 PM]

Al-Battani

of where it joins the Balikh River (on which Harran stands). The Fihrist (Index) was a work compiled by the bookseller Ibn an-Nadim in 988. It gives a full account of the Arabic literature which was available in the 10th century and it describes briefly some of the authors of this literature. The Fihrist describes al-Battani as (see for example [1]):... one of the famous observers and a leader in geometry, theoretical and practical astronomy, and astrology. He composed a work on astronomy, with tables, containing his own observations of the sun and moon and a more accurate description of their motions than that given in Ptolemy's "Almagest". In it moreover, he gives the motions of the five planets, with the improved observations he succeeded in making, as well as other necessary astronomical calculations. Some of his observations mentioned in his book of tables were made in the year 880 and later on in the year 900. Nobody is known in Islam who reached similar perfection in observing the stars and scrutinising their motions. Apart from this, he took great interest in astrology, which led him to write on this subject too: of his compositions in this field I mention his commentary on Ptolemy's Tetrabiblos. Other information about al-Battani contained in the Fihrist is that he observed between the years 877 and 918 and that his star catalogue is based on the year 880. It also describes the end of his life which seems to have occurred during a journey he made to Baghdad to protest on behalf of a group of people from ar-Raqqah because they had been unfairly taxed. Al-Battani reached Baghdad and put his arguments but died on the return journey to ar-Raqqah. The Fihrist also quotes a number of works by al-Battani. There is his Kitab al-Zij which is his major work on astronomy with tables, referred to above. We shall examine this in more detail in a moment. There is also the commentary on Ptolemy's Tetrabiblos referred to above and two other titles: On ascensions of the signs of the zodiac and On the quantities of the astrological applications. One of the chapters of the Kitab al-Zij has the title "On ascensions of the signs of the zodiac" and so the Fihrist may be wrong in thinking this is a separate work. This point still appears unclear. Al-Battani's Kitab al-Zij is by far his most important work and we should examine briefly the topics which it covered. The work contained 57 chapters. It begins with a description of the division of the celestial sphere into the signs of the zodiac and into degrees. The necessary background mathematical tools are then introduced such as the arithmetical operations on sexagesimal fractions and the trigonometric functions. Chapter 4 contains data from al-Battani's own observations. Chapters 5 to 26 discuss a large number of different astronomical problems following to some extent material from the Almagest. The motions of the sun, moon and five planets are discussed in chapters 27 to 31, where the theory given is that of Ptolemy but for al-Battani the theory appears less important than the practical aspects. After giving results to allow data given for one era to be converted to another era, al-Battani then gives 16 chapters which explain how his tables are to be read. Chapters 49 to 55 cover problems in astrology, while chapter 56 discusses the construction of a sundial and the final chapter discusses the construction of a number of astronomical instruments. What are the main achievements of al-Battani's Zij? He catalogued 489 stars. He refined the existing values for the length of the year, which he gave as 365 days 5 hours 48 minutes 24 seconds, and of the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Battani.html (2 of 4) [2/16/2002 10:56:37 PM]

Al-Battani

seasons. He calculated 54.5" per year for the precession of the equinoxes and obtained the value of 23 35' for the inclination of the ecliptic. Rather than using geometrical methods, as Ptolemy had done, al-Battani used trigonometrical methods which were an important advance. For example he gives important trigonometric formulas for right angled triangles such as b sin(A) = a sin(90 - A). Al-Battani showed that the farthest distance of the Sun from the Earth varies and, as a result, annular eclipses of the Sun are possible as well as total eclipses. However, as Swerdlow points out in [8], the influence of Ptolemy was remarkably strong on all medieval authors, and even a brilliant scientist like al-Battani probably did not dare to claim a different value of the distance from the Earth to the Sun from that given by Ptolemy. This was despite the fact that al-Battani could deduce a value for the distance from his own observations that differed greatly from Ptolemy's. In [1] Hartner gives a somewhat different opinion of the way that al-Battani is influenced by Ptolemy. He writes:While al-Battani takes no critical attitude towards the Ptolemaic kinematics in general, he evidences ... a very sound scepticism in regard to Ptolemy's practical results. Thus, relying on his own observations, he corrects - be it tacitly, be it in open words - Ptolemy's errors. This concerns the main parameters of planetary motion no less than erroneous conclusions drawn from insufficient or faulty observations, such as the invariability of the obliquity of the ecliptic or of the solar apogee. Al-Battani is important in the development of science for a number of reasons, but one of these must be the large influence his work had on scientists such as Tycho Brahe, Kepler, Galileo and Copernicus. In [5] there is a discussion on how al-Battani managed to produce more accurate measurements of the motion of the sun than did Copernicus. The author suggests that al-Battani obtained much more accurate results simply because his observations were made from a more southerly latitude. For al-Battani refraction had little effect on his meridian observations at the winter solstice because, at his more southerly site of ar-Raqqah, the sun was higher in the sky. Al-Battani's Kitab al-Zij was translated into Latin as De motu stellarum (On the motion of the stars) by Plato of Tivoli. This appeared in 1116 while a printed edition of Plato of Tivoi's translation appeared in 1537 and then again in 1645. A Spanish translation was made in the 13th century and both it and Plato of Tivoli's Latin translation have survived. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Battani.html (3 of 4) [2/16/2002 10:56:37 PM]

Al-Battani

Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Other references in MacTutor

Chronology: 900 to 1100

Honours awarded to Al-Battani (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Albategnius

Other Web sites

1. Muslim scientists 2. Muslims online 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Battani.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Battani.html (4 of 4) [2/16/2002 10:56:37 PM]

Al'Biruni

Al'Biruni This biography is now under Al-Biruni. You should be automatically forwarded to the correct page. If your browser leaves you on this page, then press HERE. JOC/EFR September 1999

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al'Biruni.html [2/16/2002 10:56:38 PM]

Al-Biruni

Abu Arrayhan Muhammad ibn Ahmad al-Biruni Born: 15 Sept 973 in Kath, Khwarazm (now Kara-Kalpakskaya, Uzbekistan) Died: 13 Dec 1048 in Ghazna (now Ghazni, Afganistan)

Click the picture above to see four larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Abu Rayhan al-Biruni was born in Khwarazm, a region adjoining the Aral Sea now known as Karakalpakstan. The two major cities in this region were Kath and Jurjaniyya. Al-Biruni was born near Kath and the town were he was born is today called Biruni after the great scholar. He lived both in Kath and in Jurjaniyya as he grew up and we know that he began studies at a very early age under the famous astronomer and mathematician Abu Nasr Mansur. Certainly by the age of seventeen al-Biruni was engaged in serious scientific work for it was in 990 that he computed the latitude of Kath by observing the maximum altitude of the sun. Other work which al-Biruni undertook as a young man was more theoretical. Before 995 (when he was 22 years old) he had written a number of short works. One which has survived is his Cartography which is a work on map projections. As well as describing his own projection of a hemisphere onto a plane, al-Biruni showed that by the age of 22 he was already extremely well read for he had studied a wide selection of map projections invented by others and he discusses them in the treatise. The comparatively quiet life that al-Biruni led up to this point was to come to a sudden end. It is interesting to speculate on how different his life, and contribution to scholarship, might have been but for the change in his life forced by the political events of 995. The end of the 10th century and beginning of the 11th century was a period of great unrest in the Islamic world and there were civil wars in the region in which al-Biruni was living. Khwarazm was at this time part of the Samanid Empire which ruled from Bukhara. Other states in the region were the Ziyarid state with its capital at Gurgan on the Caspian sea. Further west, the Buwayhid dynasty ruled over the area http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Biruni.html (1 of 7) [2/16/2002 10:56:40 PM]

Al-Biruni

between the Caspian sea and the Persian Gulf, and over Mesopotamia. Another kingdom which was rapidly rising in influence was the Ghaznavids whose capital was at Ghazna in Afghanistan, a kingdom which was to play a major role in al-Biruni's life. The Banu Iraq were the rulers of the Khwarazm region and Abu Nasr Mansur, al-Biruni's teacher, was a prince of that family. In 995 the rule by the Banu Iraq was overthrown in a coup. Al-Biruni fled at the outbreak of the civil war but it is less clear what happened to his teacher Abu Nasr Mansur at this stage. Describing these events later al-Biruni wrote [1]:After I had barely settled down for a few years, I was permitted by the Lord of Time to go back home, but I was compelled to participate in worldly affairs, which excited the envy of fools, but which made the wise pity me. Exactly where al-Biruni went when he fled from Khwarazm is unclear. He might have gone to Rayy (near to where the city of Tehran stands today) at this time, but certainly he was there at some time during the following few years. He writes that he was without a patron when in Rayy, and lived in poverty. al-Khujandi was an astronomer who was working with a very large instrument he had built on the mountain above Rayy to observe meridian transits of the sun near the solstices. He made observations on 16 and 17 June 994 for the summer solstice and 14 and 17 December 994 for the winter solstice. From these values he calculated the obliquity of the ecliptic, and the latitude of Rayy but neither are particularly accurate. Al-Khujandi discussed these observations, and his large sextant, with al-Biruni who later reported on them in his Tahdid where he claimed that the aperture of the sextant settled by about one span in the course of al-Khujandi's observations due to the weight of the instrument. Al-Biruni is almost certainly correct in pinpointing the cause of al-Khujandi's errors. Since al-Khujandi died in 1000, we can be fairly certain that al-Biruni spent part of the time between 995 and 997 at Rayy. He must also have spent part of this time in Gilan, which is bordered by the Caspian Sea on the north, for around this time he dedicated a work to the ruler of Gilan, ibn Rustam, who had connections with the Ziyarid state. We know certain dates in al-Biruni's life with certainty for he describes astronomical events in his works which allow accurate dates and places to be determined. His description of an eclipse of the moon on 24 May 997 which he observed at Kath means that he had returned to his native country by this time. The eclipse was an event that was also visible in Baghdad and al-Biruni had arranged with Abu'l-Wafa to observe it there. Comparing their timings enabled them to calculate the difference in longitude between the cities. We know that al-Biruni moved around frequently during this period for by 1000 he was at Gurgan being supported by Qabus, the ruler of the Ziyarid state. He dedicated his work Chronology to Qabus around 1000 and he was still in Gurgan on 19 February 1003 and 14 August 1003 when he observed eclipses of the moon there. We should record that in the Chronology al-Biruni refers to seven earlier works which he had written: one on the decimal system, one on the astrolabe, one on astronomical observations, three on astrology, and two on history. By 4 June 1004 al-Biruni was back in his homeland, for on that day he observed another eclipse of the moon from Jurjaniyya. Ali ibn Ma'mun had ruled over Khwarazm and he remained at the court when his brother Abu'l Abbas Ma'mun succeeded him as ruler. Both the Ma'mun brothers married sisters of the ruler Mahmud from the powerful state at Ghazna which would eventually take control of Abu'l Abbas Ma'mun's kingdom.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Biruni.html (2 of 7) [2/16/2002 10:56:40 PM]

Al-Biruni

Both Ali ibn Ma'mun and Abu'l Abbas Ma'mun were patrons of the sciences and supported a number of top scientists at their court. By 1004 Abu'l Abbas Ma'mun was ruler and he provided generous support for al-Biruni's scientific work. Not only did al-Biruni work there but Abu Nasr Mansur, his former teacher also worked there, allowing the pair to renew their collaboration. With Abu'l Abbas Ma'mun's support al-Biruni built an instrument at Jurjaniyya to observe solar meridian transits and he made 15 such observations with the instrument between 7 June 1016 and 7 December 1016. Wars in the region were to disrupt the scientific work of al-Biruni and Abu Nasr Mansur and eventually both left Khwarazm in about 1017. Mahmud was extending his influence over the region from his base in Ghazna and made a demand of Abu'l Abbas Ma'mun in 1014 to have his name inserted into the Friday prayers. This was a signal that he wanted an end to Ma'mun's rule and he was making a bid for the region to come under his control. After Ma'mun had at least partially agreed to Mahmud's demands, he was killed by his own army for what they considered to be an act of treachery. Following this Mahmud marched his army into the region and gained control of Kath on 3 July 1017. Both al-Biruni and Abu Nasr Mansur left with the victorious Mahmud, perhaps as his prisoners. There follows a strange period during which there is evidence in al-Biruni's own writings that he suffered great hardships but he also seems to have been supported by Mahmud for some scientific work. Some reports that Mahmud was cruel to al-Biruni may have some basis despite the limited patronage al-Biruni received from the ruler. Some dates and places from this period can again be deduced from descriptions of astronomical events recorded by al-Biruni. He was in Kabul on 14 October 1018 but, despite having no instruments with which to observe, he was able to make an observation with an ingenious instrument he made from materials at hand. At Lamghan, north of Kabul, on 8 April 1019 he observed an eclipse of the sun, writing [2]:... at sunrise we saw that approximately one-third of the sun was eclipsed and that the eclipse was waning. Between 1018 and 1020, supported by Mahmud, al-Biruni made observations from Ghazna which allowed an accurate determination of its latitude. On 17 September 1019 there was a lunar eclipse observed by al-Biruni from Ghazna and [2]:He gives precise details of the exact altitude of various well known stars at the moment of first contact. The relationship between Mahmud and al-Biruni is interesting. It is likely that al-Biruni was essentially a prisoner of Mahmud and was not free to leave. However Mahmud's military excursions into India meant that al-Biruni was taken to that country, and there can have been few experiences that al-Biruni would have enjoyed more. He may have wished for better treatment from Mahmud but al-Biruni's scientific work certainly benefited. From around 1022 Mahmud's armies began to have success in taking control of the northern parts of India and in 1026 his armies marched to the Indian Ocean. Al-Biruni seems only to have been in the northern parts of India, and we are uncertain how many visits he made, but observations he made there enabled him to determine the latitudes of eleven towns around the Punjab and the borders of Kashmir. His most famous work India was written as a direct result of the studies he made while in that country. The India is a massive work covering many different aspects of the country. Al-Biruni describes the religion and philosophy of India, its caste system and marriage customs. He then studies the Indian systems of writing and numbers before going on to examine the geography of the country. The book also

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Biruni.html (3 of 7) [2/16/2002 10:56:40 PM]

Al-Biruni

examines Indian astronomy, astrology and the calendar. Al-Biruni studied Indian literature in the original, translating several Sanskrit texts into Arabic. He also wrote several treatises devoted to certain aspects of Indian astronomy and mathematics which were of particular interest to him. Al-Biruni was amazingly well read, having knowledge of Sanskrit literature on topics such as astrology, astronomy, chronology, geography, grammar, mathematics, medicine, philosophy, religion, and weights and measures. See [65] for further details. Mahmud died in 1030 and he was succeeded by his eldest son Mas'ud, although not before a difficult political situation in which the two sons of Mahmud each tried to follow their father as ruler. Clearly al-Biruni was unsure who would succeed for he chose not to give a dedication in his India which appeared at this time. Better to have no dedication than to choose the wrong one! Mas'ud proved to be a ruler who treated al-Biruni more kindly than his father had done. If al-Biruni had been a virtual prisoner before, he now seems to have become free to travel as he pleased. Mas'ud was murdered in 1040 and succeeded by his son Mawdud who ruled for eight years. By this time al-Biruni was an old man but he continued his enormous output of scientific works right up to the time of his death. The total number of works produced by al-Biruni during his lifetime is impressive. Kennedy. writing in [1], estimates that he wrote around 146 works with a total of about 13,000 folios (a folio contains about the same amount as a printed page from a modern book). We have mentioned some of the works above, but the range of al-Biruni's works cover essentially the whole of science at his time. Kennedy writes [1]:... his bent was strongly towards the study of observable phenomena, in nature and in man. Within the sciences themselves he was attracted by those fields then susceptible of mathematical analysis. We have mentioned al-Biruni's astronomical observations many time above. It is worth noting that he had a better feel for errors than did Ptolemy. In [66] the author comments that Ptolemy's attitude was to select the observations which he thought most reliable (often that meant fitting in with his theory), and not to tell the reader about observations that he was discarding. Al-Biruni, on the other hand, treats errors more scientifically and when he does chose some to be more reliable than others, he also gives the discarded observations. He was also very conscious of rounding errors in calculations, and always attempted to observe quantities which required the minimum manipulation to produce answers. One of the most important of al-Biruni's many texts is Shadows which he is thought to have written around 1021. Rosenfel'd has written extensively on this work of al-Biruni (see for example [52], [55], and [59]). The contents of the work include the Arabic nomenclature of shade and shadows, strange phenomena involving shadows, gnomonics, the history of the tangent and secant functions, applications of the shadow functions to the astrolabe and to other instruments, shadow observations for the solution of various astronomical problems, and the shadow-determined times of Muslim prayers. Shadows is an extremely important source for our knowledge of the history of mathematics, astronomy, and physics. It also contains important ideas such as the idea that acceleration is connected with non-uniform motion, using three rectangular coordinates to define a point in 3-space, and ideas that some see as anticipating the introduction of polar coordinates. The book [5] details the mathematical contributions of al-Biruni. These include: theoretical and practical arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, Archimedes' theorems, trisection

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Biruni.html (4 of 7) [2/16/2002 10:56:40 PM]

Al-Biruni

of the angle and other problems which cannot be solved with ruler and compass alone, conic sections, stereometry, stereographic projection, trigonometry, the sine theorem in the plane, and solving spherical triangles. Important contributions to geodesy and geography were also made by al-Biruni. He introduced techniques to measure the earth and distances on it using triangulation. He found the radius of the earth to be 6339.6 km, a value not obtained in the West until the 16th century (see [50]). His Masudic canon contains a table giving the coordinates of six hundred places, almost all of which he had direct knowledge. Not all, however, were measured by al-Biruni himself, some being taken from a similar table given by al-Khwarizmi. The author of [27] remarks that al-Biruni seemed to realise that for places given by both al-Khwarizmi and Ptolemy, the value obtained by al-Khwarizmi is the more accurate. Al-Biruni also wrote a treatise on time-keeping, wrote several treatises on the astrolabe and describes a mechanical calendar. He makes interesting observations on the velocity of light, stating that its velocity is immense compared with that of sound. He also describes the Milky Way as ... a collection of countless fragments of the nature of nebulous stars. Topics in physics that were studied by al-Biruni included hydrostatics and made very accurate measurements of specific weights. He described the ratios between the densities of gold, mercury, lead, silver, bronze, copper, brass, iron, and tin. Al-Biruni displayed the results as combinations of integers and numbers of the form 1/n, n = 2, 3, 4, ... , 10. Many of al-Biruni's ideas were worked out in discussions and arguments with other scholars. He had a long-standing collaboration with his teacher Abu Nasr Mansur, each asking the other to undertake specific pieces of work to support their own. He corresponded with Avicenna, in a rather confrontational fashion, about the nature of heat and light. In [4], eighteen letters which Avicenna sent to al-Biruni in answer to questions that he had posed are given. These letters cover topics such as philosophy, astronomy and physics. Al-Biruni also corresponded with al-Sijzi. The paper [10] contains a letter that al-Biruni wrote to al-Sijzi (translated into English in [63]) which contains proofs of both the plane and spherical versions of the sine theorem. Al-Biruni says were due to his teacher Abu Nasr Mansur. Finally we should say a little about the personality of this great scholar. In contrast with the works of many others, we find out a lot about al-Biruni from his writings. Despite the fact that no more than one fifth of his works have survived, we get a clear picture of the great scientist. We see a man who was not a great innovator of original theories, mathematical or otherwise, but rather a careful observer who was a leading exponent of the experimental method. He was a great linguist who was able to read first hand an amazing number of the treatises that existed and he clearly saw the development of science as part of a historical process which he is always careful to put in proper context. His writings are therefore of great interest to historians of science. It appears clear that, despite his many works on astrology, al-Biruni did not believe in the 'science' but used it as a means to support his serious scientific work. A devout Muslim, he did write religious texts to suit his patrons particular sect. He shows no prejudice against different religious sects or races, but he does have strong words to say about various acts they committed. For example the Arab conquerors of Khwarazm destroyed ancient texts - what sin could be worse than that to the scholar as dedicated to learning and history as was al-Biruni. On the Christian faith al-Biruni considered the doctrine of forgiveness, writing in India [1]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Biruni.html (5 of 7) [2/16/2002 10:56:40 PM]

Al-Biruni

Upon my life, this is a noble philosophy, but the people of this world are not all philosophers. ... And indeed, ever since Constantine the Victorious became a Christian, both sword and whip have been ever employed. An indication of the sarcasm that he employed against those he saw to be foolish we give the reply that he made to a religious man who objected to the fact that an instrument which al-Biruni was showing him to determine the time for prayers had Byzantine months engraved on it. Al-Biruni reports in Shadows that he said to him:The Byzantines also eat food. Then do not imitate them in this! Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (72 books/articles)

Some Quotations (3)

A Poster of Al-Biruni

Mathematicians born in the same country

Cross-references to History Topics

1. Longitude and the Académie Royale 2. Arabic mathematics : forgotten brilliance? 3. Arabic numerals 4. Indian numerals

Other references in MacTutor

Chronology: 900 to 1100

Honours awarded to Al-Biruni (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Al-Biruni

Other Web sites

1. Muslim scientists 2. Muslims online 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Biruni.html (6 of 7) [2/16/2002 10:56:40 PM]

Al-Biruni

JOC/EFR November 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Biruni.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Biruni.html (7 of 7) [2/16/2002 10:56:40 PM]

Al'Haitam

Al'Haitam This biography is now under Al-Haytham. You should be automatically forwarded to the correct page. If your browser leaves you on this page, then press HERE. JOC/EFR November 1999

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al'Haitam.html [2/16/2002 10:56:40 PM]

Al-Haytham

Abu Ali al-Hasan ibn al-Haytham Born: 965 in (possibly) Basra, Persia (now Iraq) Died: 1040 in (possibly) Cairo, Egypt

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ibn al-Haytham is sometimes called al-Basri, meaning from the city of Basra in Iraq, and sometimes called al-Misri, meaning that he came from Egypt. He is often known as Alhazen which is the Latinised version of his first name "al-Hasan". In particular this name occurs in the naming of the problem for which he is best remembered, namely Alhazen's problem: Given a light source and a spherical mirror, find the point on the mirror were the light will be reflected to the eye of an observer. We shall discuss this problem, and ibn al-Haytham's other work, after giving some biographical details. In contrast to our lack of knowledge of the lives of many of the Arabic mathematicians, we have quite a number of details of ibn al-Haytham's life. However, although these details are in broad agreement with each other, they do contradict each other in several ways. We must therefore try to determine which are more likely to be accurate. It is worth commenting that an autobiography written by ibn al-Haytham in 1027 survives, but it says nothing of the events his life and concentrates on his intellectual development. Since the main events that we know of in ibn al-Haytham's life involve his time in Egypt, we should set the scene regarding that country. The Fatimid political and religious dynasty took its name from Fatimah, the daughter of the Prophet Muhammad. The Fatimids headed a religious movement dedicated to taking over the whole of the political and religious world of Islam. As a consequence they refused to recognise the 'Abbasid caliphs. The Fatimid caliphs ruled North Africa and Sicily during the first half of the 10th century, but after a number of unsuccessful attempts to defeat Egypt, they began a major advance into that country in 969 conquering the Nile Valley. They founded the city of Cairo as the capital of their new empire. These events were happening while ibn al-Haytham was a young boy growing up in Basra. We know little of ibn al-Haytham's years in Basra. In his autobiography he explains how, as a youth, he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Haytham.html (1 of 6) [2/16/2002 10:56:42 PM]

Al-Haytham

thought about the conflicting religious views of the various religious movements and came to the conclusion that none of them represented the truth. It appears that he did not devote himself to the study of mathematics and other academic topics at a young age but trained for what might be best described as a civil service job. He was appointed as a minister for Basra and the surrounding region. However, ibn al-Haytham became increasingly unhappy with his deep studies of religion and made a decision to devote himself entirely to a study of science which he found most clearly described in the writings of Aristotle. Having made this decision, ibn al-Haytham kept to it for the rest of his life devoting all his energies to mathematics, physics, and other sciences. Ibn al-Haytham went to Egypt some considerable time after he made the decision to give up his job as a minister and to devote himself to science, for he had made his reputation as a famous scientist while still in Basra. We do know that al-Hakim was Caliph when ibn al-Haytham reached Egypt. Al-Hakim was the second of the Fatimid caliphs to begin his reign in Egypt; al-Aziz was the first of the Fatimid caliphs to do so. Al-Aziz became Caliph in 975 on the death of his father al-Mu'izz. He was very involved in military and political ventures in northern Syria trying to expand the Fatimid empire. For most of his 20 year reign he worked towards this aim. Al-Aziz died in 996 while organising an army to march against the Byzantines and al-Hakim, who was eleven years old at the time, became Caliph. Al-Hakim, despite being a cruel leader who murdered his enemies, was a patron of the sciences employing top quality scientists such as the astronomer ibn Yunus. His support for science may have been partly because of his interest in astrology. Al-Hakim was highly eccentric, for example he ordered the sacking of the city of al-Fustat, he ordered the killing of all dogs since their barking annoyed him, and he banned certain vegetables and shellfish. However al-Hakim kept astronomical instruments in his house overlooking Cairo and built up a library which was only second in importance to that of the House of Wisdom over 150 years earlier. Our knowledge of ibn al-Haytham's interaction with al-Hakim comes from a number of sources, the most important of which is the writings of al-Qifti. We are told that al-Hakim learnt of a proposal by ibn al-Haytham to regulate the flow of water down the Nile. He requested that ibn al-Haytham come to Egypt to carry out his proposal and al-Hakim appointed him to head an engineering team which would undertake the task. However, as the team travelled further and further up the Nile, ibn al-Haytham realised that his idea to regulate the flow of water with large constructions would not work. Ibn al-Haytham returned with his engineering team and reported to al-Hakim that they could not achieve their aim. Al-Hakim, disappointed with ibn al-Haytham's scientific abilities, appointed him to an administrative post. At first ibn al-Haytham accepted this but soon realised that al-Hakim was a dangerous man whom he could not trust. It appears that ibn al-Haytham pretended to be mad and as a result was confined to his house until after al-Hakim's death in 1021. During this time he undertook scientific work and after al-Hakim's death he was able to show that he had only pretended to be mad. According to al-Qifti, ibn al-Haytham lived for the rest of his life near the Azhar Mosque in Cairo writing mathematics texts, teaching and making money by copying texts. Since the Fatimids founded the University of Al-Azhar based on this mosque in 970, ibn al-Haytham must have been associated with this centre of learning. A different report says that after failing in his mission to regulate the Nile, ibn al-Haytham fled from Egypt to Syria where he spent the rest of his life. This however seems unlikely for other reports certainly make it certain that ibn al-Haytham was in Egypt in 1038. One further complication is the title of a work

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Haytham.html (2 of 6) [2/16/2002 10:56:42 PM]

Al-Haytham

ibn al-Haytham wrote in 1027 which is entitled Ibn al-Haytham's answer to a geometrical question addressed to him in Baghdad. Several different explanations are possible, the simplest of which being that he visited Baghdad for a short time before returning to Egypt. He may also have spent some time in Syria which would partly explain the other version of the story. Yet another version has ibn al-Haytham pretending to be mad while still in Basra. Ibn al-Haytham's writings are too extensive for us to be able to cover even a reasonable amount. He seems to have written around 92 works of which, remarkably, over 55 have survived. The main topics on which he wrote were optics, including a theory of light and a theory of vision, astronomy, and mathematics, including geometry and number theory. We will give at least an indication of his contributions to these areas. A seven volume work on optics, Kitab al-Manazir, is considered by many to be ibn al-Haytham's most important contribution. It was translated into Latin as Opticae thesaurus Alhazeni in 1270. The previous major work on optics had been Ptolemy's Almagest and although ibn al-Haytham's work did not have an influence to equal that of Ptolemy's, nevertheless it must be regarded as the next major contribution to the field. The work begins with an introduction in which ibn al-Haytham says that he will begin "the inquiry into the principles and premises". His methods will involve "criticising premises and exercising caution in drawing conclusions" while he aimed "to employ justice, not follow prejudice, and to take care in all that we judge and criticise that we seek the truth and not be swayed by opinions". Also in Book I, ibn al-Haytham makes it clear that his investigation of light will be based on experimental evidence rather than on abstract theory. He notes that light is the same irrespective of the source and gives the examples of sunlight, light from a fire, or light reflected from a mirror which are all of the same nature. He gives the first correct explanation of vision, showing that light is reflected from an object into the eye. Most of the rest of Book I is devoted to the structure of the eye but here his explanations are necessarily in error since he does not have the concept of a lens which is necessary to understand the way the eye functions. His studies of optics did led him, however, to propose the use of a camera obscura, and he was the first person to mention it. Book II of the Optics discusses visual perception while Book III examines conditions necessary for good vision and how errors in vision are caused. From a mathematical point of view Book IV is one of the most important since it discusses the theory of reflection. Ibn al-Haytham gave [1]:... experimental proof of the specular reflection of accidental as well as essential light, a complete formulation of the laws of reflection, and a description of the construction and use of a copper instrument for measuring reflections from plane, spherical, cylindrical, and conical mirrors, whether convex or concave. Alhazen's problem, quoted near the beginning of this article, appears in Book V. Although we have quoted the problem for spherical mirrors, ibn al-Haytham also considered cylindrical and conical mirrors. The paper [36] gives a detailed description of six geometrical lemmas used by ibn al-Haytham in solving this problem. Huygens reformulated the problem as:To find the point of reflection on the surface of a spherical mirror, convex or concave, given the two points related to one another as eye and visible object. Huygens found a good solution which Vincenzo Riccati and then Saladini simplified and improved. Book VI of the Optics examines errors in vision due to reflection while the final book, Book VII, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Haytham.html (3 of 6) [2/16/2002 10:56:42 PM]

Al-Haytham

examines refraction [1]:Ibn al-Haytham does not give the impression that he was seeking a law which he failed to discover; but his "explanation" of refraction certainly forms part of the history of the formulation of the refraction law. The explanation is based on the idea that light is a movement which admits a variable speed (being less in denser bodies) ... Ibn al-Haytham's study of refraction led him to propose that the atmosphere had a finite depth of about 15 km. He explained twilight by refraction of sunlight once the Sun was less than 19 below the horizon. Abu al-Qasim ibn Madan was an astronomer who proposed questions to ibn al-Haytham, raising doubts about some of Ptolemy's explanations of physical phenomena. Ibn al-Haytham wrote a treatise Solution of doubts in which he gives his answers to these questions. They are discussed in [43] where the questions are given in the following form:What should we think of Ptolemy's account in "Almagest" I.3 concerning the visible enlargement of celestial magnitudes (the stars and their mutual distances) on the horizon? Is the explanation apparently implied by this account correct, and if so, under what physical conditions? How should we understand the analogy Ptolemy draws in the same place between this celestial phenomenon and the apparent magnification of objects seen in water? ... There are strange contrasts in ibn al-Haytham's work relating to Ptolemy. In Al-Shukuk ala Batlamyus (Doubts concerning Ptolemy), ibn al-Haytham is critical of Ptolemy's ideas yet in a popular work the Configuration, intended for the layman, ibn al-Haytham completely accepts Ptolemy's views without question. This is a very different approach to that taken in his Optics as the quotations given above from the introduction indicate. One of the mathematical problems which ibn al-Haytham attacked was the problem of squaring the circle. He wrote a work on the area of lunes, crescents formed from two intersecting circles, (see for example [10]) and then wrote the first of two treatises on squaring the circle using lunes (see [14]). However he seems to have realised that he could not solve the problem, for his promised second treatise on the topic never appeared. Whether ibn al-Haytham suspected that the problem was insoluble or whether he only realised that he could not solve it, in an interesting question which will never be answered. In number theory al-Haytham solved problems involving congruences using what is now called Wilson's theorem (John Wilson): if p is prime then 1+(p - 1) ! is divisible by p . In Opuscula ibn al-Haytham considers the solution of a system of congruences. In his own words (using the translation in [7]):To find a number such that if we divide by two, one remains; if we divide by three, one remains; if we divide by four, one remains; if we divide by five, one remains; if we divide by six, one remains; if we divide by seven, there is no remainder. Ibn al-Haytham gives two methods of solution:The problem is indeterminate, that is it admits of many solutions. There are two methods to find them. One of them is the canonical method: we multiply the numbers mentioned that http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Haytham.html (4 of 6) [2/16/2002 10:56:42 PM]

Al-Haytham

divide the number sought by each other; we add one to the product; this is the number sought. Here ibn al-Haytham gives a general method of solution which, in the special case, gives the solution (7-1)! + 1. Using Wilson's theorem, this is divisible by 7 and it clearly leaves a remainder of 1 when divided by 2, 3, 4, 5, and 6. Ibn al-Haytham's second method gives all the solutions to systems of congruences of the type stated (which of course is a special case of the Chinese Remainder Theorem). Another contribution by ibn al-Haytham to number theory was his work on perfect numbers. Euclid, in the Elements, had proved: If, for some k > 1, 2k - 1 is prime then 2k-1(2k - 1) is a perfect number. The converse of this result, namely that every even perfect number is of the form 2k-1(2k - 1) where 2k - 1 is prime, was proved by Euler. Rashed ([7], [8] or [27]) claims that ibn al-Haytham was the first to state this converse (although the statement does not appear explicitly in ibn al-Haytham's work). Rashed examines ibn al-Haytham's attempt to prove it in Analysis and synthesis which, as Rashed points out, is not entirely successful [7]:But this partial failure should not eclipse the essential: a deliberate attempt to characterise the set of perfect numbers. Ibn al-Haytham's main purpose in Analysis and synthesis is to study the methods mathematicians use to solve problems. The ancient Greeks used analysis to solve geometric problems but ibn al-Haytham sees it as a more general mathematical method which can be applied to other problems such as those in algebra. In this work ibn al-Haytham realises that analysis was not an algorithm which could automatically be applied using given rules but he realises that the method requires intuition. See [18] and [26] for more details. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (50 books/articles) A Poster of Ibn al-Haytham Cross-references to History Topics

Mathematicians born in the same country 1. Squaring the circle 2. Perfect numbers 3. Arabic mathematics : forgotten brilliance?

Other references in MacTutor

1. Wilson's theorem 2. Chronology: 900 to 1100

Honours awarded to Ibn al-Haytham (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Alhazen

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Haytham.html (5 of 6) [2/16/2002 10:56:42 PM]

Al-Haytham

Other Web sites

1. Eric's treasure troves (More about Alhazen's problem) 2. The Daily Telegraph (The solution of Alhazen's problem) 3. Muslim scientists 4. West Chester University 5. Muslims online 6. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Haytham.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Haytham.html (6 of 6) [2/16/2002 10:56:42 PM]

Al'Kashi

Al'Kashi This biography is now under Al-Kashi. You should be automatically forwarded to the correct page. If your browser leaves you on this page, then press HERE. JOC/EFR September 1999

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al'Kashi.html [2/16/2002 10:56:43 PM]

Al-Kashi

Ghiyath al-Din Jamshid Mas'ud al-Kashi Born: about 1380 in Kashan, Iran Died: 22 June 1429 in Samarkand, Transoxania (now Uzbek)

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Details of Jamshid al-Kashi's life and works are better known than many others from this period although details of his life are sketchy. One of the reasons we is that he dated many of his works with the exact date on which they were completed, another reason is that a number of letters which he wrote to his father have survived and give fascinating information. Al-Kashi was born in Kashan which lies in a desert at the eastern foot of the Central Iranian Range. At the time that al-Kashi was growing up Timur (often known as Tamburlaine) was conquering large regions. He had proclaimed himself sovereign and restorer of the Mongol empire at Samarkand in 1370 and, in 1383, Timur began his conquests in Persia with the capture of Herat. Timur died in 1405 and his empire was divided between his two sons, one of whom was Shah Rokh. While Timur was undertaking his military campaigns, conditions were very difficult with widespread poverty. al-Kashi lived in poverty, like so many others at this time, and devoted himself to astronomy and mathematics while moving from town to town. Conditions improved markedly when Shah Rokh took over after his father's death. He brought economic prosperity to the region and strongly supported artistic and intellectual life. With the changing atmosphere, al-Kashi's life also improved markedly. The first event in al-Kashi's life which we can date accurately is his observation of an eclipse of the moon which he made in Kashan on 2 June 1406. It is reasonable to assume that al-Kashi remained in Kashan where he worked on astronomical texts. He was certainly in his home town on 1 March 1407 when he completed Sullam Al-sama the text of which has survived. The full title of the work means The Stairway of Heaven, on Resolution of Difficulties Met by Predecessors in the Determination of Distances and Sizes (of the heavenly bodies). At this time it was necessary for scientists to obtain patronage from their kings, princes or rulers. Al-Kashi played this card

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Kashi.html (1 of 4) [2/16/2002 10:56:45 PM]

Al-Kashi

to his advantage and brought himself into favour in the new era where patronage of the arts and sciences became popular. His Compendium of the Science of Astronomy written during 1410-11 was dedicated to one of the descendants of the ruling Timurid dynasty. Samarkand, in Uzbekistan, is one of the oldest cities of Central Asia. The city became the capital of Timur's empire and Shah Rokh made his own son, Ulugh Beg, ruler of the city. Ulugh Beg, himself a great scientist, began to build the city into a great cultural centre. It was to Ulugh Beg that Al-Kashi dedicated his important book of astronomical tables Khaqani Zij which was based on the tables of Nasir al-Tusi. In the introduction al-Kashi says that without the support of Ulugh Beg he could not have been able to complete it. In this work there are trigonometric tables giving values of the sine function to four sexagesimal digits for each degree of argument with differences to be added for each minute. There are also tables which give transformations between different coordinate systems on the celestial sphere, in particular allowing ecliptic coordinates to be transformed into equatorial coordinates. See [14] for a detailed discussion of this work. The Khaqani Zij also contains [1]:... detailed tables of the longitudinal motion of the sun, the moon, and the planets. Al-Kashi also gives the tables of the longitudinal and latitudinal parallaxes for certain geographical latitudes, tables of eclipses, and tables of the visibility of the moon. Al-Kashi had certainly found the right patron in Ulugh Beg since he founded a university for the study of theology and science at Samarkand in about 1420 and he sought out the best scientists to help with his project. Ulugh Beg invited Al-Kashi to join him at this school of learning in Samarkand, as well as around sixty other scientists including Qadi Zada. There is little doubt that al-Kashi was the leading astronomer and mathematician at Samarkand and he was called the second Ptolemy by an historian writing later in the same century. Letters which al-Kashi wrote in Persian to his father, who lived in Kashan, have survived. These were written from Samarkand and give a wonderful description of the scientific life there. In 1424 Ulugh Beg began the construction of an observatory in Samarkand and, although the letters by al-Kashi are undated they were written at a time when construction of the observatory had begun. The contents of one of these letters has only recently been published, see [8]. In the letters al-Kashi praises the mathematical abilities of Ulugh Beg but of the other scientists in Samarkand, only Qadi Zada earned his respect. Ulugh Beg led scientific meetings where problems in astronomy were freely discussed. Usually these problems were too difficult for all except al-Kashi and Qadi Zada and on a couple of occasions only al-Kashi succeeded. It is clear that al-Kashi was the best scientist and closest collaborator of Ulugh Beg at Samarkand and, despite al-Kashi's ignorance of the correct court behaviour and lack of polished manners, he was highly respected by Ulugh Beg. After Al-Kashi's death, Ulugh Beg described him as (see for example [1]):... a remarkable scientist, one of the most famous in the world, who had a perfect command of the science of the ancients, who contributed to its development, and who could solve the most difficult problems. Although al-Kashi had done some fine work before joining Ulugh Beg at Samarkand, his best work was done while in that city. He produced his Treatise on the Circumference in July 1424, a work in which he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Kashi.html (2 of 4) [2/16/2002 10:56:45 PM]

Al-Kashi

calculated 2 to nine sexagesimal places and translated this into sixteen decimal places. This was an achievement far beyond anything which had been obtained before, either by the ancient Greeks or by the Chinese (who achieved 6 decimal places in the 5th century). It would be almost 200 years before van Ceulen surpassed Al-Kashi's accuracy with 20 decimal places. Al-Kashi's most impressive mathematical work was, however, The Key to Arithmetic which he completed on 2 March 1427. The work is a major text intended to be used in teaching students in Samarkand, in particular al-Kashi tries to give the necessary mathematics for those studying astronomy, surveying, architecture, accounting and trading. The authors of [1] describe the work as follows:In the richness of its contents and in the application of arithmetical and algebraic methods to the solution of various problems, including several geometric ones, and in the clarity and elegance of exposition, this voluminous textbook is one of the best in the whole of medieval literature; it attests to both the author's erudition and his pedagogical ability. Dold-Samplonius has discussed several aspects of al-Kashi's Key to Arithmetic in [11], [12], and [13]. (see also [3]). For example the measurement of the muqarnas refers to a type of decoration used to hide the edges and joints in buildings such as mosques and palaces. The decoration resembles a stalactite and consists of three-dimensional polygons, some with plane surfaces, and some with curved surfaces. Al-Kashi uses calculates decimal fractions in calculating the total surface area of types of muqarnas. The qubba is the dome of a funerary monument for a famous person. Al-Kashi finds good methods to approximate the surface area and the volume of the shell forming the dome of the qubba. We mentioned above al-Kashi's use of decimal fractions and it is through his use of these that he has attained considerable fame. The generally held view that Stevin had been the first to introduce decimal fractions was shown to be false in 1948 when P Luckey (see [4]) showed that in the Key to Arithmetic al-Kashi gives as clear a description of decimal fractions as Stevin does. However, to claim that al-Kashi is the inventor of decimal fractions, as was done by many mathematicians following the work of Luckey, would be far from the truth since the idea had been present in the work of several mathematicians of al-Karaji's school, in particular al-Samawal. Rashed (see [5] or [6]) puts al-Kashi's important contribution into perspective. He shows that the main advances brought in by al-Kashi are:(1) The analogy between both systems of fractions; the sexagesimal and the decimal systems. (2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as . Rashed also writes (see [5] or [6]):... Al-Kashi can no longer be considered as the inventor of decimal fractions; it remains nonetheless, that in his exposition the mathematician, far from being a simple compiler, went one step beyond al-Samawal and represents an important dimension in the history of decimal fractions. There are other major results in the work of al-Kashi which were pointed out by Luckey. He found that al-Kashi had an algorithm for calculating nth roots which was a special case of the methods given many centuries later by Ruffini and Horner. In later work Rashed shows (see for example [5] or [6]) that Al-Kashi was again describing methods which were present in the work of mathematicians of al-Karaji's

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Kashi.html (3 of 4) [2/16/2002 10:56:45 PM]

Al-Kashi

school, in particular al-Samawal. The last work by al-Kashi was The Treatise on the Chord and Sine which may have been unfinished at the time of his death and then completed by Qadi Zada. In this work al-Kashi computed sin 1 to the same accuracy as he had computed in his earlier work. He also considered the equation associated with the problem of trisecting an angle, namely a cubic equation. He was not the first to look at approximate solutions to this equation since al-Biruni had worked on it earlier. However, the iterative method proposed by al-Kashi was [1]:... one of the best achievements in medieval algebra. ... But all these discoveries of al-Kashi's were long unknown in Europe and were studied only in the nineteenth and twentieth centuries by ... historians of science.... Let us end with one final comment on the al-Kashi's work in astronomy. We mentioned earlier the astronomical tables Khaqani Zij produced by al-Kashi. It is worth noting that Ulugh Beg also produced astronomical tables and sine tables, and it is almost certain that these tables were based on al-Kashi's tables and almost certainly produced with al-Kashi's help. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (17 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Pi through the ages 2. A chronology of pi

Other references in MacTutor

Chronology: 1300 to 1500

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Kashi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Kashi.html (4 of 4) [2/16/2002 10:56:45 PM]

Al'Khwarizmi

Al'Khwarizmi This biography is now under Al-Khwarizmi. You should be automatically forwarded to the correct page. If your browser leaves you on this page, then press HERE. JOC/EFR September 1999

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al'Khwarizmi.html [2/16/2002 10:56:46 PM]

Al-Khwarizmi

Abu Ja'far Muhammad ibn Musa Al-Khwarizmi Born: about 780 in Baghdad (now in Iraq) Died: about 850

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

We know few details of Abu Ja'far Muhammad ibn Musa al-Khwarizmi's life. One unfortunate effect of this lack of knowledge seems to be the temptation to make guesses based on very little evidence. In [1] Toomer suggests that the name al-Khwarizmi may indicate that he came from Khwarizm south of the Aral Sea in central Asia. He then writes:But the historian al-Tabari gives him the additional epithet "al-Qutrubbulli", indicating that he came from Qutrubbull, a district between the Tigris and Euphrates not far from Baghdad, so perhaps his ancestors, rather than he himself, came from Khwarizm ... Another epithet given to him by al-Tabari, "al-Majusi", would seem to indicate that he was an adherent of the old Zoroastrian religion. ... the pious preface to al-Khwarizmi's "Algebra" shows that he was an orthodox Muslim, so Al-Tabari's epithet could mean no more than that his forebears, and perhaps he in his youth, had been Zoroastrians. However, Rashed [7], put a rather different interpretation on the same words by Al-Tabari:... Al-Tabari's words should read: "Muhammad ibn Musa al-Khwarizmi and al-Majusi al-Qutrubbulli ...", (and that there are two people al-Khwarizmi and al-Majusi al-Qutrubbulli): the letter "wa" was omitted in the early copy. This would not be worth mentioning if a series of conclusions about al-Khwarizmi's personality, occasionally even the origins of his knowledge, had not been drawn. In his article ([1]) G J Toomer, with naive confidence, constructed an entire fantasy on the error which cannot be denied the merit of making amusing reading. This is not the last disagreement that we shall meet in describing the life and work of al-Khwarizmi.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khwarizmi.html (1 of 8) [2/16/2002 10:56:48 PM]

Al-Khwarizmi

However before we look at the few facts about his life that are known for certain, we should take a moment to set the scene for the cultural and scientific background in which al-Khwarizmi worked. Harun al-Rashid became the fifth Caliph of the Abbasid dynasty on 14 September 786, about the time that al-Khwarizmi was born. Harun ruled, from his court in the capital city of Baghdad, over the Islam empire which stretched from the Mediterranean to India. He brought culture to his court and tried to establish the intellectual disciplines which at that time were not flourishing in the Arabic world. He had two sons, the eldest was al-Amin while the younger was al-Mamun. Harun died in 809 and there was an armed conflict between the brothers. Al-Mamun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Mamun became Caliph and ruled the empire from Baghdad. He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. He also built up a library of manuscripts, the first major library to be set up since that at Alexandria, collecting important works from Byzantium. In addition to the House of Wisdom, al-Mamun set up observatories in which Muslim astronomers could build on the knowledge acquired by earlier peoples. Al-Khwarizmi and his colleagues the Banu Musa were scholars at the House of Wisdom in Baghdad. Their tasks there involved the translation of Greek scientific manuscripts and they also studied, and wrote on, algebra, geometry and astronomy. Certainly al-Khwarizmi worked under the patronage of Al-Mamun and he dedicated two of his texts to the Caliph. These were his treatise on algebra and his treatise on astronomy. The algebra treatise Hisab al-jabr w'al-muqabala was the most famous and important of all of al-Khwarizmi's works. It is the title of this text that gives us the word "algebra" and, in a sense that we shall investigate more fully below, it is the first book to be written on algebra. Rosen's translation of al-Khwarizmi's own words describing the purpose of the book tells us that al-Khwarizmi intended to teach [11] (see also [1]):... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned. Now this does not sound like the contents of an algebra text and indeed only the first part of the book is a discussion of what we would today recognise as algebra. However it is important to realise that the book was intended to be highly practical and that algebra was introduced to solve real life problems that were part of everyday life in the Islam empire at that time. Early in the book al-Khwarizmi describes the natural numbers in terms that are almost funny to us who are so familiar with the system, but it is important to understand the new depth of abstraction and understanding here [11]:When I consider what people generally want in calculating, I found that it always is a number. I also observed that every number is composed of units, and that any number may be divided into units. Moreover, I found that every number which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled just as before the units were: thus arise twenty, thirty, etc. until a hundred: then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand; ... so forth to the utmost limit of numeration. Having introduced the natural numbers, al-Khwarizmi introduces the main topic of this first section of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khwarizmi.html (2 of 8) [2/16/2002 10:56:48 PM]

Al-Khwarizmi

his book, namely the solution of equations. His equations are linear or quadratic and are composed of units, roots and squares. For example, to al-Khwarizmi a unit was a number, a root was x, and a square was x2. However, although we shall use the now familiar algebraic notation in this article to help the reader understand the notions, Al-Khwarizmi's mathematics is done entirely in words with no symbols being used. He first reduces an equation (linear or quadratic) to one of six standard forms: 1. Squares equal to roots. 2. Squares equal to numbers. 3. Roots equal to numbers. 4. Squares and roots equal to numbers; e.g. x2 + 10 x = 39. 5. Squares and numbers equal to roots; e.g. x2 + 21 = 10 x. 6. Roots and numbers equal to squares; e.g. 3 x + 4 = x2. The reduction is carried out using the two operations of al-jabr and al-muqabala. Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. For example, using one of al-Khwarizmi's own examples, "al-jabr" transforms x2 = 40 x - 4 x2 into 5 x2 = 40 x. The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation. For example, two applications of "al-muqabala" reduces 50 + 3 x + x2 = 29 + 10 x to 21 + x2 = 7 x (one application to deal with the numbers and a second to deal with the roots). Al-Khwarizmi then shows how to solve the six standard types of equations. He uses both algebraic methods of solution and geometric methods. For example to solve the equation x2 + 10 x = 39 he writes [11]:... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khwarizmi.html (3 of 8) [2/16/2002 10:56:48 PM]

Al-Khwarizmi

The geometric proof by completing the square follows. Al-Khwarizmi starts with a square of side x, which therefore represents x2 (Figure 1). To the square we must add 10x and this is done by adding four rectangles each of breadth 10/4 and length x to the square (Figure 2). Figure 2 has area x2 + 10 x which is equal to 39. We now complete the square by adding the four little squares each of area 5/2 5/2 = 25/4. Hence the outside square in Fig 3 has area 4 25/4 + 39 = 25 + 39 = 64. The side of the square is therefore 8. But the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3.

These geometrical proofs are a matter of disagreement between experts. The question, which seems not to have an easy answer, is whether al-Khwarizmi was familiar with Euclid's Elements. We know that he could have been, perhaps it is even fair to say "should have been", familiar with Euclid's work. In al-Rashid's reign, while al-Khwarizmi was still young, al-Hajjaj had translated Euclid's Elements into Arabic and al-Hajjaj was one of al-Khwarizmi's colleagues in the House of Wisdom. This would support Toomer's comments in [1]:... in his introductory section al-Khwarizmi uses geometrical figures to explain equations, which surely argues for a familiarity with Book II of Euclid's "Elements". Rashed [9] writes that al-Khwarizmi's:... treatment was very probably inspired by recent knowledge of the "Elements". However, Gandz in [6] (see also [23]), argues for a very different view:Euclid's "Elements" in their spirit and letter are entirely unknown to [al-Khwarizmi]. Al-Khwarizmi has neither definitions, nor axioms, nor postulates, nor any demonstration of the Euclidean kind. I [EFR] think that it is clear that whether or not al-Khwarizmi had studied Euclid's Elements, he was influenced by other geometrical works. As Parshall writes in [35]:... because his treatment of practical geometry so closely followed that of the Hebrew text, Mishnat ha Middot, which dated from around 150 AD, the evidence of Semitic ancestry exists. Al-Khwarizmi continues his study of algebra in Hisab al-jabr w'al-muqabala by examining how the laws http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khwarizmi.html (4 of 8) [2/16/2002 10:56:48 PM]

Al-Khwarizmi

of arithmetic extend to an arithmetic for his algebraic objects. For example he shows how to multiply out expressions such as (a + b x) (c + d x) although again we should emphasise that al-Khwarizmi uses only words to describe his expressions, and no symbols are used. Rashed [9] sees a remarkable depth and novelty in these calculations by al-Khwarizmi which appear to us, when examined from a modern perspective, as relatively elementary. He writes [9]:Al-Khwarizmi's concept of algebra can now be grasped with greater precision: it concerns the theory of linear and quadratic equations with a single unknown, and the elementary arithmetic of relative binomials and trinomials. ... The solution had to be general and calculable at the same time and in a mathematical fashion, that is, geometrically founded. ... The restriction of degree, as well as that of the number of unsophisticated terms, is instantly explained. From its true emergence, algebra can be seen as a theory of equations solved by means of radicals, and of algebraic calculations on related expressions... If this interpretation is correct, then al-Khwarizmi was as Sarton writes:... the greatest mathematician of the time, and if one takes all the circumstances into account, one of the greatest of all time.... In a similar vein Rashed writes [9]:It is impossible to overstress the originality of the conception and style of al-Khwarizmi's algebra... but a different view is taken by Crossley who writes [4]:[Al-Khwarizmi] may not have been very original... and Toomer who writes in [1]:... Al-Khwarizmi's scientific achievements were at best mediocre. In [23] Gandz gives this opinion of al-Khwarizmi's algebra:Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers. The next part of al-Khwarizmi's Algebra consists of applications and worked examples. He then goes on to look at rules for finding the area of figures such as the circle and also finding the volume of solids such as the sphere, cone, and pyramid. This section on mensuration certainly has more in common with Hindu and Hebrew texts than it does with any Greek work. The final part of the book deals with the complicated Islamic rules for inheritance but require little from the earlier algebra beyond solving linear equations. Al-Khwarizmi also wrote a treatise on Hindu-Arabic numerals. The Arabic text is lost but a Latin translation, Algoritmi de numero Indorum in English Al-Khwarizmi on the Hindu Art of Reckoning gave rise to the word algorithm deriving from his name in the title. Unfortunately the Latin translation (translated into English in [19]) is known to be much changed from al-Khwarizmi's original text (of which even the title is unknown). The work describes the Hindu place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The first use of zero as a place holder in positional base notation was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khwarizmi.html (5 of 8) [2/16/2002 10:56:48 PM]

Al-Khwarizmi

probably due to al-Khwarizmi in this work. Methods for arithmetical calculation are given, and a method to find square roots is known to have been in the Arabic original although it is missing from the Latin version. Toomer writes [1]:... the decimal place-value system was a fairly recent arrival from India and ... al-Khwarizmi's work was the first to expound it systematically. Thus, although elementary, it was of seminal importance. Seven twelfth century Latin treatises based on this lost Arabic treatise by al-Khwarizmi on arithmetic are discussed in [17]. Another important work by al-Khwarizmi was his work Sindhind zij on astronomy. The work, described in detail in [48], is based in Indian astronomical works [47]:... as opposed to most later Islamic astronomical handbooks, which utilised the Greek planetary models laid out in Ptolemy's "Almagest"... The Indian text on which al-Khwarizmi based his treatise was one which had been given to the court in Baghdad around 770 as a gift from an Indian political mission. There are two versions of al-Khwarizmi's work which he wrote in Arabic but both are lost. In the tenth century al-Majriti made a critical revision of the shorter version and this was translated into Latin by Adelard of Bath. There is also a Latin version of the longer version and both these Latin works have survived. The main topics covered by al-Khwarizmi in the Sindhind zij are calendars; calculating true positions of the sun, moon and planets, tables of sines and tangents; spherical astronomy; astrological tables; parallax and eclipse calculations; and visibility of the moon. A related manuscript, attributed to al-Khwarizmi, on spherical trigonometry is discussed in [39]. Although his astronomical work is based on that of the Indians, and most of the values from which he constructed his tables came from Hindu astronomers, al-Khwarizmi must have been influenced by Ptolemy's work too [1]:It is certain that Ptolemy's tables, in their revision by Theon of Alexandria, were already known to some Islamic astronomers; and it is highly likely that they influenced, directly or through intermediaries, the form in which Al-Khwarizmi's tables were cast. Al-Khwarizmi wrote a major work on geography which give latitudes and longitudes for 2402 localities as a basis for a world map. The book, which is based on Ptolemy's Geography, lists with latitudes and longitudes, cities, mountains, seas, islands, geographical regions, and rivers. The manuscript does include maps which on the whole are more accurate than those of Ptolemy. In particular it is clear that where more local knowledge was available to al-Khwarizmi such as the regions of Islam, Africa and the Far East then his work is considerably more accurate than that of Ptolemy, but for Europe al-Khwarizmi seems to have used Ptolemy's data. A number of minor works were written by al-Khwarizmi on topics such as the astrolabe, on which he wrote two works, on the sundial, and on the Jewish calendar. He also wrote a political history containing horoscopes of prominent persons. We have already discussed the varying views of the importance of al-Khwarizmi's algebra which was his most important contribution to mathematics. Let us end this article with a quote by Mohammad Kahn, given in [3]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khwarizmi.html (6 of 8) [2/16/2002 10:56:48 PM]

Al-Khwarizmi

In the foremost rank of mathematicians of all time stands Al-Khwarizmi. He composed the oldest works on arithmetic and algebra. They were the principal source of mathematical knowledge for centuries to come in the East and the West. The work on arithmetic first introduced the Hindu numbers to Europe, as the very name algorism signifies; and the work on algebra ... gave the name to this important branch of mathematics in the European world... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (53 books/articles)

A Quotation

A Poster of Al-Khwarizmi

Mathematicians born in the same country

Some pages from publications

A facsimile of the frontispiece from Algebra A facsimile of another page and a translation.

Cross-references to History Topics

1. Quadratic, cubic and quartic equations 2. Arabic mathematics : forgotten brilliance? 3. Arabic numerals 4. The fundamental theorem of algebra 5. Pi through the ages 6. A chronology of pi 7. A history of Zero

Other references in MacTutor

Chronology: 500 to 900

Honours awarded to Al-Khwarizmi (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Al-Khwarizmi

Other Web sites

1. Karen H Parshall 2. Muslim scientists 3. Muslims online 4. Brent Byars 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khwarizmi.html (7 of 8) [2/16/2002 10:56:48 PM]

Al-Khwarizmi

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Khwarizmi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khwarizmi.html (8 of 8) [2/16/2002 10:56:48 PM]

Al-Baghdadi

Abu Mansur ibn Tahir Al-Baghdadi Born: about 980 in Baghdad, Iraq Died: 1037 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Baghdadi is sometimes known as Ibn Tahir. His full name is Abu Mansur Abr al-Qahir ibn Tahir ibn Muhammad ibn Abdallah al-Tamini al-Shaffi al-Baghdadi. We can deduce from al-Baghdadi's last two names that he was descended from the Bani Tamim tribe, one of the Sharif tribes of ancient Arabia, and that he belonged to the Madhhab Shafi'i school of religious law. This school of law, one of the four Sunni schools, took its name from the teacher Abu 'Abd Allah as-Shafi'i (767-820) and was based on both the divine law of the Qur'an or Hadith and on human logical reasoning when no divine teachings were given. We have a few details of al-Baghdadi's life. He was born and brought up in Baghdad but left that city to go to Nishapur (sometimes written Neyshabur in English) in the Tus region of northeastern Iran. He did not go to Nishapur alone, but was accompanied by his father who must have been a man of considerable wealth, for al-Baghdadi, without any apparent income himself, was able to spend a great deal of money on supporting scholarship and men of learning. At this time Nishapur was, like the whole of the region around it, a place where there was little political stability as various tribes and religious groups fought with each other. When riots broke out in Nishapur, al-Baghdadi decided that he required a more peaceful place to continue his life as an academic so he moved to Asfirayin. This town was quieter and al-Baghdadi was able to teach and study in more peaceful surroundings. He was certainly considered as one of the great teachers of his time and the people of Nishapur were sad to lose the great scholar from their city. In Asfirayin, al-Baghdadi taught for many years in the mosque. Always having sufficient wealth, he took no payment for his teachings, devoting his life to the pursuit of learning and teaching for its own sake. His writings were mainly concerned with theology, as we must assume were his teachings. However, he wrote at least two books on mathematics. One, Kitab fi'l-misaha, is relatively unimportant. It is concerned with the measurement of lengths, areas and volumes. The second is, however, a work of major importance in the history of mathematics. This treatise, al-Takmila fi'l-Hisab, is a work in which al-Baghdadi considers different systems of arithmetic. These systems derive from counting on the fingers, the sexagesimal system, and the arithmetic of the Indian numerals and fractions. He also considers the arithmetic of irrational numbers and business arithmetic. In this work al-Baghdadi stresses the benefits of each of the systems but seems to favour the Indian numerals.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Baghdadi.html (1 of 3) [2/16/2002 10:56:49 PM]

Al-Baghdadi

Several important results in number theory appear in the al-Takmila as do comments which allow us to obtain information on certain texts of al-Khwarizmi which are now lost. We shall discuss the number theory results in more detail below, but first let us comment on the light which the al-Takmila sheds on the problem of why Renaissance mathematicians were divided into "abacists" and "algorists" and exactly what is captured by these two names. It seems clear that those using Indian numerals used an abacus and were then called "abacists". The "algorists" followed the methods of al-Khwarizmi's lost work which, contrary to what was originally thought, is not a work on Indian numerals but rather a work on finger counting methods. This becomes clear from the references to the lost work by al-Baghdadi. Let us now consider the number theory in al-Takmila. Al-Baghdadi gives an interesting discussion of abundant numbers, deficient numbers, perfect numbers and equivalent numbers. Suppose that, in modern notation, S(n) denotes the sum of the aliquot parts of n, that is the sum of its proper quotients. First al-Baghdadi defines perfect numbers (those number n with S(n) = n), abundant numbers (those number n with S(n) > n), and deficient numbers (those number n with S(n) < n). Of course these properties of numbers had been studied by the ancient Greeks. Al-Baghdadi gives some elementary results and then states that 945 is the smallest odd abundant number, a result usually attributed to Bachet in the early 17th century. Nicomachus had made claims about perfect numbers in around 100 AD which were accepted, seemingly without question, in Europe up to the 16th century. However, al-Baghdadi knew that certain claims made by Nicomachus were false. Al-Baghdadi wrote (see for example [2] or [3]):He who affirms that there is only one perfect number in each power of 10 is wrong; there is no perfect number between ten thousand and one hundred thousand. He who affirms that all perfect numbers end with the figure 6 or 8 are right. Next al-Baghdadi goes on to define equivalent numbers, and appears to be the first to study them. Two numbers m and n are called equivalent if S(m) = S(n). He then considers the problem: given k, find m, n with S(m) = S(n) = k. The method he gives is a pretty one. He then gives the example k = 57, obtaining S(159) = 57 and S(559) = 57. However, he missed 703, for S(703) = 57 as well. The results that al-Baghdadi gives on amicable numbers are only a slight variations on results given earlier by Thabit ibn Qurra. In modern notation, m and n are amicable if S(n) = m, and S(m) = n. Thabit ibn Qurra's theorem is as follows: for n > 1, let pn = 3.2n-1 and qn = 9.22n-1-1. Then if pn-1, pn, and qn are prime, then a = 2n pn-1 pn and b = 2nqn are amicable numbers while a is abundant and b is deficient. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Arabic mathematics : forgotten brilliance? 2. Arabic numerals

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Baghdadi.html (2 of 3) [2/16/2002 10:56:49 PM]

Al-Baghdadi

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Baghdadi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Baghdadi.html (3 of 3) [2/16/2002 10:56:49 PM]

Al-Banna

al-Marrakushi ibn Al-Banna Born: 29 Dec 1256 in Marrakesh, Morocco Died: 1321 in Marrakesh, Morocco Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ibn al-Banna is also known as Abu'l-Abbas Ahmad ibn Muhammad ibn Uthman al-Azdi. It is a little unclear whether al-Banna was born in the city of Marrakesh or whether it was the region of Marrakesh which was named Morocco by Europeans. There is a claim that al-Banna was born in Granada in Spain and moved to North Africa for his education. What is certain is that he spent most of his life in Morocco. The Marinids tribe were allies of the Umayyad caliphs of Córdoba. The tribe lived in eastern Morocco then, under their ruler Abu Yahya, they began to conquer the region. The Marinids captured Fez in 1248 and made it their capital. They captured Marrakesh from the ruling Almohads tribe in 1269, thus taking control of the whole of Morocco. Having conquered Morocco, the Marinids tried to help Granada to prevent the Christian advance through their country. The strong link between Granada and Morocco may account for the confusion as to which country al-Banna was a native. Morocco was certainly the country that al-Banna was educated in, learning the leading mathematical skills of the period. He studied geometry in general, and Euclid's Elements in particular. He also studied fractional numbers and learnt much of the impressive contributions that the Arabs had made to mathematics over the preceding 400 years. The Marinids had a strong culture for learning and Fez became their centre of learning. At the university in Fez Al-Banna taught all branches of mathematics, which at this time included arithmetic, algebra, geometry and astronomy. Fez was a thriving city with a new quarter being built housing the Royal Palace and the adjoining Great Mosque. Many students studied under al-Banna in this thriving academic community. It is clear that al-Banna wrote a large number of works, in fact 82 are listed by Renaud (see for example [9]). Not all are on mathematics, but the mathematical texts included an introduction to Euclid's Elements, an algebra text and various works on astronomy. One difficulty with the works on mathematics is knowing how much of the material which al-Banna presents is original and how much is simply his version of work by earlier Arab mathematicians which has been lost. We should certainly say that al-Banna does not claim any originality and, indeed, the style of his writing would suggest that he is collecting together ideas that he has learnt from other mathematicians. Two "firsts" for al-Banna are that he seems to have been the first to consider a fraction as a ratio between two numbers (see [12] for more details) and he is the first to use the expression almanakc (in Arabic al-manakh meaning weather) in a work containing astronomical and meteorological data. Perhaps al-Banna's most famous work is Talkhis amal al-hisab (Summary of arithmetical operations) and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Banna.html (1 of 3) [2/16/2002 10:56:51 PM]

Al-Banna

the Raf al-Hijab which is al-Banna's own commentary on the Talkhis amal al-hisab. It is in this work that al-Banna introduces some mathematical notation which has led certain authors to believe that algebraic symbolism was first developed in Islam by ibn al-Banna and al-Qalasadi (see for example [6]). We refer the reader to the biography of al-Qalasadi where we present arguments to show that neither al-Banna nor al-Qalasadi were the inventors of mathematical notation. There are, however, many interesting mathematical ideas and results which appear in the Raf al-Hijab. For example it contains continued fractions and they are used to compute approximate square roots. Other interesting results on summing series are the results 13 + 33 + 53 + ... + (2n-1)3 = n2(2n2 - 1) and 12 + 32 + 52 + ... + (2n-1)2 = (2n + 1)2n(2n - 1)/6. Perhaps the most interesting of all is the work on binomial coefficients which is described in detail in [2] and [3]. If we denote the binomial coefficient p choose k by pCk then al-Banna shows that pC2

= p(p-1)/2

and then that pC3 = pC2 (p-2)/3. He writes (see for example [2] or [3]):... the ternary combination is thus obtained by multiplying the third of the third term preceding the given number; and so we always multiply the combination that precedes the combination sought by the number that precedes the given number, and whose distance to it is equal to the number of combinations sought. From the product, we take the part that names the number of combinations. Although this is a little difficult to interpret, what al-Banna is stating here is that pCk = pCk-1(p - (k - 1) )/k. He then goes on to give the familiar (to us) result pCk = p(p - 1)(p - 2)...(p - k + 1)/(k !) As Rashed points out in [2], this is only a small step from the Pascal triangle results given three hundred years earlier by al-Karaji, then still one hundred years before al-Banna by al-Samawal. However Rashed writes:... in our opinion, there is something more fundamental than [the Pascal triangle] results; it is precisely the combinatorial appearance of ibn al-Banna's exposition, together with the relation he partially establishes between polygonal numbers and combinations. It concern, in the first place, triangular numbers and combinations of p objects in twos, and then polygonal numbers of order 4 and combinations of p objects in threes. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Banna.html (2 of 3) [2/16/2002 10:56:51 PM]

Al-Banna

Mathematicians born in the same country Cross-references to History Topics

Arabic numerals

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Banna.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Banna.html (3 of 3) [2/16/2002 10:56:51 PM]

Al-Farisi

Kamal al-Din Abu'l Hasan Muhammad Al-Farisi Born: about 1260 Died: about 1320 in Tabriz, Iran Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Farisi is also known as Kamal al-din. His full name is Kamal al-din Abu'l Hasan Muhammad ibn al-Hasan al-Farisi. He made two major contributions to mathematics, one on light, the other on number theory. His work on light, colour and the rainbow is discussed in [1] but no mention of his work on number theory (nor mention of any other work at all by al-Farisi) occurs in that article written by R Rashed. On the other hand his contributions to number theory are discussed the references [2], [3], [4], [7], [8], and [9], most of which are also written by R Rashed but relate to discoveries made after the article [1] was written. Al-Farisi was a pupil of the astronomer and mathematician Qutb al-Din al-Shirazi (1236 - 1311), who in turn was a pupil of Nasir al-Din al-Tusi. His work on light was prompted by a question put to him concerning the refraction of light. Al-Shirazi advised him to consult the Optics of ibn al-Haytham and al-Farisi made such a deep study of this treatise that al-Shirazi suggested that he write what is essentially a revision of that major work. Al-Shirazi himself was writing a commentary on works of Avicenna at the time. Now al-Farisi went much further, for he undertook a project to study all the optical work of ibn al-Haytham. His major work the Tanqih (which means revision) was far more than a commentary on ibn al-Haytham's optical writings. Al-Farisi does not seek merely to explain the works of a master in a more elementary form, rather he is quite prepared to suggest that some of ibn al-Haytham's theories are incorrect and to propose alternative theories himself. The most important part of this work by al-Farisi is his theory of the rainbow. Ibn al-Haytham had indeed proposed a theory, but al-Farisi considered both this theory and another proposed by Avicenna before giving his own. The theory proposed by al-Farisi was the first mathematically satisfactory explanation of the rainbow. Ibn al-Haytham had proposed that light from the sun is reflected by a cloud before reaching the eye. It was a theory which did not allow for a possible experimental verification. Al-Farisi, on the other hand, proposed a model where the ray of light from the sun was refracted twice by a water droplet, one or more reflections occurring between the two refractions. This model did allow an experiment to be conducted

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Farisi.html (1 of 3) [2/16/2002 10:56:52 PM]

Al-Farisi

with a transparent sphere filled with water. Of course this introduced two additional sources for refraction, namely at the surface between the glass container and the water. Al-Farisi was able to show that the approximation obtained by his model was good enough to allow him to ignore the effects of the glass container. In order to explain the colours in the rainbow, however, al-Farisi had to produce some new ideas about how colours were formed. The view before al-Farisi was that colours were produced a mixing darkness with light. This could not explain the rainbow so, based on the experimental evidence of the colours that he had observed with his transparent sphere experiment, al-Farisi proposed that the colours occurred because of the superimposition of different forms of the image on a dark background. He wrote (see for example [1]):... If the images then interpenetrate, the light is again intensified and produces a bright yellow. Next, the blended image diminishes and becomes a darker and darker red until it disappears when the sun is outside the cone of rays refracted after one reflection. There have been arguments between modern scholars as to whether al-Farisi's theory of the rainbow was due to him or whether it was a theory proposed by his teacher al-Shirazi. Boyer writes in [5]:... the discovery of the theory should presumably be ascribed to [al-Shirazi], its elaboration to [al-Farisi]. Rashed discusses the claims of Boyer and others that the innovation in the theory of the rainbow was from al-Shirazi, but gives sound arguments for his claim that ascribing the theory to al-Shirazi is unconvincing. Al-Farisi made a number of important contributions to number theory. He noted the impossibility of giving an integer solution to the equation x4 + y4 = z4 but he attempted no proof of this case of Fermat's Last Theorem. Al-Farisi's most impressive work in number theory is on amicable numbers. Suppose that, in modern notation, S(n) denotes the sum of the aliquot parts of n, that is the sum of its proper quotients. The numbers m and n are called amicable if S(n) = m, and S(m) = n. In Tadhkira al-ahbab fi bayan al-tahabb (Memorandum for friends on the proof of amicability) al-Farisi gave a new proof of the following theorem by Thabit ibn Qurra on amicable numbers: For n > 1, let pn = 3.2n - 1 and qn = 9.22n-1 - 1. If pn-1, pn, and qn are prime numbers, then a = 2n pn-1 pn and b = 2nqn are amicable numbers. It was not a simple modification that al-Farisi made. Rather he produced a major new approach to a whole area of number theory, introducing ideas concerning factorisation and combinatorial methods. In fact al-Farisi's approach is based on the unique factorisation of an integer into powers of prime numbers, and, according to Rashed, he states and attempts to prove this, the so-called fundamental theorem of arithmetic, in this work. Whether al-Farisi proved or attempted to prove the fundamental theorem of arithmetic is also discussed in [4]. At the end of his treatise al-Farisi gives the pairs of amicable numbers 220, 284 and 17296, 18416, obtained from using Thabit's rule with n = 2 and n = 4 respectively. To check that Thabit's theorem gives amicable numbers with n = 4, al-Farisi has to show that p3, p4, and q4 are prime numbers. Now p3 = 23, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Farisi.html (2 of 3) [2/16/2002 10:56:52 PM]

Al-Farisi

p4 = 47 and q4 =1151 and, to show that 1151 is prime al-Farisi uses a number of lemmas including an application of the sieve of Eratosthenes. The pair of amicable number 17296, 18416 are known as Euler's amicable pair. There is no doubt that al-Farisi proved these to be amicable numbers long before Euler. However, al-Farisi was probably not the first to discover these amicable numbers. In [6] Hogendijk argues that they were known to Thabit ibn Qurra himself. Al-Farisi saw the relation between polygonal numbers and the binomial coefficients and he presented arguments, using an early type of mathematical induction, which showed a relation between triangular numbers, the sums of triangular numbers, the sums of the sums of triangular number, etc., and the combinations of n objects taken k at a time. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Farisi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Farisi.html (3 of 3) [2/16/2002 10:56:52 PM]

Al-Haitam

Al-Haitam This biography is now under Al-Haytham. You should be automatically forwarded to the correct page. If your browser leaves you on this page, then press HERE. JOC/EFR November 1999

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Haitam.html [2/16/2002 10:56:52 PM]

Al-Jawhari

al-Abbas ibn Said Al-Jawhari Born: about 800 in possibly Baghdad, Iraq Died: about 860 in possibly Baghdad, Iraq Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

We know little of al-Jawhari's life except that he was associated with the remarkable House of Wisdom that was set up in Baghdad by the Caliph al-Ma'mun. It is worth looking at the events which led up to the founding of this important centre for learning. Harun al-Rashid became the fifth Caliph of the Abbasid dynasty on 14 September 786, and ruled from his court in the capital city of Baghdad over the Islam empire which stretched from the Mediterranean to India. He brought culture to his court and tried to establish the intellectual disciplines which at that time were not flourishing in the Arabic world. He had two sons, the eldest was al-Amin while the younger was al-Ma'mun. Harun died in 809 and there was an armed conflict between the brothers. Al-Ma'mun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Ma'mun became Caliph and ruled the empire from Baghdad. He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. He also built up a library of manuscripts, the first major library to be set up since that at Alexandria, collecting important works from Byzantium. In addition to the House of Wisdom, al-Ma'mun set up observatories in which Muslim astronomers could build on the knowledge acquired by earlier peoples. Al-Jawhari was employed in the service of al-Ma'mun in Baghdad, although we do not know exactly when he began his work there. Mathematicians such as al-Kindi, al-Khwarizmi, Hunayn ibn Ishaq, Thabit ibn Qurra and the Banu Musa brothers were also appointed by al-Ma'mun to the House of Wisdom, so a truly remarkable collection of scholars worked there. There are very few instances in the history of mathematics when a larger number of world class mathematicians gathered together and took part in research. Al-Jawhari, although best known as a geometer, made observations in Baghdad from 829 to 830 while working for al-Ma'mun. He left Baghdad before the death of al-Ma'mun in 833, for he was observing in Damascus in 832-33. The main work by al-Jawhari was Commentary on Euclid's Elements which is listed in the Fihrist (Index), a work compiled by the bookseller Ibn an-Nadim in 988. Commentary on Euclid's Elements is almost the same work described by Nasir al-din al-Tusi (although al-Tusi gives a slightly different title for al-Jawhari's work: Emendation of the Elements ). This work contained nearly fifty propositions additional to those given by Euclid and included an attempt by al-Jawhari to prove the parallel postulate. The proof followed similar lines to that attempted by Simplicius but it is certainly not a copy of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Jawhari.html (1 of 2) [2/16/2002 10:56:53 PM]

Al-Jawhari

Simplicius's proof, containing several original ideas. Al-Tusi quotes six of the nearly fifty propositions which together form what al-Jawhari believed was a proof of the parallel postulate. This means that, as far as we are aware, al-Jawhari was the first Arabic mathematician to attempt such a proof. The fact that the proof fails was certainly noted by al-Tusi. The paper [3] discusses a thirteenth century commentary on a short treatise by al-Jawhari. In the short treatise al-Jawhari presents three additions to Book V of Euclid's Elements, which are meant prove Definition 5 which defines equal ratio, and Definition 7 which defines greater ratio. Al-Jawhari's "proofs" are examples of early attempts by Muslim mathematicians to understand the difficult concepts in Euclid's Elements. Berggren, reviewing [3], expresses surprise, not at al-Jawhari's fallacious arguments, but rather the fact that they were still being repeated 400 years later:One can only wonder, however, at the survival of such ill-conceived alterations of Euclid's "Elements" and their incorporation, so many centuries later, in an Arabic edition of the "Elements" composed late in the thirteenth century. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Jawhari.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Jawhari.html (2 of 2) [2/16/2002 10:56:53 PM]

Al-Jayyani

Abu Abd Allah Muhammad ibn Muadh Al-Jayyani Born: 989 in Cordoba, Spain Died: after 1079 in possibly Jaén, Andalusia (now Spain) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Little is known of al-Jayyani's life. Even the identification of al-Jayyani the mathematician with al-Jayyani the Spanish scholar who was born in Cordoba in 989 is not absolutely certain. Everything points to this identification being correct except one (possible) problem. The Spanish scholar who was born in Cordoba has exactly the same name as the mathematician, and the Spanish scholar is described as an expert in the Qur'an, also being knowledgeable in Arabic philology, inheritance laws and arithmetic. Al-Jayyani, the mathematician, is described as a judge and a jurist in one of his treatises. The only possible problem to the identification is that al-Jayyani wrote a treatise on the total solar eclipse which occurred in Jaén on 1 July 1079. The identification means that he was over ninety years old when he wrote this treatise which, although certainly not impossible, casts a slight doubt. The only other facts known about al-Jayyani's life are that he lived in Cairo from 1012 to 1017 and that he must have undertaken most of his work in Jaén, the city at the centre of the Moorish principality of Jayyan. This can not only be deduced from his name "al-Jayyani" which means "from Jaén", but also from the fact that the astronomical tables that he produced were for the longitude of Jaén. Certainly he observed the solar eclipse in Jaén in 1079. Al-Jayyani's work On ratio is almost certainly his most interesting mathematical work. An English translation of this remarkable treatise is given in [2]. In this work al-Jayyani sets out to defend Euclid's Elements Book V. In [7] Vahabzadeh writes:Euclid's definition, in Book V of his "Elements", of the proportionality of four magnitudes gave rise to numerous commentaries. Of these we have selected two [one being al-Jayyani's] whose goal was not to criticise Euclid's point of view but rather to justify it by trying to make explicit the assumptions underlying Euclid's argument. Al-Jayyani states that he is writing the treatise On ratio (see for example [1]):... to explain what may not be clear in the fifth book of Euclid's writing to such as are not satisfied with it.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Jayyani.html (1 of 3) [2/16/2002 10:56:55 PM]

Al-Jayyani

There are five magnitudes that, according to al-Jayyani, are used in geometry; number, line, surface, angle, and solid. Neither Euclid nor any other Greek mathematician would have considered "number" as a geometrical magnitude, but al-Jayyani needs the notion for his definition of ratio which follows the Arabic idea of number. After assuming that every intelligent person has a basic understanding of ratio, al-Jayyani deduces further properties based on this "commonly understood definition". To justify his approach he writes:There is no method to make clear what is already clear in itself. He then connects this idea of ratio with that given by Euclid. The authors of [1] write:Al-Jayyani shows here an understanding comparable with that of Isaac Barrow, who is customarily regarded as the first to have really understood Euclid's Book V. Another work of great importance is al-Jayyani's The book of unknown arcs of a sphere, the first treatise on spherical trigonometry. The work, which is published together with a Spanish translation and a commentary in [3], contains formulas for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle. Proofs are sometimes only given as sketches. Debarnot, in his review of [3], argues however that Villuendas:... in his commentary ... fails to take the originality of the Determination of the magnitudes sufficiently into account. Al-Jayyani was to have a strong influence on European mathematics. In addition to translations of his works from the Arabic, his work influenced certain European mathematicians. The article [4] argues that one of Regiomontanus sources was The book of unknown arcs of a sphere. Among the similarities between al-Jayyani's treatise and that of Regiomontanus are the definition of ratios as numbers, the lack of a tangent function, and a similar method of solving a spherical triangle when all sides are unknown. However, the author of [4] remarks that there are some marked differences in approach between al-Jayyani and Regiomontanus, such as the proof of the spherical sign law. Although it is certain that Regiomontanus based his treatise on Arabic works on spherical trigonometry it may well be that al-Jayyani's work was only one of many such sources. The article [6] describes the treatise Kitab al-asrar fi nata'ij al-Afkar (The book of secrets about the results of thoughts), attributed to al-Jayyani on the basis of internal evidence together with its date. The work studies hydraulics and water clocks. Work by al-Jayyani on astronomy was also important. He wrote on the morning and evening twilight, computing the fairly accurate value of 18 for the angle of the sun below the horizon at the start on morning twilight and at the end of the evening twilight. In the Tabulae Jahen al-Jayyani gave data to enable the calculation of the time of day, the calendar, the new moon, eclipses and information required for the timing and direction for prayers. As was common at this time, not only was there astronomical information in the work but also astrological information on horoscopes. Al-Jayyani seems to have considerable respect for al-Khwarizmi's astronomical data, which he freely used, but he rejects the ideas of al-Khwarizmi on astrology. Much of al-Jayyani's astrology is based on Hindu sources. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Jayyani.html (2 of 3) [2/16/2002 10:56:55 PM]

Al-Jayyani

List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Jayyani.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Jayyani.html (3 of 3) [2/16/2002 10:56:55 PM]

Al-Karaji

Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji Born: 13 April 953 in Baghdad (now in Iraq) Died: about 1029 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

The first comment that we must make regards al-Karaji's name. It appears both as al-Karaji and as al-Karkhi but this is not a simple matter of two different transliterations of the same Arabic name. The significance is that Karaj is a city in Iran and if the mathematician's name is al-Karaji then certainly his family were from that city. On the other hand Karkh is one of the original suburbs of Baghdad which grew up outside the southern gate of the original city. The name al-Karkhi would indicate that the mathematician came from the suburb of Baghdad. Historians seem divided as to which of these interpretations is correct. The version al-Karkhi was proposed by Woepcke (see [7] or [8]) but al-Karaji, the version which is most often used in texts today, was suggested as most likely by della Vida in 1933. Rashed comments (see [1] or [5]):In the present state of our knowledge delle Vida's argument is plausible but not decisive. On the basis of the manuscripts consulted it is far from easy to decide in favour of either name. Certainly we know that al-Karaji lived in Baghdad for most of his life and that his chief mathematical works were written during the time when he lived in that city. His important treatise on algebra Al-Fakhri was dedicated to the ruler of Baghdad and was written in the city. However, at some later point in his career, al-Karaji left Baghdad to live in what are described as the "mountain countries". He seems to have given up mathematics at this time and concentrated on engineering topics such as the drilling of wells. The importance of al-Karaji in the development of mathematics is viewed rather differently by different authors. The reason for this, rather in the same spirit as the different views on al-Khwarizmi, depends on the significance one attaches to the style of his mathematics. Some consider that his work is merely reworking ideas from earlier mathematicians while others see him as the first person to completely free algebra from geometrical operations and replace them with the arithmetical type of operations which are at the core of algebra today. Crossley [3] sounds relatively unimpressed by al-Karaji's contributions (although he describes the content accurately):[Al-Karaji] gives rules for the arithmetic operations including (essentially) the multiplication of polynomials. ... al-Karaji usually gives a numerical example for his rules but does not give any sort of proof beyond giving geometrical pictures. Often he explicitely

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Karaji.html (1 of 5) [2/16/2002 10:56:57 PM]

Al-Karaji

says that he is giving a solution in the style of Diophantus. He does not treat equations above the second degree except for ones which can easily be reduced to at most second degree equations followed by the extraction of roots. The solutions of quadratics are based explicitly on the Euclidean theorems ... Woepcke in [7] (see also the reprint [8]) was the first historian to realise the importance of al-Karaji's work and later historians mostly agree with his interpretation. He describes it as the first appearance of a:... theory of algebraic calculus ... . Rashed (see [5] which contains Rashed's article from [1] and other writings by Rashed on al-Karaji) agrees with Woepcke's interpretation and perhaps goes even further in stressing al-Karaji's importance. He writes:... the more-or-less explicit aim of [al-Karaji's] exposition was to find the means of realising the autonomy and specificity of algebra, so as to be in a position to reject, in particular, the geometric representation of algebraic operations. To give another quote from Rashed's description of al-Karaji's contribution:Al-Karaji's work holds an especially important place in the history of mathematics. ... the discovery and reading of the arithmetical work of Diophantus, in the light of the algebraic conceptions and methods of al-Khwarizmi and other Arab algebraists, made possible a new departure in algebra by Al-Karaji ... So what was this new departure in algebra? Perhaps it is best described by al-Samawal, one of al-Karaji's successors, who described it as [5]:... operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known. What al-Karaji achieved in Al-Fakhri was first to define the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and to give rules for products of any two of these. So what he achieved here was defining the product of these terms without any reference to geometry. In fact he almost gave the formula xn xm = xm+n for all integers n and m but he failed to make the definition x0 = 1 so he fell just a little short. Having given rules for multiplication and division of monomials al-Karaji then looked at "composite quantities" or sums of monomials. For these he gave rules for addition, subtraction and multiplication but not for division in the general case, only giving rules for the division of a composite quantity by a monomial. He was able to give a rule for finding the square root of a composite quantity which is not completely general since it required the coefficients to be positive, but it is still a remarkable achievement. Al-Karaji also uses a form of mathematical induction in his arguments, although he certainly does not give a rigorous exposition of the principle. Basically what al-Karaji does is to demonstrate an argument for n = 1, then prove the case n = 2 based on his result for n = 1, then prove the case n = 3 based on his result for n = 2, and carry on to around n = 5 before remarking that one can continue the process indefinitely. Although this is not induction proper, it is a major step towards understanding inductive proofs.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Karaji.html (2 of 5) [2/16/2002 10:56:57 PM]

Al-Karaji

One of the results on which al-Karaji uses this form of induction comes from his work on the binomial theorem, the binomial coefficients and the Pascal triangle. In Al-Fakhri al-Karaji computed (a+b)3 and in Al-Badi he computed (a-b)3 and (a+b)4. The general construction of the Pascal triangle was given by al-Karaji in work described in the later writings of al-Samawal. In the translation by Rashed and Ahmad (see for example [5]) al-Samawal writes:Let us now recall a principle for knowing the necessary number of multiplications of these degrees by each other, for any number divided into two parts. Al-Karaji said that in order to succeed we must place 'one' on a table and 'one' below the first 'one', move the first 'one' into a second column, add the first 'one' to the 'one' below it. Thus we obtain 'two', which we put below the transferred 'one' and we place the second 'one' below the 'two'. We have therefore 'one', 'two', and 'one'. To see how the second column of 1,2,1 corresponds to squaring a+b al-Samawal continues to describe Al-Karaji's work writing:This shows that for every number composed of two numbers, if we multiple each of them by itself once - since the two extremes are 'one' and 'one' - and if we multiply each one by the other twice - since the intermediate term is 'two' - we obtain the square of this number. This is a beautiful description of the binomial theorem using the Pascal triangle. The description continues up to the binomial coefficients which give (a+b)5 but we shall only quote how al-Karaji constructs the third column from the second:If we transfer the 'one' in the second column into a third column, then add 'one' from the second column to 'two' below it, we obtain 'three' to be written under the 'one' in the third column. If we then add 'two' from the second column to ''one' below it we have 'three' which is written under the 'three', then we write 'one' under this 'three'; we thus obtain a third column whose numbers are 'one', 'three', 'three', and 'one'. The table al-Karaji constructed looks like the Pascal triangle on its side.

Other results obtained by al-Karaji include summing the first n natural numbers, the squares of the first n natural numbers and the cubes of these numbers. He proved that the sum of the first n natural numbers was n(1/2 + n/2). He also gave (in Rashed and Ahmad's translation, see for example [5]):The sum of the squares of the numbers that follow one another in natural order from one is http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Karaji.html (3 of 5) [2/16/2002 10:56:57 PM]

Al-Karaji

equal to the sum of these numbers and the product of each of them by its predecessor. In modern notation this result is i2 = i + i(i-1). Al-Karaji also considered sums of the cubes of the first n natural numbers writing (in Rashed and Ahmad's translation, see for example [5]):If we want to add the cubes of the numbers that follow one another in their natural order we multiply their sum by itself. In modern notation i3 = ( i)2. Al-Karaji showed that (1 + 2 + 3 + ... + 10)2 was equal to 13 + 23 + 33 + ... + 103. He did this by first showing that (1 + 2 + 3 + ... + 10)2 = (1 + 2 + 3 + ... + 9)2 + 103. He could now use the same rule on (1 + 2 + 3 + ... + 9)2, then on (1 + 2 + 3 + ... + 8)2 etc. to get ( 1 + 2 + ... + 10)2 = (1 + 2 + 3 + ... + 8)2 + 93 + 103 = (1 + 2 + 3 + ... + 7)2 + 83 + 93 + 103 =... = 13 + 23 + 33 + ... + 103. Finally we should mention the influence of Diophantus on al-Karaji. The first five books of Diophantus's Arithmetica had been translated into Arabic by ibn Liqa around 870 and these were studied by al-Karaji. Woepcke in his introduction to Al-Fakhri ([7] or [8]) writes that he found:... more than a third of the problems of the first book of Diophantus, the problems of the second book starting with the eighth, and virtually all the problems of the third book were included by al-Karaji in his collection. Al-Karaji also invented many new problem of his own but even those of Diophantus were certainly not just taken without further development. He always tried to generalise Diophantus's results and to find methods which were more generally applicable. It was not only to algebra that al-Karaji contributed. The paper [9] discusses some of his geometrical work. This occurs in a chapter entitled On measurement and balances for measuring of buildings and structures. al-Karaji defines points, lines, surfaces, solids and angles. He also gives rules for measuring both plane and solid figures, often using arches as examples. He also gives methods of weighing different substances. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (21 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Karaji.html (4 of 5) [2/16/2002 10:56:57 PM]

Al-Karaji

Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Karaji.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Karaji.html (5 of 5) [2/16/2002 10:56:57 PM]

Al-Khalili

Shams al-Din Abu Abdallah Al-Khalili Born: about 1320 in possibly Damascus, Syria Died: about 1380 in possibly Damascus, Syria Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Khalili's full name is Shams al-din Abu Abdallah Muhammad ibn Muhammad al-Khalili. As can be seen from the list of references, much of the study of the work of al-Khalili has been done by David A King, who also wrote the article in [1]. Note that the articles [3], [4] and [5] are reprinted in [2]. King writes:Al-Khalili was an astronomer associated with the Umayyad Mosque in Damascus in the latter half of the fourteenth century, who compiled an extensive corpus of tables for timekeeping by the sun and regulating the astronomically defined time of Muslim prayer ... Of course, giving tables for timekeeping using astronomical events, requires a thorough understanding of geometry on the sphere and the work by al-Khalili can be seen as the end-product of the work of the Arabs on this mathematical topic. Of course, it is interesting to realise that Muslim mathematicians had to solve this type of problem for religious reasons, and the religious requirements made them delve much more deeply into this area of mathematics than was necessary to solve the much less critical problems of the agricultural calendar. The tables, which were not really studied by historians of mathematics until King worked on them in the 1970s, were used for many centuries in Damascus, Cairo and Istanbul. They consist of [1]:... tables for reckoning time by the sun, for the latitude of Damascus; tables for regulating the time of Muslim prayer, for the latitude of Damascus; tables of auxiliary mathematical functions for timekeeping by the sun for all latitudes; tables of auxiliary mathematical functions for solving the problems of spherical astronomy for all latitudes; a table for displaying ... the direction of Mecca, as a function of terrestrial latitude and longitude; and tables for converting lunar ecliptic coordinates to equatorial coordinates. Of course, al-Khalili did not do all this work without basing some of it on the work of earlier mathematicians, to see the magnitude of his task note that one table alone contains over 13000 entries. Tables for reckoning time by the sun and tables for regulating the time of Muslim prayer, computed for the latitude of Cairo, had been earlier computed by ibn Yunus. The astronomer al-Mizzi spent his early life in Egypt, then moved to Damascus where he converted ibn Yunus's table for use there. Al-Mizzi died around 1350 and the first two of al-Khalili's tables were improved versions of the ones produced by al-Mizzi, where al-Khalili had taken more accurate values for the terrestrial coordinates of Damascus. Al-Khalili's tables for solving the problems of spherical astronomy can be seen to be tables which solve spherical triangles using a method similar to the modern cosine rule. The tables are remarkable for their http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khalili.html (1 of 2) [2/16/2002 10:56:58 PM]

Al-Khalili

accuracy and Van Brummelen in [6] uses:.... computer-based tests to determine, if possible, the methods of computation used by al-Khalili in the construction of his auxiliary tables. This paper suggests a possible interpolation scheme used by al-Khalili and shows up a deep understanding that al-Khalili must have had regarding errors in his calculations which show [6]:... a curious lack of concern for accuracy at an early stage of the calculation, followed by a careful computation at a later stage where the calculation is sensitive to error. The calculation of the direction of Mecca, as a function of terrestrial latitude and longitude, was one of the hardest of all problems of spherical trigonometry for which Islam required a solution. There is a puzzle which has not yet been explained. The tables produced by al-Khalili for the direction of Mecca must have been calculated using his own auxiliary tables (which would be the most accurate available). However, the tables giving the direction of Mecca are remarkable for their accuracy having errors of around 0.1 . This is a greater degree of accuracy than would result if al-Khalili used his auxiliary tables in their present form. One possible solution is that al-Khalili had computed more accurate auxiliary tables before calculating his tables for the direction of Mecca but these are now lost. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Khalili.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khalili.html (2 of 2) [2/16/2002 10:56:58 PM]

Al-Khazin

Abu Jafar Muhammad ibn al-Hasan Al-Khazin Born: about 900 in Khurasan (eastern Iran) Died: about 971 in possibly Rayy Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Abu Jafar al-Khazin may have worked on both astronomy and number theory or there may have been two mathematicians both working around the same period, one working on astronomy and one on number theory. As far as this article is concerned we will assume that al-Khazin worked on both topics. There seems no way of being certain which position is correct. Al-Khazin's family were from Saba, a kingdom in southwestern Arabia, perhaps better known as Sheba from the biblical story of King Solomon and the Queen of Sheba. In the Fihrist, a tenth century survey of Islamic culture, he is described Al-Khurasani which means that he came from Khurasan in eastern Iran. The Buyid dynasty, ruling in western Iran and Iraq, reach its peak around the time that al-Khazin lived. It undertook public schemes such as building hospitals and dams, as well as patronising the arts and sciences. Rayy, situated southeast of present day Tehran, was one of the major cultural centres of the Buyid dynasty. Islamic writers described Rayy as:... a city of extraordinary beauty, built largely of fired brick and brilliantly ornamented with blue faience (glazed earthenware). Al-Khazin was one of the scientists brought to the court in Rayy by the ruler of the Buyid dynasty, Adud ad-Dawlah, who ruled from 949 to 983. We know that in 959/960 al-Khazin was required by the vizier of Rayy, who was appointed by Adud ad-Dawlah, to measure the obliquity of the ecliptic (the angle which the plane in which the sun appears to move makes with the equator of the earth). He is said to have made the measurement:... using a ring of about 4 meters. One of al-Khazin's works Zij al-Safa'ih (Tables of the disks of the astrolabe) was described by his successors as the best work in the field and they make many reference to it. The work describes some astronomical instruments, in particular it describes an astrolabe fitted with plates inscribed with tables and a commentary on the use of these. A copy of this instrument was made but vanished in Germany at the time of World War II. A photograph of this copy was taken and the article [5] examines this. Al-Khazin wrote a commentary on Ptolemy's Almagest which was criticised by al-Biruni for being too verbose. Only one fragment of this commentary has survived and a translation of it is given in [6]. The fragment which has survived contains a discussion by al-Khazin of Ptolemy's argument that the universe http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khazin.html (1 of 3) [2/16/2002 10:56:59 PM]

Al-Khazin

is spherical. Ptolemy wrote [6]:.. of different figures of equal perimeter, the one with more angles is greater in capacity, and therefore it is necessary that a circle is the greatest of surfaces (i.e. of all plane figures with a constant perimeter) and the sphere the greatest of solids. Al-Khazin gives 19 propositions relating to this statement by Ptolemy. The most interesting results show, with a very ingenious proof, that an equilateral triangle has a greater area than any isosceles or scalene triangle with the same perimeter. When he tries to generalise this result to polygons, however, al-Khazin gives incorrect proofs. Other results among the 19 are based on propositions given by Archimedes in On the sphere and cylinder. The author of [6] argues that the ingenious results on triangles are unlikely to be due to al-Khazin but are probably taken by him from some unknown source. The suggestion in [6] that al-Khazin is a third rate mathematician is somewhat doubtful given his work on number theory but as we stated at the beginning of this article, it is possible that there were two mathematicians of the same name. The papers [4], [9] and [7] all look at this number theory work by al-Khazin (see also [2] and [3]). The work of al-Khazin which is described seems to have been motivated by work of a mathematician by the name of al-Khujandi. Al-Khujandi claimed to have proved that x3 + y3 = z3 is impossible for whole numbers x, y, z which of course is the n = 3 case of Fermat's Last Theorem. In a letter al-Khazin wrote:I demonstrate earlier ... that what Abu Muhammad al-Khujandi advanced - may God have mercy on him - in his demonstration that the sum of two cubic numbers is not a cube is defective and incorrect. This seems to have motivated further correspondence on number theory between al-Khazin and other Arabic mathematicians. Results by al-Khazin here are interesting indeed. His main result is to:... show how, if we are given a number, to find a square number so that if the given number were added to it or subtracted from it the result would be square. In modern notation the problem is given a natural number a, find natural numbers x, y, z so that x2 + a = y2 and x2 - a = z2. Al-Khazin proves that the existence of x, y, z with these properties is equivalent to the existence of natural numbers u, v with a = 2uv, and u2 + v2 is a square (in fact u2 + v2 = x2). The smallest example of a satisfying these conditions is 24 which al-Khazin gives 52 + 24 = 72, 52 - 24 = 12. He also gives a = 96 with 102 + 96 = 142, 102 - 96 = 22 although, rather strangely, he seems to discount this case by another of his statements. Rashed suggests this may be because 96 = 2 48 = 2 6 8 and 62 + 82 = 102 is not a primitive Pythagorean triple. There is a mystery which Rashed notes in [7] (also in [2] and [3]). This relates to the quote above by al-Khazin regarding the false proof by al-Khujandi of the impossibility of proving x3 + y3 = z3. Rashed has discovered a manuscript which appears to be by al-Khazin, yet contains exactly what he had attributed to al-Khujandi. Although al-Khazin could have realised the error in al-Khujandi's proof and attempted a similar proof himself which he believed correct, there is no really satisfactory explanation of these facts.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khazin.html (2 of 3) [2/16/2002 10:56:59 PM]

Al-Khazin

Finally we should mention that al-Khazin proposed a different solar model from that of Ptolemy. Ptolemy had the sun moving in uniform circular motion about a centre which was not the earth. Al-Khazin was unhappy with this model since he claimed that if this were the case then the apparent diameter of the sun would vary throughout the year and observation showed that this were not the case. Of course the apparent diameter of the sun does vary but by too small an amount to be observed by al-Khazin. To get round this problem, al-Khazin proposed a model in which the sun moved in a circle which was centred on the earth, but its motion was not uniform about the centre, rather it was uniform about another point (called the excentre). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Khazin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khazin.html (3 of 3) [2/16/2002 10:56:59 PM]

Al-Khujandi

Abu Mahmud Hamid ibn al-Khidr Al-Khujandi Born: about 940 in Khudzhand, Tajikistan Died: 1000 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Few facts about al-Khujandi's life are known. What little we know comes through his writings which have survived and also some comments made by Nasir al-din al-Tusi. From al-Tusi's comments we can be fairly certain that al-Khujandi came from the city of Khudzhand. The city lies along both banks of the Syrdarya river, at the entrance to the fertile Fergana Valley, and it was captured by the Arabs in the 8th century. Al-Tusi says that al-Khujandi was one of the rulers of the Mongol tribe in that region so he must have come from the nobility. Al-Khujandi was supported in his scientific work for most of his life by members of the Buyid dynasty. The dynasty came to power in 945 when Ahmad ad-Dawlah occupied the 'Abbasid capital of Baghdad. Members of Ahmad ad-Dawlah's family became rulers in different provinces and there was never a great deal of cohesion in the Buyid empire. Al-Khujandi received patronage from Fakhr ad-Dawlah who ruled from 976 to 997. It was Fakhr ad-Dawlah who supported al-Khujandi in his major project to construct a huge mural sextant for his observatory at Rayy, which is near modern Tehran. It was believed by many Arabic scientists that the larger an instrument was, the more accurate were the results obtained. In fact al-Khujandi's mural sextant was his own invention and it did break new ground in having a scale which indicated seconds, a level of accuracy never before attempted. During the year 994 al-Khujandi used the very large instrument to observe a series of meridian transits of the sun near the solstices. He used these observations, made on 16 and 17 June 994 for the summer solstice and 14 and 17 December 994 for the winter solstice, to calculate the obliquity of the ecliptic, and the latitude of Rayy. He described his measurements in detail in a treatise On the obliquity of the ecliptic and the latitudes of the cities. From his observations he obtained 23 32' 19" for the obliquity of the ecliptic. This value was lower than values obtained previously [1]:Al-Khujandi says that the Indians found the greatest obliquity of the ecliptic, 24 ; Ptolemy 23 51' ; himself 23 32' 19". These divergent values cannot be due to defective instruments. Actually the obliquity of the ecliptic is not constant; it is a decreasing quantity.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khujandi.html (1 of 3) [2/16/2002 10:57:00 PM]

Al-Khujandi

There is, however, an error in al-Khujandi's value for the obliquity of the ecliptic; it is about two minutes too low. The error was discussed by al-Biruni in his Tahdid where he claimed that the aperture of the sextant settled about one span in the course of al-Khujandi's observations due to the weight of the instrument. Al-Biruni is almost certainly correct in pinpointing the cause of the error. However, al-Khujandi's latitude for Rayy, 35 34' 38.45", despite being calculated using his erroneous value for the obliquity of the ecliptic, is accurate to the nearest minute of arc. It remains for us to discuss the claim that al-Khujandi discovered the sine theorem. The claim was made by al-Tusi who gives al-Khujandi's proof of the result for spherical triangles in his Shakl al-qatta. Although there is no reason to doubt al-Tusi that the proof he gives does indeed come from al-Khujandi there is quite a few reason to believe that one of Abu'l-Wafa or Abu Nasr Mansur was the original discoverer. Both Abu'l-Wafa and Abu Nasr Mansur claim to have discovered the sine theorem while, as far as we are aware, al-Khujandi makes no such claim. Also al-Khujandi was more of a designer of astronomical instruments and an astronomical observer than he was theoretician. Finally, although this really proves little, the theorem appears many times in the writings of Abu Nasr Mansur both his writings on geometry as well as those on astronomy. We should make one final comment on the mathematical contributions of al-Khujandi. He stated Fermat's Last Theorem in the case n = 3 although, not surprisingly, his proof is wrong. Al-Khazin wrote:I demonstrated earlier ... that what Abu Muhammad al-Khujandi advanced - may God have mercy on him - in his demonstration that the sum of two cubic numbers is not a cube, is defective and incorrect. It is certainly interesting that al-Khujandi, despite his practical rather than theoretical achievements, should be interested in this number theory result. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khujandi.html (2 of 3) [2/16/2002 10:57:00 PM]

Al-Khujandi

JOC/EFR November 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Khujandi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Khujandi.html (3 of 3) [2/16/2002 10:57:00 PM]

Al-Kindi

Abu Yusuf Yaqub ibn Ishaq al-Sabbah Al-Kindi Born: about 801 in Kufah, Iraq Died: 873 in Baghdad, Iraq Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Kindi was born and brought up in Kufah, which was a centre for Arab culture and learning in the 9th century. This was certainly the right place for al-Kindi to get the best education possible at this time. Although quite a few details (and legends) of al-Kindi's life are given in various sources, these are not all consistent. We shall try to give below details which are fairly well substantiated. According to [3], al-Kindi's father was the governor of Kufah, as his grandfather had been before him. Certainly all agree that al-Kindi was descended from the Royal Kindah tribe which had originated in southern Arabia. This tribe had united a number of tribes and reached a position of prominence in the 5th and 6th centuries but then lost power from the middle of the 6th century. However, descendants of the Royal Kindah continued to hold prominent court positions in Muslim times. After beginning his education in Kufah, al-Kindi moved to Baghdad to complete his studies and there he quickly achieved fame for his scholarship. He came to the attention of the Caliph al-Ma'mun who was at that time setting up the "House of Wisdom" in Baghdad. Al-Ma'mun had won an armed struggle against his brother in 813 and became Caliph in that year. He ruled his empire, first from Merv then, after 818, he ruled from Baghdad where he had to go to put down an attempted coup. Al-Ma'mun was a patron of learning and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. Al-Kindi was appointed by al-Ma'mun to the House of Wisdom together with al-Khwarizmi and the Banu Musa brothers. The main task that al-Kindi and his colleagues undertook in the House of Wisdom involved the translation of Greek scientific manuscripts. Al-Ma'mun had built up a library of manuscripts, the first major library to be set up since that at Alexandria, collecting important works from Byzantium. In addition to the House of Wisdom, al-Ma'mun set up observatories in which Muslim astronomers could build on the knowledge acquired by earlier peoples. In 833 al-Ma'mun died and was succeeded by his brother al-Mu'tasim. Al-Kindi continued to be in favour and al-Mu'tasim employed al-Kindi to tutor his son Ahmad. Al-Mu'tasim died in 842 and was succeeded by al-Wathiq who, in turn, was succeeded as Caliph in 847 by al-Mutawakkil. Under both these Caliphs al-Kindi fared less well. It is not entirely clear whether this was because of his religious views or because of internal arguments and rivalry between the scholars in the House of Wisdom. Certainly al-Mutawakkil persecuted all non-orthodox and non-Muslim groups while he had synagogues and churches in Baghdad http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Kindi.html (1 of 3) [2/16/2002 10:57:01 PM]

Al-Kindi

destroyed. However, al-Kindi's [6]:... lack of interest in religious argument can be seen in the topics on which he wrote. ... he appears to coexist with the world view of orthodox Islam. In fact most of al-Kindi's philosophical writings seem designed to show that he believed that the pursuit of philosophy is compatible with orthodox Islam. This would seem to indicate that it is more probably that al-Kindi became [1]:... the victim of such rivals as the mathematicians Banu Musa and the astrologer Abu Ma'shar. It is claimed that the Banu Musa brothers caused al-Kindi to lose favour with al-Mutawakkil to the extent that he had him beaten and gave al-Kindi's library to the Banu Musa brothers. Al-Kindi was best known as a philosopher but he was also a mathematician and scientist of importance [3]:To his people he became known as ... the philosopher of the Arabs. He was the only notable philosopher of pure Arabian blood and the first one in Islam. Al-Kindi "was the most leaned of his age, unique among his contemporaries in the knowledge of the totality of ancient scientists, embracing logic, philosophy, geometry, mathematics, music and astrology. Perhaps, rather surprisingly for a man of such learning whose was employed to translate Greek texts, al-Kindi does not appear to have been fluent enough in Greek to do the translation himself. Rather he polished the translations made by others and wrote commentaries on many Greek works. Clearly he was most influenced most strongly by the writings of Aristotle but the influence of Plato, Porphyry and Proclus can also be seen in al-Kindi's ideas. We should certainly not give the impression that al-Kindi merely borrowed from these earlier writer, for he built their ideas into an overall scheme which was certainly his own invention. Al-Kindi wrote many works on arithmetic which included manuscripts on Indian numbers, the harmony of numbers, lines and multiplication with numbers, relative quantities, measuring proportion and time, and numerical procedures and cancellation. He also wrote on space and time, both of which he believed were finite, 'proving' his assertion with a paradox of the infinite. Garro gives al-Kindi's 'proof' that the existence of an actual infinite body or magnitude leads to a contradiction in [7]. In his more recent paper [8], Garro formulates the informal axiomatics of al-Kindi's paradox of the infinite in modern terms and discusses the paradox both from a mathematical and philosophical point of view. In geometry al-Kindi wrote, among other works, on the theory of parallels. He gave a lemma investigating the possibility of exhibiting pairs of lines in the plane which are simultaneously non-parallel and non-intersecting. Also related to geometry was the two works he wrote on optics, although he followed the usual practice of the time and confused the theory of light and the theory of vision. Perhaps al-Kindi's own words give the best indication of what he attempted to do in all his work. In the introduction to one of his books he wrote (see for example [1]):It is good ... that we endeavour in this book, as is our habit in all subjects, to recall that concerning which the Ancients have said everything in the past, that is the easiest and shortest to adopt for those who follow them, and to go further in those areas where they have not said everything ... http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Kindi.html (2 of 3) [2/16/2002 10:57:01 PM]

Al-Kindi

Certainly al-Kindi tried hard to follow this path. For example in his work on optics he is critical of a Greek description by Anthemius of how a mirror was used to set a ship on fire during a battle. Al-Kindi adopts a more scientific approach (see for example [1]):Anthemius should not have accepted information without proof ... He tells us how to construct a mirror from which twenty-four rays are reflected on a single point, without showing how to establish the point where the rays unite at a given distance from the middle of the mirror's surface. We, on the other hand, have described this with as much evidence as our ability permits, furnishing what was missing, for he has not mentioned a definite distance. Much of al-Kindi's work remains to be studied closely or has only recently been subjected to scholarly research. For example al-Kindi's commentary on Archimedes' The measurement of the circle has only received careful attention as recently as the 1993 publication [10] by Rashed. Article by: J J O'Connor and E F Robertson List of References (12 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Kindi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Kindi.html (3 of 3) [2/16/2002 10:57:01 PM]

Al-Maghribi

Muhyi l'din al-Maghribi Born: about 1220 in Spain Died: about 1283 in Maragha, Iran Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Muhyi l'din al-Maghribi was an eminent astronomer who was born in Spain, but who first worked in Damascus in Syria. His life seems to have been greatly affected by the wars of the period and he seems to have found favour with the winning side eventually working with al-Tusi at the Mongol observatory at Maragha, Iran. In 1256 the castle of Alamut was attacked by the forces of the Mongol leader Hulegu, a grandson of Genghis Khan, who was at that time set on extending Mongol power in Islamic areas. Some claim that al-Tusi, who was in the castle at this time, betrayed the defences of Alamut to the invading Mongols. Certainly Hulegu's forces destroyed Alamut and since Hulegu was himself interested in science, he treated al-Tusi with great respect. Hulegu attacked Baghdad in 1258, laid siege to the city, and entered it in February 1258. Hulegu, however, had made Maragha, in the Azerbaijan region of northwestern Iran, his capital. Muhyi l'din went to Maragha in 1258 as a guest of Hulegu. Al-Tusi and Muhyi l'din were involved in the construction of an Observatory. Work began in 1259 west of Maragha, and traces of it can still be seen there today. The observatory at Maragha became operational in 1262. There is a unique manuscript by Muhyi l'din in which he lists precise observations made at the Maragha Observatory between 1262 and 1274. The author of [4] discusses the three observations of the sun and the mathematical methods which Muhyi l'din used to find the solar eccentricity and apogee. Perhaps Muhyi l'din is most famous for his work on trigonometry. He wrote Book on the theorem of Menelaus and Treatise on the calculation of sines. In this second work he used interpolation to calculate an approximate value for the sine of one degree. He did this by two different methods, then compared the values he obtained achieving an accuracy of 4 places. A more accurate value was not obtained until the work of Qadi Zada and al-Kashi. In doing this work Muhyi l'din also found an approximate value for which he compared with the bounds obtained by Archimedes using 96 inscribed and circumscribed polygons. Muhyi l'din also considered the classical problem of doubling the cube which he approached by Hippocrates' method of finding two mean proportionals between two given lines. Another important aspect of Muhyi l'din's work was the critical commentaries which he produced on some of the classic Greek works such as Euclid's Elements, Apollonius's Conics, Theodosius's Spherics, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Maghribi.html (1 of 2) [2/16/2002 10:57:03 PM]

Al-Maghribi

and Menelaus's Spherics. A particularly important commentary by Muhyi l'din is that on Book XV of the Elements (which was not written by Euclid). Hypsicles added a Book XIV to the Elements which dealt with the mensuration of the regular dodecahedron and icosahedron. Later Book XV was written in Arabic by an unknown author, perhaps using Greek works which are now lost. Book XV has common features with Book XIV by Hypsicles but contains considerably more. The original Arabic version of Book XV is lost but there are four surviving manuscripts containing Muhyi l'din's commentary on it. We know that there was more than one version of the Arabic Book XV, for recently a Hebrew translation of Book XV has been discovered which has been translated from a different version to that which Muhyi l'din used for his commentary. Muhyi l'din's Book XV contains [3]:... the ratios of (1) the edges, (2) the faces, (3) the surface areas, (4) the perpendicular distances from the centre to a face and (5) the volumes of the five regular polyhedra inscribed in one sphere. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Maghribi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Maghribi.html (2 of 2) [2/16/2002 10:57:03 PM]

Al-Mahani

Abu Abd Allah Muhammad ibn Isa Al-Mahani Born: about 820 in Mahan, Kerman, Persia (now Iran) Died: 880 in Baghdad, Iraq Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

There is very little information regarding al-Mahani's life. We do know a little about al-Mahani's work in astronomy from Ibn Yunus's astronomical handbook al-Zij al-Hakimi al-kabir. In this work Ibn Yunus quotes from writings by al-Mahani, which have since been lost, which describe observations which al-Mahani made between the years 853 and 866. At least we have accurate dating of al-Mahani's life from this source. Ibn Yunus writes that al-Mahani observed lunar eclipses and [1]:... he calculated their beginnings with an astrolabe and that the beginnings of three consecutive eclipses were about half an hour later than calculated. The Fihrist (Index) was a work compiled by the bookseller Ibn an-Nadim in 988. It gives a full account of the Arabic literature which was available in the 10th century and in particular mentions al-Mahani, not for his work in astronomy, but rather for his work in geometry and arithmetic. However the work which al-Mahani did in mathematics may well have been motivated by various problems of an astronomical nature. We know that some of al-Mahani's work in algebra was motivated by trying to solve problems due to Archimedes. The problem of Archimedes which he attempted to solve in a novel way was that of cutting a sphere by a plane so that the two resulting segments had volumes of a given ratio. It was Omar Khayyam, giving an important historical description of algebra, who puts al-Mahani's work into context. Omar Khayyam writes (see for example [2] or [3]):Al-Mahani was one of the modern authors who conceived the idea of solving the auxiliary theorem used by Archimedes in the fourth proposition of the second book of his treatise on the sphere and the cylinder algebraically. However, he was led to an equation involving cubes, squares and numbers which he failed to solve after giving it lengthy meditation. Therefore, this solution was declared impossible until the appearance of Ja'far al-Khazin who solved the equation with the help of conic sections. Omar Khayyam is quite correct to rate this work highly. It would be too easy to say that since al-Mahani has proposed a method of solution which he could not carry through then his work was of little value. However, this, as Omar Khayyam is well aware, is not so at all and the fact that al-Mahani conceived the idea of reducing problems such as duplicating the cube to problems in algebra was an important step

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Mahani.html (1 of 2) [2/16/2002 10:57:04 PM]

Al-Mahani

forward. A number of works by al-Mahani have survived, in particular commentaries which he wrote on parts of Euclid's Elements. In particular his work on ratio and irrational ratios which are contained in commentaries he gave on Books V and X of the Elements survive as does his attempt to clarify difficult parts of Book XIII. He also wrote a work which gives those 26 propositions in Book I which can be proved without using a reductio ad absurdum argument but this work has been lost. Also lost is his work attempting to improve the descriptions given by Menelaus in his Spherics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Mahani.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Mahani.html (2 of 2) [2/16/2002 10:57:04 PM]

Al-Nasawi

Abu l'Hasan Ali ibn Ahmad Al-Nasawi Born: about 1010 in possibly Nasa, Khurasan Died: about 1075 in possibly Baghdad, Iraq Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nothing was known about al-Nasawi in Europe until 1863 when Woepcke published information on a manuscript containing a work by al-Nasawi on elementary arithmetic. Al-Nasawi had prepared an original version of it in Persian for the library of the Iranian prince Majd al-Dawlah, of the Buyid dynasty. Before the work was completed, however, Majd al-Dawlah was deposed as ruler so, on completion of the work, al-Nasawi presented it to Sharaf al-Muluk who was the vizier of Jalal ad-Dawlah (Jalal ad-Dawlah was the ruler of Baghdad from 1025 to 1044). Sharaf al-Muluk ordered al-Nasawi to rewrite the work in Arabic, and this he did. The Arabic version has survived and it is this which Woepcke studied in 1863. From this description, and from the fact that al-Nasawi dedicated another work to a Shi'ite leader in Baghdad, we at least can deduce that al-Nasawi worked for part of his life in Baghdad. A few more details of his life have become known recently. A paragraph about al-Nasawi's life has been found in a manuscript and it tells us that he spent time in Rayy, and was visited by ibn Sina. The authors of [3] give an analysis of this mid-12th century manuscript which once contained 80 tracts, but of these only 43 survive. Tract 26 is a summary of Euclid's Elements by al-Nasawi. The reasons which al-Nasawi gives for writing this summary are two-fold. On the one hand he says that it will act as an introduction to the Elements while on the other hand it will provide all the necessary background in geometry for anyone wanting to read Ptolemy's Almagest. He does not meet the first of these aims very successfully for the tract is nothing more than a copy of the first six books of the Elements together with Book XI. All al-Nasawi appears to have done is to omit some constructions and change a few of the proofs. This work is interesting historically for our understanding of the way that the Elements was transmitted in Arabic countries but has little significance for its contributions to mathematics. There were three different types of arithmetic used in Arab countries around this period: (i) a system derived from counting on the fingers with the numerals written entirely in words; this finger-reckoning arithmetic was the system used by the business community, (ii) the sexagesimal system with numerals denoted by letters of the Arabic alphabet, and (iii) the arithmetic of the Indian numerals and fractions with the decimal place-value system. The arithmetic book by al-Nasawi is of this third "Indian numeral" type. The book is composed of four separate treatises, each dealing with a particular class of numbers. The first deals with integers, the second with proper common fractions, the third with improper fractions, and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Nasawi.html (1 of 2) [2/16/2002 10:57:05 PM]

Al-Nasawi

finally the fourth with sexagesimals. In each of the four cases al-Nasawi explains the four elementary arithmetical operations. He also explains doubling, halving, taking square roots, and taking cube roots. Each method for each of the four types is illustrated with worked examples and a checking procedure is explained which usually involves usually casting out nines The method al-Nasawi gives for taking cube roots is the same as the method described in the Chinese Mathematics in Nine Books, but quite how he learnt of this method is unknown. Al-Nasawi is critical of works on arithmetic written by earlier authors. However, looking at the texts which he criticises that we can examine because they have survived, we can see now that his criticisms are not valid. In fact, in some respects, al-Nasawi does not rate too highly as a mathematician. There seems nothing original in any of his works and, more significantly, there are several places where al-Nasawi presents pieces of mathematics which he fails to properly understand. For example he fails to understand the principle of "borrowing" when doing subtraction. Two other works by al-Nasawi have survived. One discusses the theorem of Menelaus while the other is [1]:... a corrected version of Archimedes' Lemmata as translated into Arabic by Thabit ibn Qurra, which was last revised by Nasir al-Din al-Tusi. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 900 to 1100

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Nasawi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Nasawi.html (2 of 2) [2/16/2002 10:57:05 PM]

Al-Nayrizi

Abu'l Abbas al-Fadl ibn Hatim Al-Nayrizi Born: about 865 in possibly Nayriz, Iraq Died: about 922 in possibly Baghdad, Iraq Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Nayrizi was probably born in Nayriz which was a small town southeast of Shiraz now in central Iran. Certainly he must have been associated with this town in his youth to have been called al-Nayrizi. Little is known of his life but we do know that he dedicated some of his works to al-Mu'tadid so he almost certainly moved to Baghdad and worked there for the caliph. The period during which al-Nayrizi was growing up was a turbulent one in the region in which he lived. Following the assassination of the caliph al-Mutawwakil in 861 there was a period of anarchy and civil war. The Caliph al-Mu'tamid and his brother al-Muwaffaq who was a military leader, reunited the empire from 870 but a rebellion was eventually put down in 883 only after many years of military campaigns by al-Muwaffaq and his brother al-Mu'tadid. Al-Mu'tamid died in 892 and, since al-Mu'tadid had forced him to disinherit his own son, al-Mu'tadid became caliph in that year. Al-Mu'tadid reorganised the administration and reformed finances, and he demonstrated great skill and ruthlessness in dealing with the many factions that had arisen. There followed a period of great cultural activity, with Baghdad home to many intellectuals. Al-Nayrizi must have worked for al-Mu'tadid during his ten year of rule, for he wrote works for the caliph on meteorological phenomena and on instruments to measure the distance to objects. If al-Mu'tadid's reign had begun with political intrigue then it seemed to end in the same way, the general opinion being that, in 902, al-Mu'tadid was poisoned by his political enemies. Al-Mu'tadid's son al-Muktafi became caliph in 902 and ruled until 908. It seems likely that al-Nayrizi would continue to work in Baghdad for the new caliph since the same support for intellectuals in Baghdad continued. The Fihrist (Index) was a work compiled by the bookseller Ibn an-Nadim in 988. It gives a full account of the Arabic literature which was available in the 10th century and in particular mentions al-Nayrizi as a distinguished astronomer. Eight works by al-Nayrizi are listed in the Fihrist. A later work, written in the 13th century, described al-Nayrizi as both a distinguished astronomer and as a leading expert in geometry. Al-Nayrizi's works on astronomy include a commentary of Ptolemy's Almagest and Tetrabiblos. Neither have survived. He is most famous for his commentary on Euclid's Elements which has survived. The Leiden manuscript referred to in the title of [3] contains the revision by al-Nayrizi of the second Arabic translation of Euclid's Elements by al-Hajjaj. The translation by al-Hajjaj has not survived and the article [3] examines to what extent al-Nayrizi changed the translation, arguing that indeed he made considerable changes. The paper [4] looks at different manuscripts containing versions of al-Nayrizi's commentary, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Nayrizi.html (1 of 2) [2/16/2002 10:57:06 PM]

Al-Nayrizi

some in Arabic, one a Latin version. In dealing with ratio and proportion in his commentary on the Elements, al-Nayrizi adopts concepts proposed by al-Mahani who had worked in Baghdad, probably before al-Nayrizi arrived there. Al-Nayrizi wrote a work on how to calculate the direction of the sacred shrine of the Ka'bah in Mecca (to was important for Muslims to be able to do this since they had to face that direction five times each day when performing the daily prayer). In this work he effectively uses the tan function, but he was not the first to use these trigonometrical ideas. The article [2] is a translation into Russian of the short treatise by al-Nayrizi on Euclid's fifth postulate. In his work on proofs of the parallel postulate, al-Nayrizi quotes work by a mathematician named Aghanis. In [1] Sabra argues convincingly that Aghanis is the Athenian philosopher Agapius who was a pupil of Proclus and Marinus and taught around 511 AD. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Nayrizi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Nayrizi.html (2 of 2) [2/16/2002 10:57:06 PM]

Al-Qalasadi

Abu'l Hasan ibn Ali al Qalasadi Born: 1412 in Bastah (now Baza), Spain Died: 1486 in Béja, Tunisia Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Qalasadi (or al-Kalasadi, as it is sometimes written) was born in Bastah, a Moorish city in Andalusia, now a part of Spain. Andalusia was derived from the Arabic name al-Andalus which was originally applied by the Muslims to the whole of present day Spain and Portugal, an area which they occupied from the 8th century. In the 11th century Christians began to retake the area, slowly moving down from the north and east. Andalusia was then the name applied to the region remaining under Muslim rule. The Christian reconquest took four hundred years. Andalusia had prospered during the 13th century and the Alhambra, a wonderful palace and fortress of the rulers of Granada, was largely completed by 1360. The Christian kingdom of Castile to the north had suffered civil strife through the 14th century, so Andalusia had prospered but, in 1407, five years before al-Qalasadi was born, Castile began a major push to conquer the whole of Spain and Portugal. Al-Qalasadi was a Muslim who was brought up in Bastah which is north-east of Granada city. It must have been a difficult period in which to live in Bastah, with a steady, yet intermittent, encroachment of Castile towards the city. Al-Qalasadi began his education in Bastah, studying law, the Qur'an and the science of fixed shares in an estate. He moved south, away from the war zone, to Granada where he continued his studies, in particular philosophy, science and Muslim law. Al-Qalasadi chose to remain in the Islamic world and he left Granada and travelled widely throughout Islamic. In particular he spent much time in the North Africa, living in Islamic countries which had supported Andalusia, both with political and with military aid in its resistance to the Christian attacks. He spent some time in Tlemcen (now in northwestern Algeria, near the Moroccan border) where he studied under teachers who taught him arithmetic and its applications. Form there al-Qalasadi went to Egypt where again he studied with some of the leading scholars. Eventually al-Qalasadi reached Mecca, the purpose of his pilgrimage, and returned to Granada. Things were in a bad way when al-Qalasadi returned to Granada. The last remaining parts of the Muslim state were under severe attack from the Christians of Aragon and Castile. However, al-Qalasadi taught and wrote some of his major works during this period but eventually the advancing Christian armies made life impossible for him. Al-Qalasadi [5]:Courageously ... exerted himself in trying to organise resistance, but he was soon forced to

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Qalasadi.html (1 of 4) [2/16/2002 10:57:07 PM]

Al-Qalasadi

join the Andalusian hordes of refugees that were spreading over the Maghrib. The defeat of the whole Muslim state in Granada finally took place until 1492, six years after al-Qalasadi's death in North Africa, when the city of Granada fell to the Christian Castile. In [2] al-Qalasadi is described as a specialist in the apportioning of inheritances who took the first steps toward the introduction of algebraic symbolism. His contributions to algebraic symbolism were in using short Arabic words, or just their initial letters, as mathematical symbols. In particular he used wa meaning "and" for + illa meaning "less" for fi meaning "times" for ala meaning "over" for j from jadah meaning "root" sh from shay meaning "thing" (x, the unknown) m from mal for x2 k form kab for x3 l from yadilu for = Al-Qalasadi wrote several books on arithmetic and one on algebra. Some of these are commentaries such as his commentary on the Talkhis amal al-hisab (Summary of arithmetical operations) by ibn al-Banna. Ibn al-Banna was a Moroccan who had died over 100 years before al-Qalasadi wrote his commentary but, perhaps surprisingly, ibn al-Banna himself had written a commentary on his own work. Certainly al-Qalasadi wrote original works. His major treatise was al-Tabsira fi'lm al-hisab (Clarification of the science of arithmetic). This was a difficult text and, perhaps to some extent following the example of ibn al-Banna, al-Qalasadi followed it up by writing a simpler version which he called Unveiling the science of arithmetic. Even this he must have considered to be too difficult to be used as a teaching book, for he wrote yet a third version Unfolding the secrets of the use of dust letters. The title of this work needs some explanation. The early methods of calculating with Hindu numerals involved the use of a dust board. A dust board was used because the methods required the moving of numbers around in the calculation and rubbing some out as the calculation proceeded. The dust board allowed this in the same sort of way that one can use a blackboard, chalk and a blackboard eraser. However, al-Uqlidisi in the tenth century had showed how to modify arithmetical techniques so that pen and paper could be used instead of the dust board. In his arithmetic texts al-Qalasadi computed n2, n3 and used the method of successive approximation to determine square roots. Both of the simpler versions of al-Qalasadi's arithmetic treatise proved popular in teaching arithmetic in North Africa and the works were in use for over 100 years. It is now certain that, despite being popular teaching books, there was little original in al-Qalasadi's work. For example, the sequences n2 and n3 had been studied by al-Samawal and al-Baghdadi, and methods for computing square roots were known to the Babylonians. However, this was poorly understood by the historians of the 19th century who first tried to understand the contributions to mathematics by the Muslims. The difficulty was that al-Qalasadi, being one of the last of the mathematicians associated with the major mathematical contributions by the Muslims and Arabs, was better known than many of the earlier contributors. Ignorance of the earlier contributions led http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Qalasadi.html (2 of 4) [2/16/2002 10:57:07 PM]

Al-Qalasadi

historians to give too much credit to al-Qalasadi who in many ways displayed the same characteristics as the later ancient Greek mathematicians. Once established, however, ideas are harder to overturn than one might imagine. J Samso-Moya, reviewing [6]. writes:The author analyses the work of the mathematicians of the Maghrib as if they were entirely independent of their predecessors in Eastern Islam. This leads him to stress the importance of the algebraic symbolism used by al-Qalasadi (1412-1486) without taking into consideration similar previous attempts both in Eastern and in Western Islam, a fact which was already known - in the second half of the 19th century - by F Woepcke. The book [2] is a reprint of Woepcke's 19th century treatise refered to by Samso-Moya. Again reviewing [3] J Samso-Moya writes:The author seems to believe that algebraic symbolism was first developed in Islam by the Spanish-Arabic mathematicians Ibn al-Banna (d. 1321, a Moroccan) and al-Qalasadi (d. 1486): the extreme rarity of algebraical symbolism in the parts dedicated to algebra in medieval Italian books on the abacus and arithmetic is possibly due to the fact that Leonardo Fibonacci (d. after 1240), whose "Liber abaci" was extremely influential in medieval Italy, was not aware of the work of Andalusian mathematicians. Certainly symbols were not the invention of al-Qalasadi. Perhaps even more telling is that the particular symbols he used were not even his own invention since the same ones had been used by other Muslim mathematicians in North Africa 100 years earlier. Symbols had been used in the east of the Muslim empire even earlier than that. We should not, however, let any of this argument detract from al-Qalasadi's contribution. We must stress that he does not clam originality - this was the incorrect invention of historians 400 years later. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Qalasadi.html (3 of 4) [2/16/2002 10:57:07 PM]

Al-Qalasadi

JOC/EFR November 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Qalasadi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Qalasadi.html (4 of 4) [2/16/2002 10:57:07 PM]

Al-Quhi

Abu Sahl Waijan ibn Rustam al-Quhi Born: about 940 in Tabaristan (now Mazanderan), Persia (now Iran) Died: about 1000 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

There are two spellings of al-Quhi's name in English which seem to appear about equally often, namely al-Quhi and al-Kuhi. We can deduce from al-Quhi's name that he came from the village of Quh in Tabaristan. He was brought up during the period that a new dynasty was being established which would rule over Iran. The Buyid Islamic dynasty ruled in western Iran and Iraq from 945 to 1055 in the period between the Arab and Turkish conquests. The period began in 945 when Ahmad Buyeh occupied the 'Abbasid capital of Baghdad. The high point of the Buyid dynasty was during the reign of 'Adud ad-Dawlah from 949 to 983. He ruled from Baghdad over all southern Iran and most of what is now Iraq. A great patron of science and the arts, 'Adud ad-Dawlah supported a number of mathematicians at his court in Baghdad, including al-Quhi, Abu'l-Wafa and al-Sijzi. In 969 'Adud ad-Dawlah ordered that observations be made of the winter and summer solstices in Shiraz. These observations of the winter and summer solstices were made by al-Quhi, al-Sijzi and other scientists in Shiraz during 969/970. Sharaf ad-Dawlah was 'Adud ad-Dawlah's son and he became Caliph in 983. He continued to support mathematics and astronomy so al-Quhi remained at the court in Baghdad working for the new Caliph. Sharaf ad-Dawlah required al-Quhi to make observations of the seven planets and in order to do this al-Quhi had an observatory built in the garden of the palace in Baghdad. The instruments in the observatory were built to al-Quhi's own design and installed once the building was complete. Al-Quhi was made director of the observatory and it was officially opened in June 988. A number of scientists were present at the opening. One in particular, the famous mathematician and astronomer Abu'l-Wafa, is worthy of mention. He was also employed at the court of Sharaf ad-Dawlah. Another who was present at the opening was Abu Ishaq al-Sabi. Al-Sabi was a high ranking official in Baghdad who was interested in mathematics. We mention later in this article correspondence between al-Quhi and al-Sabi. Some accurate observations were made but the observatory ceased work in 989 on the death of Sharaf ad-Dawlah. The Buyid dynasty was by this stage beginning to lose control of the empire. The economy was on a downward path, and rebellions in the army made the ruler's life difficult. Fine cultural activities such as an observatory took a lower priority. Our description of al-Quhi's life has highlighted his work in astronomy. However, it is in mathematics that he is more famous, being the leading figure in a revival and continuation of Greek higher geometry http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Quhi.html (1 of 3) [2/16/2002 10:57:09 PM]

Al-Quhi

in the Islamic world. The geometric problems that al-Quhi studied usually led to quadratic or cubic equations. Nasir al-Din al-Tusi described one of the problems considered by al-Quhi writing (see for example [1]):To construct a sphere segment equal in volume to a given sphere segment and equal in area to a second sphere segment - a problem similar to but more difficult than related problems solved by Archimedes - Al-Quhi constructed the two unknown lengths by intersecting an equilateral hyperbola with a parabola and rigorously discussed the conditions under which the problem is soluble. Al-Quhi's solution to the problem is given in [5]. It is a classical style of solution using results from Euclid's Elements, Apollonius's Conics and Archimedes' On the sphere and cylinder. If a solution exists, al-Quhi showed that it will have coordinates which lie on a particular rectangular hyperbola that he has constructed. Of course, al-Quhi does not express the mathematics in these modern terms but rather in the usual classical geometry of ancient Greek mathematics. Next al-Quhi introduces the "cone of the surface" which, after many deductions, leads to showing that the solution has coordinates lying on a parabola. The problem is then beautifully solved as the intersection of the two curves. In another treatise On the construction of an equilateral pentagon in a known square al-Quhi solves the problem given in the title again using the intersection of two conic sections, this time two hyperbolas. Although it is impossible to inscribe a regular pentagon in a square, an equilateral pentagon can be inscribed in two ways. One, which requires the solution of a quadratic equation, had been found by Abu Kamil in the ninth century. The other, which requires the solution of a quartic equation, is the one presented by al-Quhi. Details of this treatise are given in [6] (see the corrections and additions of [7]), and [8]. Al-Quhi also described a conic compass, a compass with one leg of variable length, for drawing conic sections in the treatise On the perfect compass [1]:... he first described the method of constructing straight lines, circles, and conic sections with this compass, and then treated the theory. He concluded that one could now easily construct astrolabes, sundials and similar instruments. Indeed al-Quhi did consider the problem of constructing astrolabes in On the construction of the astrolabe. The astrolabe was an instrument used to observe altitudes, and it provided a mechanical means to transform celestial coordinates between an equatorial system and one based on the horizon. This treatise is in two Books, the first being divided into four chapters, the second book into seven chapters. There are a number of difficult mapping problems solved by al-Quhi in this work. In particular, using a method resembling descriptive geometry, he maps circles on the sphere into the equatorial plane. After manipulation, they are mapped back again onto the sphere in a remarkable piece of visualisation. Despite the appearance of the work being of practical use in constructing an astrolabe, it would appear that al-Quhi was more interested in the mathematics for its own sake than he was in giving a practical manual. Finally we should mention the correspondence between al-Quhi and al-Sabi which we mentioned above. It is known that there were at least six letters exchanged but only details of four survive. These are given in both Arabic and English in [3]. Topics covered are quite varied, ranging from a discussion of what "known" means to solutions of specific problems such as the following Suppose we are given a circle and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Quhi.html (2 of 3) [2/16/2002 10:57:09 PM]

Al-Quhi

two intersecting straight lines l and m. Suppose the tangent to the circle at a point T meets l at L and m at M. How can one choose T so that TL : TM is equal to a given ratio? Perhaps the most interesting parts of the correspondence are six theorems given by al-Quhi concerning the centres of gravity of various figures. Five of the six results are correct but the sixth is false. It states that the centre of gravity of a semicircle divides the radius in the ratio 3 : 7. From this false result al-Quhi deduces the equally false result that = 28/9. Even the best mathematicians can make mistakes! Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Other Web sites

Georgia (Constuction of tangent circles)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Quhi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Quhi.html (3 of 3) [2/16/2002 10:57:09 PM]

Al-Samarqandi

Shams al-Din ibn Ashraf Al-Samarqandi Born: about 1250 in Samarqand, Uzbekistan, Russia Died: about 1310 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Shams al-Din al-Samarqandi's full name is Shams al-din Muhammad ibn Ashraf al-Husayni al-Samarqandi. Nothing is known of al-Samarqandi's life except that he composed his most important works around 1276. He wrote works on theology, logic, philosophy, mathematics and astronomy which have proved important in their own right and also in giving information about the works of other scientists of his period. Al-Samarqandi wrote a work Risala fi adab al-bahth which discussed the method of intellectual investigation of reasoning using dialogue. Such methods of enquiry were much used by the ancient Greeks. He also wrote Synopsis of astronomy and produced a star catalogue for the year 1276-77. In mathematics al-Samarqandi is famous for a short work of only 20 pages which discusses 35 of Euclid's propositions. Although a short work, al-Samarqandi consulted widely the works of other Arab mathematicians before writing it. For example he refers to writings by al-Haytham, Omar Khayyam, al-Jawhari, Nasir al-Din al-Tusi, and al-Abhari. Not only did al-Samarqandi consult the writings of many other mathematicians before writing his short work, in turn several later mathematicians read al-Samarqandi's 20 page work and referred to it in their own writings. For example Qadi Zada mentions al-Samarqandi's short work on Euclid's propositions. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Samarqandi.html (1 of 2) [2/16/2002 10:57:10 PM]

Al-Samarqandi

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Samarqandi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Samarqandi.html (2 of 2) [2/16/2002 10:57:10 PM]

Al-Samawal

Ibn Yahya al-Maghribi Al-Samawal Born: about 1130 in Baghdad, Iraq Died: about 1180 in Maragha, Iran Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Samawal's father was Abul-Abbas Yahya al-Maghribi, a Jewish scholar of religion and literature. Abul-Abbas was born in Fez in Morocco and later moved to Baghdad where he was living at the time of al-Samawal's birth. Al-Samawal's mother, Anna Isaac Levi, had moved from Basra in Iraq. Certainly al-Samawal was brought up in a family where learning was highly valued and the first topic which interested him was medicine. Perhaps the main attraction of this topic came from the fact that he had an uncle who was a medical doctor. At about the same time as he began to study medicine, al-Samawal also began to study mathematics. He was about thirteen years old when he began serious study, beginning with Hindu methods of calculation and a study of astronomical tables. Baghdad at this time was not a great centre for mathematical learning, and al-Samawal had soon mastered all the mathematics which his teachers knew. These teachers had covered topics including an introduction to surveying, elementary algebra, and the geometry of the first few books of Euclid's Elements. In order to take his mathematical studies further, al-Samawal had to study on his own. He read the works of Abu Kamil, al-Karaji and others so that by the time he was eighteen years old he had read almost all the available mathematical literature. The work which most impressed him was that of al-Karaji, yet he found himself less than completely satisfied with it and began to work out improvements for himself. His most famous treatise al-Bahir fi'l-jabr, meaning The brilliant in algebra, was written when al-Samawal was only nineteen years old. It is a work of great importance both for the original ideas which it contains and also for the information that it records concerning works by al-Karaji which are now lost. The al-Bahir was published in an edition with notes and an introduction in [2]. Details of the work are also given in [3] and [4]. The treatise consists of four books: (1) On premises, multiplication, division and extraction of roots, (2) On extraction of unknown quantities, (3) On irrational magnitudes, and (4) On classification of problems. Al-Samawal's predecessors had begun to develop what has been called by historians today the "arithmetisation of algebra". In fact al-Samawal was the first to give this development a precise description when he wrote that it was concerned (see for example [3] or [4]):... with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known. This will strongly suggest to mathematicians today that al-Samawal was developing the study of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Samawal.html (1 of 4) [2/16/2002 10:57:12 PM]

Al-Samawal

polynomial rings and indeed this is a fair description of the work he was undertaking. In the first book of the al-Bahir he defines powers x, x2, x3, ... , x-1, x-2, x-3, ... . After defining polynomials, al-Samawal describes addition, subtraction, multiplication and division of polynomials. He also gave methods for the extraction of the roots of polynomials. Al-Samawal could not have described arithmetic operations on powers of the unknown without having developed an understanding of negative numbers. He had refined the ideas of his predecessors into a form which would not be given by European mathematicians until many centuries later. He also used 0 in his calculations writing [1]:If we subtract a positive number from an empty power, the same negative number remains. By this al-Samawal meant, in modern notation, 0 - a = -a. He continued:... if we subtract the negative number from an empty power, the same positive number remains. Again in modern notation this is 0 - (-a) = a. Multiplication of negative numbers was also completely understood by al-Samawal. He wrote [3]:... the product of a negative number by a positive number is negative, and by a negative number is positive. In Book 2 of al-Bahir al-Samawal describes the theory of quadratic equations but, rather surprisingly, he gave geometric solutions to these equations despite algebraic methods having been fully described by al-Khwarizmi, al-Karaji, and others. Al-Samawal also described the solution of indeterminate equations such as finding x so that a xn is a square, and finding x so that axn + bxn-1 is a square. Also in this book is al-Samawal's description of the binomial theorem where the coefficients are given by the Pascal triangle. The method is attributed by al-Samawal to al-Karaji and provides the only surviving account of this remarkable work. Perhaps one of the most remarkable achievements appearing in Book 2 is al-Samawal's use of an early form of induction. What he does is to demonstrate an argument for n = 1, then prove the case n = 2 based on his result for n = 1, then prove the case n = 3 based on his result for n = 2, and carry on to around n = 5 before remarking that one can continue the process indefinitely. Although this is not induction proper, it is a major step towards understanding inductive proofs. We should also comment that he was not the first to use this form of recursive reasoning, since al-Karaji had used similar methods. The result in Book 2 which al-Samawal himself was most proud of (and rightly so) is 12 + 22 + 32 + ... + n2 = n(n+1)(2n+1)/6 which does not appear in earlier texts. Book 3 contains a description of how to carry out arithmetic with irrational numbers. It follows Book X of Euclid's Elements and, although a very fine exposition of these ideas, contains little that is original. One result here which again particularly pleased al-Samawal was his calculation of how to rationalise 30/( 2+ 5+ 6). Al-Karaji had failed to solve this problem, which explains why al-Samawal was particularly pleased to solve it. [My computer algebra package saves me having to do the thinking

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Samawal.html (2 of 4) [2/16/2002 10:57:12 PM]

Al-Samawal

al-Samawal had to do and gives (5 6+2 5+6 15-20 2)/13.] The final book of al-Bahir contains an interesting example of a problem in combinatorics, namely to find ten unknowns given the 210 equations which give their sums taken 6 at a time. Of course such a system of 210 equations need not be consistent and al-Samawal gave the 504 conditions which are necessary for the system to be consistent. In Book 4 al-Samawal also classifies problems into necessary problems, namely ones which can be solved; possible problems, namely ones where it is not known whether a solution can be found or not, and impossible problems which [3]:... if one could assume the existence of their solution, this existence would lead to an absurdity. After writing the al-Bahir al-Samawal travelled in many countries including Iraq, Syria, Kohistan (a mountainous area in Pakistan and Afghanistan) and Azerbaijan (northwestern of Iran). We know from his own writings that he was in Maragheh in Azerbaijan on 8 November 1163, for on that date al-Samawal made a commitment to the faith of Islam. This decision was not taken without a great deal of thought by al-Samawal. He had put much effort into testing the validity of the claims made by the major religions and he reports that on 8 November 1163 he decided that Islam was the most satisfactory. He wrote a work Decisive refutation of the Christians and Jews which has survived. Of course al-Samawal's father, being Jewish, would have found his son's conversion to Islam a painful experience and al-Samawal, not wishing to hurt his father, delayed his conversion for four years. After this time al-Samawal wrote to his father setting out his reasons for changing his religion from the Jewish faith to Islam. At this time the much travelled al-Samawal was in Aleppo, in northern Syria, and his father set out at once to see him on receiving the letter. However, al-Samawal's father died on the journey before seeing his son. We mentioned that al-Samawal was trained in medicine in his youth. In fact he practised his medical skills on his journeys and became quite famous for this expertise in this area. Several rulers, always keen to have the best possible doctors, became patients of al-Samawal. He relates in his writings that he developed some medicines which were almost miraculous cures. Unfortunately, no details of these have survived. The only medical work by al-Samawal which has survived is essentially a sex manual which includes many erotic stories. The work exhibits the fact that al-Samawal was a good scientific observer in his descriptions of various diseases. In particular al-Samawal shows that he is interested in the psychological aspects of disease. His remedy for depression is [1]:... well-lighted houses, the sight of running water and verdure, warm baths and music. Most of the works of al-Samawal have not survived, but he is reported to have written 85 books or articles. Some other of al-Samawal's mathematical writings have survived but these are elementary works of relatively little importance. They contain work on fractions with examples such as showing how to 10) as the sum of fractions with numerators 1. He gives express 80/(3 7 9 80/(3 7 9 10) = 1/(3 10) + 1/(3 7 9) + 1/(3 9 10) The work here is carried out using the sexagesimal system, showing that, although mathematicians of this period favoured the decimal system, commercial use still favoured the sexagesimal system. Al-Samawal's elementary texts were clearly teaching books.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Samawal.html (3 of 4) [2/16/2002 10:57:12 PM]

Al-Samawal

Another of al-Samawal's surviving works is The exposure of the errors of the astrologers which, as the title suggests, argues against the scientific value of astrology. The introduction to this work has been translated into English in [6]. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Arabic mathematics : forgotten brilliance? 2. A history of Zero

Other references in MacTutor

Chronology: 1100 to 1300

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Samawal.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Samawal.html (4 of 4) [2/16/2002 10:57:12 PM]

Al-Sijzi

Abu Said Ahmad ibn Muhammad Al-Sijzi Born: about 945 in Sijistan, Persia Died: about 1020 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Sijzi's full name is Abu Said Ahmad ibn Muhammad ibn Abd al-Jalil al-Sijzi. Very little is known about his life but we can give fairly accurate dates for his life since we know that he corresponded with al-Biruni and quoted results by him in his own work. He dedicated works to a prince of Balkh, then the capital of Khorasan. Another work he dedicated to 'Adud ad-Dawlah who ruler over all southern Iran and most of what is now Iraq from 949 to 983. It is quite possible that 'Adud ad-Dawlah was al-Sijzi's patron since he was a leader well known for patronising the arts and science. We also know that al-Sijzi worked in Shiraz making astronomical observations during 969-970. It was certainly at Shiraz at this time that he wrote some of his mathematical works. As well as writing original works he copied other mathematical works and they were dated 969 at Shiraz. In particular he copied, and dated the copy 969, Thabit ibn Qurra's treatise on complete quadrilaterals. We mentioned above that al-Sijzi corresponded with al-Biruni. The paper [3] contains a letter that al-Biruni wrote to Abu Said, who is almost certainly al-Sijzi. The letter contains proofs of both the plane and spherical versions of the sine theorem, which al-Biruni says were due to his teacher Abu Nasr Mansur ibn Ali ibn Iraq. An English translation of the letter appears in [10]. In [1] Y Dold-Samplonius writes:Al-Sijzi's main scientific activity was in astrology, and he had a vast knowledge of the older literature. He usually compiled and tabulated, adding his own critical commentary. ... Al-Sijzi's mathematical papers are less numerous but more significant than his astrological ones, and he is therefore better known as a geometer. The book [2] contains an English translation (as well as the Arabic text) of al-Sijzi's treatise on geometrical problem solving. Among the problems al-Sijzi discusses are the following. Given a circle, find a point outside the circle where the tangent to the circle and diameter produced, have a given ratio. Given a triangle and three given numbers, find a point inside the triangle where the lines to the three vertices divide the triangle into three triangles having areas proportional to the three given numbers. A treatise on spheres by al-Sijzi Book of the measurement of spheres by spheres is of considerable interest. The treatise, dated by al-Sijzi 969, contains twelve theorems investigating a large sphere containing between one and three smaller spheres. The small spheres are mutually tangent and tangent to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Sijzi.html (1 of 3) [2/16/2002 10:57:13 PM]

Al-Sijzi

the big sphere. Al-Sijzi finds the volume inside the large sphere which is outside the small ones inside it. He expresses this volume as that of a sphere of a particular radius which he computes in terms of the radii of the spheres in the given system. The authors of [9] claim that the main interest of the work lies in the last two propositions in which al-Sijzi considered four-dimensional spheres. In [5] the author again suggests that in these propositions al-Sijzi is dealing with spheres in a space of four dimensions. However J P Hogendijk reviewing [9] writes:One could also assume that the crucial identity ... is due to an oversight made by Al-Sijzi, who does not use four-dimensional spheres anywhere else in his treatise. We note that the treatise was written around 969 AD, at a time when al-Sijzi was a very young and perhaps inexperienced geometer. Another short work by al-Sijzi is the Treatise on how to imagine the two lines which approach but do not meet when they are produced indefinitely, which the excellent Apollonius mentioned in the second Book of the Conics. In this treatise al-Sijzi classifies geometrical theorems into five types, one of which is [7]:... propositions which are difficult to imagine even though the proof of them is correct. In work on geometrical algebra al-Sijzi proves geometrically that (a + b)3 = a3 + 3ab(a + b) + b3. He does this by decomposing a cube of side a + b into the sum of two cubes of sides a and b and a number of parallelepipeds of total volume 3ab(a + b). This is considered by most historians to be a three-dimensional extension by al-Sijzi of the geometrical algebra propositions in Book 2 of Euclid's Elements. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic numerals

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Sijzi.html (2 of 3) [2/16/2002 10:57:13 PM]

Al-Sijzi

JOC/EFR November 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Sijzi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Sijzi.html (3 of 3) [2/16/2002 10:57:13 PM]

Al-Tusi_Nasir

Nasir al-Din al-Tusi Born: 18 Feb 1201 in Tus, Khorasan (now Iran) Died: 26 June 1274 in Kadhimain (near Baghdad now in Iraq)

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Although usually known as Nasir al-Din al-Tusi, his proper name was Muhammad ibn Muhammad ibn al-Hasan al-Tusi. In fact al-Tusi was known by a number of different names during his lifetime such as Muhaqqiq-i Tusi, Khwaja-yi Tusi and Khwaja Nasir. Al-Tusi was born in Tus, which lies close to Meshed in northeastern Iran high up in the valley of the Kashaf River. He was born at the beginning of a century which would see conquests across the whole of the Islamic world from close to China in the east to Europe in the west. It was the era when the vast military power of the Mongols would sweep across the vast areas of the Islamic world displaying a bitter animosity towards Islam and cruelly massacring people. This was a period in which there would be little peace and tranquillity for great scholars to pursue their works, and al-Tusi was inevitably drawn into the conflict engulfing his country. In Tus, al-Tusi's father was a jurist in the Twelfth Imam School. The Twelfth Imam was the main sect of Shi'ite Muslims and the school where al-Tusi was educated was mainly a religious establishment. However, while studying in Tus, al-Tusi was taught other topics by his uncle which would have an important influence on his intellectual development. These topics included logic, physics and metaphysics while he also studied with other teachers learning mathematics, in particular algebra and geometry. In 1214, when al-Tusi was 13 years old, Genghis Khan, who was the leader of the Mongols, turned away from his conquests in China and began his rapid advance towards the west. It would not be too long before al-Tusi would see the effects of these conquests on his own regions, but before that happened he was able to study more advanced topics. From Tus, al-Tusi went to Nishapur which is 75 km west of Tus. Nishapur was a good choice for al-Tusi to complete his education since it was an important centre

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Tusi_Nasir.html (1 of 5) [2/16/2002 10:57:15 PM]

Al-Tusi_Nasir

of learning. There al-Tusi studied philosophy, medicine and mathematics. In particular he was taught mathematics by Kamal al-Din ibn Yunus, who himself had been a pupil of Sharaf al-Din al-Tusi. While in Nishapur al-Tusi began to acquire a reputation as an outstanding scholar and became well known throughout the area. The Mongol invasion reached the area of Tus around 1220 and there was much destruction. Genghis Khan turned his attention again towards the east leaving his generals and sons in the west to continue his conquests. There was, amid the frequent fighting in the region, peaceful havens which attracted al-Tusi. The Assassins, who practised an intellectual form of extremist Shi'ism, controlled the castle of Alamut in the Elburz Mountains, and other similar impregnable forts in the mountains. When invited by the Isma'ili ruler Nasir ad-Din 'Abd ar-Rahim to join the service of the Assassins, al-Tusi accepted and became a highly regarded member of the Isma'ili Court. Whether he would have been able to leave, had he wished to, is not entirely clear. However, al-Tusi did some of his best work while moving round the different strongholds, and during this period he wrote important works on logic, philosophy, mathematics and astronomy. The first of these works, Akhlaq-i nasiri, was written in 1232. It was a work on ethics which al-Tusi dedicated to the Isma'ili ruler Nasir ad-Din 'Abd ar-Rahim. In 1256 al-Tusi was in the castle of Alamut when it was attacked by the forces of the Mongol leader Hulegu, a grandson of Genghis Khan, who was at that time set on extending Mongol power in Islamic areas. Some claim that al-Tusi betrayed the defences of Alamut to the invading Mongols. Certainly Hulegu's forces destroyed Alamut and, Hulegu himself being himself interested in science, he treated al-Tusi with great respect. It may be that indeed al-Tusi felt that he was being held in Alamut against his will, for certainly he seemed enthusiastic in joining the victorious Mongols who appointed him as their scientific advisor. He was also put in charge of religious affairs and was with the Mongol forces under Hulegu when they attacked Baghdad in 1258. Al-Musta'sim, the last Abbasid caliph in Baghdad, was a weak leader and he proved no match for Hulegu's Mongol forces when they attacked Baghdad. After having laid siege to the city, the Mongols entered it in February 1258 and al-Musta'sim together with 300 of his officials were murdered. Hulegu had little sympathy with a city after his armies had won a battle, so he burned and plundered the city and killed many of its inhabitants. Certainly al-Tusi had made the right move as far as his own safety was concerned, and he would also profit scientifically by his change of allegiance. Hulegu was very pleased with his conquest of Baghdad and also pleased that such an eminent scholar as al-Tusi had joined him. So, when al-Tusi presented Hulegu with plans for the construction of a fine Observatory, Hulegu was happy to agree. Hulegu had made Maragheh his capital . Maragheh was in the Azerbaijan region of northwestern Iran, and it was at Maragheh that the Observatory was to be built. Construction of the Observatory began in 1259 west of Maragheh, and traces of it can still be seen there today. The observatory at Maragheh became operational in 1262. Interestingly the Persians were assisted by Chinese astronomers in the construction and operation of the observatory. It had various instruments such as a 4 metre wall quadrant made from copper and an azimuth quadrant which was the invention of Al-Tusi himself. Al-Tusi also designed other instruments for the Observatory which was far more than a centre for astronomy. It possessed a fine library with books on a wide range of scientific topics, while work on science, mathematics and philosophy were vigorously pursued there.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Tusi_Nasir.html (2 of 5) [2/16/2002 10:57:15 PM]

Al-Tusi_Nasir

Al-Tusi put his Observatory to good use, making very accurate tables of planetary movements. He published Zij-i ilkhani (the Ilkhanic Tables), written first in Persian and later translated into Arabic, after making observations for 12 years. This work contains tables for computing the positions of the planets, and it also contains a star catalogue. This was not the only important work which al-Tusi produced in astronomy. It is fair to say that al-Tusi made the most significant development of Ptolemy's model of the planetary system up to the development of the heliocentric model in the time of Copernicus. In al-Tusi's major astronomical treatise, al-Tadhkira fi'ilm al-hay'a (Memoir on astronomy) he [17]:... devised a new model of lunar motion, essentially different from Ptolemy's. Abolishing the eccentric and the centre of prosneusis, he founded it exclusively on the principle of eight uniformly rotating spheres and thereby succeeded in representing the irregularities of lunar motion with the same exactness as the "Almagest". His claim that the maximum difference in longitude between the two theories amounts to 10 proves perfectly true. In his model Nasir, for the first time in the history of astronomy, employed a theorem invented by himself which, 250 years later, occurred again in Copernicus, "De Revolutionibus", III 4. The theorem referred to in this quotation concerns the famous "Tusi-couple" which resolves linear motion into the sum of two circular motions. The aim of al-Tusi with this result was to remove all parts of Ptolemy's system that were not based on the principle of uniform circular motion. Many historians claim that the Tusi-couple result was used by Copernicus after he discovered it in Al-Tusi's work, but not all agree; see for example [38] where it is claimed that Copernicus took the result from Proclus's Commentary on the first book of Euclid and not from al-Tusi. Among numerous other contributions to astronomy, al-Tusi calculated the value of 51' for the precession of the equinoxes. He also wrote works on astronomical instruments, for example on constructing and using an astrolabe. In logic al-Tusi followed the teachings of ibn Sina (Avicenna). He wrote five works on the subject, the most important of which is one on inference. In [33] Street describes this as follows:Tusi, a thirteenth century logician writing in Arabic, uses two logical connectives to build up molecular propositions: 'if-then', and 'either-or'. By referring to a dichotomous tree, Tusi shows how to choose the proper disjunction relative to the terms in the disjuncts. He also discusses the disjunctive propositions which follow from a conditional proposition. Al-Tusi wrote many commentaries on Greek texts. These included revised Arabic versions of works by Autolycus, Aristarchus, Euclid, Apollonius, Archimedes, Hypsicles, Theodosius, Menelaus and Ptolemy. In particular he wrote a commentary on Menelaus's Spherics (see [41] for details), and Archimedes' On the sphere and cylinder (see [21] for details). In the latter work al-Tusi discussed objections raised by earlier mathematicians to comparing lengths of straight lines and of curved lines. Al-Tusi argues that comparisons are legitimate, despite the objections that, being different entities, they are incomparable. Ptolemy's Almagest was one of the works which Arabic scientists studied intently. In 1247 al-Tusi wrote Tahrir al-Majisti (Commentary on the Almagest) in which he introduced various trigonometrical techniques to calculate tables of sines; see [5] for details. As in the Zij-i Ilkhahi al-Tusi gave tables of sines with entries calculated to three sexagesimal places for each half degree of the argument. One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Tusi_Nasir.html (3 of 5) [2/16/2002 10:57:15 PM]

Al-Tusi_Nasir

mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. As stated in [1]:This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth. This work also contains the famous sine formula for plane triangles: a/sin A = b/sin B = c/sin C. Another mathematical contribution was al-Tusi's manuscript, dated 1265, concerning the calculation of n-th roots of an integer; see [6] for details of a copy of this manuscript made in 1413. This work by al-Tusi is almost certainly not original but rather it is his version of methods developed by al-Karaji's school. In the manuscript al-Tusi determined the coefficients of the expansion of a binomial to any power giving the binomial formula and the Pascal triangle relations between binomial coefficients. We should mention briefly other fields in which al-Tusi contributed. He wrote a famous work on minerals which contains an interesting theory of colour based on mixtures of black and white, and included chapters on jewels and perfumes. He also wrote on medicine, but his medical works are among his least important. Much more important were al-Tusi's contributions to philosophy and ethics. In particular in philosophy he asked important questions on the nature of space. Al-Tusi had a number of pupils, one of the better known being Nizam al-a'Raj who also wrote a commentary on the Almagest. Another of his pupils Qutb ad-Din ash-Shirazi gave the first satisfactory mathematical explanation of the rainbow. al-Tusi's influence, which continued through these pupils, is summed up in [1] as follows:Al-Tusi's influence, especially in eastern Islam, was immense. Probably, if we take all fields into account, he was more responsible for the revival of the Islamic sciences than any other individual. His bringing together so many competent scholars and scientists at Maragheh resulted not only in the revival of mathematics and astronomy but also in the renewal of Islamic philosophy and even theology. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (41 books/articles) Mathematicians born in the same country Cross-references to History Topics Other Web sites

Arabic mathematics : forgotten brilliance? 1. C Qajar 2. Muslims online 3. CWI, Netherlands 4. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Tusi_Nasir.html (4 of 5) [2/16/2002 10:57:15 PM]

Al-Tusi_Nasir

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Tusi_Nasir.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Tusi_Nasir.html (5 of 5) [2/16/2002 10:57:15 PM]

Al-Tusi_Sharaf

Sharaf al-Din al-Muzaffar al-Tusi Born: about 1135 in Tus, Khorasan (now Iran) Died: 1213 in Iran Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sharaf al-Din al-Tusi's full name is Sharaf al-Din Al-Muzaffar ibn Muhammad ibn Al-Muzaffar al-Tusi. Very little is known of his life but a few details can be reconstructed from references that occur in works about other scientists of the time. We can certainly deduce from his name that he was born in the region of Tus. This region, in northeastern Iran, includes the towns of Tus and the close-by town of Meshed, both high up in the valley of the Kashaf River. Nishapur, which is 75 km west of Tus, is in the same region and it would be impossible without discovering further information to be precise about which town of the Tus region that he was born in. What is certain is that he spent a large part of his life teaching in different towns over quite a wide area. The Seljuq Turks had captured Damascus in Syria in 1154 and made it the capital of their large empire. The city prospered and many, including al-Tusi, were attracted to it. Certainly around 1165 Al-Tusi was in Damascus for there he taught Abu'l Fadl about the works of Euclid and Ptolemy. Abu'l Fadl was an interesting person, for he had started out as a carpenter before studying mathematics with al-Tusi. From Damascus it would appear that Al-Tusi remained in Syria, going from the largest to the second largest city of Syria, namely Aleppo. Al-Tusi must have taught in Aleppo for at least three years, and it is interesting that there he taught an important member of the Jewish community of that city. Aleppo contained both a Jewish and Muslim community and around 50 years earlier it had played a major part in the Muslim resistance to the crusaders, who had unsuccessfully besieged the city. In Aleppo al-Tusi taught various mathematical topics including the science of numbers, astronomical tables and astrology. From Aleppo, al-Tusi must have gone to Mosul, a city in northwestern Iraq, situated on the right bank of the Tigris River. At this time the city was at its political high point, being under the rule of the Zangid dynasty. In Mosul Al-Tusi taught his most famous pupil Kamal al-Din ibn Yunus. In turn Kamal al-Din ibn Yunus went on to teach Nasir al-Din al-Tusi, one of the most famous of all the Islamic scholars of the period. By this time al-Tusi seems to have acquired an outstanding reputation as a teacher of mathematics for some travelled long distances hoping to become his students. Al-Tusi was probably still in Mosul when Saladin (who had himself been brought up in Mosul) moved his forces into Syria to begin his policy of uniting, partly by force and partly by diplomacy, the area of Syria, Mesopotamia, Palestine, and Egypt. Saladin captured Damascus in 1174 and at about this time http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Tusi_Sharaf.html (1 of 3) [2/16/2002 10:57:17 PM]

Al-Tusi_Sharaf

al-Tusi left Mosul and returned to Iran. He taught in Baghdad towards the end of his life and it was during this period that he wrote his famous work on algebra which we shall describe below. We do have a number of works by al-Tusi which are important in the development of mathematics. The most important is described by Sarton [5] as:... a treatise on algebra ... [written] in 1209 [which] is only known through a commentary by an unknown author. Sarton's use of the word "commentary" is a little misleading, since the unknown author of the manuscript writes (see for example [4]):In this work I wanted to summarise the art of algebra and al-muqabala, adapt what has survived from the great philosopher Sharaf al-Din al-Muzaffar ibn al-Muzaffar ibn Muhammad al-Tusi, and reduce his over lengthy exposition to a moderate size; I eliminated the tables he drew up to make his computations and solve his problems. What is in this Treatise on equations by al-Tusi? Basically it is a treatise on cubic equations, but it does not follow the general development that came through al-Karaji's school of algebra. Rather, as Rashed writes in [2]:... it represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry. In the treatise equations of degree at most three are divided into 25 different types. First al-Tusi discusses twelve types of equation of degree at most two. He then looks at eight types of cubic equation which always have a positive solution, then five types which may have no positive solution. The method which al-Tusi used is quite remarkable. We illustrate the method by showing how al-Tusi examined one of the five types of equation which under certain conditions has a solution, namely the equation x3 + a = bx, where a, b are positive. We use, of course, modern notation to make the solution easy to understand, while al-Tusi would express all his mathematics in words. Now al-Tusi's first comment is that if t is a solution to this equation then t3 + a = bt and, since a > 0, t3 < bt so t < b. Next al-Tusi notes that bx - x3 = a and he then finds where the maximum of y = bx - x3 occurs. Basically using the derivative of this expression, al-Tusi finds the maximum occurs at x = (b/3) and then finds the maximum value for y of 2(b/3)3/2 by substituting x = (b/3) back into y = bx - x3. Thus the equation bx x3 = a has a solution if a 2(b/3)3/2. Then Al-Tusi deduces that the equation has a positive root if D = b3/27 - a2/4 0 where D is the discriminant of the equation. Of course al-Tusi's use of the derivative of a function, without of course saying so, is very interesting. The paper [11] attempts to discover the origin of this implicit use of the derivative, which the author claims arises from algebraic proofs based on analytical procedures. The paper [12] suggests that a rather different approach, not one analogous to the modern derivative, lay behind Al-Tusi's method. The papers [10] and [14] contribute to this discussion; see also [2], [3] and [4] for further insights. Al-Tusi then went on to give what we would essentially call the Ruffini-Horner method for approximating the root of the cubic equation. Although this method had been used by earlier Arabic mathematicians to find approximations for the nth root of an integer, al-Tusi is the first that we know http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Tusi_Sharaf.html (2 of 3) [2/16/2002 10:57:17 PM]

Al-Tusi_Sharaf

who applied the method to solve general equations of this type. Another famous work by al-Tusi is one in which he describes the linear astrolabe, sometimes called the "staff of al-Tusi", which he invented. It was [1]:... a simple wooden rod with graduated markings but without sights. It was furnished with a plumb line and a double chord for making angular measurements and bore a perforated pointer. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Tusi_Sharaf.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Tusi_Sharaf.html (3 of 3) [2/16/2002 10:57:17 PM]

Al-Umawi

Abu Abdallah Yaish ibn Ibrahim Al-Umawi Born: about 1400 in possibly Andalusia, Spain Died: 1489 in possibly Damascus, Syria Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Umawi's full name is Abu Abdallah Yaish ibn Ibrahim ibn Yusuf ibn Simak al-Umawi. There are clear problems with his date of birth and death date. It is claimed that he died in 1489, but there is a marginal note on one of his works allowing the copyist to teach the material and this note is dated 1373 (this is not strictly true but the date given corresponds to 1373 after changing calendars). The copyist records the date he completed making the copy, also 1373, and the place in which he made the copy which is Mount Qasyun in Damascus, Syria. If al-Umawi wrote his manuscript before 1373 he cannot have lived to 1489 so one date must be incorrect but there is no other evidence as to which is correct and which is wrong. It is usual to regard al-Umawi as a 14th century mathematician and we have given rough dates based on the assumption that the manuscript date is correct. Although al-Umawi lived in Damascus in Syria, he came from Andalusia in the south of Spain. The name Andalusia comes from the Arabic "Al-Andalus" given to this district by the Muslims who conquered it in the 8th century. The unified Spanish Muslim state broke up in the early 11th century but Muslims from Africa kept Spanish Islam strong into the 14th century. Indeed al-Umawi was a Muslim but the mathematical scholarship of the Muslim world at this time was certainly not uniform. There were differences in the numerals used in western areas (which al-Umawi came from) and those used in the east. Indeed some scholars find it surprising that al-Umawi as a westerner wrote an arithmetic text for those in the east. The usual perception is that, at this time. the arithmetical skills of the east exceeded those of the west. Two texts by al-Umawi which have survived are Marasim al-intisab fi'ilm al-hisab (On arithmetical rules and procedures), and Raf'al-ishkal fi ma'rifat al-ashkal which is a work on mensuration. It is the first of these two works which contains the 1373 date referred to in the first paragraph and it is the most interesting of the two texts. Before describing the Marasim we should make some brief comments about al-Umawi's work calculating lengths and areas. In it al-Umawi gives rules for calculating: lengths of chords and lengths of arcs of circles (using Pythagoras's theorem); areas of circles, areas of segments of circles, areas of triangles and quadrilaterals; volumes of spheres, volumes of cones and volumes of prisms. It is not a work of any great importance and Saidan, writes in [1] that:... it is a small treatise of seventeen folios in which we find nothing on mensuration that the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Umawi.html (1 of 3) [2/16/2002 10:57:18 PM]

Al-Umawi

arithmeticians of the East did not know. Let us now return to the more important treatise on arithmetical rules and procedures. This is the earliest surviving arithmetical treatise written by an Arab from Spain, so it is interesting to see the content of the work. After describing the very briefly the basic arithmetical operations of addition and multiplication, al-Umawi moves on to discuss the summation of series. Among the series al-Umawi considers are arithmetic and geometric series. He considers the sum of the first n polygonal numbers, that is 1 + (r - 1)d summed from r = 1 to r = n. These sums of polygonal numbers are called pyramidal numbers and al-Umawi then considers the sums of the first n pyramidal numbers. In discussing r3, (2r+1)3, and (2r)3 al-Umawi was giving results which al-Karaji had proved geometrically 400 years earlier. Al-Umawi then describes casting out sevens, eights, nines, and elevens. Although he only gives these special cases, the general rule which they all obey is the following: take a number n written in decimal notation as n = aq +10a1 + 102a2 + 103a3 + ... Let rj = 10j (mod t) where, as far as al-Umawi is concerned, t = 7, 8, 9, or 11. Then if Sajrj is divisible by t so is n. This theorem is attributed to Pascal three hundred years after al-Umawi, and indeed al-Umawi only gives the special cases mention here. However, he does note that the sequence r1 , r2 , r3 , r4 , r5 , ... recurs after finitely many steps in each of the cases he considers. Some results appearing in this work by al-Umawi are not found in any other Arabic arithemetics. He gives some interesting conditions for the decimal representation of a number n to be a square: n must either end in 00, 1, 4, 5, 6, or 9; if n ends in 6, the 10's place is odd, otherwise the 10's place is even; if n ends in 5 then the 10's place must be 2; n must leave a remainder of 0, 1, 2, or 4 on division by 7; n must leave a remainder of 0, 1, or 4 on division by 8; n must leave a remainder of 0, 1, 4, or 7 on division by 9. Al-Umawi gives similar results for n to be a cube including: n must leave a remainder of 0, 1, or 6 on division by 7; n must leave a remainder of 0, 1, 3, 5, or 7 on division by 8; n must leave a remainder of 0, 1, or 8 on division by 9. None of these results are hard to prove today (try them!) with our understanding of the decimal representation of numbers. One has to remember that these results are about decimal representations rather than about numbers themselves and show how an understanding of the decimal system was progressing at a time when Christian Europe (if I may call it that) had little interest in anything beyond the mathematics of the ancient Greeks.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Umawi.html (2 of 3) [2/16/2002 10:57:18 PM]

Al-Umawi

If you have enjoyed proving these results due to al-Umawi then here is one more he gives in the Marasim. If the integer n is a square and its final digit is 1, then either both the 100's place and 1/2 the 10's place are both even or they are both odd. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Umawi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Umawi.html (3 of 3) [2/16/2002 10:57:18 PM]

Al-Uqlidisi

Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi Born: about 920 in possibly Damascus, Syria Died: about 980 in possibly Damascus, Syria Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Al-Uqlidisi is a mathematician who is only known to us through two manuscripts on arithmetic, Kitab al-fusul fi al-hisab al-Hindi and Kitab al-hajari fi al-hisab. Despite this he is a figure of some importance and has prompted an interesting scholarly argument among historians of science. The manuscript of the Kitab al-fusul fi al-hisab al-Hindi which has survived is a copy of the original which was made in 1157. An English translation of this work has been published by Saidan [4]. The manuscript gives al-Uqlidisi's full name on the front page as well as the information that he composed the text in Damascus in 952-53. In the introduction al-Uqlidisi writes that he travelled widely and learnt from all the mathematicians he met on his travels. He also claimed to have read all the available texts on arithmetic. Other than being able to deduce a little of al-Uqlidisi's character from his writing, we have no other information on his life. The Kitab al-fusul fi al-hisab al-Hindi of al-Uqlidisi is the earliest surviving book that presents the Hindu system. In it al-Uqlidisi argues that the system is of practical value [4]:Most arithmeticians are obliged to use it in their work: since it is easy and immediate, requires little memorisation, provides quick answers, demands little thought ... Therefore, we say that it is a science and practice that requires a tool, such as a writer, an artisan, a knight needs to conduct their affairs; since if the artisan has difficulty in finding what he needs for his trade, he will never succeed; to grasp it there is no difficulty, impossibility or preparation. This treatise on arithmetic is in four parts. The aim of the first part is to introduce the Hindu numerals, to explain a place value system and to describe addition, multiplication and other arithmetic operations on integers and fractions in both decimal and sexagesimal notation. The part second collects arithmetical methods given by earlier mathematicians and converts them in the Indian system. For example the method of casting out nines is described. The third part of the treatise tries to answer to the standard type of questions that are asked by students: why do it this way ... ?, how can I ... ?, etc. There is plenty of evidence here that al-Uqlidisi must have been a teacher, for only a teacher would know understand the type of problem that a beginning student would encounter.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Uqlidisi.html (1 of 3) [2/16/2002 10:57:20 PM]

Al-Uqlidisi

The fourth part has considerable interest for it claims that up to this work by al-Uqlidisi the Indian methods had been used with a dust board. A dust board was used because the methods required the moving of numbers around in the calculation and rubbing some out as the calculation proceeded. The dust board allowed this in the same sort of way that one can use a blackboard, chalk and a blackboard eraser. However, al-Uqlidisi showed how to modify the methods for pen and paper use. Al-Uqlidisi's work is historically important as it is the earliest known text offering a direct treatment of decimal fractions. It is here that the scholarly argument referred to above arises. At one time it was thought that Stevin was the first to propose decimal fractions. Further research showed that decimal fractions appeared in the work of al-Kashi, who was then credited with this extremely important contribution. When Saidan studied al-Uqlidisi's Kitab al-fusul fi al-hisab al-Hindi in detail he wrote [6]:The most remarkable idea in this work is that of decimal fraction. Al-Uqlidisi uses decimal fractions as such, appreciates the importance of a decimal sign, and suggests a good one. Not al-Kashi (d. 1436/7) who treated decimal fractions in his "Miftah al-Hisab", but al-Uqlidisi, who lived five centuries earlier, is the first Muslim mathematician so far known to write about decimal fractions. Following Saidan's paper, some historians went even further in attributing to al-Uqlidisi the complete credit for giving the first complete description and applications of decimal fractions. Rashed, however, although he does not wish to minimise the importance of al-Uqlidisi's contribution to decimal fractions, sees it as [2]:... preliminary to its history, whereas al-Samawal's text already constitutes the first chapter. The argument depends on how one interprets the following passage in al-Uqlidisi's treatise. He explains how to raise a number by one tenth five times [4]:... we want to raise a number by its tenth five times. We write down this number as usual; write it down again below moved one place to the right; we therefore know its tenth, which we add to it. So was have added its tenth to this number. We put the resulting fraction in front of this number and we move it to the unit place after marking it [with the ' sign he uses for the decimal point] thus. We add its tenth and so on five times. Saidan (writing in [1]) sees in this passage that al-Uqlidisi has fully understood the idea of decimal fractions, saying that earlier authors:... rather mechanically transformed the decimal fraction obtained into the sexagesimal system, without showing any sign of comprehension of the decimal idea. ... In all operations where powers of ten are involved in the numerator or the denominator, [al-Uqlidisi] is well at home. On the other hand Rashed sees this passage rather differently [2]:... unlike al-Samawal, al-Uqlidisi never formulates the idea of completing the sequence of powers of ten by that of their inverse after having defined the zero power. That said, in the passage just quoted, three basic ideas emerge whose intuitive resonance may have misled historians; what they thought was a theoretical exposition was merely understood implicitly, and, as a result, they have overestimated the author's contribution to decimal fractions. The two points of view are almost impossible to decide between since what we are looking at is the development of the idea of decimal fractions by different mathematicians, each contributing to its understanding. To take a particular text as the one where the idea appears for the first time in its entirety http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Uqlidisi.html (2 of 3) [2/16/2002 10:57:20 PM]

Al-Uqlidisi

must always be a somewhat arbitrary decision. There is no disagreement on the fact that al-Uqlidisi made a major step forward. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic numerals

Other references in MacTutor

Chronology: 900 to 1100

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Al-Uqlidisi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Al-Uqlidisi.html (3 of 3) [2/16/2002 10:57:20 PM]

Albanese

Giacomo Albanese Born: 11 July 1890 in Geraci Siculo (near Palermo), Italy Died: 8 June 1948 in Sao Paulo, Brasil Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giacomo Albanese studied at the Institute of Physics in Palermo, graduating in 1903. He received his doctorate from the Scola Normale Superiore in 1913 having the distinction of receiving the 'Ulisse Dini' prize for his doctoral dissertation. From 1913 to 1920 Albanese taught at Pisa, having leave for military service during the years 1917 and 1918. In 1920 he moved to Padua to become Severi's assistant. Although he was only to serve in this post for a few months it was to have a very significant effect both on the direction of his research and on his research output which increased markedly. Albanese left Padua still in the year 1920 to take up a professorship in analysis and algebra at the Naval Academy in Livorno. Three years later he moved to the University of Catania, then in 1927 he returned to Palermo. From 1929 until 1936 he held a chair at Pisa, moving to Sao Paulo in Brazil in 1936. He spent the rest of his life in Sao Paulo and, according to [1]:There he created a very good mathematical library, especially rich in books of algebraic geometry. A Weil and O Zariski taught in Sao Paulo some years later and said that the library was a source of inspiration to both of them. Albanese's research involved examining curves on algebraic surfaces and the genus of an algebraic variety. He considered the problem of resolution of singularities, a major problem in algebraic geometry, and produced some elegant results. He pioneered investigations related to the Riemann-Roch problem and studied the rational equivalence of 0-cycles on surfaces. His name is remembered today for Albanese varieties used as a standard tool in algebraic geometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Albanese.html (1 of 2) [2/16/2002 10:57:20 PM]

Albanese

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Albanese.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Albanese.html (2 of 2) [2/16/2002 10:57:20 PM]

Albert

Albert of Saxony Born: 1316 in Helmstedt, Lower Saxony (now Germany) Died: 8 July 1390 in Halberstadt, Saxony (now Germany) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Albert studied at Prague and then at Paris. He taught at Paris from 1351 to 1362 becoming rector there in 1353. Albert became rector of the University of Vienna in 1365 and Bishop of Halberstadt from 1366 until his death. Albert was mainly a transmitter of good mathematical ideas but he did contribute his own work to these. He wrote about the ideas of Bradwardine, Ockham, Oresme and others. His books on logic are his best where he examined 254 logical paradoxes while his work on projectiles is, as such work was at that time, incorrect. Albert believed that a projectile fired horizontally will travel horizontally for a certain distance, then follow a curved path for a while, then fall vertically. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Albert.html (1 of 2) [2/16/2002 10:57:21 PM]

Albert

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Albert.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Albert.html (2 of 2) [2/16/2002 10:57:21 PM]

Albert_Abraham

Abraham Adrian Albert Born: 9 Nov 1905 in Chicago, Illinois, USA Died: 6 June 1972 in Chicago, Illinois, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

A Adrian Albert's parents were Russian. His father, Elias Albert, came to the United States from England and had set up a retail business. His mother, Fannie Fradkin, had come to the United States from Russia. Adrian was the second of Elias and Fannie's three children, but he also had both a half-brother and half-sister from his mother's side. It was in Chicago that Adrian undertook most of his education; in fact this was his home town for most of his life. He began his education there entering primary school in 1911. In 1914, however, the family moved to Iron Mountain in Michigan where he continued his schooling until the family returned to Chicago in 1916. Back in the town of his birth, Adrian entered the Theodore Herzl Elementary School where he studied until 1919 then, in that year, he entered the John Marshall High School. In 1922 he graduated from the High School and began his studies at the University of Chicago. Albert completed his S.B. degree in 1926 and was awarded his Master's degree in the following year. He remained at the University of Chicago undertaking research under L E Dickson's supervision. That Dickson, the leading American mathematician in the fields of number theory and algebra, was on the Chicago faculty was a piece of good fortune for Albert. Not only did Dickson strongly influence the course of all Albert's later research, but also his style as a teacher and academic rubbed off on Albert. He was awarded his Ph.D. in 1928 for a doctoral dissertation entitled Algebras and their Radicals and Division Algebras. By the time that he received his doctorate Albert was a married man, having married Freda Davis on 18 December 1927. The economic situation in the United States was deteriorating at this time with the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Albert_Abraham.html (1 of 4) [2/16/2002 10:57:23 PM]

Albert_Abraham

advent of the depression. Herstein writes in [5] (see also [4]):Shortly after he got his Ph.D., the great economic depression started. Sensitive as he was to the suffering of others, he deeply felt the economic hardship that so many of his friends were undergoing. he, too, did not have an easy time of it economically. In addition he was beset by a series of illnesses ... In his doctoral thesis Albert had made considerable progress in classifying division algebras. It was impressive piece of work and it led to him being awarded a National Research Council Fellowship to enable him to undertake postdoctoral study at Princeton. He spent nine months at Princeton in 1928-29 and this was an important period for Albert since during his time there Lefschetz suggested that he look at open problems in the theory of Riemann matrices. These matrices arise in the theory of complex manifolds and Albert went on to write an important series of papers on these questions over the following years. Albert was then offered a post as an instructor at Columbia University and he worked there for two years from 1929 to 1931. His first paper A determination of all normal division algebras in sixteen units was published in 1929. It was based on the second half of his doctoral thesis but Albert had, by this time, pushed the ideas further classifying division algebras of dimension 16 over their centres. The case of dimension 9, the next smaller case, had been solved by Wedderburn. Albert returned to the University of Chicago in 1931 where he was appointed as assistant professor. He remained on the staff there for the rest of his life being promoted to assistant professor in 1937 and full professor in 1941. During the years 1958 to 1962 he was chairman of the Chicago Department. Kaplansky writes [6]:The main stamp he left on the Department was a project dear to his heart; maintaining a lively flow of visitors and research instructors, for whom he skilfully got support in the form of research grants. Shortly after beginning his second three year term as Chairman of the Department Albert was asked to take on the post of Dean of Physical Sciences. He served Chicago for 9 year in the role until 1971. Herstein writes [4]:He dearly loved Chicago as a city, more especially the Hyde Park area surrounding the university, and most especially, the university itself. He was an integral part of the university, and the university was an integral part of his life. He knew and was known by almost everybody at the university, and his influence went well beyond the realm of the physical sciences. Of all the honours and responsibilities that came to him, the one that he probably enjoyed most, and which meant the most to him, was that of being Dean of Physical Sciences in his beloved university. We should say a little about Albert's family life which was filled with great happiness until the tragic death of his son Roy. He was one of their three children (the other two being Alan and Nancy) and Roy's death at the age of 23 in 1958 brought a deep sadness which Albert and his wife never got over. One should say, though, that it was in his nature not to allow his grief to be too visible publicly. The blow was softened, if ever the loss of a child could ever really be softened, by the happiness that Albert and his wife enjoyed from their other two children and from their five grand-children. We have already said a little above about Albert's mathematical contributions but we should now give

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Albert_Abraham.html (2 of 4) [2/16/2002 10:57:23 PM]

Albert_Abraham

further details. His main work was on associative algebras, non-associative algebras, and Riemann matrices. He worked on classifying division algebras building on the work of Wedderburn but Brauer, Hasse and Emmy Noether got the main result first. Albert's major contribution is, however, detailed in a joint paper with Hasse. Albert's book Structure of Algebras, published in 1939, remains a classic. The contents of this treatise was the basis of the Colloquium Lectures which he gave to the American Mathematical Society in 1939. We should also mention Albert's other fine text on algebra, Modern Higher Algebra, which was published two years before Structure of Algebras. Albert's work on Riemann matrices was, as we mentioned above, a consequence of suggestions made by Lefschetz. For his papers on the construction of Riemann matrices published in the Annals of Mathematics in 1934 and 1935 Albert received the Cole prize in algebra from the American Mathematical Society in 1939. These important papers had been a direct consequence of Albert's 1928-29 visit to Princeton and when he spent the academic year 1933-34 at the Institute for Advanced Study at Princeton he again received a stimulus which would lead him to further important results. Lectures by Weyl on Lie algebras were particularly stimulating but perhaps even more important was his introduction to Jordan algebras. These algebras had been introduced by Pascual Jordan as being related the quantum theory. Jordan had worked with von Neumann and Wigner on the structure of these algebras but they had left open certain fundamental questions. Albert was able to use his expertise in structural questions regarding algebras to solve some of the problems in his 1934 paper On certain algebras of quantum mechanics. His work on Jordan algebras did not end there for he published three further fundamental papers on their structure in 1946, 1947 and 1950. During the Second World War Albert contributed to the war effort as associate director of the Applied Mathematics Group at Northwestern University which tackled military problems. Another interest of Albert's, which appears to have been prompted by the War, was that of cryptography. He lectured to the American Mathematical Society on Some mathematical aspects of cryptography at the Society's meeting in November 1941. Lectures by Weyl on Lie algebras in 1934-35 introduced Albert to the theory of non-associative algebras. It was not until 1942, however, that he published his first major work on non-associative algebras. Kaplansky writes in [7]:Albert investigated just about every aspect of non-associative algebras. At times a particular line of attack failed to fulfil the promise it had shown; he would then exercise his sound instinct and good judgement by shifting the assault to a different area. In fact, he repeatedly displayed an uncanny knack for selecting projects which later turned out to be well conceived ... Albert received many honours for his outstanding achievements. He was elected to the National Academy of Sciences in 1943, the Brazilian Academy of Sciences in 1952, and the Argentine Academy of Sciences in 1963. He served as chairman of the Mathematics Section of the National Research Council from 1958 to 1961, and President of the American Mathematical Society in 1965-66. Herstein says this about Albert as a person ([5] or [6]):What characterised him best as a person was his intense loyalty to his friends and to his profession. He viewed the profession of mathematician with a great deal of pride and he did

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Albert_Abraham.html (3 of 4) [2/16/2002 10:57:23 PM]

Albert_Abraham

everything he could to have it recognised as he felt it deserved. He constantly fought for the improvement of working conditions, salaries, and student support in his chosen field. Although he had a strong set of principles in life and a definite attitude to moral and professional behaviour, he was endowed with an enormous tolerance for the changes that were taking place around him ... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Other references in MacTutor

1. Chronology: 1930 to 1940 2. Chronology: 1940 to 1950

Honours awarded to A Adrian Albert (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1965 - 1966

AMS Colloquium Lecturer

1939

AMS Cole Prize

Awarded 1939

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Albert_Abraham.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Albert_Abraham.html (4 of 4) [2/16/2002 10:57:23 PM]

Alberti

Leone Battista Alberti Born: 18 Feb 1404 in Genoa, Italy Died: 3 April 1472 in Rome, Italy

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

As a child Leone Alberti received his mathematical education from his father. He attended a school in Padua then the University of Bologna where he studied law but did not enjoy this topic. Alberti lived mainly in Rome and Florence working within the Roman Catholic Church, by 1432 he was following a literary career as a secretary in the Papal Chancery in Rome writing biographies of the saints in elegant Latin. Alberti studied the representation of 3-dimensional objects and wrote the first general treatise Della Pictura on the laws of perspective in 1435. It was printed in 1511. He said Nothing pleases me so much as mathematical investigations and demonstrations, especially when I can turn them to some useful practice drawing from mathematics the principles of painting perspective and some amazing propositions on the moving of weights. Alberti also worked on maps (again involving his skill at geometrical mappings) and he collaborated with Toscanelli who supplied Columbus with the maps for his first voyage. He also wrote the first book on cryptography which contains the first example of a frequency table. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) A Poster of Leone Alberti

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1300 to 1500

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.html (1 of 2) [2/16/2002 10:57:25 PM]

Alberti

Other Web sites

1. The Catholic Encyclopedia 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Alberti.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Alberti.html (2 of 2) [2/16/2002 10:57:25 PM]

Albertus

Saint Albertus Magnus Born: 1200 in Lauingen an der Donau, Swabia (now Germany) Died: 15 Nov 1280 in Cologne, Prussia

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Albert (or Albertus) studied at Padua and joined the Dominican Order. Then he studied and taught at Padua, Bologna, Cologne and other German convents. He then was sent to the University of Paris where he read the translations of the Arabic and Greek texts of Aristotle. While in Paris Albert began the task of presenting the entire body of knowledge, natural science, logic, rhetoric, mathematics, astronomy, ethics, economics, politics and metaphysics. He wrote commentaries on all of Aristotle's works with his own observations and experiments. By 'experiment' Albert meant 'observing, describing and classifying'. Albert was made a Saint in 1931 and, in 1941, was made patron of natural scientists. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles)

A Quotation

A Poster of Albertus

Mathematicians born in the same country

Other Web sites

1. The Catholic Encyclopedia 2. Albert Pinto 3. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Albertus.html (1 of 2) [2/16/2002 10:57:27 PM]

Albertus

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Albertus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Albertus.html (2 of 2) [2/16/2002 10:57:27 PM]

Alcuin

Alcuin of York Born: 735 in York, Yorkshire, England Died: 19 May 804 in Tours, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alcuin of York was born into a high ranking family who lived near the East Coast of England. He was sent to York where he became a pupil at York cathedral school, Archbishop Ecgberht's School. After being a pupil at Archbishop Ecgberht's School, Alcuin remained there as a teacher, becoming headmaster of the school in 778. During his time as a teacher at this school in York Alcuin built up a fine library, one of the best in Europe, and made the school one of the most important centres of learning in Europe. He wrote a long poem describing the men associated with York's history before he left for the continent. In 781 Alcuin accepted an invitation from Charlemagne to go to Aachen to a meeting of the leading scholars of the time. Following this meeting, he was appointed head of Charlemagne's Palace School at Aachen and there he developed the Carolingian minuscule, a clear script which has become the basis of the way the letters of the present Roman alphabet are written. Before leaving Aachen, Alcuin was responsible for the most precious of Carolingian codices, now called the Golden Gospels. These were a series of illuminated masterpieces written largely in gold, often on purple coloured vellum. The development of Carolingian minuscule had, although somewhat indirectly, a large impact on the history of mathematics. It was a script which was much more readable than the old unspaced capital script which was in use before this and, as a consequence, most of the mathematical works were freshly copied into this new script in the 9th century. Most of the works of the ancient Greek mathematicians which have survived do so because of this copying process and it is the 'latest' version written in minuscule script which has survived.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Alcuin.html (1 of 2) [2/16/2002 10:57:28 PM]

Alcuin

Not only was Alcuin headmaster of Charlemagne's Palace School at Aachen but he also was a personal friend to Charlemagne and became the teacher of his two sons. In fact Alcuin lived in Aachen for two periods, during the years 782 to 790 and then again from 793 to 796. In 796 Alcuin retired from Charlemagne's Palace School at Aachen and became abbot of the Abbey of St Martin at Tours, where he had his monks continue to work with the Carolingian minuscule script. While in Tours Alcuin arranged for some of his pupils to go to York to bring some of the rarer works that he had collected there back to Tours. He wrote:I say this that you may agree to send some of our boys to get everything we need from there and bring the flowers of Britain back to France that as well as the walled garden in York there may be off-shoots of paradise bearing fruit in Tours. Alcuin wrote elementary texts on arithmetic, geometry and astronomy at a time when there was just beginning a renaissance in learning in Europe, a renaissance mainly led by Alcuin himself. His lesson books were written in a question - and - answer format. However his work in this area, unlike the inspired calligraphy he developed, shows little originality. Late in his life Alcuin summed up his own career with a rather beautiful description:In the morning, at the height of my powers, I sowed the seed in Britain, now in the evening when my blood is growing cold I am still sowing in France, hoping both will grow, by the grace of God, giving some the honey of the holy scriptures, making others drunk on the old wine of ancient learning... Article by: J J O'Connor and E F Robertson List of References (6 books/articles) A Poster of Alcuin

Mathematicians born in the same country

Other references in MacTutor

Chronology: 500 to 900

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Alcuin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Alcuin.html (2 of 2) [2/16/2002 10:57:28 PM]

Aleksandrov

Pavel Sergeevich Aleksandrov Born: 7 May 1896 in Bogorodsk (also called Noginsk), Russia Died: 16 Nov 1982 in Moscow, USSR

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Like most Russian mathematicians there are different ways to transliterate Aleksandrov's name into the Roman alphabet. The most common way, other than Aleksandrov, is to write it as Alexandroff. Pavel Sergeevich Aleksandrov's father Sergi Aleksandrovich Aleksandrov was a medical graduate from Moscow University who had decided not to follow an academic career but instead had chosen to use his skills in helping people and so he worked as a general practitioner in Yaroslavskii. Later he worked in more senior positions in a hospital in Bogorodskii, which is where he was when Pavel Sergeevich was born. When Pavel Sergeevich was one year old his father moved to Smolensk State hospital, where he was to earn the reputation of being a very fine surgeon, and the family lived from this time in Smolensk. The city of Smolensk is on the Dnieper River 420 km west of Moscow. Pavel Sergeevich's early education was from his mother, Tsezariya Akimovna Aleksandrova, who applied all her considerable talents to bringing up and educating her children. It was from her that Aleksandrov learnt French and also German. His home was one that was always filled with music as his brothers and sisters all great talent in that area. The fine start which his mother gave him meant that he always excelled at the grammar school in Smolensk which he attended. His mathematics teacher Alexsander Romanovich Eiges soon realised that his pupil had a remarkable talent for the subject and ([3] and [4]):... at grammar school he studied celestial mechanics and mathematical analysis. But his interest was mainly directed towards fundamental problems of mathematics: the foundations

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aleksandrov.html (1 of 5) [2/16/2002 10:57:30 PM]

Aleksandrov

of geometry and non-euclidean geometry. Eiges had a proper appreciation of his pupil and exerted a decisive influence on his choice of a career in mathematics. In 1913 Aleksandrov graduated from the grammar school being dux of the school and winning the gold medal. Certainly at this time he had already decided on a career in mathematics, but he had not set his sights as high as a university teacher, rather he was aiming to become a secondary school teacher of mathematics. Eiges was the role model who he was aspiring to match at this stage, for Eiges had done more than teach Aleksandrov mathematics, he had also influenced his tastes in literature and the arts. Aleksandrov entered Moscow University in 1913 and immediately he was helped by Stepanov. Stepanov, who was working at Moscow University, was seven years older than Aleksandrov but his home was also in Smolensk and he often visited the Aleksandrov home there. Stepanov was an important influence on Aleksandrov at this time and suggested that Aleksandrov join Egorov's seminar even in the first year of his studies in Moscow. In Aleksandrov's second year of study he came in contact with Luzin who had just returned to Moscow. Aleksandrov wrote (see for example [3] or [4]):After Luzin's lecture I turned to him for advice on how best to continue my mathematical studies and was struck most of all by Luzin's kindness to the man addressing him - an 18-year old student ... I then became a student of Luzin, during his most creative period ... To see Luzin in those years was to see a display of what is called an inspired relationship to science. I learnt not only mathematics from him, I received also a lesson in what makes a true scholar and what a university professor can and should be. Then, too, I saw that the pursuit of science and the raining of young people in it are two facets of one and the same activity - that of a scholar. Aleksandrov proved his first important result in 1915, namely that every non-denumerable Borel set contains a perfect subset. It was not only the result which was important for set theory, but also the methods which Aleksandrov used which turned out to be one of the most useful methods in descriptive set theory. After Aleksandrov's great successes Luzin did what many a supervisor might do, he realised that he had one of the greatest mathematical talents in Aleksandrov so he thought that it was worth asking him to try to solve the biggest open problem in set theory, namely the continuum hypothesis. After Aleksandrov failed to solve the continuum hypothesis (which is not surprising since it can neither be proved or disproved as was shown by Cohen in the 1960s) he thought he was not capable of a mathematical career. Aleksandrov went to Novgorod-Severskii and became a theatre producer. He then went to Chernikov where, in addition to theatrical work, he lectured on Russian and foreign languages, becoming friends with poets, artists and musicians. After a short term in jail in 1919 at the time of the Russian revolution, Aleksandrov returned to Moscow in 1920. Luzin and Egorov had built up an impressive research group at the University of Moscow which the students called 'Luzitania' and they, together with Privalov and Stepanov, were very welcoming to Aleksandrov on his return. It was not an immediate return to Moscow for Aleksandrov, however, for he spent 1920-21 back home in Smolensk where he taught at the University. During this time he worked on his research, going to Moscow about once every month to keep in touch with the mathematicians there and to prepare himself for his examinations. At around this time Aleksandrov became friendly with Urysohn, who was a member of 'Luzitania', and the friendship would soon develop into a major mathematical collaboration.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aleksandrov.html (2 of 5) [2/16/2002 10:57:30 PM]

Aleksandrov

After taking his examinations in 1921, Aleksandrov was appointed as a lecturer at Moscow university and lectured on a variety of topics including functions of a real variable, topology and Galois theory. In July 1922 Aleksandrov and Urysohn went to spend the summer at Bolshev, near to Moscow, where they began to study concepts in topology. Hausdorff, building on work by Fréchet and others, had created a theory of topological and metric spaces in his famous book Grundzüge der Mengenlehre published in 1914. Aleksandrov and Urysohn now began to push the theory forward with work on countably compact spaces producing results of fundamental importance. The notion of a compact space and a locally compact space is due to them. In the summers of 1923 and 1924 Aleksandrov and Urysohn visited Göttingen and impressed Emmy Noether, Courant and Hilbert with their results. The mathematicians in Göttingen were particularly impressed with their results on when a topological space is metrisable. In the summer of 1924 they also visited Hausdorff in Bonn and he was fascinated to hear the major new directions that the two were taking in topology. However while visiting Hausdorff in Bonn ([3] and [4]):Every day Aleksandrov and Urysohn swam across the Rhine - a feat that was far from being safe and provoked Hausdorff's displeasure. Aleksandrov and Urysohn then visited Brouwer in Holland and Paris in August 1924 before having a holiday in the fishing village of Bourg de Batz in Brittany. Of course mathematicians continue to do mathematics while on holiday and they were both working hard. On the morning of 17 August Urysohn began to write a new paper but tragically he drowned while swimming in the Atlantic later that day. Aleksandrov determined that no ideas of his great friend and collaborator should be lost and he spent part of 1925 and 1926 in Holland working with Brouwer on preparing Urysohn's paper for publication. The atmosphere in Göttingen had proved very helpful to Aleksandrov, particularly after the death of Urysohn, and he went there every summer from 1925 until 1932. He became close friends with Hopf and the two held a topological seminar in Göttingen. Of course Aleksandrov also taught in Moscow University and from 1924 he organised a topology seminar there. At Göttingen, Aleksandrov also lectured and participated in Emmy Noether's seminar. In fact Aleksandrov always included Emmy Noether and Hilbert among his teachers, as well as Brouwer in Amsterdam and Luzin and Egorov in Moscow. From 1926 Aleksandrov and Hopf were close friends working together. They spent some time in 1926 in the south of France with Neugebauer. Then Aleksandrov and Hopf spent the academic year 1927-28 at Princeton in the United States. This was an important year in the development of topology with Aleksandrov and Hopf in Princeton and able to collaborate with Lefschetz, Veblen and Alexander. During their year in Princeton, Aleksandrov and Hopf planned a joint multi-volume work on Topology the first volume of which did not appear until 1935. This was the only one of the three intended volumes to appear since World War II prevented further collaboration on the remaining two volumes. In fact before the joint work with Hopf appeared in print, Aleksandrov had begun yet another important friendship and collaboration. In 1929 Aleksandrov's friendship with Kolmogorov began and they ([3] and [4]):... journeyed a lot along the Volga, the Dnieper, and other rivers, and in the Caucuses, the Crimea, and the south of France. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aleksandrov.html (3 of 5) [2/16/2002 10:57:30 PM]

Aleksandrov

The year 1929 marks not only the beginning of the friendship with Kolmogorov but also the appointment of Aleksandrov as Professor of Mathematics at Moscow University. In 1935 Aleksandrov went to Yalta with Kolmogorov, then finished the work on his Topology book in the nearby Crimea and the book was published in that year. The 'Komarovski' period also began in that year ([3] and [4]):Over the last forty years, many of the events in the history of mathematics in the University of Moscow have been linked with Komarovka, a small village outside Moscow. Here is the house owned since 1935 by Aleksandrov and Kolmogorov. Many famous foreign mathematicians also visited Komarovka - Hadamard, Fréchet, Banach, Hopf, Kuratowski, and others. In 1938-1939 a number of leading mathematicians from the Moscow University, among them Aleksandrov, joined the Steklov Mathematical Institute of the Russian Academy of Sciences but at the same time they kept their positions at the University. Aleksandrov wrote about 300 scientific works in his long career. As early as 1924 he introduced the concept of a locally finite covering which he used as a basis for his criteria for the metrisability of topological spaces. He laid the foundations of homology theory in a series of fundamental papers between 1925 and 1929. His methods allowed arguments of combinatorial and algebraic topology to be applied to point set topology and brought together these areas. Aleksandrov's work on homology moved forward with his homological theory of dimension around 1928-30 Aleksandrov was the first to use the phrase 'kernel of a homomorphism' and around 1940-41 he discovered the ingredients of an exact sequence. He worked on the theory of continuous mappings of topological spaces. In 1954 he organised a seminar on this last topic aimed at first year students at Moscow University and in this he showed one of the aspects of his career which was of major importance to him, namely the education of students. This is described in ([3] and [4]):To the training of these students and those who came after them, Aleksandrov literally devoted all his strength. His influence on the class of young men studying topology under him was never purely mathematical, however real and significant that was. There were physical days exercise on topological walks, in long outings lasting several days by boat, ... in swimming across the Volga or other broad stretches of water, in skiing excursions lasting for hours on the slopes outside Moscow, slopes to which Aleksandrov gave striking, fantastic names... Many honours were given to Aleksandrov for his outstanding contribution to mathematics. He was president of the Moscow Mathematical Society from 1932 to 64, vice president of the International Congress of Mathematicians from 1958 to 62, a corresponding member of the Soviet Academy of Sciences from 1929 and a full member from 1953. Many other societies elected Aleksandrov to membership including the Göttingen Academy of Sciences, the Austrian Academy of Sciences, the Leopoldina Academy in Halle, the Polish Academy of Sciences, the National Academy of Sciences of the United States, the London Mathematical Society, the American Philosophical Society, and the Dutch Mathematical Society. He edited several mathematical journals, in particular the famous Soviet Journal Uspekhi Matematicheskikh Nauk, and he received many Soviet awards, including the Stalin Prize in 1943 and five Orders of Lenin.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aleksandrov.html (4 of 5) [2/16/2002 10:57:30 PM]

Aleksandrov

Today the Department of General Topology and Geometry of Moscow State University is Russia's leading centre of research in set-theoretic topology. After Aleksandrov's death in November 1982, his colleagues from the Department of Higher Geometry and Topology, in which he had held the chair, sent a letter to Moscow University's rector A A Logunov proposing that one of Aleksandrov's former students should become Head of the Department, to preserve Aleksandrov's scientific school. On 28 December 1982 the rector issued a circular creating the Department of general topology and Geometry. Vitaly Vitalievich Fedorchuk was elected Head of the Department. Also in memory of Aleksandrov's contributions to topology at Moscow University and his work with the Moscow Mathematical Society, there is an annual topological symposium Aleksandrov Proceedings held every May. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1940 to 1950

Other Web sites

1. Akron (More pictures and links) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Aleksandrov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aleksandrov.html (5 of 5) [2/16/2002 10:57:30 PM]

Aleksandrov_Aleksandr

Aleksandr Danilovic Aleksandrov Born: 4 Aug 1912 in Volyn, Ryazan, Russia Died: 27 July 1999

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aleksandr Danilovic Aleksandrov's father was the headmaster of a secondary school in St Petersburg and his mother was a teacher at the same school. In fact, although he was born in the village of Volyn, he lived in St Petersburg from a very young age. Of course this is not strictly true, since when Aleksandrov was two years old St Petersburg changed its name to Petrograd. Aleksandrov attended school in Petrograd but while he was at school the name of the city in which he lived changed yet again and so by the time he left school in 1928 it was a Leningrad school from which he graduated. When Aleksandrov left school he did not intend to study mathematics but rather his interests were in physics. Therefore when he entered Leningrad University in 1929 he set out on a theoretical physics course in the Faculty of Physics. In 1930, while still only 18 years of age, he began original work on optics in the Optics Institute. However Aleksandrov was taught mathematics in the Faculty of Physics by B N Delone. Delone's interests in the geometry of numbers and the structure of crystals soon began to attract Aleksandrov at least as much as his work in physics which was supervised by V A Fok. In 1932 Aleksandrov moved from the Optics Institute to Physics Research Institute of Leningrad University where he worked on the theoretical side of the subject. He graduated with a degree in theoretical physics in 1933 and continued his research, working with two supervisors in Fok and Delone. The influence of these two are clearly seen in Aleksandrov's first few publications which appeared in 1933 and 1934 and represented research largely carried out while he was still an undergraduate. In 1933 he published A theorem on convex polyhedra and An elementary proof of the existence of a centre of symmetry in a three-dimensional convex polyhedron. Then, in 1934, he published a book

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aleksandrov_Aleksandr.html (1 of 3) [2/16/2002 10:57:32 PM]

Aleksandrov_Aleksandr

Mathematical foundations of the structural analysis of crystals jointly written with Delone and N N Padurov. These first three works were all as a result of his mathematical work with Delone but also in 1934 he published two physics papers on quantum mechanics On the calculation of the energy of a bivalent atom by Fok's method and Remark on the commutation rule in Schrödinger's equation. His fourth work in 1934 was on geometry in the area of his first three papers. While continuing to work at the Physics Research Institute, Aleksandrov began to teach in the Faculty of Mathematics and Mechanics from 1933. Two years later he presented a thesis on geometry for a Master's degree which was on the topic of mixed volumes of convex bodies and he published six papers over the next couple of years on the results of this thesis. This work generalises classical problems in differential geometry. Aleksandrov's doctoral thesis (more of a habilitation thesis) was presented in 1937 and in it he studied the topics of additive set functions and the geometrical theory of weak convergence. Aleksandrov was appointed as Professor of Geometry at Leningrad University in 1937. He was also appointed to the Steklov Mathematical Institute of the Russian Academy of Sciences which, as a part of the Steklov Institute of Physics and Mathematics had been set up in the early 1920s. In 1934 the Steklov Mathematical Institute had been set up but moved to Moscow and Delone, Aleksandrov's supervisor, had moved with it to head the Algebra Department. In 1940 the Leningrad Branch of the Mathematical Institute was founded and it had among its members Aleksandrov, Kantorovich, Linnik, and Faddeev. However the Moscow part of the Steklov Mathematical Institute was moved to Kazan at the beginning of World War II and, in 1942 Aleksandrov went to Kazan to continue his research within the Mathematical Institute. In 1944 Aleksandrov returned to the University of Leningrad where he was Professor of Geometry. In 1952 he became Rector of the University of Leningrad. It was a period in which he worked hard to recreate the mathematical activity in Leningrad which had been associated with the St Petersburg Mathematical Society. The St Petersburg Mathematical Society was founded in 1890 and was the third oldest mathematical society in Russia (Moscow founded 1867 and the Kharkov founded 1879 are older). It had ceased to exist in 1917 due to the Revolution, but was recreated after initiatives from Steklov as the Petrograd Physical and Mathematical Society in 1921. Both of Aleksandrov's supervisors, Fok and Delone, played major roles in the Physical and Mathematical Society. However the Society was again closed down due to political pressure. Then Smirnov organised the Leningrad Mathematical Seminar in 1953 which went some way to filling the gap left but both Aleksandrov and Smirnov worked hard to restart the Leningrad Mathematical Society. They succeeded in 1959 when the Leningrad Mathematical Society again began to hold meetings. In 1964 Aleksandrov left Leningrad and moved to Novosibirsk where he was appointed as Head of the Department of Geometry of the University of Novosibirsk. He also became Head of the Department of Geometry of the Mathematical Institute of the Siberian Branch of the Academy of Sciences of the USSR. In [9] and [10] Aleksandrov's work in geometry is put into perspective:[Aleksandrov] approached the differential geometry of surfaces [by extending the notion of the objects studied], extending the class of regular convex surfaces to the class of all convex surfaces ... . In order to solve concrete problems Aleksandrov had to replace the Gaussian http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aleksandrov_Aleksandr.html (2 of 3) [2/16/2002 10:57:32 PM]

Aleksandrov_Aleksandr

geometry of regular surfaces by a much more general theory. In the first place the intrinsic properties (i.e. those properties that appear as a result of measurements carried out on the surface) of an arbitrary convex surface had to be studied, and methods found for the proof of theorems on the connection between intrinsic and exterior properties of convex surfaces. Aleksandrov constructed a theory of intrinsic geometry of convex surfaces on that basis. Because of the depth of this theory, the importance of its applications and the breadth of its generality, Aleksandrov comes second only to Gauss in the history of the development of the theory of surfaces. Aleksandrov's work in physics did not stop in his student days. He published on optics, quantum mechanics, and relativity. He often lectured on the history of mathematical ideas, a topic which greatly fascinated him. In addition he wrote encyclopaedia articles and wrote chapters on methodology. Finally let us note some of Aleksandrov's interests outside mathematics. He loved mountaineering and in fact it is noted in ([9] and [10]) that he spent his fiftieth birthday on a mountaineering expedition to Pamir. His other interests, noted in ([4] and [5]), include questions of education, morals, and other questions of community interest. Aleksandrov received many awards for his major contributions to geometry. In 1942 he received the State Prize for his work in geometry, then in 1946 he was elected a Corresponding Member of the Academy of Sciences. In 1951 he received the international Lobachevsky Prize. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Aleksandrov_Aleksandr.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Aleksandrov_Aleksandr.html (3 of 3) [2/16/2002 10:57:32 PM]

Alexander

James Waddell Alexander Born: 19 Sept 1888 in Sea Bright, New Jersey, USA Died: 23 Sept 1971 in Princeton, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James Alexander's father was John White Alexander and his mother was Elizabeth Alexander. The name Waddell came through his grandfather on his mother's side, John Waddell Alexander who was the President of the Equitable Life Assurance Society. John White Alexander was quite famous in his own right as an artist and painter of murals. Alexander studied mathematics and physics at Princeton, where he was a student of Veblen, obtaining a B.S. degree in 1910 and an M.S. degree in 1911. From 1911 to 1912 he served as an instructor in the mathematics department at Princeton. Then in 1912 he went to Europe to further his studies. During this period abroad, Alexander studied at Paris and Bologna. He returned to Princeton where he submitted his dissertation Functions which map the interior of the unit circle upon simple regions and, in 1915, was awarded his Ph.D. He was appointed as a instructor at Princeton in 1915, being made a lecturer in 1916. Of course this was the time when the United States required the expertise of mathematicians to solve problems which related to the military needs created by World War I. In 1917, after marrying Natalia Levitzkaja on 15 January, later that year Alexander served as a lieutenant in the U.S. Army Ordnance Office at the Aberdeen Proving Ground. This was a military weapons testing site, established in 1917 in Aberdeen, Maryland. By the end of the war Alexander had reached the rank of captain. Leaving military service, he returned to Princeton where he was an assistant professor from 1920, being promoted to associate professor in 1926 and full professor in 1928. From 1933 until he retired in 1951 he was a member of the Institute for Advanced Studies in Princeton. Alexander, however, never drew a salary from the Institute for Advanced Studies. He had become a millionaire through inherited wealth and, a rich man, was in no need of a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Alexander.html (1 of 3) [2/16/2002 10:57:34 PM]

Alexander

salary. During World War II, Alexander again did war work, but on this occasion he did not rejoin the army. He worked during World War II as a civilian for the U.S. Army Air Force at their Office of Scientific Research and Development. During the early 1950s senator Joseph R McCarthy whipped up strong feelings against communism in the United States. Alexander, who held left-wing political views, began to come under suspicion from authorities who saw imaginary problems everywhere. Alexander had virtually become a recluse after he retired in 1951 and the McCarthy era resulted in his disappearance from public life. The last time he was seen in public was July 1954 when he signed a statement of support for J Robert Oppenheimer who had lost his security clearance. In a collaboration with Veblen, he showed that the topology of manifolds could be extended to polyhedra. Before 1920 he had shown that the homology of a simplicial complex is a topological invariant. Alexander's work around this time went a long way to put the intuitive ideas of Poincaré on a more rigorous foundation. Also before 1920 Alexander had made fundamental contributions to the theory of algebraic surfaces and to the study of Cremona transformations. Soon after arriving in Princeton, Alexander generalised the Jordan curve theorem and continued his work, now exclusively on topology, with an important paper on the Jordan-Brouwer separation theorem. This latter paper contains the Alexander Duality Theorem and Alexander's lemma on the n-sphere. In 1924 he introduced the now famous Alexander horned sphere. In 1928 he discovered the Alexander polynomial which is much used in knot theory. In the same year the American Mathematical Society awarded Alexander the Bôcher Prize for his memoir, Combinatorial analysis situs published in the Transactions of the American Mathematical Society two years earlier. Knot theory and the combinatorial theory of complexes were the main topics on which he worked over the following few years. The theory which is now called the Alexander-Spanier cohomology theory, was introduced in 1935 by Alexander but was generalised by Spanier in 1948 to the form seen today. Also around 1935 Alexander discovered cohomology theory, at essentially the same time as Kolmogorov, and the theory was announced in the 1936 Moscow Conference. Zund, in [4] writes:A mathematician of unusual depth and power, Alexander was a principal figure in the American development of algebraic/combinatorial topology. ... His papers were very carefully written and were very influential in the United States and abroad. Much of his work was of such a basic character that it became common knowledge in topology, with its discoverer being forgotten as a result... Alexander's character is also described in [4], where he is said to have been:... an imposing figure who possessed great charm and a very "youthful" view of mathematics, being one of the first American mathematicians to fully appreciate the use of modern algebraic methods in topology. Colleagues remember his great fondness for limericks and his passion for mountain climbing. Among the many honours bestowed on Alexander was his election to the American Academy of Sciences in 1930. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Alexander.html (2 of 3) [2/16/2002 10:57:34 PM]

Alexander

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1920 to 1930

Honours awarded to James Alexander (Click a link below for the full list of mathematicians honoured in this way) AMS Bôcher Prize

Awarded 1928

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Alexander.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Alexander.html (3 of 3) [2/16/2002 10:57:34 PM]

Alexander_Archie

Archie Alphonso Alexander Born: 14 May 1888 in Ottumwa, Iowa, USA Died: 4 Jan 1958 in Des Moines, Iowa, USA Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Archie Alexander's father was Price Alexander who was a janitor in Ottumwa while his mother was Mary Hamilton. Archie was born into a African-American family that lived in an area of Ottumwa which was set aside for poor people, and of course this meant that he would not be expected to have an education. It took the remarkable person that Archie was, together with strong support from his parents, to overcome these disadvantages, both from prejudice against someone of his race and against someone from a poor working class background, and go on to a life filled with achievements. When Archie was eleven years old his family moved from Ottumwa to live on a small farm on the outskirts of Des Moines. He had the advantage of being able to attend a school in Des Moines which was open to both white and black students and in 1905, at the age of seventeen, he graduated from Oak Park High School in Des Moines. Alexander's parents were far too poor to be able to afford to send him to College to study further, yet Alexander was determined to continue his education. To allow him to study at Highland Park College in Des Moines, Alexander took on some poorly paid part-time jobs and his parents also helped out as much as they could. In addition he attended Cummins Art College in Des Moines before entering the University of Iowa in 1908 to study engineering. It is important to realise what an achievement it was for Alexander to enter the College of Engineering in Iowa City which was attached to the University of Iowa. He was the only student of his race in the College and, when he achieved sporting success by becoming a member of the university football team, he was the only member of the team who was not white. We should explain that, in the title of [3], "Alexander the Great" refers to Alexander's nickname as a football player. How he found time for sport in addition to his studies, while he still had to spend long hours undertaking poorly paid part-time work to support himself, says much for his determination to succeed in everything he undertook. One might have expected someone from Alexander's background to have been encouraged to undertake university studies, but this had certainly not been the case. His advisor at university had bluntly told him that [2]:... a Negro could not hope to succeed as an engineer... and after he graduated in 1912 and began to seek a position as an engineer he discovered that his advisor had simply been telling him the truth. Prejudice against someone of his race prevented Alexander from being appointed to any of the engineering posts to which he made application. However, one haapy event in 1913 was that he married Audra A Linzy. The couple had one child who sadly died a few years later.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Alexander_Archie.html (1 of 3) [2/16/2002 10:57:35 PM]

Alexander_Archie

Alexander was not one to give up when he encountered such difficulties. He decided that if he could not find an engineering post then he would join a firm as a labourer and work his way up. He took a manual labouring job with the Marsh Engineering Company and in two year he had shown that he could overcome the prejudice by rising through the Company until he was in charge of the Marsh Engineering Company's bridge building programme in Iowa and Minnesota. One might have thought that Alexander, having achieved remarkable success for someone of his background, might have been content with his position. However, he was a man with ambitions and after two years with the Marsh Engineering Company, he left to form his own engineering company. Now, of course, he had to fight against prejudice much harder than before. Few would willingly give a major engineering contract to a firm run by an African-American if there were other firms able to do the work. As a consequence, Alexander's company ended up with the jobs for which no other firm competed. While at the Marsh Engineering Company, Alexander had become friendly with another engineer George F Higbee. Alexander took Higbee on a partner in 1917 and the partnership was only ended in 1925 when Higbee was tragically killed in a construction accident. For four years Alexander continued to run the company on his own, gaining a reputation as a talented construction engineer building fine bridges, viaducts and tunnels. Then in 1929 he took on a new partner, Maurice A Repass, who had been a fellow student with Alexander at the Iowa College of Engineering. Their firm was now named Alexander and Repass and were so successful that they were called in the press:... the nation's most successful interracial business. Alexander did not confine his talents to developing his engineering firm. He also took part in the political life of Iowa, serving as the assistant chairman of the Iowa Republican State Committee in 1932 and again in 1940. His outstanding contribution to the Republican Party was rewarded in 1954 when he was appointed as governor of the Virgin Islands. Three of the Virgin Islands were purchased by the United States from Denmark in 1917 and they became an unincorporated territory of the United States. In 1927 the islanders became US citizens and various acts established the government of the Islands. The Revised Organic Act of 1954 created a central government and Alexander was appointed as Governor in April of that year, see [1]. It was to turn out to be a disastrous appointment. As Wynes writes in [2]:Dogmatic, paternalistic, undemocratic, and with an openly stated contempt for the easygoing Virgin Islanders ... Alexander's period as governor lasted only sixteen months before he was forced to resign. It was a sad episode which almost certainly hastened Alexander's death and must have left him with much sadness after a life filled with so many achievements against all the odds that were stacked against him. Wynes sums up his life saying:Engineer, businessman, loyal and active member of the Republican Party, civil rights and interracial leader, Alexander was, ironically, a failure only in the world of state diplomacy. The one failure is ironic in that surely he had to be a diplomat in the largely white world in which he lived and worked. In 1975, on the death of Alexander's wife, the University of Iowa, Tuskegee Institute in Alabama, and Howard University each received a substantial sum for engineering scholarships from a trust fund set up by Alexander in his will. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Alexander_Archie.html (2 of 3) [2/16/2002 10:57:35 PM]

Alexander_Archie

List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Alexander_Archie.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Alexander_Archie.html (3 of 3) [2/16/2002 10:57:35 PM]

Ampere

André Marie Ampère Born: 20 Jan 1775 in Lyon, France Died: 10 June 1836 in Marseilles, France

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

André-Marie Ampère's father, Jean-Jacques Ampère, was a prosperous man who owned a home in Lyon and a country house in Poleymieux, which is only 10 km from Lyon. Up till André-Marie was seven years old the family spent most of the year in Lyon except the summer months which were spent at Poleymieux. However, in 1782, the home at Poleymieux became their main residence since André-Marie's father wished to spend more time on his son's education. Only a short time in winter was spent at Lyon where André-Marie's father saw to his business interests. Despite not attending school, André-Marie was to be given an excellent education. He describes this education in autobiographical writings (rather strangely referring to himself in the third person):His father, who had never ceased to cultivate Latin and French literature, as well as several branches of science, raised him himself in the country near the city where he was born. He never required him to study anything, but he knew how to inspire in him a desire to know. Before being able to read, the young Ampère's greatest pleasure was to listen to passages from Buffon's natural history. Ampère read articles from L'Encyclopédie many of which, Arago remarked many years later, he could recite in full in later life. Arago also claims that Ampère read the Encyclopédie starting at volume 1 and reading the articles in alphabetical order. Whether Ampère's later desire for classification in all subjects arose from this education, or whether he enjoyed Buffon and the Encyclopédie because of a natural liking for classifying, is hard to say. It has been claimed that Ampère had mastered all known mathematics by the age of twelve years but this

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ampere.html (1 of 6) [2/16/2002 10:57:37 PM]

Ampere

seems somewhat of an exaggeration since, by Ampère's own account, he did not start to read elementary mathematics books until he was 13 years old. However Ampère was always one to feel very confident in his own abilities and he certainly began to develop his own mathematical ideas very quickly and he began to write a treatise on conic sections. Ampère had no contacts with anyone with any depth of mathematical knowledge so it is not surprising that he felt that his ideas were original. While still only 13 years old Ampère submitted his first paper to the Académie de Lyon. This work attempted to solve the problem of constructing a line of the same length as an arc of a circle. His method involves the use of infinitesimals but since Ampère had not studied the calculus the paper was not found worthy of publication. Shortly after writing the article Ampère began to read d'Alembert's article on the differential calculus in the Encyclopédie and realised that he must learn more mathematics. After taking a few lessons in the differential and integral calculus from a monk in Lyon, Ampère began to study works by Euler and Bernoulli. He then acquired a copy of the 1788 edition of Lagrange's Mécanique analytique and began serious study of the work. Ampère writes (again writing about himself in the third person):... the reading of [Mécanique analytique] had animated him with a new ardour. He repeated all the calculations in it ... However his life was soon to be shattered. The French Revolution began with the storming of the Bastille on 14 July 1789 but the effect on the Poleymieux region was not very great at first. Ampère's father kept out of trouble until late in 1791 when he accepted the position of Justice of the Peace in Lyon. This post made it virtually impossible for him to avoid trouble but the first tragedy to hit the family was in 1792 when André-Marie's sister died. The city of Lyon refused to carry out instructions from Paris and the city was besieged for two months. On the fall of the city Ampère's father was arrested for issuing an arrest warrant for the Jacobin Chevalier who had then been put to death. Ampère's father went to the guillotine with remarkable composure writing to Ampère's mother from his cell:I desire my death to be the seal of a general reconciliation between all our brothers; I pardon those who rejoice in it, those who provoked it, and those who ordered it.... The effect on Ampère of his father's death was devastating. He gave up his studies of Mécanique analytique and did not return to the study of mathematics for 18 months. He only returned to something like his old self when he met a girl, Julie, who he fell deeply in love with. Julie seemed less attracted to Ampère:He has no manners; he is awkward, shy and presents himself poorly. Despite this coolness they were engaged to be married in 1797 and Ampère decided he better show that he could earn a living so began tutoring mathematics in Lyon. He married Julie in 1799 and their son Jean-Jacques was born in 1800. Ampère continued tutoring mathematics until 1802 when he was appointed professor of physics and chemistry at Bourg Ecole Centrale. This was a difficult time for Ampère since Julie became ill before he made the move to Bourg leaving her at Poleymieux. While Ampère was in Bourg he spent much time teaching physics and chemistry but his research was in mathematics. This research resulted in him composing a treatise on probability, The Mathematical Theory of Games, which he submitted to the Paris Academy in 1803. Laplace noticed an error, explaining the error to Ampère in a letter, which Ampère was able to correct and the treatise was reprinted. In fact the treatise was modified a number of times and Ampère was reluctant to call it

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ampere.html (2 of 6) [2/16/2002 10:57:37 PM]

Ampere

completed for fear that further changes might be required. This work was followed by one on the calculus of variations in 1803. After a year in Bourg, Ampère moved closer to Poleymieux being appointed to a mathematics position at the Lycée in Lyon on Delambre's recommendation. His time spent in Lyon had been made difficult due to the continuing decline in his wife's health. Mathematically he continued to produce good work, this time an interesting treatise on analytic geometry. Like a number of other mathematicians, Ampère seemed able to concentrate on his theorems despite the personal tragedy around him and, sadly, this would be required of him throughout his unhappy life. After his wife died in July 1803, Ampère was left with feelings of guilt for he had lived apart from his wife during much of their short marriage. He decided to leave Lyon for Paris. Hofman writes in [4] regarding his feelings following his wife's death:His subsequent depression contributed to his decision to take the earliest opportunity to leave Lyon for new surroundings in Paris. Later he would regret this decision. The Lyon friends who attempted to fill the emotional void left by Julie's death were missed painfully. Although Ampère gradually adjusted to the priority disputes and infighting of the Parisian scientific community, he always longed for a return to the intellectual life he experienced in Lyon. By this time Ampère had a fair reputation as both a teacher of mathematics and as a research mathematician and on the strength of this reputation he was appointed répétiteur (basically a tutor) in analysis at the Ecole Polytechnique in 1804. Without a formal education and formal qualifications his appointment is surprising but shows that his potential was recognised at this stage. His life, already containing many tragedies, did not improve and he embarked on a disastrous marriage. Lagrange and Delambre attended his wedding to Jenny on 1 August 1806 but, before the birth of their daughter on 6 July 1807, the couple were living apart and were not on speaking terms. They were legally separated in 1808 and Ampère was given custody of their daughter Albine. Appointed professor of mathematics at the Ecole Polytechnique in 1809 he held posts there until 1828. Ampère and Cauchy shared the teaching of analysis and mechanics and there was a great contrast between the two with Cauchy's rigorous analysis teaching leading to great mathematical progress but found extremely difficult by students who greatly preferred Ampère's more conventional approach to analysis and mechanics. Ampère was appointed to a chair at Université de France in 1826 which he held until his death. In Paris Ampère worked on a wide variety of topics. Although a mathematics professor, his interests included, in addition to mathematics, metaphysics, physics and chemistry. In mathematics he worked on partial differential equations, producing a classification which he presented to the Institut in 1814. This seems to have been a crucial step in his election to the Institut National des Sciences in November 1814 when he defeated Cauchy, receiving 28 of the 56 votes cast. Ampère was also making significant contributions to chemistry. In 1811 he suggested that an anhydrous acid prepared two years earlier was a compound of hydrogen with an unknown element, analogous to chlorine, for which he suggested the name fluorine. After concentrating on mathematics as he sought admission to the Institut, Ampère returned to chemistry after his election in 1814 and produced a classification of elements in 1816. Ampère also worked on the theory of light, publishing on refraction of light in 1815. By 1816 he was a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ampere.html (3 of 6) [2/16/2002 10:57:37 PM]

Ampere

strong advocate of a wave theory of light, agreeing with Fresnel and opposed to Biot and Laplace who advocated a corpuscular theory. Fresnel became a good friend of Ampère's and lodged at Ampère's home from 1822 until his death in 1827. In the early 1820's, Ampère attempted to give a combined theory of electricity and magnetism after hearing about experimental results by the Danish physicist Hans Christian Orsted. Ampère formulated a circuit force law and treated magnetism by postulating small closed circuits inside the magnetised substance. It is worth commenting on how quickly Ampère produced this theory, the inspiration striking him immediately he heard of Orsted's experimental results. Orsted's work was reported the Academy in Paris on 4 September 1820 by Arago and a week later Arago repeated Orsted's experiment at an Academy meeting. Ampère demonstrated various magnetic / electrical effects to the Academy over the next weeks and he had discovered electrodynamical forces between linear wires before the end of September. He spoke on his law of addition of electrodynamical forces at the Academy on 6 November 1820 and on the symmetry principle in the following month. Ampère wrote up the work he had described to the Academy with remarkable speed and it was published in the Annales de Chimie et de Physique. Ampère was assisted over the next few years in his work by Felix Savary whose help in getting Ampère to write up his results was invaluable [4]:... beginning with the memoir he completed early in 1823, Savary now made much more creative contributions. But more than his creativity, it was Savary's discipline and ability to concentrate at length on specific problems that proved especially valuable to Ampère. There is room to speculate that, without Savary's aid. Ampère might never have found time to complete the detailed calculations required to apply his force law to magnetic phenomena. However Ampère was not the only one to react quickly to Arago's report of Orsted's experiment. Biot, with his assistant Savart, also quickly conducted experiments and reported to the Academy in October 1820. This led to the Biot-Savart Law. Another who worked on magnetism at this time was Poisson who insisted on treating magnetism without any reference to electricity. Poisson had already written two important memoirs on electricity and he published two on magnetism in 1826. Ampère's most important publication on electricity and magnetism was also published in 1826. It is called Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience and contained a mathematical derivation of the electrodynamic force law and describes four experiments. Maxwell, writing about this Memoir in 1879, says:We can scarcely believe that Ampère really discovered the law of action by means of the experiments which he describes. We are led to suspect, what, indeed, he tells us himself, that he discovered the law by some process which he has not shown us, and that when he had afterwards built up a perfect demonstration he removed all traces of the scaffolding by which he had raised it. Ampère's theory became fundamental for 19th century developments in electricity and magnetism. Faraday discovered electromagnetic induction in 1831 and, after initially believing that he had himself discovered the effect in 1822, Ampère agreed that full credit for the discovery should go to Faraday. Weber also developed Ampère's ideas as did Thomson and Maxwell.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ampere.html (4 of 6) [2/16/2002 10:57:37 PM]

Ampere

In 1826 Ampère began to teach at the Collège de France. Here he was in a position to teach courses of his own design, rather than at the Ecole Polytechnique were the topics were set down. Ampère therefore taught electrodynamics at the Collège de France and this course was taken by Liouville in 1826-27. This was the second time Ampère had taught Liouville since Liouville had taken Ampère's courses at the Ecole Polytechnique in the previous session. Liouville made an important contribution to Ampère's electrodynamics course by editing a set of notes taken from Ampère's lectures. Given the tragedy in Ampère's life it might have been hoped that his children would bring him some happiness. His son certainly achieved fame as a historian and philologist who studied the cultural origins of western European languages. He was appointed to a chair of history of foreign literature at the Sorbonne in 1830. However his relationship with his father was difficult. Hofmann in [4] writes:Both men were temperamental and subject to long periods of brooding followed by explosive outbursts of anger. Ampère's home simply was not expansive to house both of them for any extended period of time. Ampère had an even more difficult time with his daughter. She married one of Napoleon's lieutenants in 1827 but he was an alcoholic and the marriage soon was in trouble. Ampère's daughter fled to her father's house in 1830 and, some days later, Ampère allowed her husband to live with him also. This proved a difficult situation, led to police intervention and much unhappiness for Ampère. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles)

A Quotation

A Poster of André-Marie Ampère

Mathematicians born in the same country

Cross-references to History Topics

Memory, mental arithmetic and mathematics

Other references in MacTutor

Chronology: 1820 to 1830

Honours awarded to André-Marie Ampère (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1827

Lunar features

Mons Ampere

Paris street names

Rue Ampère (17th Arrondissement)

Commemorated on the Eiffel Tower Other Web sites

1. The Catholic Encyclopedia 2. West Chester University 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ampere.html (5 of 6) [2/16/2002 10:57:37 PM]

Ampere

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ampere.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ampere.html (6 of 6) [2/16/2002 10:57:37 PM]

Amringe

John Howard Van Amringe Born: 3 April 1835 in Philadelphia, Pennsylvania , USA Died: 10 Sept 1915 in Morristown, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Howard Van Amringe was educated at home by his father until he entered Montgomery Academy. There he prepared to enter Yale University which he did in 1854. In 1856 he left and became a teacher of mathematics but returned to his studies in 1858 when he entered Columbia College in New York City. The College was to become Columbia University in 1912. Van Amringe received his A.B. in 1860 and his A.M. in 1863. Van Am, as he was known, spent his entire career at Columbia. He was professor of mathematics in the School of Mines from 1865 until 1873 when he was appointed professor of mathematics in the School of Arts. He was head of mathematics from 1892 until he retired in 1910. Amringe was a good teacher of mathematics, Thomas [4] writes:... probably no other teacher of his day was so loved and revered... but he was not a research mathematician of any quality. He did not publish any research papers on mathematics but he is important in his role in the founding of the New York Mathematical Society which quickly changed its name to the American Mathematical Society. He was the first president of the Society from 1888 to 1890. At the first meeting after his term as president had ended, on 5 December 1890, Van Amringe proposed that the Society should publish a Bulletin. Burgess [2] describes Van Amringe in colourful terms:He was ... the ideal college patriot, and consequently the idol of the students and alumni of the college, although he was quite a disciplinarian in the classroom. ... He was always having some accident, such as breaking an arm or a leg. ... he was a great smoker and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Amringe.html (1 of 2) [2/16/2002 10:57:39 PM]

Amringe

frequenter of clubs. He was also something of a politician .. he was a good, staunch, reliable friend and very agreeable in social intercourse. No one could know the man and not love him. As Archibald writes about the American Mathematical Society in [1]:... it was only natural that one of his prominence, occupying the position that he did, should have become our Societies first president. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Howard Van Amringe (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1889 - 1890

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Amringe.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Amringe.html (2 of 2) [2/16/2002 10:57:39 PM]

Amsler

Jacob Amsler Born: 16 Nov 1823 in Stalden bei Brugg, Switzerland Died: 3 Jan 1912 in Schaffhausen, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jacob Amsler's father was a farmer. He grew up in Stalden, a town in the Vispa valley six kilometres south of the main Rhône valley. His school education was in the local schools from which he graduated with the intention of studying theology. Amsler studied theology first at the University of Jena and then at the University of Königsberg. It was at Königsberg that Amsler changed his course from theology to mathematics and physics. The reason, as is so often in such cases, was inspiring teaching. The teacher who changed Amsler's future was Franz Neumann whose lectures and laboratory sessions he attended at the University of Königsberg. In 1848 Amsler was awarded his doctorate from the University of Königsberg and he then returned to Switzerland to continue his education. Back in Switzerland, Amsler worked for a year at the observatory in Geneva. He completed his studies at Zurich where he was awarded his habilitation, thus gaining the right to teach in universities. In 1850-51 he taught at the university in Zurich, teaching courses on mathematics and on mathematical physics topics. In 1851, wishing to have more time to devote to his research, Amsler took a position at the Gymnasium in Schaffhausen, a town in northern Switzerland situated on the bank of the Rhine. Indeed the Gymnasium did allow him to find more time to undertake research and as a consequence he published a number of article on mathematical physics over the next few years, in particular writing papers on magnetism, heat conduction and on the attraction of ellipsoids. It is worth mentioning this last result in more detail for he worked on a problem which had quite a famous history. That was the problem of the attraction of an ellipsoid, which was first studied in depth by

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Amsler.html (1 of 3) [2/16/2002 10:57:41 PM]

Amsler

Ivory whose solution was later generalised by Poisson. Amsler extended the theorems of both Ivory and Poisson on this topic. It was a promising start to his research career in mathematical physics. In 1854 Amsler married and this may have been the turning point in his career. His wife, Elsie Laffon, was the daughter of a well known Swiss scientist. Amsler seems to have been pleased to be connected with this family for he indicated that he wished to be known as Amsler-Laffon from this time on. Elsie and Jacob Amsler-Laffon's children were, however, always known by the name Amsler rather than Amsler-Laffon. This suggests that Amsler was trying to gain some advantage from the connections acquired by marriage. Shortly after his marriage Amsler changed his research interests and his career. He began to study the construction of precision mathematical instruments and quite quickly he had an idea for the design of a new type of planimeter. He invented the polar planimeter, a device for measuring areas enclosed by plane curves. It was based on polar coordinates whereas earlier instruments were based on cartesian coordinates. In 1856 Amsler published a paper Uber das Planimeter in which he gave details of his idea. As Mahoney writes in [1], Amsler's planimeter:... adapted easily to the determination of static and inertial moments and to the coefficients of Fourier series: it proved especially useful to shipbuilders and railway engineers. In order to make money from his invention, Amsler set up a workshop in Schaffhausen in 1854 specially designed to produce his polar planimeter. Three years later he had given up al his other interests to concentrate fully on producing instruments in the workshop. His shop produced 50 000 such instruments during his lifetime. Amsler did not rest his fame on this single inspired idea but continued to invent new precision instruments. None of his other inventions came close to the polar planimeter in importance, but they were of sufficient quality to win him prizes at the world exhibition at Vienna in 1873, at Paris in 1881, and again in Paris in 1889. His brilliance was recognised with election to the Paris Académie des Sciences in 1892. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Amsler.html (2 of 3) [2/16/2002 10:57:41 PM]

Amsler

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Amsler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Amsler.html (3 of 3) [2/16/2002 10:57:41 PM]

Anaxagoras

Anaxagoras of Clazomenae Born: 499 BC in Clazomenae (30 km west of Izmir), Lydia (now Turkey) Died: 428 BC in Lampsacus, Mysia (now Turkey)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Anaxagoras of Clazomenae was described by Proclus, the last major Greek philosopher, who lived around 450 AD as (see for example [4]):After [Pythagoras] Anaxagoras of Clazomenae dealt with many questions in geometry... Anaxagoras was an Ionian, born in the neighbourhood of Smyrna in what today is Turkey. We know few details of his early life, but certainly he lived the first part of his life in Ionia where he learnt about the new studies that were taking place there in philosophy and the new found enthusiasm for a scientific study of the world. He came from a rich family but he gave up his wealth. As Heath writes in [4]:He neglected his possessions, which were considerable, in order to devote himself to science. Although Ionia had produced philosophers such as Pythagoras, up to the time of Anaxagoras this new study of knowledge had not spread to Athens. Anaxagoras is famed as the first to introduce philosophy to the Athenians when he moved there in about 480 BC. During Anaxagoras's stay in Athens, Pericles rose to power. Pericles, who was about five years younger than Anaxagoras, was a military and political leader who was successful in both developing democracy and building an empire which made Athens the political and cultural centre of Greece. Anaxagoras and Pericles became friends but this friendship had its drawbacks since Pericles' political opponents also set themselves against Anaxagoras. In about 450 BC Anaxagoras was imprisoned for claiming that the Sun was not a god and that the Moon reflected the Sun's light. This seems to have been instigated by opponents of Pericles. Russell in [6] http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anaxagoras.html (1 of 4) [2/16/2002 10:57:42 PM]

Anaxagoras

writes:The citizens of Athens ... passed a law permitting impeachment of those who did not practice religion and taught theories about 'the things on high'. Under this law they persecuted Anaxagoras, who was accused of teaching that the sun was a red-hot stone and the moon was earth. We should examine this teaching of Anaxagoras about the sun more closely for, although it was used as a reason to put him in prison, it is a most remarkable teaching. It was based on his doctrine of "nous" which is translated as "mind" or "reason". Initially "all things were together" and matter was some homogeneous mixture. The nous set up a vortex in this mixture. The rotation [4]:... began in the centre and then gradually spread, taking in wider and wider circles. The first effect was to separate two great masses, one consisting of the rare, hot, dry, called the "aether", the other of the opposite categories and called "air". The aether took the outer, the air the inner place. From the air were next separated clouds, water, earth and stones. The dense, the moist, the dark and cold, and all the heaviest things, collected in the centre as a result of the circular motion, and it was from these elements when consolidated that the earth was formed; but after this, in consequence of the violence of the whirling motion, the surrounding fiery aether tore stones away from the earth and kindled them into stars. There are remarkable insights in this description. The idea of differentiation of matter which plays a large role in modern theories of creation of the solar system is present. Anaxagoras also shows an understanding of centrifugal force which again shows the major scientific insights that he possessed. Anaxagoras proposed that the moon shines by reflected light from the "red-hot stone" which was the sun, the first such recorded claim. Showing great genius he was also then able to take the next step and become the first to explain correctly the reason for eclipses of the sun and moon. His explanation of eclipses of the sun is completely correct but he did spoil his explanation of eclipses of the moon by proposing that in addition to being caused by the shadow of the earth, there were other dark bodies between the earth and the moon which also caused eclipses of the moon. It is a little unclear why he felt it necessary to postulate the existence of these bodies but it does not detract from this major breakthrough in mathematical astronomy. There is also other evidence to suggest that Anaxagoras had applied geometry to the study of astronomy. As to the structure of matter, Anaxagoras postulated an infinite number of elements, or basic building blocks. He claimed:... there is a portion of every thing, i.e. of every elemental stuff, in every thing...[but] each is and was most manifestly those things of which there is most in it. However, it was the power of nous, or mind, that not only created the world but also was the driving force in its day to day processes. For example [2]:The growth of living things, according to Anaxagoras, depends on the power of mind within the organisms that enables them to extract nourishment from surrounding substances. Aristotle both found much to praise in Anaxagoras's theory of nous. Both Plato and Aristotle, however, were critical of the fact that the driving force of the nous as proposed by Anaxagoras was not ethical. They wanted nous to always act in the best interests of the world. In fact the nous of Anaxagoras does provide a mechanical explanation of the world after the non-mechanical start when the vortex is produced. It is worth noting that Newton's mechanical universe would have more in common with http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anaxagoras.html (2 of 4) [2/16/2002 10:57:42 PM]

Anaxagoras

Anaxagoras's views than the continuing ethical intelligence proposed by Plato and Aristotle. We can obtain some clues to the mathematics that Anaxagoras studied but, unfortunately, very little remains in the records to allow us to know of definite results which he may have proved. While in prison he tried to solve the problem of squaring the circle, that is constructing with ruler and compasses a square with area equal to that of a given circle. This is the first record of this problem being studied and this problem, and other similar problems, were to play a major role in the development of Greek mathematics. One other intriguing piece of information comes from the writing of Vitruvius, a Roman architect, engineer, and author who lived in the first century BC. He records information about the painting of stage scenes for the plays which were performed in Athens and says that Anaxagoras wrote a treatise on how to paint scenes so that some objects appeared to be in the foreground while other appeared in the background. This fascinating comment must mean that Anaxagoras wrote a treatise on perspective, but sadly no such work survives. Anaxagoras was saved from prison by Pericles but had to leave Athens. He returned to Ionia where he founded a school at Lampsacus. This Greek city on the Asiatic shore of the Hellespont was the place for the worship of Priapus, a god of procreation and fertility. Anaxagoras died there and the anniversary of his death became a holiday for schoolchildren. The best that we can hope to learn of Anaxagoras's personality is from the story that when once asked what as the point of being born he replied [4]:The investigation of sun. moon, and heaven. Even if this story is fictitious, it is likely to be based on the way that Anaxagoras lived his life and so tells us something of the personality of this remarkable scientist who gave a description of the creation of the solar system that took 2000 years to improve upon. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles)

A Quotation

Mathematicians born in the same country Cross-references to History Topics

Squaring the circle

Honours awarded to Anaxagoras (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Anaxagoras

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anaxagoras.html (3 of 4) [2/16/2002 10:57:42 PM]

Anaxagoras

Other Web sites

1. G Don Allen 2. Internet Encyclopedia of Philosophy 3. Drury College 4. S M Cohen 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Anaxagoras.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anaxagoras.html (4 of 4) [2/16/2002 10:57:42 PM]

Anderson

Oskar Johann Viktor Anderson Born: 2 Aug 1887 in Minsk, Belarus Died: 12 Feb 1960 in Munich, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oskar Anderson is sometimes said to be Russian, sometimes German. His name appears both as Oskar Nikolaevich Anderson and Oskar Johann Viktor Anderson. The reason for the confusion is that he was born in Kazan, which is in western Russia on the edge of Siberia, but he was born into a family which was ethnically German. His father Nikolas Anderson was professor of Finno-Ugric languages at the University of Kazan. Oskar attended Kazan Gymnasium and graduated with the gold medal in 1906, being one year ahead of Lobachevsky who graduated from the gymnasium the following year. Anderson spent one year studying mathematics at the University of Kazan before going to St Petersburg where he studied economics at the Polytechnic Institute. From 1907 to 1915 he was A A Chuprov's assistant and his dissertation was on variance-difference methods for analysing time series. It developed the ideas of his supervisor Chuprov on correlation and was published in Biometrika in 1914. See [3] for details of Chuprov and his work with Anderson. While studying at the Polytechnic Institute of St Petersburg, and acting as assistant to Chuprov, he took on further duties in 1912 when he began lecturing at a commercial school in St Petersburg. As well as teaching at this school, Anderson graduated with a law degree. Johnson and Kotz write in [5]:Among his other activities at this time, he organised and participated in an expedition in 1915 to Turkestan to carry out an agricultural survey in the area around the Syr Darya river. This survey was on a large scale, and possessed a representativity ahead of contemporary surveys in Europe and the USA. In 1917 Anderson left St Petersburg and moved to Kiev. There he held two posts, one in the Commercial http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anderson.html (1 of 3) [2/16/2002 10:57:44 PM]

Anderson

Institute where he first studied statistics and then taught, the other in the Demographic Institute of the Kiev Academy of Sciences. Slutsky was teaching in the Kiev Institute of Commerce when Anderson arrived and they would remain colleagues for three years. Events in 1920 were to have a profound influence on Anderson's life. The Russian civil war began in 1918 after the end of World War I. There were three sides, the Red Army led by Trotsky, the Whites Army who were anti-communists led by former imperial officers and the Greens who were anarchists opposed to the Reds and strongest in Ukraine. Lenin came to power but he did not favour moving quickly toward a socialist economy since the necessary economic skills were lacking. There were disputes over economic policy, some wishing to move quickly to a socialist economy. The policies which were put in place left the work force feeling cheated since they had expected to gain control of industry, and there were strikes and unrest in the country. Lenin approached Anderson and offered him a high position in the economic running of Russia. Anderson, like many Russians at this time, had political views which were leftist. However he was unhappy with the direction of the country and particularly unhappy about the way that some of his colleagues had been treated. He left Russia in 1920 with his family; they were effectively refugees. Wold writes in [10]:The course of outer events in Oskar Anderson's life reflects the turbulence and agonies of a Europe torn by wars and revolution. Indeed this turbulence was to have a dramatic effect on Anderson's life. He lost three children over the following period, first a daughter, then a little while later a son, and finally a second son died serving in World War II. But we have got ahead of the events that we were describing, so returning to the period after Anderson left Russia, he went to Budapest and in 1921 he became a school teacher there. After a few years he moved to Bulgaria and from 1924 until 1933 he was a professor at the Commercial Institute of Varna. In 1933 Anderson was awarded a Rockefeller Scholarship which enabled him to travel to England and Germany. He held the scholarship until 1935 and during this time he wrote his first book, which was on mathematical statistics, which was published in Vienna in 1935. In this book Anderson tried to present the latest statistical methods assuming only that his readers had covered school level mathematics. Returning to Bulgaria in 1935 Anderson was appointed professor at the University of Sofia. He had been involved in several important pieces of work for the Bulgarian government such as the use of sampling techniques in the 1926 census of population and manufacture, Bulgarian agricultural production in 1931-32, and crop statistics in 1936. He was sent to Germany in 1940 by the Bulgarian government to study rationing in war time, and in 1942 he accepted a chair at the University of Kiel. Two years after the end of World War II, at the age of sixty, Anderson accepted the chair of economics at the University of Munich. He remained in Munich for the rest of his life. Fels writes in [2]:At the time of his death, his authority in German statistical circles was unrivalled. It was mainly through his efforts that the statistical training for economists at German universities was improved ... Tintner, in [9], describes Anderson as "perhaps the most widely known statistician in Central Europe". He goes on to write that:... through his origin in the flourishing Russian school of probability, ... Anderson belongs to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anderson.html (2 of 3) [2/16/2002 10:57:44 PM]

Anderson

the so-called 'continental' school of statistics, and worked in the tradition of Lexis and Bortkiewicz. He might be the last representative of this approach .... He applied mathematical statistics to economics, using nonparametric methods. An overview of his work is presented in [10]:His scientific work, always marked by personal involvement, is of sufficient stature to be of lasting interest... Some of Anderson's endeavours were ahead of his time... Thus his emphasis on casual analysis of non-experimental data is a reminder that this important sector of applied statistics is far less developed than descriptive statistics and experimental analysis.... The main strength of Anderson's scientific work lies, I think, in the systematic coordination of theory and application. In [1] his contribution is summed up as follows:He especially believed that statistics, based on the law of large numbers and the sorting out of random deviations, is the only substitute for experimentation, which is impossible in economics. Sensibly estimating the difficulties inherent in economics as a science, Anderson was opposed to the use of "refined" statistical methods and to accepting preconditions regarding laws of distribution. This led him to nonparametric methods and to the necessity of casual analysis in economics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Anderson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anderson.html (3 of 3) [2/16/2002 10:57:44 PM]

Andreev

Konstantin Andreev Born: 26 March 1848 in Moscow, Russia Died: 29 Oct 1921 in Moscow, Russia Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Konstantin Andreev taught at the University of Kharkov from 1873 to 1898. While at Kharkov, he was promoted to professor of mathematics in 1879. Andreev took a leading role in the foundation and development of the Kharkov Mathematical Society. This Society is one of the early mathematics societies and was founded in 1879, the year Andreev became a professor. In 1898 Andreev left Kharkov to take up an appointment in Moscow where he was professor of mathematics. He remained in Moscow for the rest of his career. Andreev is best known for his work on geometry, although he also made contributions to analysis. In the area of geometry he did major pieces of work on projective geometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Andreev.html (1 of 2) [2/16/2002 10:57:45 PM]

Andreev

JOC/EFR December 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Andreev.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Andreev.html (2 of 2) [2/16/2002 10:57:45 PM]

Angeli

Stefano degli Angeli Born: 11 Sept 1623 in Venice, Italy Died: 11 Oct 1697 in Venice, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stephano Angeli studied mathematics at the University of Bologna. He taught literature, philosophy and theology at Ferrara from 1644, then transferred to Bologna in 1645 where he came under the influence of Cavalieri. After leaving Bologna, Angeli continued his contacts with Cavalieri by correspondence. He also corresponded with a number of other mathematicians including Torricelli and Viviani. Angeli was Rector of a religious establishment in Rome (the Jesuate order) from 1647 to 1652. Then he moved to a monastery of the same religious order in Venice before being appointed professor of mathematics at University of Padua in 1662. In fact the Jesuate order was disbanded in 1668 but Angeli continued in the priesthood while holding the chair of mathematics. He remained in this chair until his death in 1697. Angeli's many works were on infinitesimals and he used them to study spirals, parabolas and hyperbolas. While in Venice he published De infinitorum parabolis (1654), De infinitorum spiralium spatiorum mensura (1660) and De infinitorum cochlearum (1661). Gregory studied with Angeli in Padua from 1664 to 1668 and learnt from him about series expansions of functions. Angeli examined fluid statics based on Archimedes' principle and Torricelli's experiments. He published Della gravita dell aria e fluidi in 1671 while holding the chair at Padua. He also considered the motion bodies falling towards a rotating Earth. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

The rise of the calculus

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Angeli.html (1 of 2) [2/16/2002 10:57:47 PM]

Angeli

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Angeli.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Angeli.html (2 of 2) [2/16/2002 10:57:47 PM]

Anstice

Robert Richard Anstice Born: 9 April 1813 in Madeley, Shropshire, England Died: 17 Dec 1853 in Wigginton (near Tring), Hertfordshire, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Anstice's parents were William and Penelope Anstice. William Anstice ran an ironworks and this family business was eventually taken over by the eldest of the Anstice's four sons who was named William after his father. The second son of the family, Joseph, attended Westminster School and then studied at Christ Church, Oxford before becoming professor of Classical Literature at King's College London when he was only 22 years old. Robert Anstice was the fourth and youngest of William and Penelope Anstice's sons. He took the same educational route as his older brother Joseph, attending Westminster School before entering Christ Church, Oxford in 1831. There he studied mathematics, graduating with a BA with first class honours in 1835 and an MA in 1837. We know that Anstice was awarded a scholarship to study mathematics after graduating at Oxford but there is then a rather strange gap in our knowledge of him for nothing is known of what he did over the following ten years. Clearly he decided to join the Church at some stage for the next event that we know of in his life was his ordination. Anstice was ordained in 1846 and the following year he became rector of Wigginton, near Tring, in the diocese of St Albans. He died after only six years as a Church of England parish minister, the parish records describing him as:a fine philosophical preacher, and greatly mourned when he died. Anstice died young, being only 40 years old, but he did survive longer than his elder brother Joseph, who died at about the age of 28 years, only six years after his appointment to the chair of Classical Literature at King's College London. Robert Anstice wrote three mathematical papers in his six years as rector in the parish of Wigginton. The first paper was On the motion of a free pendulum but the next two are the ones of real interest. They are both on combinatorics and they each have the same title namely On a problem in combinations. These two papers on combinatorics deal with Kirkman triple systems. The authors of [2] write:During his time at Wigginton, Anstice became interested in the mathematical work of another rector, Kirkman, who had written on the subject of Steiner triple systems (as they are now called). In one of his papers Kirkman gave an elegant construction of a resolvable http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anstice.html (1 of 3) [2/16/2002 10:57:48 PM]

Anstice

Steiner triple system on 15 elements (the famous Kirkman 15 schoolgirls problem), making use of what are now known as a Room square of order 8 and the Fano plane. Kirkman stated that the generalisation of this construction seemed very hard. Anstice achieved a brilliant generalisation to a resolvable Steiner triple system on 2p + 1 elements for all primes p congruent to 1 modulo 6. He gave infinite families of cyclic Steiner triple systems 40 years before Netto (who did so in 1893). Anstice also constructed an infinite number of Room squares a hundred years before T G Room wrote his paper in which it was thought for many years that the squares first appeared and from which they were named. Perhaps even more remarkable is the fact that the method of differences which Anstice used has become one of the standard methods of design construction. Anstice also gave examples of 2-rotational Kirkman triple systems which remained the only ones known until 1971. His remarkable results seem to have been little noticed until the paper [1] by Anderson who summarised his paper as follows:We reveal the contents of two remarkable papers by R R Anstice, and thereby rewrite part of the early history of combinatorial designs. Infinite families of cyclic Steiner triple systems and Room squares are constructed [in the papers]. Anstice, however, appears to have failed to realise the importance of his own work as he ends one paper with the comment:But too much space has been already devoted to such a trifle. However, as the authors of [2] comment:[Anstice] is buried beside his parents, almost forgotten by the mathematical community. He deserves greater recognition. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anstice.html (2 of 3) [2/16/2002 10:57:48 PM]

Anstice

JOC/EFR July 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Anstice.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anstice.html (3 of 3) [2/16/2002 10:57:48 PM]

Anthemius

Anthemius of Tralles Born: about 474 in (possibly) Tralles (near Aydin in southwest Turkey) Died: about 534

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Anthemius of Tralles's father was a doctor by the name of Stephanus. Anthemius came from a well educated family with two brothers who were doctors, one brother who was a lawyer and another who was described as a man of learning. Anthemius is known both a mathematician and an architect. As an architect he is best known for replacing the old church of Hagia Sophia at Constantinople in 532. His skills seem also to have extended to engineering for he is said to have been employed to repair flood defences at Daras. You can see an article about Hagia Sophia. He described the construction of an ellipse with a string fixed at the two foci. His famous book On Burning Mirrors also describes the focal properties of a parabola. Heath [2] gives one of his problems which leads to the ellipse construction:To contrive that a ray of the sun (admitted through a small hole or window) shall fall in a given spot, without moving away at any hour and season. Heath [2] gives Anthemius's solution:This is contrived by constructing an elliptical mirror one focus of which is at the point where the ray of the sun is admitted while the other is at the point to which the ray is required to be reflected at all times. Anthemius studied the focal properties of the parabola and proves that [2]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anthemius.html (1 of 2) [2/16/2002 10:57:50 PM]

Anthemius

... parallel rays can be reflected to one single point from a parabolic mirror of which the point is the focus. The directrix is used in the construction, which follows, mutatis mutandis, the same course as the above construction in the case of the ellipse. He compiled a survey of remarkable mirror configurations in his work On remarkable mechanical devices which was known to certain of the Arab mathematicians such as al-Haytham. There are a number of stories told of Anthemius which may be totally fictitious but, as is often the case with such stories, they may give an indication of his character. Huxley writes [1] (see also [3]):Anthemius persecuted a neighbour and rival Zenon by reflecting sunlight into his house. He also produced the impression of an earthquake in Zenon's house by the use of steam led under pressure through pipes connected to a boiler. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 500 to 900

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Anthemius.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Anthemius.html (2 of 2) [2/16/2002 10:57:50 PM]

Antiphon

Antiphon the Sophist Born: 480 BC in (possibly) Athens, Greece Died: 411 BC in Athens, Greece Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Antiphon was an orator and statesman who took up rhetoric as a profession. He was a Sophist and a contemporary of Socrates. These definite assertions are, however, disputed by some historians. The problem seems to revolve round whether there was one Sophist philosopher named Antiphon who lived around this time or whether there are two, or as some experts claim, three distinct Antiphons. In what follows we shall assume that at least the orator named Antiphon was the same person as the Sophist who made the mathematical advances. This is the same line as taken in [1] while in [2] only Antiphon as an orator is discussed without reference to the philosophical or mathematical works. In [7] the hypothesis that Antiphon is one, or several different men is discussed without any definite view being preferred either way. A number of speeches which were written by Antiphon have been preserved. Three of these speeches were real speeches made by Antiphon as the prosecutor in murder trials. Twelve speeches are specimen speeches written by Antiphon for use in teaching students the skills of prosecuting and defending clients in cases. The speeches come as three collections of four; two prosecution speeches and two defence speeches for each of three different cases. Antiphon published a number of works on philosophy which have been lost except for a small number of fragments which have been discovered together with some quotations from the works in the writings of other authors. These works include On Truth, On Concord, The Statesman, and On Interpretation of Dreams. The work On Truth is written to support the views of Parmenides who believed that there was a single sole reality and that the apparent world of many things was unreal. In this work Antiphon is defending the same philosophical ideas which Zeno of Elea supported with his paradoxes. In On Concord Antiphon [1]:... defends the authority of the community as a safeguard against anarchy and recommends the ideals of concord and self-restraint both within communities and within the individual soul. Most probably he was only concerned to criticise the laws of a city by asking whether or not they satisfy the "natural" needs of the individual. Hobbs in [7] notes that:... some have doubted whether the same man could have written "On Truth" and the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Antiphon.html (1 of 3) [2/16/2002 10:57:51 PM]

Antiphon

conventional gnomic utterances of "On Concord". In [7] three reasons are given to support at least the same author for these two philosophical works:(1) "On Truth" is not as radical as it appears, but simply a plea for legal reform; (2) its doctrines, although radical, are not endorsed by Antiphon; (3) Antiphon changed his mind. Finally while discussing which works were written by Antiphon it is worth remarking that some historians reject the idea that Antiphon wrote the two other works attributed to him The Statesman and On Interpretation of Dreams. Antiphon made an early and important contribution to mathematics when he made an attempt to square the circle. In doing so he became the first to propose a method of exhaustion although it is not entirely clear how well he understood his own proposal. He proposed successively doubling the number of sides of a regular polygon inscribed in a circle so that the difference in areas would eventually become exhausted. We know of his work via Aristotle and his commentators. Aristotle claims that a geometer only needs to show that false arguments are false if they are based on geometry, otherwise he can ignore them. Aristotle writes in his Physics (see for example [4]):... thus it is the geometer's business to refute the quadrature by means of segments, but it is not his business to refute that of Antiphon. In case the reader is wondering what Aristotle refers to with his phrase 'quadrature by means of segments' then it is almost certain that he means the method of lunes of Hippocrates. However Simplicius failed to properly understand what Antiphon was doing. He thought that Antiphon was claiming to have squared the circle. He wrote (translation by Heath given in [4]):Antiphon thought that in this way the area of the circle would be used up, and we should some time have a polygon inscribed in the circle the sides of which, owing to their smallness, coincide with the circumference of the circle. And as we can make a square equal to any polygon ... we shall be in a position to make a square equal to a circle. However, according to Heath, this was not what Antiphon claimed [4]:Antiphon therefore deserves an honourable place in the history of geometry as having originated the idea of exhausting an area by means of inscribed regular polygons with an ever increasing number of sides, an idea upon which ... Eudoxus founded his epoch-making method of exhaustion. Kerferd in [1] suggests that Antiphon may have regarded a circle as a polygon with a large number of sides:In modern times it has often been supposed that Antiphon was simply making a bad mistake in geometry by supposing that any approximation could ever amount to coincidence between a polygon with however many sides and a continuously curved circumference of a curved circle. ... This may not be the right view to take. Antiphon appears to have believed that complete coincidence could be achieved by his method ... This may mean that Antiphon

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Antiphon.html (2 of 3) [2/16/2002 10:57:51 PM]

Antiphon

regarded the circle as a polygon with a very large (or possibly infinite) number of sides. Antiphon was involved in an anti-democratic revolution which failed. Thucydides' in his famous History believes that Antiphon was the leader of the revolution [2]:[Antiphon] conceived the whole matter and the means by which it was brought to pass. Despite his profession as a writer of defence speeches, his brilliant speech, described by Thucydides as:... the greatest ever made by a man on trial for his life... failed to save him when he was tried for treason and he was executed. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Pi through the ages 2. Squaring the circle 3. The rise of the calculus

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Antiphon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Antiphon.html (3 of 3) [2/16/2002 10:57:51 PM]

Apastamba

Apastamba Born: about 600 BC in India Died: about 600 BC in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

To write a biography of Apastamba is essentially impossible since nothing is known of him except that he was the author of a Sulbasutra which is certainly later than the Sulbasutra of Baudhayana. It would also be fair to say that Apastamba's Sulbasutra is the most interesting from a mathematical point of view. We do not know Apastamba's dates accurately enough to even guess at a life span for him, which is why we have given the same approximate birth year as death year. Apastamba was neither a mathematician in the sense that we would understand it today, nor a scribe who simply copied manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in mathematics for its own sake, merely interested in using it for religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for religious rites and to improve and expand on the rules which had been given by his predecessors. Apastamba would have been a Vedic priest instructing the people in the ways of conducting the religious rites he describes. The mathematics given in the Sulbasutras is there to enable the accurate construction of altars needed for sacrifices. It is clear from the writing that Apastamba, as well as being a priest and a teacher of religious practices, would have been a skilled craftsman. He must have been himself skilled in the practical use of the mathematics he described as a craftsman who himself constructed sacrificial altars of the highest quality. The Sulbasutras are discussed in detail in the article Indian Sulbasutras. Below we give one or two details of Apastamba's Sulbasutra. This work is an expanded version of that of Baudhayana. Apastamba's work consisted of six chapters while the earlier work by Baudhayana contained only three. The general linear equation was solved in the Apastamba's Sulbasutra. He also gives a remarkably accurate value for 2 namely 1 + 1/3 + 1/(3 4) - 1/(3 4 34). which gives an answer correct to five decimal places. A possible way that Apastamba might have reached this remarkable result is described in the article Indian Sulbasutras. As well as the problem of squaring the circle, Apastamba considers the problem of dividing a segment into 7 equal parts. The article [3] looks in detail at a reconstruction of Apastamba's version of these two problems.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Apastamba.html (1 of 2) [2/16/2002 10:57:52 PM]

Apastamba

Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. An overview of Indian mathematics 2. The Indian Sulbasutras

Other references in MacTutor

Chronology: 30000BC to 500BC

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Apastamba.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Apastamba.html (2 of 2) [2/16/2002 10:57:52 PM]

Apollonius

Apollonius of Perga Born: about 262 BC in Perga, Pamphylia, Greek Ionia (now Murtina, Antalya, Turkey) Died: about 190 BC in Alexandria, Egypt

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Apollonius of Perga was known as 'The Great Geometer'. Little is known of his life but his works have had a very great influence on the development of mathematics, in particular his famous book Conics introduced terms which are familiar to us today such as parabola, ellipse and hyperbola. Apollonius of Perga should not be confused with other Greek scholars called Apollonius, for it was a common name. In [1] details of others with the name of Apollonius are given: Apollonius of Rhodes, born about 295 BC, a Greek poet and grammarian, a pupil of Callimachus who was a teacher of Eratosthenes; Apollonius of Tralles, 2nd century BC, a Greek sculptor; Apollonius the Athenian, 1st century BC, a sculptor; Apollonius of Tyana, 1st century AD, a member of the society founded by Pythagoras; Apollonius Dyscolus, 2nd century AD, a Greek grammarian who was reputedly the founder of the systematic study of grammar; and Apollonius of Tyre who is a literary character. The mathematician Apollonius was born in Perga, Pamphylia which today is known as Murtina, or Murtana and is now in Antalya, Turkey. Perga was a centre of culture at this time and it was the place of worship of Queen Artemis, a nature goddess. When he was a young man Apollonius went to Alexandria where he studied under the followers of Euclid and later he taught there. Apollonius visited Pergamum where a university and library similar to Alexandria had been built. Pergamum, today the town of Bergama in the province of Izmir in Turkey, was an ancient Greek city in Mysia. It was situated 25 km from the Aegean Sea on a hill on the northern side of the wide valley of the Caicus River (called the Bakir river today). While Apollonius was at Pergamum he met Eudemus of Pergamum (not to be confused with Eudemus of Rhodes who wrote the History of Geometry) and also Attalus, who many think must be King Attalus I of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Apollonius.html (1 of 5) [2/16/2002 10:57:54 PM]

Apollonius

Pergamum. In the preface to the second edition of Conics Apollonius addressed Eudemus (see [4] or [5]):If you are in good health and things are in other respects as you wish, it is well; with me too things are moderately well. During the time I spent with you at Pergamum I observed your eagerness to become aquatinted with my work in conics. The only other pieces of information about Apollonius's life is to be found in the prefaces of various books of Conics. We learn that he had a son, also called Apollonius, and in fact his son took the second edition of book two of Conics from Alexandria to Eudemus in Pergamum. We also learn from the preface to this book that Apollonius introduced the geometer Philonides to Eudemus while they were at Ephesus. We are in a somewhat better state of knowledge concerning the books which Apollonius wrote. Conics was written in eight books but only the first four have survived in Greek. In Arabic, however, the first seven of the eight books of Conics survive. First we should note that conic sections to Apollonius are by definition the curves formed when a plane intersects the surface of a cone. Apollonius explains in his preface how he came to write his famous work Conics (see [4] or [5]):... I undertook the investigation of this subject at the request of Naucrates the geometer, at the time when he came to Alexandria and stayed with me, and, when I had worked it out in eight books, I gave them to him at once, too hurriedly, because he was on the point of sailing; they had therefore not been thoroughly revised, indeed I had put down everything just as it occurred to me, postponing revision until the end. Books 1 and 2 of the Conics began to circulate in the form of their first draft, in fact there is some evidence that certain translations which have come down to us have come from these first drafts. Apollonius writes (see [4] or [5]):... it happened that some persons also, among those who I have met, have got the first and second books before they were corrected.... Conics consisted of 8 books. Books one to four form an elementary introduction to the basic properties of conics. Most of the results in these books were known to Euclid, Aristaeus and others but some are, in Apollonius's own words:... worked out more fully and generally than in the writings of others. In book one the relations satisfied by the diameters and tangents of conics are studied while in book two Apollonius investigates how hyperbolas are related to their asymptotes, and he also studies how to draw tangents to given conics. There are, however, new results in these books in particular in book three. Apollonius writes of book three (see [4] or [5]):... the most and prettiest of these theorems are new, and it was their discovery which made me aware that Euclid did not work out the syntheses of the locus with respect to three and four lines, but only a chance portion of it, and that not successfully; for it was not possible for the said synthesis to be completed without the aid of the additional theorems discovered by me. Books five to seven are highly original. In these Apollonius discusses normals to conics and shows how many can be drawn from a point. He gives propositions determining the centre of curvature which lead

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Apollonius.html (2 of 5) [2/16/2002 10:57:54 PM]

Apollonius

immediately to the Cartesian equation of the evolute. Heath writes that book five [5]:... is the most remarkable of the extant Books. It deals with normals to conics regarded as maximum and minimum straight lines drawn from particular points to the curve. Included in it are a series of propositions which, though worked out by the purest geometrical methods, actually lead immediately to the determination of the evolute of each of the three conics; that is to say, the Cartesian equations of the evolutes can be easily deduced from the results obtained by Apollonius. There can be no doubt that the Book is almost wholly original, and it is a veritable geometrical tour de force. The beauty of Apollonius's Conics can readily be seen by reading the propositions as given by Heath, see [4] or [5]. However, Heath explains in [5] how difficult the original text is to read:... the treatise is a great classic which deserves to be more known than it is. What militates against its being read in its original form is the great extent of the exposition (it contains 387 separate propositions), due partly to the Greek habit of proving particular cases of a general proposition separately from the proposition itself, but more to the cumbersomeness of the enunciations of complicated propositions in general terms (without the help of letters to denote particular points) and to the elaborateness of the Euclidean form, to which Apollonius adheres throughout. Pappus gives some indications of the contents of six other works by Apollonius. These are Cutting of a ratio (in two books), Cutting an area (in two books), On determinate section (in two books), Tangencies (in two books), Plane loci (in two books), and On verging constructions (in two books). Cutting of a ratio survives in Arabic and we are told by the 10th century bibliographer Ibn al-Nadim that three other works were translated into Arabic but none of these survives. To illustrate how far Apollonius had taken geometric constructions beyond that of Euclid's Elements we consider results which are known to have been contained in Tangencies. In the Elements Book III Euclid shows how to draw a circle through three given points. He also shows how to draw a tangent to three given lines. In Tangencies Apollonius shows how to construct the circle which is tangent to three given circles. More generally he shows how to construct the circle which is tangent to any three objects, where the objects are points or lines or circles. In [11] Hogendijk reports that two works of Apollonius, not previously thought to have been translated into Arabic, were in fact known to Muslim geometers of the 10th century. These are the works Plane loci and On verging constructions. In [11] some results from these works which were not previously known to have been proved by Apollonius are described. From other sources there are references to still further books by Apollonius, none of which have survived. Hypsicles refers to a work by Apollonius comparing a dodecahedron and an icosahedron inscribed in the same sphere, which like Conics appeared in two editions. Marinus, writing a commentary on Euclid's Data, refers to a general work by Apollonius in which the foundations of mathematics such as the meaning of axioms and definitions are discussed. Apollonius also wrote a work on the cylindrical helix and another on irrational numbers which is mentioned by Proclus. Eutocius refers to a book Quick delivery by Apollonius in which he obtained an approximation for better than the 223/

71

0, subject to an initial condition of the form u(x,0) = f(x) for all real x. They considered numerical methods which find an approximate solution on a grid of values of x and t, replacing ut(x,t) and uxx(x,t) by finite difference approximations. One of the simplest such replacements was proposed by L F Richardson in 1910. Richardson's method yielded a numerical solution which was very easy to compute, but alas was numerically unstable and thus useless. The instability was not recognised until lengthy numerical computations were carried out by Crank, Nicolson and others. Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level. Article by: G M Phillips, St Andrews

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Crank.html (1 of 2) [2/16/2002 11:05:56 PM]

Crank

Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Crank.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Crank.html (2 of 2) [2/16/2002 11:05:56 PM]

Crelle

August Leopold Crelle Born: 11 March 1780 in Eichwerder (near Wriezen), Germany Died: 6 Oct 1855 in Berlin, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

August Crelle's father was a builder who had little in the way of income to be able to give his son a good education. Crelle was therefore largely self-taught, studying civil engineering. He then secured a job as a civil engineer in the service of the Prussian Government. He worked for the Prussian Ministry of the Interior on the construction and planning of roads and the one of the first railways in Germany (completed in 1838) between Berlin and Potsdam. Had his family had the resources, then Crelle would have studied mathematics at university. He always had a love of the subject but earning money was a necessity for him. However, he was always one to be prepared to study on his own and indeed he spent a great deal of time working on mathematics. He achieved a remarkable level of mathematics considering that he had never been formally taught, and when he was 36 years old he submitted a thesis De calculi variabilium in geometria et arte mechanica usu to the University of Heidelberg and was duly awarded a doctorate. Crelle was certainly not a great original mathematician, but he had three qualities which made him as important for the subject as any great researcher might have been. These three qualities were firstly his great enthusiasm for the subject, secondly his organisational ability, and thirdly his ability to spot exceptional talent in young mathematicians. This last gift is described in [1] as follows:Crelle had a unique sensitivity to mathematical genius and in [12] as:Crelle had an extraordinary intuition for judging the qualities of young talents, and for encouraging then with their research work.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Crelle.html (1 of 3) [2/16/2002 11:05:57 PM]

Crelle

He founded a journal devoted entirely to mathematics Journal für die reine und angewandte Mathematik in 1826. Although not the first such journal, it was organised quite differently from journals that existed at that time since these other journals were basically reporting meetings of Academies and Learned Societies where papers were read. Crelle was very much in control of the journal, and he acted as editor-in-chief for the first 52 volumes. He did not want an exclusive work but, as he put it in the first volume, a journal which:... must endeavour to offer itself to a larger public so as to first and foremost ensure its longevity and the possibility to perfect itself. Crelle realised the importance of Abel's work and published several articles by him in this first volume, including his proof of the insolubility of the quintic equation by radicals. In fact Abel and Steiner had strongly encouraged Crelle in his founding of the journal and Steiner was also a major contributor to the first volume of Crelle's Journal. Other young mathematicians had their first papers published in Crelle's journal, largely due to his genius in spotting the importance of their research. In addition to Abel, mathematicians such as Dirichlet, Eisenstein, Grassmann, Hesse, Jacobi, Kummer, Lobachevsky, Möbius, Plücker, von Staudt, Steiner, and Weierstrass all had their early works made famous by publication in Crelle's journal. In 1828 Crelle left the service of the Prussian Ministry of the Interior and joined the Prussian Ministry of Education and Cultural Affairs. There he used his mathematical skills and connections, advising on policy for teaching mathematics in schools and technical colleges. He spent a spell in the summer of 1830 in France studying the teaching methods used by the French. He wrote a report on his return to Germany which praised highly the way that mathematics teaching was organised in France, but he was critical of the French having such a strong emphasis on the applications of mathematics rather than, what Crelle believed in, the importance of mathematical learning in its own right. Crelle wrote (see for example [13]):The real purpose of mathematics is to be the means to illuminate reason and to exercise spiritual forces. However, he became keen to bring the model of the Ecole Polytechnique to Germany for this was the French route to train high quality teachers. One of the outcomes of his involvement with teaching of mathematics in schools was that he published a large number of textbooks and published multiplication tables that went through many editions. We have mentioned above Crelle's reaction to pure and applied mathematics. His original intention when he began his Journal für die reine und angewandte Mathematik was, as the title indicates, to deal equally with both pure and applied mathematics. He changed his view of this equality of balance when he found it impossible to find applied mathematics articles of the same intellectual depth to those on pure mathematics. The solution was simple, even if it required a change in policy, and that was to have a second journal for more practical mathematics and this he moved to a second journal which he started in 1829, the Journal für die Bankunst. This journal published 30 volumes but ended its run in 1851, a few years before Crelle's death. Crelle was elected to the Berlin Academy of Sciences in 1827 with the strong support of Alexander von Humboldt. In [5] Eccarius looks at:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Crelle.html (2 of 3) [2/16/2002 11:05:57 PM]

Crelle

... the recommendations Crelle wrote for prospective members of the Academy [and] the mathematical papers he read there, as well as the prize-problems he proposed and evaluated for the Academy ... We should say a little about Crelle's personal character and lifestyle which also proved important in his successful ventures. Abel visited Crelle in Berlin not long before Journal für die reine und angewandte Mathematik began publication. Abel wrote to Holmboe in January 1826:You cannot imagine what an excellent man [Crelle] is, exactly as one should be, thoughtful and yet not horribly polite like so many people, quite honest, for that matter. I am with him on as good terms as I am with you or other very good friends. In another letter, this time to Hansteen, Abel wrote (see for example [12]):There is at his place some kind of meeting where music is mainly discussed, of which unfortunately I do not understand much. I enjoy it all the same since I always meet there some young mathematicians with whom to talk. At Crelle's house there used to be a weekly meeting of mathematicians, but he had to suspend it because of a certain [Martin Ohm, the brother of Georg Ohm] with whom nobody could get along due to his terrible arrogance. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1820 to 1830

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Crelle.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Crelle.html (3 of 3) [2/16/2002 11:05:57 PM]

Cremona

Antonio Luigi Gaudenzio Giuseppe Cremona Born: 7 Dec 1830 in Pavia, Lombardy (now Italy) Died: 10 June 1903 in Rome, Italy

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Luigi Cremona was educated at the Ginnasio in Pavia. Luigi's father died when Luigi was 11 years old and this nearly prevented him from having a proper education. However, his stepfather supported him through school so that he was able to complete his school education and enter the University of Pavia. Political events, however, were to have a major effect on Cremona's life and the first impact came from the revolution of 1848 which attempted to achieve a new and more liberal constitution. The Austrian government arrested opposition leaders of the revolution in Venice and Milan, and suppressed student demonstrations in Padua and Pavia. Cremona, all his life an ardent Italian nationalist, immediately joined the 'Free Italy' battalion. Within a few days the Austrian army lost nearly all of Lombardy - Venetia and retreated. Sardinia Piedmont declared war on Austria. After some initial successes by the Italians, the Austrian armies began to win victories. In Lombardy the Austrian reconquest of Brescia in March 1849, after 10 days of fighting, left Venice isolated. Cremona, by this time a sergeant, was with the troops defending Venice. The city resisted Austrian forces until 24 August 1849. The bravery of the defending forces had been such that the Austrian attackers allowed them to leave the city with honour. Cremona was able to return to Pavia. Back in Pavia, Cremona discovered that his mother had died while he had been fighting to free Italy. Again he was financially unable to return to university without the support of his family and again his family rallied round and provided the necessary support. He entered the University of Pavia on 27 http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cremona.html (1 of 5) [2/16/2002 11:05:59 PM]

Cremona

November 1849 to study for a degree in civil engineering. There he was taught by Bordoni, Casorati and Brioschi but he was most influenced by Brioschi. Cremona later wrote, see [6]:The years that I passed with Brioschi as pupil and later as colleague are a grand part of my life; in the first portion of these years I learned to love science and in the other how to transfer it to a large circle of auditors. On 9 May 1853 Cremona was awarded his doctorate in civil engineering from the University of Pavia. By this time Camillo Benso di Cavour had become the head of the cabinet, and the unification of Italy had begun. But Austria still controlled the region and this made life hard for Cremona who had fought against the Austrian occupation. As Greitzer writes in [1]:His record of military service against Austrian rule prevented him from obtaining an official teaching post in the education system, so his first employment was as a private tutor to several families in Pavia. Cremona was by this time engaged in mathematical research and his first paper appeared in March 1855. This must have had some effect in helping him to gain permission to teach physics on a temporary basis at the Ginnasio in Pavia where he had himself trained for university. In May 1856 his second mathematical paper appeared and this, together with his good teaching, seems to have helped him secure the position of associate teacher at the Ginnasio in December 1856. On 17 January 1857, Cremona was appointed a full teacher at the Ginnasio in Cremona. Cremona, the capital of the Cremona region of Lombardy was situated on the north bank of the Po River southeast of Milan. Cremona was to remain at the school there for three years and during this time he wrote a number of mathematical articles. They are not of great importance except that some of them were examining curves using projective methods, techniques which would be characteristic of Cremona's later important mathematics. While the mathematician Cremona taught in the town of Cremona, political events were taking place which would have a large effect on his future. In July 1858 Cavour made an agreement with Napoleon III of France for his support of Piedmont. When Austria declared war on Piedmont on 26 April 1859, France honoured its alliance with Piedmont. After a number of battles the Austrians were in retreat and a treaty was signed at Villafranca by Napoleon III accepting the cession of Lombardy from Austria, which he then passed to Piedmont. Lombardy, liberated from Austrian rule, were quick to see that Cremona should no longer be held back for political reasons. On 28 November 1859 he was appointed as teacher at the Lycée St Alexandre in Milan. With events moving quickly towards a unified Italy under Victor Emmanuel II as King, Cremona was appointed by Royal decree as an ordinary professor at the University of Bologna on 10 June 1860. The Kingdom of Italy was officially proclaimed on 17 March 1861, by a parliament sitting in Turin. Cremona was to remain in Bologna until October 1867. In Cremona's Complete Works there appear 45 articles which he published while at Bologna. Sixteen of these articles are answers to questions posed in the Nouvelles Annales. There are eight book reviews and four historical articles intended to encourage research into geometry. Also included are Cremona's important work on transformations of plane curves, which were published during this period in 1863 and 1865. It was this work which won him the Steiner Prize for 1866, the prize being awarded jointly to Cremona and Rudolf Sturm. Also while at Bologna Cremona developed the theory of birational

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cremona.html (2 of 5) [2/16/2002 11:05:59 PM]

Cremona

transformations, later known as Cremona transformations, and wrote a series of papers on twisted cubic surfaces. White describes this period of Cremona's work in [11]:Chasles was the type on whom at first Cremona modelled his own work. The geometrical purism of von Staudt was a later influence. Hence it is natural to find metric foundations for geometric definitions instead of, or interchangeably with, projective. The apparatus of algebraic geometry is built upon polars, and these upon distances. The geometric method is principally a use of terms or descriptive relations instead of equations. And the beginnings of enumerative geometry are here... Greitzer, writing in [1], describes the importance of Cremona transformations which he introduced in his Bologna period:Cremona transformations have been used for studying rational surfaces, for the resolution of singularities of plane and space curves, and for the study of elliptic integrals and Riemann surfaces. They are effective in the reduction of singularities of curves to double points with distinct tangents. In October 1867, on Brioschi's recommendation, another Royal Decree was issued, this time appointing Cremona to the Polytechnic Institute of Milan. He received the title of Professor in 1872 while at Milan. Greitzer writes in [1]:The period in Milan, where he remained until 1873, was the time of Cremona's greatest creativity. He wrote articles on such diverse topics as twisted cubics, developable surfaces, the theory of conics, the theory of plane curves, third- and fourth-degree surfaces, statics and projective geometry. Cremona's work in statics is of great importance and gives, in a clearer form, some theorems due to Maxwell. In a paper of 1872 Cremona took an idea of Maxwell's on forces in frame structures that had appeared in an engineering journal in 1867 and interpreted Maxwell's notion of reciprocal figures as duality in projective 3-space. These reciprocal figures, for example, have three forces in equilibrium in one figure represented by a triangle while in the reciprocal figure they are represented by three concurrent lines. In 1873 Cremona was offered a political post as secretary general of the new Italian Government. This was a fitting tribute to the highly patriotic Cremona yet his mathematics researches were of such interest that he refused to accept the political post. Instead he moved to Rome on 9 October 1873, again appointed by Royal Decree, as director of the newly established Polytechnic School of Engineering. In addition he was appointed professor of graphic statics. However, if he had hoped that refusing the political post would leave him able to continue his mathematical research he soon found that his administration and teaching duties put an effective end to research. In November 1877, Cremona was appointed to the chair of higher mathematics at the University of Rome. However political pressures made him feel that he should serve the new Italian State. On 16 March 1879 he was appointed a senator and at this point his mathematical work ended completely. He became Minister of Public Education and ended his political career as Vice-President of the Senate. Greitzer writes in [1]:On 10 June 1903, after leaving a sickbed to act on some legislation, Cremona succumbed to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cremona.html (3 of 5) [2/16/2002 11:05:59 PM]

Cremona

a heart attack. Cremona had a large influence on geometry in Italy. Many of Steiner's proofs on synthetic geometry were revised and improved by Cremona. One of the themes which were present in almost all his work throughout his career was projective geometry. But, Greitzer [1], states:Cremona made no startling discoveries in this area: but he did derive many properties of projectively related figures, and he did present the subject to has classes in a manner calculated to clarify and bring out relationships most simply. In fact Cremona had a fine reputation as a lecturer [1]:Cremona was an excellent lecturer: calm, rigorous, yet interesting and even exciting. White [11] writes:... it is evident from his writings that it was the innate love of his chosen science that moved him to teaching and to the preparation of the books through which his name is most widely known. Cremona was extremely fair in citing the works of others, something which was not too common among many mathematicians at that time. Again we quote White [11]:Cremona himself was conscientious and indefatigable in searching out the work of his predecessors upon matters that he himself was investigating. In one of his works, after two pages of historical summary, he gives a footnote excusing:... his previous ignorance of, and failure to cite, the works of Möbius and Seydewitz to which the essay of Schroeter had now directed his attention. Cremona had many pupils who were to make major contributions to geometry, for example Bertini, Veronese and Guccia. In 1879, the year he became a Senator, he received the scientific honour of becoming a corresponding member of the Royal Society of London. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) A Poster of Luigi Cremona

Mathematicians born in the same country

Cross-references to History Topics

A comment from Thomas Hirst's diary

Honours awarded to Luigi Cremona (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1879

Lunar features

Crater Cremona

Other Web sites

Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cremona.html (4 of 5) [2/16/2002 11:05:59 PM]

Cremona

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Cremona.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cremona.html (5 of 5) [2/16/2002 11:05:59 PM]

Crighton

David George Crighton Born: 15 Nov 1942 in Llandudno, Wales Died: 12 April 2000 in Cambridge, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

David Crighton was the son of George Wolfe Johnson Crighton and Violet Grace Garrison. He was born in Llandudno, Wales because, due to the bombing of London during World War II, his mother had been sent there so that her child might be born in safety. His secondary education was at Watford Grammar School, but there his first interests were not in mathematics but in classics. Only in his final two years at secondary school did he became interested in mathematics and this was brought about by one of the masters at Watford Grammar School who said:... whatever else, he will never be any good at mathematics. Crighton related later in his life that it was this challenge which made him change from his intended career in classics, and switch to mathematics and physics for his main school subjects over his final two years at secondary school. After completing his studies at Watford Grammar School, Crighton entered St John's College, Cambridge, in 1961. However he only completed the first two parts of the Mathematical Tripos for, despite obtaining a First in both parts, he preferred to end his education in 1964 with a BA and to take a lecturing position at Woolwich Polytechnic. We should note that Woolwich Polytechnic has since then became the University of Greenwich, but at the time Crighton joined it, it did not have university status. Pedley writes in [2]:There he taught a broad spectrum of mathematics for up to 23 hours a week, and learned the techniques of crowd control - the evening class on subsidiary mathematics for engineering included several leather-clad "mature students" bearing knives. A chance meeting with John Ffowcs Williams, who was an expert in aeroacoustics on the mathematics http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Crighton.html (1 of 4) [2/16/2002 11:06:02 PM]

Crighton

staff at Imperial College, London, changed Crighton's career. He left Woolwich Polytechnic to become Williams' research assistant at Imperial College, taking a huge cut in salary in the process, and he also studied for his doctorate while at Imperial College. He was awarded his doctorate in 1969 and remained at Imperial until he was appointed as a Research Fellow in the Department of Engineering at the University of Cambridge five years later. Crighton's first publication was in 1970 when Radiation from turbulence near a composite flexible boundary appeared in the Proceedings of the Royal Society of London. This paper Crighton studied the sound wave associated with a turbulent fluid flow over a discontinuous surface formed by two semi-infinite flexible planes. He gave a mathematical model in which the problem reduce to solving two singular integral equations with Cauchy-type kernels, and with variable coefficients. Solving the equations he showed that he boundary converts the energy stored in the turbulent boundary layer into the sound waves which generate noise. Despite being appointed to the Department of Engineering at Cambridge in 1974 he never took up the post for, at the instigation of Sir James Lighthill, he was appointed to the chair of Applied Mathematics at the University of Leeds. Under Crighton's leadership the Department at Leeds was [1]:... was transformed over the next 12 years into one of the top departments of its kind in the country. Crighton's successive spells as head of department, chairman of school, and chairman of the science board (effectively Dean), left in every case indelible marks of his imagination and effectiveness, tough decisions being invariably coupled with a genuine concern for the individuals affected. There is a nice story told by Pedley in [2] about one of Crighton's fifteen research students at Leeds:When he had been there only about a year, a new research student asked if they could fix a time for a regular weekly meeting. David looked at his diary and said: "How about Mondays at 7 o'clock?" "Am or pm," the student inquired - to which the reply was: "Whichever you prefer". In 1986 Batchelor retired as professor of applied mathematics at Cambridge and Crighton was appointed to succeed him. At this time he was elected as a Fellow of St John's College. Crighton became Head of the Department of Applied Mathematics and Theoretical Physics at Cambridge in 1991. He made new appointments to the areas of nonlinear dynamics and to solid mechanics. The department achieved great things under his leadership for he had a [2]:... fierce determination that the department should not only remain by far the strongest mathematical sciences department in Britain and Europe, but that it should rank at least equal with the much more generously funded departments in America. In 1993 Crighton was elected to a Fellowship of the Royal Society. However, he was not the only member of his department to achieve this honour and it is certainly a great tribute to his leadership that, remarkably, fifteen members of his department had been so honoured by 1999. We have described above the contents of Crighton's first research paper. We gave some details of this work for it is typical of the applications of mathematics in which he worked throughout his life. He was particularly concerned in applying his ideas to noise control in aircraft design. His work on the air flow over the wings and fuselage of an aircraft was again done with the aim of achieving a reduction in the noise produced particularly in takeoff and landing.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Crighton.html (2 of 4) [2/16/2002 11:06:02 PM]

Crighton

I [EFR] last heard Crighton speak on 27 November 1998 at a meeting in University College London to remember the life and work of Sir James Lighthill. At that meeting Crighton talked on Lighthill and acoustics. It was a superb lecture, given by a mathematician at the height of his powers, describing the work of another remarkable mathematician. In fact Crighton's mathematical work was very much in the same area as that of Lighthill and he shared with Lighthill another passion, namely music and opera. Other than his passion for research, achieving the highest world ranking for his department, and his family, Crighton loved music [1]:The operas of Wagner were a particular enthusiasm. He never missed an opportunity to attend the Bayreuth Festival, and would always seek to incorporate at least one opera in each of his many lecturing engagements around the world. A final achievement, that gave him great personal satisfaction and came just a few weeks before his death, was to conduct the college orchestra in a moving performance of the Overture to Tannhäuser. The piano was another love. While at Leeds he became a great admirer of the formidable Russian pianist Tatiana Nikolayeva, who had been on the jury of the city's International Piano Competition in 1984. When they met in the breakfast queue at a Leningrad hotel they struck up a friendship that lasted until her death in 1993. More recently he had championed the Slovenian pianist Dobravka Tomsic. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to David Crighton (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1993

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Crighton.html (3 of 4) [2/16/2002 11:06:02 PM]

Crighton

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Crighton.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Crighton.html (4 of 4) [2/16/2002 11:06:02 PM]

Cunha

José Anastácio da Cunha Born: 1744 in Lisbon, Portugal Died: 1 Jan 1787 in Lisbon, Portugal Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Anastácio da Cunha's father was Lorenzo da Cunha, who was a painter, and his mother was Jacinta Ignes. Da Cunha was educated in the town of his birth, studying at the Congregation of the Oratory in Lisbon. However, he did not learn much of mathematics and physics during his formal education and it was these topics which he studied on his own. It is certainly fair to say that in these areas he was self-educated, a fact which may have led to his later work being particularly innovative. During the period of his education Lisbon was hit by an earthquake on 1 November 1755 and two-thirds of the city was reduced to rubble. However rebuilding the old town soon created one of the most beautiful of European cities. When da Cunha was nineteen years old, in 1763, he volunteered for military service and he became a lieutenant of artillery giving ten years' of service. He spent most of his years in the army at Valença do Minho. Subsequent events were greatly influenced by the politics of Portugal during the period. In 1750 Joseph I had been crowned king of Portugal. Joseph appointed Sebastiao de Carvalho marques de Pombal as a minister and soon de Pombal came to have such an influence over the running of Portuguese affairs that his powers became almost absolute. De Pombal put through a series of major reforms, and around 1758-59 he moved against the Jesuits and the Society of Jesus. Da Cunha supported these religious reforms by de Pombal and also the later reforms he put in place such as the reform of university education, the beginning of commercial education, the creation of trading companies, and the reorganisation of the army. Da Cunha [1]:... became known as a progressive thinker, talented poet, and author of an original memoir on ballistics. As part of his university reforms de Pombal appointed da Cunha professor of geometry in 1773. His period in the Faculty of Mathematics at Coimbra University was, however, short. King Joseph I died on 24 February 1777 and suddenly de Pombal lost all his powers. Queen Maria I assumed the throne and she set political prisoners free. De Pombal was accused of having abused his powers, was found guilty by a judicial tribunal, and then banished from Lisbon. Da Cunha was arrested and imprisoned by the Inquisition. In October 1778 he was sentenced by the General Council of the Inquisition in Lisbon to three years in prison for being a follower of Voltaire and supporting heretical views. He was freed in 1781 but prison had ruined his health. Although he was appointed as Professor of Mathematics in the College of Sao Lucas and took up his mathematical research again, he died a few years later.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cunha.html (1 of 3) [2/16/2002 11:06:03 PM]

Cunha

Da Cunha wrote a 21 part encyclopaedia of mathematics Principios Mathemáticos which he began to publish in parts from 1782 (it was published as a complete work in 1790) which contained a rigorous exposition of mathematics, in particular a rigorous exposition of the calculus. The book contained the elements of geometry and algebra in addition to the calculus. In all areas da Cunha paid unusual attention to methodology as well as rigour. Struik, reviewing [11] writes:His importance for the history of mathematics is due to his Principios Mathemáticos, published posthumously in 1790 and translated into French by J M d'Abreu [Racle, Bordeaux, 1811]. This book is characterised by the attempts at rigor, especially in the calculus. Da Cunha develops a criterion for the convergence of a series which he uses to define the exponential function in a rather modern way, and from these develops the binomial series. His definition of the differential of a function y = f(x) anticipates that of Cauchy, and, written in our present notation, amounts to this: if Dy = f(x+Dx) - f(x) can be represented in the form | Dy| = A Dx + e Dx, where A is independent of Dx and e 0 when Dx 0, then A Dx is called the differential of y = f(x). In Principios Matemáticos da Cunha also gave a definition of the convergence of a series which is equivalent to Cauchy's convergence criterion. However, this was not realised until comparatively recently since most historians of mathematics studied the French translation of 1811 which is inaccurate at the crucial place where this definition is given. There is a second publication of a mathematical work by da Cunha discussed by Bogolyubov in [3] where he studies:... the Portuguese mathematician J A da Cunha's work ["Essay on the principles of mechanics"(Spanish), London, 1807; Amsterdam 1808], which contains interesting approaches to the foundations of mechanics, similar in many respects to contemporary ones. Many historians discuss the influence of da Cunha's remarkable, yet little known, work. Yushkevich, in [11], claims that da Cunha should rank with Bolzano, Cauchy, Abel and others for his contributions to the principles of the calculus. In [9] Yushkevich notes that da Cunha was not quite as unknown as had previously been thought since an anonymous, but unfavourable, review of Principios Matemáticos appeared in 1811 in the Gottingische gelehrte Anzeigen. In the same year, Gauss wrote a letter to Bessel in which he commented positively on da Cunha's definitions of the exponential and logarithmic functions. Despite this da Cunha's work was not widely known and, sadly, had little influence on the development of mathematics. Article by: J J O'Connor and E F Robertson List of References (11 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cunha.html (2 of 3) [2/16/2002 11:06:03 PM]

Cunha

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Cunha.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cunha.html (3 of 3) [2/16/2002 11:06:03 PM]

Cunningham

Ebenezer Cunningham Born: 7 May 1881 in Hackney, London, England Died: 12 Feb 1977

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ebenezer Cunningham was educated at St John's College, Cambridge which he entered in 1899. His lecturers at Cambridge included Baker, Larmor, J G Leathem and R Pendlebury. Baker was his director of studies. His main interests outside mathematics were choral music and rowing. He became a pacifist while at Cambridge through the Boer War years 1899 to 1902. After graduating as Senior Wrangler in 1902 he worked for a Smith's prize. Results similar to those he obtained were, unfortunately, published in a French journal before he had submitted. He started work on a new topic submitting a winning entry on matrices for the Smith's prize of 1904. In 1904 not only was he elected to a Fellowship at St John's but he also became a lecturer at Liverpool University. While at Liverpool he collaborated with Bateman. Until 1907 he worked both in Liverpool and in Cambridge. Then he moved to University College London where he worked under Pearson. He wrote on linear differential equations, prompted by Pearson's work and other work related to statistics. Although Cunningham's early papers were on analysis, he was soon to change topic. While at Cambridge, he had read Larmor's famous book Aether and Matter and then, in 1905, after reading Einstein's paper on special relativity, he began to work on that topic. Cunningham published The Principle of Relativity in 1914, the first English book on the topic. Many papers on relativity followed. In fact Cunningham had returned to St John's College Cambridge in 1911, at the invitation of Baker. His

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cunningham.html (1 of 2) [2/16/2002 11:06:04 PM]

Cunningham

work in Cambridge was interrupted by World War I when he worked on the land rather than join the army. Of course during this period he found it hard to keep in touch with developments in relativity theory which took place in Germany. After this he never returned to major research projects and spent the rest of his career as an enthusiastic teacher of mathematics at Cambridge. Cunningham himself blamed the administrative work for his lack of research, saying that it for some years came between me and any freedom to follow up and keep abreast of the extremely rapid advance of science. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Cunningham.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cunningham.html (2 of 2) [2/16/2002 11:06:04 PM]

Curry

Haskell Brooks Curry Born: 12 Sept 1900 in Millis, Massachusetts, USA Died: 1 Sept 1982 in State College, Pennsylvania , USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Haskell Curry was educated at Harvard and received a doctorate from Göttingen in 1930 for a thesis, supervised by Hilbert, entitled Grundlagen der kombinatorischen Logik. He taught at Harvard, Princeton, then for 35 years at Pennsylvania State University. During World War II Curry researched in applied physics at Johns Hopkins University. In 1966 he accepted the chair of mathematics at Amsterdam. Curry's main work was in mathematical logic with particular interest in the theory of formal systems and processes. He formulated a logical calculus using inferential rules. His works include Combinatory Logic (1958) (with Robert Feys) and Foundations of Mathematical Logic (1963). Article by: J J O'Connor and E F Robertson List of References (5 books/articles) A Poster of Haskell B Curry

Mathematicians born in the same country

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Curry.html (1 of 2) [2/16/2002 11:06:06 PM]

Curry

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Curry.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Curry.html (2 of 2) [2/16/2002 11:06:06 PM]

Cusa

Nicholas of Cusa Born: 1401 in Kues, Trier (now Germany) Died: 11 Aug 1464 in Todi, Papal States (now Italy)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nicholas of Cusa was ordained in 1440 and was a cardinal in Brixon (now Bressanone) and in 1450 became bishop there. He was interested in geometry and logic. He contributed to the study of infinity, studying the infinitely large and the infinitely small. He looked at the circle as the limit of regular polygons and used it in his religious teaching to show how one can approach truth but never reach it completely. Cusa is best known as a philosopher who argued the incomplete nature of man's knowledge of the universe. He claimed that the search for truth was equal to the task of squaring the circle. In 1444 he became interested in astronomy and purchased sixteen books on astronomy, a wooden celestial globe, a copper celestial globe and various astronomical instruments including an astrolabe. His interest in astronomy certainly led him to certain theories which were true and others which may still prove to be true. For example he claimed that the Earth moved round the Sun. He also claimed that the stars were other suns and that space was infinite. He also believed that the stars had other worlds orbiting them which were inhabited. He got much right that perhaps this will also be found to be true one day! Cusa published improvements to the Alfonsine Tables which gave a practical method to find the position of the Sun, Moon and planets using Ptolemy's model. These tables had originally been compiled in 1272 with the support of King Alfonso X of Castile. Like many learned men of his time, Cusa also wrote on calendar reform.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cusa.html (1 of 2) [2/16/2002 11:06:08 PM]

Cusa

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Cross-references to History Topics

Squaring the circle

Other references in MacTutor

Chronology: 1300 to 1500

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Cusa.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Cusa.html (2 of 2) [2/16/2002 11:06:08 PM]

D'Alembert

Jean Le Rond d'Alembert Born: 17 Nov 1717 in Paris, France Died: 29 Oct 1783 in Paris, France

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean d'Alembert's father was an artillery officer, Louis-Camus Destouches and his mother was Mme de Tencin. She had been a nun but had received a papal dispensation in 1714 which allowed her to begin [4]:... a brilliant social career in which political intrigues and amorous liaisons contended for first place; a timely participation in the famous John Law Scheme allowed her to pursue these activities in complete financial security. [John Law was a Scottish monetary reformer who founded a bank in Paris in 1716 with authority to issue notes. It was highly successful at first, the time when Mme de Tencin made her money, but collapsed in 1720.] D'Alembert was the illegitimate son from one of Mme de Tencin 'amorous liaisons'. His father, Louis-Camus Destouches, was out of the country at the time of d'Alembert's birth and his mother left the newly born child on the steps of the church of St Jean Le Rond. The child was quickly found and taken to a home for homeless children. He was baptised Jean Le Rond, named after the church on whose steps he had been found. When his father returned to Paris he made contact with his young son and arranged for him to be cared for by the wife of a glazier, Mme Rousseau. She would always be d'Alembert's mother in his own eyes, particularly since his real mother never recognised him as her son, and he lived in Mme Rousseau's house until he was middle-aged. The first school that d'Alembert attended was a private school, his education being arranged by his father.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/D'Alembert.html (1 of 6) [2/16/2002 11:06:10 PM]

D'Alembert

His father died in 1726 when d'Alembert was nine years old and he left him just enough money to give him security. The Destouches family continued to look after d'Alembert's education and they arranged for him to enter the Jansenist Collège des Quatre Nations. He enrolled in the name of Jean-Baptiste Daremberg but soon changed his name to Jean d'Alembert. The Collège des Quatre Nations was an excellent place for d'Alembert to study mathematics even though the course was elementary. The mathematics course, given by Professor Carron, was based on Varignon's lectures and d'Alembert was able to make use of the excellent mathematics library at the Collège. As well as the mathematical training, he learnt about Descartes' physical ideas at the Collège but, when he formed his own ideas later in his life, he would have little respect for the views of Descartes. The main aim of the Jansenist Collège des Quatre Nations was to produce scholars who could become experts in theology and argue the Jansenist case against the Jesuits. However, d'Alembert was turned off the study of theology at the Collège. After graduating in 1735 he decided that he would make a career in law but his real passion was for mathematics and he continued to work in his spare time on that subject. In 1738 d'Alembert qualified as an advocate but he seems to have decided that this was not the career for him. The following year d'Alembert studied medicine but this was a topic that he found even worse than theology. Of all the topics he had studied the one that he had real enthusiasm for was mathematics and his progress in this was quite remarkable, particularly given that he had studied almost exclusively on his own and at a time when he was supposed to be studying foe other qualifications. In July 1739 d'Alembert read his first paper to the Paris Academy of Science on some errors he had found in Reyneau's standard text Analyse démontrée which were not of great significance but marked the start of his mathematical career. In 1740 he submitted a second work on the mechanics of fluids which was praised by Clairaut. In May 1741 d'Alembert was admitted to the Paris Academy of Science, on the strength of these and papers on the integral calculus. It took some determination on his part, submitting three unsuccessful applications in quick succession, before his appointment. Before discussing d'Alembert's contributions it is useful to discuss his personality, which was to have a major effect on the way his scientific work was to develop. In one sense d'Alembert's life was uneventful. He travelled little and worked at the Paris Academy of Science and the French Academy all his life. On another level his life was one of great drama as he argued with almost everyone around him. As stated in [5]:D'Alembert was always surrounded by controversy. ... he was a lightning rod which drew sparks from all the foes of the philosophes. ... Unfortunately he carried this... pugnacity into his scientific research and once he had entered a controversy, he argued his cause with vigour and stubbornness. He closed his mind to the possibility that he might be wrong... Despite this tendency to quarrel with all around him, his contributions were truly outstanding. D'Alembert helped to resolve the controversy in mathematical physics over the conservation of kinetic energy by improving Newton's definition of force in his Traité de dynamique which he published in 1743. This also contains d'Alembert's principle of mechanics. This is an important work and the preface contains a clear statement by d'Alembert of an attempt to lay a firm foundation for mechanics. In [5] d'Alembert's ideas, as presented in this preface, are described:... d'Alembert was a mathematician, not a physicist, and he believed mechanics was just as much a part of mathematics as geometry or algebra. Rational mechanics was a science based on simple necessary principles from which all particular phenomenon could be http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/D'Alembert.html (2 of 6) [2/16/2002 11:06:10 PM]

D'Alembert

deduced by rigorous mathematical methods. ... d'Alembert thought mechanics should be made into a completely rationalistic mathematical system. D'Alembert had begun to read parts of his Traité de dynamique to the Academy in late 1742 but soon afterwards Clairaut began to read his own work on dynamics to the Academy. Clearly a rivalry quickly sprung up and d'Alembert stopped reading the work to the Academy and rushed into print with the treatise. The two mathematicians had come up with similar ideas and indeed the rivalry was to become considerably worse in the next few years. D'Alembert stated his position clearly that he believed mechanics to be based on metaphysical principles and not on experimental evidence. He seems not to have realised in his reading of Newton's Principia how strongly Newton based his laws of motion on experimental evidence. For d'Alembert these laws of motion were logical necessities. In 1744 d'Alembert applied his results to the equilibrium and motion of fluids and published Traité de l'equilibre et du mouvement des fluides. This work gave an alternative treatment of fluids to the one published by Daniel Bernoulli. D'Alembert thought it a better approach, of course, as one might expect, Daniel Bernoulli did not share this view. D'Alembert became unhappy at the Paris Academy, almost certainly because of his rivalry with Clairaut and disagreements with others. His position became even less happy in 1745 when Maupertuis left Paris to take up the post of head of the Berlin Academy where, at that time, Euler was working. In around 1746 d'Alembert's life took a rather sudden change. This is described in [4] as follows:Until [1746] he had been satisfied to lead a retired but mentally active existence at the house of his foster-mother. In 1746 he was introduced to Mme Geoffrin, the rich, imperious, unintellectual but generous founder of a salon to which d'Alembert was suddenly invited. He soon entered a social life in which, surprisingly enough, he began to enjoy great success and popularity. Around the same time d'Alembert began to become involved in a major project, namely editing the Encyclopédie with Diderot. He was contracted as an editor to cover mathematics and physical astronomy but his work covered a wider field. When the first volume appeared in 1751 it contained a Preface written by d'Alembert which was widely acclaimed as a work of great genius. Buffon said that:It is the quintessence of human knowledge... D'Alembert worked on the Encyclopédie for many years. In fact he wrote most of the mathematical articles in this 28 volume work. However, he continued his mathematical work while working on the Encyclopédie. He was a pioneer in the study of partial differential equations and he pioneered their use in physics. His work on this topic first appeared in an article which he submitted for the 1747 prize of the Prussian Academy Réflexions sur la cause générale des vents which indeed he won the prize. Euler, however, saw the power of the methods introduced by d'Alembert and soon developed these far further than had d'Alembert. In fact this work by d'Alembert on the winds suffers from a defect which was typical of all of his work, namely it was mathematically very sound but was based on rather poor physical evidence. In this case, for example, d'Alembert assumed that the winds were generated by tidal effects on the atmosphere and heating of the atmosphere played only a very minor role. Clairaut attacked d'Alembert's methods [5]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/D'Alembert.html (3 of 6) [2/16/2002 11:06:10 PM]

D'Alembert

In order to avoid delicate experiments or long tedious calculations, in order to substitute analytical methods which cost them less trouble, they often make hypotheses which have no place in nature; they pursue theories that are foreign to their object, whereas a little constancy in the execution of a perfectly simple method would have surely brought them to their goal. A heated argument between d'Alembert and Clairaut resulted in the two fine mathematicians trading insults in the scientific journals of the day. The year 1747 was an important one for d'Alembert in that a second important work of his appeared in that year, namely his article on vibrating strings. The article contains the first appearance of the wave equation in print but again suffers from the defect that he used mathematically pleasing simplifications of certain boundary conditions which led to results which were at odds with observation. Euler had learnt of d'Alembert's work in around 1743 through letters from Daniel Bernoulli. However, Daniel Bernoulli became highly critical of d'Alembert after reading his Traité de l'equilibre et du mouvement des fluides for reasons we noted above. When d'Alembert won the prize of the Prussian Academy with his essay on winds he produced a work which Euler considered superior to that of Daniel Bernoulli. Certainly at this time Euler and d'Alembert were on very good terms with Euler having high respect for d'Alembert's work and the two corresponded on many topics of mutual interest. However relations between Euler and d'Alembert soon took a turn for the worse after the dispute in the Berlin Academy involving Samuel König which began in 1751. The situation became more relevant to d'Alembert in 1752 when he was invited to became President of the Berlin Academy. Another reason for d'Alembert to feel angry with Euler was that he felt that Euler was stealing his ideas and not giving him due credit. In one sense d'Alembert was justified but on the other hand his work was usually so muddled that Euler could not follow it and resorted to starting from scratch to clarify the problem being solved. The Paris Academy had not been a place for d'Alembert to publish after he fell out with colleagues there and he was sending his mathematical papers to the Berlin Academy during the 1750s. However Euler was unhappy to publish these works and d'Alembert stopped publishing his mathematical articles, collecting them together and publishing them as Opuscules mathématique which appeared in eight volumes between 1761 and 1780. Again Frederick II, the King of Prussia, tried to persuade d'Alembert to accept the presidency of the Berlin Academy. Euler was strongly opposed to this and wrote to Lagrange (see [5]):... d'Alembert has tried to undermine [my solution to the vibrating strings problem] by various cavils, and that for the sole reason that he did not get it himself. ... He thinks he can deceive the semi-learned by his eloquence. ... He wished to publish in our journal not a proof, but a bare statement that my solution is defective. ... From this you can judge what an uproar he would let loose if he were to become our president. Euler need not have feared however, for d'Alembert visited Frederick II for three months in 1764, turned down the offer of the presidency again, and tried to persuade Frederick II to made Euler president. This was not the only offer d'Alembert turned down. He also turned down an invitation from Catherine II to go to Russia as a tutor for her son.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/D'Alembert.html (4 of 6) [2/16/2002 11:06:10 PM]

D'Alembert

D'Alembert made other important contributions to mathematics which we have not yet mentioned. In an article entitled Différentiel in volume 4 of Encyclopédie written in 1754, he suggested that the theory of limits be put on a firm foundation. He was one of the first to understand the importance of functions and, in this article, he defined the derivative of a function as the limit of a quotient of increments. His ideas on limits led him to the test for convergence, known today as d'Alembert's ratio test, which appears in Volume 5 of Opuscules mathématique. In the latter part of his life d'Alembert turned more towards literature and philosophy. D'Alembert's philosophical works appear mainly in the five volume work Mélanges de littérature et de philosophie which appeared between 1753 and 1767. In this work he sets out his skepticism concerning metaphysical problems. He accepts the argument in favour of the existence of God, based on the belief that intelligence cannot be a product of matter alone. However, although he took this public view in his books, evidence from his friends showed that he was persuaded by Diderot towards materialism before 1770. D'Alembert was elected to the French Academy on 28 November 1754. In 1772 he was elected perpetual secretary of the French Academy and spent much time writing obituaries for the academy [1]:He became the academy's most influential member, but, in spite of his efforts, that body failed to produce anything noteworthy in the way of literature during his pre-eminence. D'Alembert complained from 1765, after a bout of illness, that his mind was no longer able to concentrate on mathematics. In 1777, in a letter to Lagrange, he showed how much he regretted this:What annoys me the most is the fact that geometry, which is the only occupation that truly interests me, is the one thing that I cannot do. All that I do in literature, although very well received in our public sessions of the French Academy, is for me only a way to fill the time for lack of anything better to do. He suffered bad health for many years and his death was as the result of a bladder illness. As a known unbeliever, d'Alembert was buried in a common unmarked grave. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (35 books/articles)

Some Quotations (6)

A Poster of Jean d'Alembert

Mathematicians born in the same country

Cross-references to History Topics

1. Matrices and determinants 2. The fundamental theorem of algebra 3. Orbits and gravitation 4. Abstract linear spaces

Other references in MacTutor

1. Chronology: 1720 to 1740 2. Chronology: 1740 to 1760 3. Chronology: 1760 to 1780

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/D'Alembert.html (5 of 6) [2/16/2002 11:06:10 PM]

D'Alembert

Honours awarded to Jean d'Alembert (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1748

Lunar features

Crater d'Alembert

Paris street names

Rue d'Alembert (14th Arrondissement)

Other Web sites

1. Rouse Ball 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/D'Alembert.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/D'Alembert.html (6 of 6) [2/16/2002 11:06:10 PM]

D'Ovidio

Enrico D'Ovidio Born: 11 Aug 1842 in Campobasso, Italy Died: 21 March 1933 in Turin, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Enrico D'Ovidio studied in Naples, southwest of his home town of Campobasso, where he prepared to enter the School of Bridges and Roads. He studied there for a time and attended lectures by Battaglini and Fergola whose influence made D'Ovidio become interested in an academic career. Already at this time D'Ovidio was beginning original research in mathematics although he had not taken a standard university course. He wrote some short articles on determinants and conics and these were published in the early volumes of Battaglini's new journal Giornale di Matematiche which was founded in 1863. D'Ovidio began school teaching but was granted an honorary degree in mathematics by the University of Naples despite never having taken a degree course. He sat no written examination papers for this degree since it was felt that he had already proved his mathematical abilities. In 1869 D'Ovidio published a geometry text for schools and then, in 1872, Beltrami persuaded him to enter the competition for the Chair of Algebra and Analytic Geometry at the University of Turin. D'Ovidio was reluctant to leave the Naples area of Italy where he was close to his family but he decided to enter the competition and he was offered the post. D'Ovidio was to work for 46 years in the University of Turin. He was chairman of the Faculty of Science in 1879-80 and rector of the University between 1880 and 1885. Another spell as chairman of the Faculty of Science between 1893 and 1907 ended when he was appointed Commissioner of the Polytechnic of Turin.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/D'Ovidio.html (1 of 2) [2/16/2002 11:06:12 PM]

D'Ovidio

Euclidean and noneuclidean geometry were the areas of special interest to D'Ovidio. He built on the geometric ideas which had been introduced by Lobachevsky, Bolyai, Riemann and Cayley. D'Ovidio's most important work is probably his paper of 1877 The fundamental metric functions in spaces of arbitrarily many dimensions with constant curvature. D'Ovidio also worked on binary forms, conics and quadrics. He had two famous assistants, Peano (1880-83) and Corrado Segre (1883-84). D'Ovidio and Corrado Segre built an important school of geometry at Turin. Many honours came D'Ovidio's way. He was elected to the Academy of Sciences of Turin in 1878 and to the Accademia dei Lincei from 1883. Kennedy writes in [1]:He was named a senator in March 1905, but there were rumours that this was due to a mix-up and that the nomination was intended for his brother Francesco, the noted philologist, who was in fact named a senator a few months later. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/D'Ovidio.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/D'Ovidio.html (2 of 2) [2/16/2002 11:06:12 PM]

Dandelin

Germinal Pierre Dandelin Born: 12 April 1794 in Le Bourget, France Died: 15 Feb 1847 in Brussels, Belgium Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Germinal Dandelin's father, who was an administrator, was French but his mother came from Hainaut, now in Belgium. Dandelin studied at Ghent, then in 1813 he entered the Ecole Polytechnique in Paris. However his career was to be very much influenced by the political events of these turbulent times. In 1813 Dandelin had volunteered to fight the British. In March 1814 the Treaty of Chaumont united Austria, Russia, Prussia and Britain in the aim of defeating Napoleon. When the allied armies arrived near Paris on 30 March 1814, Dandelin was in the opposing French army and was wounded on that day. Napoleon abdicated on 6 April, but in the following year he returned for the 100 days. During Napoleon's time back in control of France, Dandelin worked at the Ministry of the Interior under the command of Carnot. After Napoleon was defeated at Waterloo, Dandelin returned to Belgium. He became a citizen of the Netherlands in 1817. Back in Belgium Dandelin continued his military career as an engineer. From 1825 he spent five years as professor of mining engineering at Liège. Then, in 1830, he was back in the thick of the Revolution which erupted that year. From 1835 he was in the Belgium army, assigned posts in charge of building fortifications in Namur, Liège and, later, in Brussels. Dandelin's early mathematical influence was Quetelet, who was two years younger than him, and his early interests were in geometry. Dandelin has an important theorem on the intersection of a cone and its inscribed sphere with a plane, discovered in 1822, named after him. This theorem shows that if a cone is intersected by a plane in a conic, then the foci of the conic are the points where this plane is touched by the spheres inscribed in the cone. In 1826 he generalised his theorem to a hyperboloid of revolution, rather than a cone, relating Pascal's hexagon, Brianchon's hexagon and the hexagon formed by the generators of the hyperboloid. Dandelin also worked on stereographic projection of a sphere on a plane (1827), statics, algebra and probability. He gave a method of approximating the roots of an algebraic equation, now named the Dandelin-Gräffe method. The history of the Dandelin-Gräffe method is discussed in [2] and [5]. Among the honours which Dandelin received was election to the Royal Academy of Sciences in Brussels in 1825. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dandelin.html (1 of 2) [2/16/2002 11:06:13 PM]

Dandelin

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Other Web sites

1. Free University of Brussels, Belgium (A colour picture illustrating Dandelin's theorem) 2. Eric's World of Mathematics (Another picture)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dandelin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dandelin.html (2 of 2) [2/16/2002 11:06:13 PM]

Danti

Egnatio Pellegrino Rainaldi Danti Born: April 1536 in Perugia, Italy Died: 19 Oct 1586 in Alatri, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

At the age of 18 Egnatio Danti entered the Dominican Order having already attended courses at the University of Perugia. In 1562 he was asked by Cosimo I de' Medici, the second duke of Florence, to prepare maps and a huge terrestrial globe which is still preserved. The maps were hung on the walls in the Palazzo Vecchio in Florence. Cosimo became first grand duke of Tuscany in 1569 and he appointed Danti to be professor of mathematics at Pisa. However Cosimo died in 1574 and Danti's position became insecure. In 1576 he had to leave Tuscany and he went to Bologna. From 1577 Danti mapped the area around Perugia, and in the same year he was appointed professor of mathematics at Bologna. He also accepted a commission to map the Papal states. In 1574 Danti detected the 11 day error in the calendar and from that time on became a leading figure in calendar reform. He designed and published work on astronomical instruments, an interest which led him to discover the 11 day error. He built an instrument to determine the true equinox so that the calendar might be corrected and constructed an astronomical quadrant. He built other instruments, namely ones to indicate the wind direction and a surveying instrument. Among Danti's mathematical publications are editions of some of Euclid's works. He ended his career back in the church being appointed Bishop of Altri in 1583. He remained there until his death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Julian and Gregorian calendars

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Danti.html (1 of 2) [2/16/2002 11:06:14 PM]

Danti

Other Web sites

1. The Galileo Project 2. The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Danti.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Danti.html (2 of 2) [2/16/2002 11:06:14 PM]

Dantzig

David van Dantzig Born: 23 Sept 1900 in Rotterdam, Netherlands Died: 22 July 1959 in Amsterdam, Netherlands Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

David van Dantzig was at secondary school when he wrote his first mathematics paper. He was only thirteen years old at the time. However, his main interest in secondary school was not mathematics, rather it was chemistry. After leaving school he continued with his studies of chemistry, but this he did not enjoy and when he was forced to give up his academic studies to help support his family van Dantzig took on a number of jobs purely to make money. By now van Dantzig knew that mathematics was the subject which he really wanted to study but he was not in a position to do so, both because he had to earn money and also because he did not have the necessary school qualifications. He put in hours of work on mathematics in the evenings after finishing his money earning tasks for the day. He took the state mathematics examinations in 1921, at a higher level the following year and again in 1923 he passed at a higher level still. Entering the University of Amsterdam to study mathematics he soon passed examinations which took him essentially to Master's Degree level. Van Dantzig became an assistant to Schouten in 1927 at Delft Technical University. Then, for a short time, he taught at a teacher training institution, but he returned to Delft as a lecturer in 1932. This was the year in which he received his doctorate from Gröningen for a thesis which he submitted in 1931 Studiën over topologische Algebra. In this work he coined the now familiar term topological algebra but the thesis is memorable in other ways too. It [1]:... is a fine example of mathematical style: it consists of a concise string of definitions and theorems organised in such a way that in this context each theorem is obvious and none needs a proof. He was promoted to extraordinary professor at Delft in 1938 and then an ordinary professor in 1940. The Dutch had tried to remain neutral when World War II broke out in 1939 but in the spring of 1940 German troops, in a strategic move on their way to attack France, entered Holland and the Dutch were defeated in a week. Van Dantzig was dismissed from his chair when the Germans occupied Holland and he was forced to move with his family from the Hague to Amsterdam. After the war ended, he was appointed professor at the University of Amsterdam in 1946. In Amsterdam he was the cofounder of the research and service institution, the Mathematisch Centrum. He played a major role in both this Centre and in the University of Amsterdam where he continued to hold his chair

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dantzig.html (1 of 2) [2/16/2002 11:06:16 PM]

Dantzig

until his death. Van Dantzig studied differential geometry, electromagnetism and thermodynamics. His most important work was in topological algebra and in addition to his doctoral thesis which we mentioned above, he wrote a whole series of papers on topological algebra. He studied metrisation of groups, rings and fields. One paper classified fields with a locally compact topology. After the Second World War, van Dantzig changed topics and worked on probability and statistics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles)

Some Quotations (3)

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dantzig.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dantzig.html (2 of 2) [2/16/2002 11:06:16 PM]

Dantzig_George

George Dantzig Born: 8 Nov 1914 in Portland, Oregon, USA

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Dantzig studied mathematics at the University of Maryland, receiving his A.B. in 1936. The following year he received an M.A. in mathematics from the University of Michigan. Dantzig worked as a Junior Statistician in the U.S. Bureau of Labor Statistics from 1937 to 1939, then, from 1941 to 1946, he was head of the Combat Analysis Branch, U.S.A.F. Headquarters Statistical Control. He received his doctorate in mathematics from the University of California, Berkeley in 1946. In that year he was appointed Mathematical Advisor for USAF Headquarters. In 1947 Dantzig made the contribution to mathematics for which he is most famous, the simplex method of optimisation. It grew out of his work with the U.S. Air Force where he become an expert on planning methods solved with desk calculators. In fact this was known as "programming", a military term that, at that time, referred to plans or schedules for training, logistical supply or deployment of men. Dantzig mechanised the planning process by introducing "linear programming", where "programming" has the military meaning explained above. The importance of linear programming methods was described, in 1980, by Laszlo Lovasz who wrote:If one would take statistics about which mathematical problem is using up most of the computer time in the world, then ... the answer would probably be linear programming. Also in 1980 Eugene Lawler wrote:[Linear programming] is used to allocate resources, plan production, schedule workers, plan investment portfolios and formulate marketing (and military) strategies. The versatility and economic impact of linear programming in today's industrial world is truly awesome. Dantzig however modestly wrote:The tremendous power of the simplex method is a constant surprise to me.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dantzig_George.html (1 of 3) [2/16/2002 11:06:18 PM]

Dantzig_George

Dantzig became a research mathematician with the RAND Corporation in 1952, then in 1960 he was appointed professor at Berkeley and Chairman of the Operations Research Center. While there he wrote Linear programming and extensions (1963). In 1966 he was appointed Professor of Operations Research and Computer Science at Stanford University. His work in a wide range of topics related to optimisation and operations research over the years has been of major importance. However, writing in 1991, Dantzig noted that:... it is interesting to note that the original problem that started my research is still outstanding - namely the problem of planning or scheduling dynamically over time, particularly planning dynamically under uncertainty. If such a problem could be successfully solved it could eventually through better planning contribute to the well-being and stability of the world. Dantzig has received many honours including the Von Neumann Theory Prize in Operational Research in 1975. His work is summarised by Stanford University as follows:A member of the National Academy of Engineering, the National Academy of Science, the American Academy of Arts and Sciences and recipient of the National Medal of Science, plus eight honorary degrees, Professor Dantzig's seminal work has laid the foundation for much of the field of systems engineering and is widely used in network design and component design in computer, mechanical, and electrical engineering. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Other references in MacTutor

Chronology: 1940 to 1950

Honours awarded to George Dantzig (Click a link below for the full list of mathematicians honoured in this way) AMS Gibbs Lecturer

1990

Other Web sites

Siam News

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dantzig_George.html (2 of 3) [2/16/2002 11:06:18 PM]

Dantzig_George

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Dantzig_George.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dantzig_George.html (3 of 3) [2/16/2002 11:06:18 PM]

Darboux

Jean Gaston Darboux Born: 14 Aug 1842 in Nimes, Gard, Languedoc, France Died: 23 Feb 1917 in Paris, France

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gaston Darboux attended the Lycée at Nimes and then the Lycée at Montpellier. In 1861 he entered the Ecole Polytechnique and then the Ecole Normale Supérieure. While he was a student his great talent for mathematics became clear to those around him. While at the Ecole Normale Supérieure, and still a student, he published his first paper on orthogonal surfaces. Darboux had studied the work of Lamé, Dupin and Bonnet on orthogonal systems of surfaces. Darboux generalised results of Kummer giving a system defined by a single equation with many interesting properties. He announced his results to the Académie des Sciences on 1 August 1864, and on the same day Moutard announced that he had also discovered the same system. These results were included in Darboux's doctoral thesis Sur les surfaces orthogonales for which he awarded his doctorate in 1866. Darboux was appointed to the Collège de France for the academic year 1866-1867, then he taught at the Lycée Louis le Grand (where Galois was educated) between 1867 and 1872. In 1872 he was appointed to the Ecole Normale Supérieure where he taught until 1881. From 1873 to 1878 he was suppléant to Liouville in the chair of rational mechanics at the Sorbonne. Then, in 1878 he became suppléant to Chasles in the chair of higher geometry, also at the Sorbonne. Two years later Chasles died and Darboux succeeded him to the chair of higher geometry, holding this chair until his death. He was dean of the Faculty of Science from 1889 to 1903. Darboux made important contributions to differential geometry and analysis. D J Struik writes in [1]:... he followed in the spirit of Gaspard Monge, and Darboux's spirit can be detected in the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Darboux.html (1 of 3) [2/16/2002 11:06:20 PM]

Darboux

work of Elie Cartan. Again Struik writes [1]:Relying on the classical results of Monge, Gauss, and Dupin, Darboux fully used, in his own creative way, the results of his colleagues Bertrand, Bonnet, Ribaucour, and others. He may now be best known for the Darboux integral which is named after him. This integral was introduced in a paper on differential equations of the second order which he wrote in 1870. In 1875 he gave his way of looking at the Riemann integral, defining upper and lower sums and defining a function to be integrable if the difference between the upper and lower sums tends to zero as the mesh size gets smaller. In 1873 Darboux wrote a paper on cyclides and between 1887 and 1896 he produced four volumes on infinitesimal geometry which included most of his earlier work it was titled Leçons sur la théorie général des surfaces et les applications géométriques du calcul infinitésimal. Included in volume four of this work is a discussion of one surface rolling on another surface. In particular he studied the geometrical configuration generated by points and lines which are fixed on the rolling surface. Eisenhart says of this work in [7]:His geometrical proofs of the theorems dealing with rolling surfaces ... are as pure as they are simple and beautiful. Darboux also studied the problem of finding the shortest path between two points on a surface. Work in this area was also done at around the same time by Zermelo and by Kneser. Darboux's success in research is discussed by Eisenhart in [7]:Darboux's ability was based on a rare combination of geometrical fancy and analytical power. He did not sympathise with those who use only geometrical reasoning in attacking geometrical problems, nor with those who feel that there is a certain virtue in adhering strictly to analytic processes. ... brilliant are his reductions of various geometrical problems to a common analytic basis, and their solution and development from a common point of view. However Darboux was also renowned as an exceptional teacher, writer and administrator. Eisenhart writes [7]:His writings possess not only content but singular finish and refinement of style. In the presentation of results the form of exposition was carefully studied. Darboux's varied powers combined with his personality in making him a great teacher, so that he always had about him a group of able students. In common with Monge he was not content with discoveries, but he felt that it was equally important to make disciples. Darboux is known for a wider range of mathematics to that described above. Struik writes in [1]:Darboux also did research in function theory, algebra, kinematics and dynamics. His appreciation of the history of science is shown in numerous addresses, many given as éloges before the Academy. He also edited Joseph Fourier's "Oeuvres" (1888-1890). Of course Darboux received many honours for his work. Lebon in [3] lists over 100 Scientific Societies which elected Darboux as a member. He was elected to the Royal Society of London in 1902, winning its

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Darboux.html (2 of 3) [2/16/2002 11:06:20 PM]

Darboux

Sylvester Medal in 1916. In 1884 he was elected to the Académie des Sciences, becoming its secretary in 1900. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) A Poster of Gaston Darboux

Mathematicians born in the same country

Honours awarded to Gaston Darboux (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1902

Royal Society Sylvester Medal

Awarded 1916

Paris street names

Rue Gaston Darboux (18th Arrondissement)

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Darboux.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Darboux.html (3 of 3) [2/16/2002 11:06:20 PM]

Darwin

George Howard Darwin Born: 9 July 1845 in Downe, Kent, England Died: 7 Dec 1912 in Cambridge, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Darwin was a son of Charles Darwin. In 1883 he became Plumian professor of astronomy and experimental philosophy at Cambridge University. He studied tidal effects on the planets. In particular, using methods introduced by Laplace and Thomson, he discussed the effects of tidal action on the Sun-Earth-Moon system. One of his theories, namely that the Moon was pulled from a molten Earth early in its history by tidal action of the Sun, is now considered incorrect. Darwin made a major study of the three-body problem in the case of the orbits of the Sun-Earth-Moon system. He also studied the stability of rotating fluids, again motivated by his interest in the Moon being formed in fluid form from a molten Earth. His conclusions that a pear shaped rotating mass is stable is today thought to be incorrect. Despite the fact that we do not accept Darwin's conclusions today, he is important in being the first to apply mathematical techniques to study the evolution of the Sun-Earth-Moon system. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) A Poster of George Darwin

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Darwin.html (1 of 2) [2/16/2002 11:06:22 PM]

Darwin

Honours awarded to George Darwin (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1879

Royal Society Copley Medal

Awarded 1911

Royal Society Royal Medal

Awarded 1884

Royal Society Bakerian lecturer

1891

Planetary features

Crater Darwin on Mars

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Darwin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Darwin.html (2 of 2) [2/16/2002 11:06:22 PM]

Dase

Johann Martin Zacharias Dase Born: 1824 in Hamburg, Germany Died: 1861 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Zacharias Dase had incredible calculating skills but little mathematical ability. He gave exhibitions of his calculating powers in Germany, Austria and England. While in Vienna in 1840 he was urged to use his powers for scientific purposes and he discussed projects with Gauss and others. Dase used his calculating ability to calculate to 200 places in 1844. This was published in Crelle's Journal for 1844. Dase also constructed 7 figure log tables and produced a table of factors of all numbers between 7 000 000 and 10 000 000. Gauss requested that the Hamburg Academy of Sciences allow Dase to devote himself full-time to his mathematical work but, although they agreed to this, Dase died before he was able to do much more work. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Cross-references to History Topics

Memory, mental arithmetic and mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dase.html (1 of 2) [2/16/2002 11:06:23 PM]

Dase

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dase.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dase.html (2 of 2) [2/16/2002 11:06:23 PM]

Davenport

Harold Davenport Born: 30 Oct 1907 in Huncoat, near Accrington, Lancashire, England Died: 9 June 1969 in Cambridge, Cambridgeshire, England

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Harold Davenport attended Accrington Grammar School. His main interests were mathematics and chemistry, and in 1924 he obtained a scholarship to attend Manchester University. He studied mathematics at Manchester being taught complex analysis by Mordell and applied mathematics by Milne. He graduated in 1927. After Manchester he went to Trinity College, Cambridge to take another 'first degree' which was a common thing to do at that time. Coxeter was among the friends he made at Cambridge. Coxeter wrote:When Davenport was working for the Tripos he seemed wonderfully relaxed. He would give me a cheerful welcome whenever I dropped in to see him under the Clock in Trinity Great Court. I would find him listening to Scheherazade on the phonograph or reading Gibbon's 'Decline and Fall' for the third time. Davenport was most attracted by Littlewood's lectures on the theory of primes and Besicovitch on almost periodic functions. Davenport wrote a Ph.D. thesis at Cambridge under Littlewood's supervision. He was awarded a Trinity fellowship in 1932 and soon after taking up the fellowship he visited Hasse in Marburg and wrote an important joint work with him. Davenport met Heilbronn while in Germany and they worked together for many years. After returning to Cambridge his research struck an incredibly rich vein and he published a great number of papers. At this time life in Cambridge was enriched by a large number of visiting mathematicians who were escaping from the Nazi threat on the continent. Those who interacted with Davenport included Richard Rado, Hirsch, Courant, Taussky (later Taussky-Todd), Kober and Mahler. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Davenport.html (1 of 3) [2/16/2002 11:06:25 PM]

Davenport

He left Cambridge in 1937, accepting an offer from Mordell of an assistant lectureship at the University of Manchester. At Manchester he was influenced by Mordell to become interested in the geometry of numbers and Diophantine approximation. While at Manchester he had Mahler, Erdös and Beniamino Segre as colleagues. In 1941 Davenport was appointed to the chair of mathematics at the University College of North Wales at Bangor. Then London in 1945 he succeeded Jeffrey as Astor professor of mathematics in London. In 1958 he returned to Cambridge as Rouse Ball professor. Davenport worked on number theory, in particular the geometry of numbers, Diophantine approximation and the analytic theory of numbers. He wrote a number of important textbooks and monographs. The higher arithmetic (1952) was a book written at a low level in an attempt to bring results in number theory before as wide an audience as possible. He wrote a monograph Analytic methods for Diophantine equations and Diophantine inequalities (1962) which includes many of his contributions extending the Hardy-Littlewood method. He also wrote an important monograph on the analytic approach to the theory of the distribution of primes Multiplicative number theory (1967). Davenport was elected a Fellow of the Royal Society in 1940 while still an assistant lecturer, receiving its Sylvester Medal in 1967:... in recognition of his many distinguished contributions to the theory of numbers. He was President of the London Mathematical Society during 1957-59, and was awarded the Berwick Prize of the Society in 1954. Davenport described his philosophy of mathematics in the following way:Mathematicians are extremely lucky, they are paid for doing what they would by nature have to do anyway. One should not have a non-teaching fellowship too long, there comes a time when one must make a contribution to society. Great mathematics is achieved by solving difficult problems not by fabricating elaborate theories in search of a problem. Always a heavy smoker (he tried to give up the habit several times but always failed), Davenport succumbed to lung cancer at a young age. His influence on those around him is summed up in [6] as follows:... the extent which he helped others can only be guessed, he was probably responsible for encouraging work at least as extensive as his own. ... He made his collaborators and colleagues his friends, and gave them generously of his humour and wisdom. He made a practice of writing helpful letters to all who approached him on mathematical matters whether they were professionals, students, amateurs or even cranks. By correspondence and by direct contact he stimulated and encouraged many mathematicians to do much of their best mathematics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Davenport.html (2 of 3) [2/16/2002 11:06:25 PM]

Davenport

List of References (7 books/articles)

A Quotation

A Poster of Harold Davenport

Mathematicians born in the same country

Honours awarded to Harold Davenport (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1940

Royal Society Sylvester Medal

Awarded 1967

London Maths Society President

1957 - 1959

LMS Berwick Prize winner

1954

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Davenport.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Davenport.html (3 of 3) [2/16/2002 11:06:25 PM]

Davidov

August Yulevich Davidov Born: 15 Dec 1823 in Libav, Russia Died: 22 Dec 1885 in Moscow, Russia

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

August Yulevich Davidov's father was a medical doctor. Davidov studied at the University of Moscow, enrolling in 1841 and graduating four years later. In 1845 Davidov submitted a thesis The theory of equilibrium of bodies immersed in a liquid for his Master's degree. It is worth noting that the Russian Master's degree was of a similar standard to the current PhD in Britain or the United States. The thesis continued work on the equilibrium of floating bodies which had been started by Euler, Poisson and Dupin. Grigorian write in [1] that in this thesis:[Davidov] was the first to give a general analytic method for determining the position of equilibrium of a floating body, applied his method to the determination of positions of equilibrium of bodies, explained the analytic theory by geometric constructions, and investigated the stability of equilibrium of floating bodies. He began teaching at Moscow University in 1850, and he obtained a doctorate in 1851 for another thesis on fluids, this time on The theory of capillary phenomena. Again we should note that the level of the Russian doctorate was close to the German habilitation. He was promoted to Professor after being awarded his doctorate and worked at Moscow university for the whole of his life. Before leaving the topic of Davidov's two theses on fluids we note that the quality of these can be judged by the fact that they were both awarded the Demidov Prize by the St Petersburg Academy of Sciences. One of Davidov's colleagues at the Moscow University was Nikolai Dmetrievich Brashman (who had himself been awarded the Demidov Prize fifteen years earlier). He had been appointed to the university in 1834 and was a senior figure there in the area of applied mathematics, in particular mechanics. He greatly influenced Davidov's career and his mathematical research.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Davidov.html (1 of 2) [2/16/2002 11:06:27 PM]

Davidov

Together Davidov and Brashman were joint founders of the Moscow Mathematical Society in 1866, the year of Brashman's death. Davidov the became the first president of the Society and held this role for nearly twenty years until his death in 1885. He also did fine work for the Faculty of Physics and Mathematics at Moscow University being its head for twelve years. As well as his work on the equilibrium of a floating body, Davidov also worked on partial differential equations, elliptic functions and the application of probability to statistics. He wrote [1]:... a number of excellent texts for secondary schools. Of these, the geometry and algebra textbooks enjoyed special success and were republished many times. Through the next half century the geometry text underwent thirty-nine editions and the algebra text twenty-four. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Davidov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Davidov.html (2 of 2) [2/16/2002 11:06:27 PM]

Davies

Evan Tom Davies Born: 24 Sept 1904 in Pencader, Carmarthenshire, Wales Died: 8 Oct 1973 in Waterloo, Ontario, Canada

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Evan Tom Davies entered the University College of Wales at Aberystwyth in 1921. He graduated in 1926 and spent a number of years working on the Continent of Europe. His first visit was to Rome where he studied under Levi-Civita. He spent two years in Rome before moving to his next European capital Paris. In Paris he spent time at the Sorbonne and at the Collège de France where he was greatly influenced by Cartan. In 1930 Davies returned from the Continent to take up a post at King's College, part of the University of London. His first appointment there was as a Lecturer but he was later promoted to Reader. The University of Southampton offered Davies the chair of mathematics in 1946 and, after accepting, he spent the rest of his career there until his retirement in 1969 at the age of 65. Retirement did not mean an end to mathematical research for Davies for, after he retired, he went to Canada to spend two years as Professor of Mathematics at the University of Calgary. In 1971 Davies, remaining in Canada, took up an appointment as Professor of Mathematics at the University of Waterloo. Davies was an editor of Aequationes Mathematicae and, on his death, his fellow editors, writing in [5], described him as:... a mathematician of great breadth. His steady stream of publications in differential geometry and the calculus of variations attests to his authority in this field.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Davies.html (1 of 2) [2/16/2002 11:06:29 PM]

Davies

Davies had many interests outside mathematics. He was a linguist who, again quoting [5]:... was fluent in five languages and delighted in the friendship of people from all walks of life in countries throughout the world. He had a passionate regard for the Welsh culture and his Celtic enthusiasm and fine spirit endeared him to us all. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Davies.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Davies.html (2 of 2) [2/16/2002 11:06:29 PM]

De_Beaune

Florimond de Beaune Born: 7 Oct 1601 in Blois, France Died: 18 Aug 1652 in Blois, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Florimond de Beaune was born in Blois which is in central France, on the Loire River and, at the time de Beaune was born there, it was almost a second capital of France. He was educated in Paris where he went on to study law. After military service he seems to have had little need for earning a living. He became a jurist to the court in Blois and was only an amateur mathematician. He inherited an estate near Blois and he spent a lot of his time on this estate. His second marriage was one which brought de Beaune a substantial amount of money. It enabled him to build up an extensive library as well as to build himself an observatory. In fact his library contained a good proportion of astronomy works as well as mathematics volumes. De Beaune produced the first important introduction to Descartes' cartesian geometry. He wrote Notes brièves which was published in 1649 as part of the first Latin edition of Descartes' Géométrie. De Beaune proved among many results that y2 = xy + bx, y2 = -dy + bx, y2 = bx - x2 represent hyperbolas, parabolas and ellipses respectively. Two other papers by de Beaune on algebra appeared as part of the second edition of Géométrie. In fact de Beaune and Descartes were good friends and corresponded frequently on mathematical topics. Many of the top scientists French scientists of the time corresponded with de Beaune, for example Mydorge, Billy and Mersenne, and friends visited him to discuss mathematical topics, for example Descartes and Bartholin. De Beaune was also interested in mechanics and optics and wrote on these topics. However his work in these areas was never published and little is known of his contribution. His interest in optics was related to his interest in astronomy for he worked on grinding lenses, in particular experimenting with non-spherical lenses. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Beaune.html (1 of 2) [2/16/2002 11:06:30 PM]

De_Beaune

Mathematicians born in the same country Cross-references to History Topics

The rise of the calculus

Other references in MacTutor

Chronology: 1625 to 1650

Honours awarded to Florimond de Beaune (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue de Beaune (7th Arrondissement)

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/De_Beaune.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Beaune.html (2 of 2) [2/16/2002 11:06:30 PM]

De_Groot

Johannes de Groot Born: 7 May 1914 in Garrelsweer, Netherlands Died: 11 Sept 1972 in Rotterdam, Netherlands

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johannes De Groot entered the University of Groningen in 1933 to study mathematics. Although mathematics was his main subject he also studied physics and philosophy as secondary subjects. After graduating, he went on to study for his doctorate which was awarded in 1942 for a thesis entitled Topological Studies. Teaching at a secondary school was de Groot's first job. Then in 1946 he was appointed as a scientific officer at the Mathematical Centre in Amsterdam. There were two universities in Amsterdam, the University of Amsterdam (founded 1632) and the Free (Vrije) University (founded 1880). The Mathematical Centre, however, was an independent institution not attached to either of these universities. The following year, de Groot was appointed a lecturer in mathematics at the University of Amsterdam. Then in 1948 he was appointed professor of mathematics at the Technological University of Delft. Four years later, in 1952, he was appointed Professor of Mathematics at the University of Amsterdam. He retained his position at the Mathematical Centre in Amsterdam and, in 1960, was appointed Head of Pure Mathematics there. In 1964 he became Dean of the Faculty of Science at the University of Amsterdam and, at this time, he gave up his position of Head of Pure Mathematics at the Mathematical Centre but remained associated with the Mathematical Centre as Advisor to Pure Mathematics [3]:... actively participating in and in many instances decisively influencing its research activity. De Groot worked in topology and group theory. In group theory one of the topics he studied was that of groups with only trivial automorphisms. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Groot.html (1 of 2) [2/16/2002 11:06:32 PM]

De_Groot

Later de Groot worked on set-theoretic topology. He introduced the concept of co-compactness and other topological concepts. A recent book, J M Aarts and T Nishiura, Dimension and extensions (1993), has been published discussing a long-standing problem of de Groot. The main conjecture made by him has recently been solved. In a description of the contents of this book the problem of de Groot is described as follows:The problem of de Groot concerned compactifications of spaces by means of an adjunction of a set of minimal dimension. This minimal dimension was called the compactness deficiency of a space. Early success in 1942 lead de Groot to invent a generalization of the dimension function, called the compactness degree of a space, with the hope that this function would internally characterize the compactness deficiency which is a topological invariant of a space that is externally defined by means of compact extensions of a space. From this, the two extension problems were spawned. De Groot received many honours, perhaps the most prestigious of which was his election in 1969 to the Royal Dutch Academy of Sciences. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/De_Groot.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Groot.html (2 of 2) [2/16/2002 11:06:32 PM]

De_L'Hopital

Guillaume François Antoine Marquis de L'Hôpital Born: 1661 in Paris, France Died: 2 Feb 1704 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Guillaume De l'Hôpital served as a cavalry officer but resigned because of nearsightedness. From that time on he directed his attention to mathematics. L'Hôpital was taught calculus by Johann Bernoulli in 1691. L'Hôpital was a very competent mathematician and solved the brachystochrone problem. The fact that this problem was solved independently by Newton, Leibniz and Jacob Bernoulli puts l'Hôpital in very good company. L'Hôpital's fame is based on his book Analyse des infiniment petits pour l'intelligence des lignes courbes (1692) which was the first text-book to be written on the differential calculus. In the introduction L'Hôpital acknowledges his indebtedness to Leibniz, Jacob Bernoulli and Johann Bernoulli but L'Hôpital regarded the foundations provided by him as his own ideas. In this book is found the rule, now known as L'Hôpital's rule, for finding the limit of a rational function whose numerator and denominator tend to zero at a point. Article by: J J O'Connor and E F Robertson List of References (6 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_L'Hopital.html (1 of 2) [2/16/2002 11:06:33 PM]

De_L'Hopital

A Poster of Guillaume De l'Hôpital

Mathematicians born in the same country

Cross-references to History Topics

1. A visit to James Clerk Maxwell's house 2. Matrices and determinants

Cross-references to Famous Curves

1. Cycloid 2. Epicycloid 3. Hypocycloid 4. Serpentine 5. Tschirnhaus's cubic

Honours awarded to Guillaume De l'Hôpital (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Boulevard de l'Hôpital (13th Arrondissement) (Actually this is probably not named after the mathematician) 1. The Catholic Encyclopedia

Other Web sites

2. The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/De_L'Hopital.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_L'Hopital.html (2 of 2) [2/16/2002 11:06:33 PM]

De_Moivre

Abraham de Moivre Born: 26 May 1667 in Vitry (near Paris), France Died: 27 Nov 1754 in London, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

After spending five years at a Protestant academy at Sedan, Abraham de Moivre studied logic at Saumur from 1682 until 1684. He then went to Paris, studying at the Collège de Harcourt and taking private lessons in mathematics from Ozanam. A French Protestant, de Moivre emigrated to England in 1685 following the revocation of the Edict of Nantes and the expulsion of the Huguenots. He became a private tutor of mathematics and hoped for a chair of mathematics, but this was not to be since foreigners were at a disadvantage. In 1697 he was elected a fellow of the Royal Society. In 1710 de Moivre was appointed to the Commission set up by the Royal Society to review the rival claims of Newton and Leibniz to be the discovers of the calculus. His appointment to this Commission was due to his friendship with Newton. The Royal Society knew the answer it wanted! De Moivre pioneered the development of analytic geometry and the theory of probability. He published The Doctrine of Chance in 1718. The definition of statistical independence appears in this book together with many problems with dice and other games. He also investigated mortality statistics and the foundation of the theory of annuities. In Miscellanea Analytica (1730) appears Stirling's formula (wrongly attributed to Stirling) which de Moivre used in 1733 to derive the normal curve as an approximation to the binomial. In the second edition of the book in 1738 de Moivre gives credit to Stirling for an improvement to the formula. De Moivre is also remembered for his formula for http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Moivre.html (1 of 3) [2/16/2002 11:06:35 PM]

De_Moivre

(cos x + i sin x)n which took trigonometry into analysis. Despite de Moivre's scientific eminence his main income was by tutoring mathematics and he died in poverty. He, like Cardan, is famed for predicting the day of his own death. He found that he was sleeping 15 minutes longer each night and from this the arithmetic progression, calculated that he would die on the day that he slept for 24 hours. He was right! Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles) A Poster of Abraham de Moivre

Mathematicians born in the same country

Some pages from publications

The title page of The Doctrine of Chances (1718).

Cross-references to History Topics

1. Mathematical games and recreations 2. The trigonometric functions

Cross-references to Famous Curves

Frequency curve

Other references in MacTutor

1. Stirling's formula 2. Chronology: 1700 to 1720 3. Chronology: 1720 to 1740

Honours awarded to Abraham de Moivre (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1697

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Moivre.html (2 of 3) [2/16/2002 11:06:35 PM]

De_Moivre

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/De_Moivre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Moivre.html (3 of 3) [2/16/2002 11:06:35 PM]

De_Morgan

Augustus De Morgan Born: 27 June 1806 in Madura, Madras Presidency, India (now Madurai, Tamil Nadu, India) Died: 18 March 1871 in London, England

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Augustus De Morgan's father John was a Lieutenant- Colonel who served in India. While he was stationed there his fifth child Augustus was born. Augustus lost the sight of his right eye shortly after birth and, when seven months old, returned to England with the family. John De Morgan died when Augustus was 10 years old. At school De Morgan did not excel and, because of his physical disability ... he did not join in the sports of other boys, and he was even made the victim of cruel practical jokes by some schoolfellows. De Morgan entered Trinity College Cambridge in 1823 at the age of 16 where he was taught by Peacock and Whewell - the three became lifelong friends. He received his BA but, because a theological test was required for the MA, something to which De Morgan strongly objected despite being a member of the Church of England, he could go no further at Cambridge being not eligible for a Fellowship without his MA. In 1826 he returned to his home in London and entered Lincoln's Inn to study for the Bar. In 1827 (at the age of 21) he applied for the chair of mathematics in the newly founded University College London and, despite having no mathematical publications, he was appointed. In 1828 De Morgan became the first professor of mathematics at University College. He gave his inaugural lecture On the study of mathematics . De Morgan was to resign his chair, on a matter of principle, is 1831. He was appointed to the chair again in 1836 and held it until 1866 when he was to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Morgan.html (1 of 3) [2/16/2002 11:06:37 PM]

De_Morgan

resign for a second time, again on a matter of principle. His book Elements of arithmetic (1830) was his second publication and was to see many editions. In 1838 he defined and introduced the term 'mathematical induction' putting a process that had been used without clarity on a rigorous basis. The term first appears in De Morgan's article Induction (Mathematics) in the Penny Cyclopedia. (Over the years he was to write 712 articles for the Penny Cyclopedia.) The Penny Cyclopedia was published by the Society for the Diffusion of Useful Knowledge, set up by the same reformers who founded London University, and that Society also published a famous work by De Morgan The Differential and Integral Calculus. In 1849 he published Trigonometry and double algebra in which he gave a geometric interpretation of complex numbers. He recognised the purely symbolic nature of algebra and he was aware of the existence of algebras other than ordinary algebra. He introduced De Morgan's laws and his greatest contribution is as a reformer of mathematical logic. De Morgan corresponded with Charles Babbage and gave private tuition to Lady Lovelace who, it is claimed, wrote the first computer program for Babbage. De Morgan also corresponded with Hamilton and, like Hamilton attempted to extend double algebra to three dimension. In a letter to Hamilton, De Morgan writes of his correspondence with Hamilton and William Hamilton. He writes Be it known unto you that I have discovered that you and the other Sir W. H. are reciprocal polars with respect to me (intellectually and morally, for the Scottish baronet is a polar bear, and you, I was going to say, are a polar gentleman). When I send a bit of investigation to Edinburgh, the W. H. of that ilk says I took it from him. When I send you one, you take it from me, generalise it at a glance, bestow it thus generalised upon society at large, and make me the second discoverer of a known theorem. In 1866 he was a co-founder of the London Mathematical Society and became its first president. De Morgan's son George, a very able mathematician, became its first secretary. In the same year De Morgan was elected a Fellow of the Royal Astronomical Society. De Morgan was never a Fellow of the Royal Society as he refused to let his name be put forward. He also refused an honorary degree from the University of Edinburgh. He was described by Thomas Hirst thus: A dry dogmatic pedant I fear is Mr De Morgan, notwithstanding his unquestioned ability. Macfarlane remarks that ... De Morgan considered himself a Briton unattached neither English, Scottish, Welsh or Irish. He also says He disliked the country and while his family enjoyed the seaside, and men of science were having a good time at a meeting of the British Association in the country he remained in the hot and dusty libraries of the metropolis. ... he had no ideas or sympathies in common with the physical philosopher. His attitude was doubtless due to his physical infirmity, which prevented him from being either an observer or an experimenter. He never voted in an http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Morgan.html (2 of 3) [2/16/2002 11:06:37 PM]

De_Morgan

election, and he never visited the House of Commons, or the Tower, or Westminster Abbey. De Morgan was always interested in odd numerical facts and writing in 1864 he noted that he had the distinction of being x years old in the year x2 (He was 43 in 1849). Anyone born in 1980 can claim the same distinction. Article by: J J O'Connor and E F Robertson List of References (19 books/articles)

Some Quotations (9)

A Poster of Augustus De Morgan

Mathematicians born in the same country

Cross-references to History Topics

1. A comment from Thomas Hirst's diary 2. The four colour theorem 3. Squaring the circle 4. Pi through the ages

Other references in MacTutor

1. Chronology: 1830 to 1840 2. Chronology: 1840 to 1850

Honours awarded to Augustus De Morgan (Click a link below for the full list of mathematicians honoured in this way) London Maths Society President

1865 - 1866

Lunar features

Crater De Morgan

Other Web sites

1. Interactive Real Analysis 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/De_Morgan.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Morgan.html (3 of 3) [2/16/2002 11:06:37 PM]

De_Prony

Gaspard Clair François Marie Riche de Prony Born: 22 July 1755 in Chamelet, Beaujolais, France Died: 29 July 1839 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gaspard de Prony's family name was Riche, the de Prony title having been bought by his parents. In fact de Prony's younger brother was always known by the name Riche. De Prony was educated at the Benedictine College at Toissei in Doubs. From there he entered the Ecole des Ponts et Chaussés in 1776 where he studied engineering. He graduated in 1779 at the top student and remained for a further year in Paris, doing as the head of the Ecole des Ponts et Chaussés told him:M de Prony ... concern yourself with acquiring a deep knowledge of your art, for you are destined to become head of the Ecole des Ponts et Chaussés. In 1780 he became an engineer with the Ecole des Ponts et Chaussés and after three years in a number of different regions of France he returned to the Ecole des Ponts et Chaussés in Paris 1783. This was the same year he published his first major work in the Académie des Sciences on the forces on arches. Monge was impressed with this paper and realised that de Prony was someone of great potential. In 1785 de Prony visited England on a project to obtain an accurate measurement of the relative positions of the Greenwich Observatory and the Paris Observatory. Two years later he was promoted to inspector at the Ecole des Ponts et Chaussés. Around this time he was involved with the work on the Louis XVI Bridge in Paris which is now called the Pont de la Concorde. Further promotion in 1790 was followed the next year by his being appointed Engineer-in-Chief of the Ecole des Ponts et Chaussés. This promotion was as a result of the opening of the Louis XVI Bridge.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Prony.html (1 of 3) [2/16/2002 11:06:39 PM]

De_Prony

Also around 1791 de Prony was working on geometry with Pierre Girard. Then in 1792, de Prony began a major task of producing logarithmic and trigonometric tables, the Cadastre. With the assistance of Legendre, Carnot and other mathematicians, and between 70 to 80 assistants, the work was undertaken over a period of years, being completed in 1801. The tables were, see [2]:... vast, with values calculated to between fourteen and twenty-nine decimal places. Each copy consisted of eighteen folio volumes together with another volume of mathematical procedures. Getting such a massive work published was another matter. Negotiations went on over a number of years until, in 1809, it seemed they would appear. The publisher wrote:The present generation would never have witnessed the end of this monumental work if M de Prony had not had the fortunate idea of applying the powerful method of division of labour, conceiving methods to reduce the long and laborious part of the production of the tables to simple additions and subtractions... However the tables were never published in full and it was near the end of the century before even a part appeared. It was just too expensive to print at a time when France was not in the best of financial states. In 1794 the Ecole Centrale des Travaux Publics was founded by and was directed by Carnot and Monge. It was renamed the Ecole Polytechnique in 1795 and de Prony was certainly one of the main lectures by this time. He is listed among the first teachers at the university as:Prony, lecturer in analysis, director of the Cadastre, member of the Institute. Annual salary 6,000 francs. Accommodation within the school... De Prony's lectures given at the Ecole Polytechnique were published, including his lectures on hydraulics. In 1798 de Prony refused Napoleon's request that he join his army of invasion to Egypt. Fourier, Monge and Malus had agreed to be part of the expeditionary force and Napoleon was angry that de Prony would not come. It did mean that de Prony was to fail to receive the honours he deserved from Napoleon but de Prony's wife was a close friend of Joséphine and this probably saved de Prony from anything worse. In 1798 de Prony achieved his ambition of being appointed director of the Ecole des Ponts et Chaussés. His desire for this post was almost certainly a main reason for his refusing to join Napoleon. As director he began producing a number of important texts on mathematical physics. He became a member of the Bureau de Longitude and, in 1810 and 1811, he produced two further major texts from his lectures at the Ecole Polytechnique, namely Leçons de Mécanique Analytique and Sommaire des Leçons du Cours de Mécanique. After Napoleon was defeated the reorganisation in France included a reorganisation of the Ecole Polytechnique which was closed during 1816. De Prony lost his position as professor there and was not part of the reorganisation committee. However, as soon as the school reopened, de Prony was asked to be an examiner so he continued his connection yet only had to work one month per year. One of de Prony's most important scientific inventions was the 'de Prony brake' which he invented in 1821 to measure the performance of machines and engines. It was based on ideas of Hachette and was a considerable improvement on a method which Pierre Girard had used two years earlier.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Prony.html (2 of 3) [2/16/2002 11:06:39 PM]

De_Prony

The last part of de Prony's career was more involved with education rather than administration. One battle he fought, without success, was against Cauchy's more towards pure mathematics and away from the more applied mathematics which de Prony firmly believed in. In [2] Margaret Bradley writes:... there had long been an increasing demand for the reform of the Ecole des Ponts et Chaussés and his lack of attention to this attracted severe criticism. He was now showing even less interest in the day to day running of the school, in favour of science. He was disillusioned by the failure of his attempts to reform mathematics teaching at the Ecole Polytechnique, where he had made energetic and determined efforts to combat the emphasis on theory of A-L Cauchy ... Prony seems to have lost heart for the continuing struggle and to have been less conscientious with regard to his duties as examiner. Article by: J J O'Connor and E F Robertson List of References (4 books/articles)

Some Quotations (2)

Mathematicians born in the same country Other references in MacTutor

Chronology: 1780 to 1800

Honours awarded to Gaspard de Prony (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1818

Paris street names

Rue de Prony (17th Arrondissement)

Commemorated on the Eiffel Tower

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/De_Prony.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Prony.html (3 of 3) [2/16/2002 11:06:39 PM]

De_Rham

Georges de Rham Born: 10 Sept 1903 in Roche, Canton Vaud, Switzerland Died: 9 Oct 1990 in Lausanne, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georges de Rham attended the secondary school Collège d'Aigle from 1914 to 1919 and then at the Gymnase classique de Lausanne from 1919 until 1921. Having graduated from secondary school with Latin and Greek as his main subjects, de Rham entered the University of Lausanne in 1921 with the intention of studying chemistry, physics and biology. He began to study mathematics in an attempt to understand questions that arose in the physics he was studying. After five semesters he gave up biology and turned to mathematics. In 1925 he obtained his Licence ès Sciences. From 1926 he studied in Paris for his doctorate, spending the winter term of 1930/31 at the University of Göttingen. He was awarded his doctorate from Paris in 1931 and became a lecturer at the University of Lausanne. There he was promoted to extraordinary professor in 1936 and to full professor in 1943. He retired and was given an honorary appointment by Lausanne in 1971. However de Rham also held a position at the University of Geneva. He was appointed there as extraordinary professor in 1936, being promoted to full professor in 1953. He retired from Geneva and was given an honorary position there in 1973. In addition to these permanent appointments de Rham held a number of visiting professorships. He visited Harvard in 1949/50 and the Institute for Advanced Study at Princeton in 1950 and again in 1957/58. He also visited the Tata Institute in Bombay in 1966. In [4] Raoul Bott describes the context of de Rham's famous theorem:In some sense the famous theorem that bears his name dominated his mathematical life, as indeed it dominates so much of the mathematical life of this whole century. When I met de http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Rham.html (1 of 3) [2/16/2002 11:06:41 PM]

De_Rham

Rham in 1949 at the Institute in Princeton he was lecturing on the Hodge theory in the context of his 'currents'. These are the natural extensions to manifolds of the distributions which had been introduced a few years earlier by Laurent Schwartz and of course it is only in this extended setting that both the de Rham theorem and the Hodge theory become especially complete. The original theorem of de Rham was most probably believed to be true by Poincaré and was certainly conjectured (and even used!) in 1928 by E Cartan. But in 1931 de Rham set out to give a rigorous proof. The technical problems were considerable at the time, as both the general theory of manifolds and the 'singular theory' were in their early formative stages. The details of the de Rham theorem are given in [4] but as far as this article is concerned it is sufficient to give the 'feel' for the type of theorem as nicely described there:The theorem is then a sort of topological form of the particle-wave equivalence of quantum mechanics, and the quest for 'truly' understanding these and analogous dualities has been one of the great motivating forces in the mathematics of the last fifty years. Of course de Rham produced much in the way of important mathematics in addition to the de Rham theorem. He gave a reducibility theorem for Riemann spaces which is fundamental in the development of Riemannian geometry. He also worked on Reidemeister torsion and his work on this topic was the beginning of rapid developments. We end with two descriptions of de Rham's character, the first by Bott and the second by Chandrasekhar. ... de Rham had a subtle charm which drew younger people to him immediately. In those early days at Princeton he would easily mingle with the boisterous postdocs, his exquisite manners contrasting amusingly with our rude ways. He was always lean and one could feel the steel in his sinews, but he never boasted of his mountaineering exploits and it was only at secondhand that the daredevil in him became apparent... The second description:Tough as steel in his adherence to principle, resilient, placable, self-less and generous beyond the dictates of fashion, steadfast in friendship, but not at the price of reason, de Rham strides the world of mathematics a happy warrior. De Rham received many honours. He was President of the International Mathematical Union from 1963 to 1966. He was elected a member of the academies of Lincei, Göttingen, and the Institute of France. He received honorary degrees from the universities of Strasbourg, Genoble, Lyon, and l'Ecole Polytechnique Fédérale Zurich. He received the Prize of the Marcel Benoist Foundation and of the City of Lausanne. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Rham.html (2 of 3) [2/16/2002 11:06:41 PM]

De_Rham

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/De_Rham.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Rham.html (3 of 3) [2/16/2002 11:06:41 PM]

De_Witt

Johan de Witt Born: 24 Sept 1625 in Dordrecht, Netherlands Died: 20 Aug 1672 in The Hague, Netherlands

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johan de Witt attended Beeckman's school in Dordrecht, then in 1641 he entered the University of Leiden to study law. At university he showed remarkable talents, especially in mathematics and law. In 1645 Johan and his elder brother Cornelius visited France, Italy, Switzerland and England, then on his return Johan lived at The Hague as an advocate. He received a doctorate in law from University of Angers in 1645. De Witt was an associate of van Schooten and lived for a while in his house. His most important work Elementa curvarum linearum (1659-61) was written before 1650, and was the first systematic development of the analytic geometry of the straight line and conic. It was published by van Schooten as part of his edition of Descartes' Géometrie (1660). The word directrix is due to de Witt. In 1650 de Witt was appointed the leader of Dordrecht's deputation in the government of Holland. Three years later he was appointed political leader of Holland. As leader of Holland, de Witt applied his mathematical knowledge to the financial and budgetary problems of the republic. He wrote The Worth of Life Annuities Compared to Redemption Bonds which applied probability to questions of state finance. De Witt brought about peace with England in 1654 and after this he was extremely successful in bringing prosperity to Holland. When war broke out again with England in 1665 de Witt was able to bring about a very satisfactory settlement at the Treaty of Breda (1667). His political skills were further seen in the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Witt.html (1 of 2) [2/16/2002 11:06:43 PM]

De_Witt

Triple Alliance (1668) between the Dutch Republic, England and Sweden. In 1672 France invaded and there were demonstrations against de Witt. His brother Cornelius was arrested on July 24 and two weeks later Johan de Witt resigned as political leader of Holland. When Johan came to visit Cornelius in prison they were attacked and killed by a large crowd. Quoting [2]:Cornelius was put to the torture and on August 19 sentenced to deprivation of his offices and banishment. His brother came to visit him in the Gevangenpoort at The Hague. A vast crowd, hearing this, collected outside and finally burst in, seized the two brothers, and tore them to pieces. Thus perished one of the greatest statesmen of his age and of Dutch history. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

Matrices and determinants

Other references in MacTutor

Chronology: 1650 to 1675

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/De_Witt.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/De_Witt.html (2 of 2) [2/16/2002 11:06:43 PM]

Dechales

Claude François Milliet Dechales Born: 1621 in Chambéry, France Died: 28 March 1678 in Turin, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Claude Dechales became a Jesuit at the age of 15 and was educated within the Jesuit Order. For a time he was a Jesuit missionary in Turkey. Dechales lectured at Jesuit colleges, first in Lyon and Chambéry. From Chambéry he went to Marseilles where King Louis XIV appointed him Royal Professor of Hydrography. In Marseilles he taught navigation, military engineering and other applications of mathematics. From Marseilles he moved to Turin where he was appointed professor of mathematics. Dechales is best remembered for Cursus seu mundus mathematicus, a complete course of mathematics. Topics covered in this wide ranging work included practical geometry, mechanics, statics, magnetism and optics as well as topics outwith the usual topics of mathematics such as geography, architecture, astronomy, natural philosophy and music. The book was widely used but it reflects his ability to teach rather than a research ability and fails to use the mathematical advances of the day. It is old-fashioned in its coverage: in algebra, for example, it owes more to Diophantus than to the algebraists of its day. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dechales.html (1 of 2) [2/16/2002 11:06:44 PM]

Dechales

Glossary index

Quotations index

Mathematicians of the day JOC/EFR December 1996

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dechales.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dechales.html (2 of 2) [2/16/2002 11:06:44 PM]

Dedekind

Julius Wihelm Richard Dedekind Born: 6 Oct 1831 in Braunschweig, duchy of Braunschweig (now Germany) Died: 12 Feb 1916 in Braunschweig, duchy of Braunschweig (now Germany)

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Richard Dedekind's father was a professor at the Collegium Carolinum in Brunswick. His mother was the daughter of a professor who also worked at the Collegium Carolinum. Richard was the youngest of four children and never married. He was to live with one of his sisters, who also remained unmarried, for most of his adult life. He attended school in Brunswick from the age of seven and at this stage mathematics was not his main interest. The school, Martino-Catharineum, was a good one and Dedekind studied science, in particular physics and chemistry. However, physics became less than satisfactory to Dedekind with what he considered an imprecise logical structure and his attention turned towards mathematics. The Collegium Carolinum was an educational institution between a high school and a university and he entered it in 1848 at the age of 16. There he was to receive a good understanding of basic mathematics studying differential and integral calculus, analytic geometry and the foundations of analysis. He entered the University of Göttingen in the spring of 1850 with a solid grounding in mathematics. Göttingen was a rather disappointing place to study mathematics at this time, and it had not yet become the vigorous research centre that it turned into soon afterwards. Mathematics was directed by M A Stern and G Ulrich. Gauss also taught courses in mathematics, but mostly at an elementary level. The physics department was directed by Listing and Wilhelm Weber. The two departments combined to initiate a seminar which Dedekind joined from its beginning. There he learnt number theory which was the most http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dedekind.html (1 of 5) [2/16/2002 11:06:46 PM]

Dedekind

advanced material he studied. His other courses covered material such as the differential and integral calculus, of which he already had a good understanding. The first course to really make Dedekind enthusiastic was, rather surprisingly, a course on experimental physics taught by Weber. More likely it was Weber who inspired Dedekind rather than the topic of the course. In the autumn term of 1850, Dedekind attended his first course given by Gauss. It was a course on least squares and [1]:... fifty years later Dedekind remembered the lectures as the most beautiful he had ever heard, writing that he had followed Gauss with constantly increasing interest and that he could not forget the experience. Dedekind did his doctoral work in four semesters under Gauss's supervision and submitted a thesis on the theory of Eulerian integrals. He received his doctorate from Göttingen in 1852 and he was to be the last pupil of Gauss. However he was not well trained in advanced mathematics and fully realised the deficiencies in his mathematical education. At this time Berlin was the place where courses were given on the latest mathematical developments but Dedekind had not been able to learn such material at Göttingen. By this time Riemann was also at Göttingen and he too found that the mathematical education was aimed at students who were intending to become secondary school teachers, not those with the very top abilities who would go on to research careers. Dedekind therefore spent the two years following the award of his doctorate learning the latest mathematical developments and working for his habilitation. In 1854 both Riemann and Dedekind were awarded their habilitation degrees within a few weeks of each other. Dedekind was then qualified as a university teacher and he began teaching at Göttingen giving courses on probability and geometry. Gauss died in 1855 and Dirichlet was appointed to fill the vacant chair at Göttingen. This was an extremely important event for Dedekind who found working with Dirichlet extremely profitable. He attended courses by Dirichlet on the theory of numbers, on potential theory, on definite integrals, and on partial differential equations. Dedekind and Dirichlet soon became close friends and the relationship was in many ways the making of Dedekind, whose mathematical interests took a new lease of life with the discussions between the two. Bachmann, who was a student in Göttingen at this time [12]:... recalled in later years that he only knew Dedekind by sight because Dedekind always arrived and left with Dirichlet and was completely eclipsed by him. Dedekind wrote in a letter in July 1856 [4]:What is most useful to me is the almost daily association with Dirichlet, with whom I am for the first time beginning to learn properly; he is always completely amiable towards me, and he tells me without beating about the bush what gaps I need to fill and at the same time he gives me the instructions and the means to do it. I thank him already for infinitely many things, and no doubt there will be many more. Dedekind certainly still continued to learn mathematics at this time as a student would by attending courses, such as those by Riemann on abelian functions and elliptic functions. Around this time Dedekind studied the work of Galois and he was the first to lecture on Galois theory when he taught a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dedekind.html (2 of 5) [2/16/2002 11:06:46 PM]

Dedekind

course on the topic at Göttingen during this period. While at Göttingen, Dedekind applied for J L Raabe's chair at the Polytechnikum in Zürich. Dirichlet supported his application writing that Dedekind was 'an exceptional pedagogue'. In the spring of 1858 the Swiss councillor who made appointments came to Göttingen and Dedekind was quickly chosen for the post. Dedekind was appointed to the Polytechnikum in Zürich and began teaching there in the autumn of 1858. In fact it was while he was thinking how to teach differential and integral calculus, the first time that he had taught the topic, that the idea of a Dedekind cut came to him. He recounts that the idea came to him on 24 November 1858. His idea was that every real number r divides the rational numbers into two subsets, namely those greater than r and those less than r. Dedekind's brilliant idea was to represent the real numbers by such divisions of the rationals. Dedekind and Riemann travelled together to Berlin in September 1859 on the occasion of Riemann's election to the Berlin Academy of Sciences. In Berlin, Dedekind met Weierstrass, Kummer, Borchardt and Kronecker. The Collegium Carolinum in Brunswick had been upgraded to the Brunswick Polytechnikum by the 1860s, and Dedekind was appointed to the Polytechnikum in 1862. With this appointment he returned to his home town and even to his old educational establishment where his father had been one of the senior administrators for many years. Dedekind remained there for the rest of his life, retiring on 1 April 1894. He lived his life as a professor in Brunswick [1]:... in close association with his brother and sister, ignoring all possibilities of change or attainment of a higher sphere of activity. The small, familiar world in which he lived completely satisfied his demands: in it his relatives completely replaced a wife and children of his own and there he found sufficient leisure and freedom for scientific work in basic mathematical research. He did not feel pressed to have a more marked effect in the outside world: such confirmation of himself was unnecessary. After he retired, Dedekind continued to teach the occasional course and remained in good health in his long retirement. The only spell of bad health which Dedekind had experienced was 10 years after he was appointed to the Brunswick Polytechnikum when he had a serious illness, shortly after the death of his father. However he completely recovered and, as we mentioned, remained in good health. Dedekind made a number of highly significant contributions to mathematics and his work would change the style of mathematics into what is familiar to us today. One remarkable piece of work was his redefinition of irrational numbers in terms of Dedekind cuts which, as we mentioned above, first came to him as early as 1858. He published this in Stetigkeit und Irrationale Zahlen in 1872. In it he wrote:Now, in each case when there is a cut (A1, A2) which is not produced by any rational number, then we create a new, irrational number a, which we regard as completely defined by this cut; we will say that this number a corresponds to this cut, or that it produces this cut. As well as his analysis of the nature of number, his work on mathematical induction, including the definition of finite and infinite sets, and his work in number theory, particularly in algebraic number fields, is of major importance.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dedekind.html (3 of 5) [2/16/2002 11:06:46 PM]

Dedekind

Dedekind loved to take his holidays in Switzerland, the Austrian Tyrol or the Black Forest in southern Germany. On one such holiday in 1874 he met Cantor while staying in the beautiful city of Interlaken and the two discussed set theory. Dedekind was sympathetic to Cantor's set theory as is illustrated by this quote from Was sind und was sollen die Zahlen (1888) regarding determining whether a given element belongs to a given set :In what way the determination comes about, or whether we know a way to decide it, is a matter of no consequence in what follows. The general laws that are to be developed do not depend on this at all. In this quote Dedekind is arguing against Kronecker's objections to the infinite and, therefore, is agreeing with Cantor's views. Among Dedekind's other notable contributions to mathematics were his editions of the collected works of Peter Dirichlet, Carl Gauss, and Georg Riemann. Dedekind's study of Dirichlet's work did, in fact, to lead to his own study of algebraic number fields, as well as to his introduction of ideals. Dedekind edited Dirichlet's lectures on number theory and published these as Vorlesungen über Zahlentheorie in 1863. It is noted in [12] that:Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death. It was in the third and fourth editions of Vorlesungen über Zahlentheorie, published in 1879 and 1894, that Dedekind wrote supplements in which he introduced the notion of an ideal which is fundamental to ring theory. Dedekind formulated his theory in the ring of integers of an algebraic number field. The general term 'ring' does not appear, it was introduced later by Hilbert. Dedekind, in a joint paper with Heinrich Weber published in 1882, applies his theory of ideals to the theory of Riemann surfaces. This gave powerful results such as a purely algebraic proof of the Riemann-Roch theorem. Dedekind's work was quickly accepted, partly because of the clarity with which he presented his ideas and partly since Heinrich Weber lectured to Hilbert on these topics at the University of Königsberg. Dedekind's notion of ideal was taken up and extended by Hilbert and then later by Emmy Noether. This led to the unique factorisation of integers into powers of primes to be generalised to ideals in other rings. In 1879 Dedekind published Über die Theorie der ganzen algebraischen Zahlen which was again to have a large influence on the foundations of mathematics. In the book Dedekind [1]:... presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space. Among other things, he provides a definition independent of the concept of number for the infiniteness or finiteness of a set by using the concept of mapping and treating the recursive definition, which is so important to the theory of ordinal numbers. Dedekind's brilliance consisted not only of the theorems and concepts that he studied but, because of his ability to formulate and express his ideas so clearly, he introduced a new style of mathematics that been a major influence on mathematicians ever since. As Edwards writes in [12]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dedekind.html (4 of 5) [2/16/2002 11:06:46 PM]

Dedekind

Dedekind's legacy ... consisted not only of important theorems, examples, and concepts, but a whole style of mathematics that has been an inspiration to each succeeding generation. Many honours were given to Dedekind for his outstanding work, although he always remained extraordinarily modest regarding his own abilities and achievements. He was elected to the Göttingen Academy (1862), the Berlin Academy (1880), the Academy of Rome, the Leopoldino-Carolina Naturae Curiosorum Academia, and the Académie des Sciences in Paris (1900). Honorary doctorates were awarded to him by the universities of Kristiania (Oslo), Zurich and Brunswick. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (30 books/articles)

A Quotation

A Poster of Richard Dedekind

Mathematicians born in the same country

Cross-references to History Topics

1. The beginnings of set theory 2. A history of group theory 3. An overview of the history of mathematics

Other references in MacTutor

1. Chronology: 1850 to 1860 2. Chronology: 1870 to 1880 3. Chronology: 1880 to 1890

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dedekind.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dedekind.html (5 of 5) [2/16/2002 11:06:46 PM]

Dee

John Dee Born: 13 July 1527 in London, England Died: Dec 1608 in Mortlake, Surrey, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Dee was educated at a school in Chelmsford in Essex, then entered St. John's College, Cambridge in 1542. He became a Fellow of St. John's College in 1545 and the next year became a Fellow of Trinity College, Cambridge. Henry VIII founded Trinity College, the largest of the Cambridge colleges, in 1546 and Dee became one of its first Fellows. Dee then travelled on the Continent (1547-1551). In a number of visits he studied with Gemma Frisius and Gerardus Mercator at the University of Louvain. In 1551 Dee was offered an appointment as professor of mathematics in Paris but declined. He also declined a lectureship in mathematics at Oxford three years later. Dee became astrologer to Queen Mary but was imprisoned for being a magician. He was released in 1555. He then found favour with Queen Elizabeth and cast horoscopes for her. He even selected the day for her coronation. In 1570 Dee edited an edition of Euclid translated by Billingsley. Dee wrote a famous preface to this edition justifying the study of mathematics. In 1573 Dee wrote Parallacticae commentationis praxosque which gives trigonometric methods which might be applied to find the distance to 'Tycho (Brahe)'s supernova' of 1572. Dee also wrote on calendar reform, on navigation, on geography and on astrology. Dee brought instruments of navigation back from the Continent when he returned in 1551. From 1555 he was a consultant to the Muscovy Company. The Muscovy Company was formed in 1555 by the navigator and explorer Sebastian Cabot together with a number of London merchants. It was granted a monopoly of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dee.html (1 of 2) [2/16/2002 11:06:47 PM]

Dee

Anglo-Russian trade and had as one of its aims the search for the Northeast Passage. Dee prepared nautical information, including charts for navigation in the polar regions, for the company during the next 32 years. Later in his career Dee became interested in astrology and alchemy, and he gave up other work for this. The lack of reaction of others to his scientific work drove him in the direction of alchemy which he saw as a quick way to glory. Dee visited Poland and Bohemia (1583-89), giving displays of magic at the courts of princes. Article by: J J O'Connor and E F Robertson List of References (7 books/articles)

Some Quotations (4)

A Poster of John Dee

Mathematicians born in the same country

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dee.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dee.html (2 of 2) [2/16/2002 11:06:47 PM]

Dehn

Max Wilhelm Dehn Born: 13 Nov 1878 in Hamburg, Germany Died: 27 June 1952 in Black Mountain, North Carolina, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Max Dehn wrote one of the first systematic expositions of topology (1907) and later formulated important problems on group presentations, namely the word problem and the isomorphism problem. Dehn studied at Göttingen under Hilbert's supervision obtaining his doctorate in 1900 for a thsesis entitled Die Legendreschen Sätze über die Winkelsumme im Dreieck. From 1921 until 1935 he held the chair of Pure and Applied Mathematics at the University of Frankfurt but he was forced to leave his post by the Nazi regime in 1938. In 1940 he emigrated to the USA, travelling there via Scandinavia, Russia and Japan. Once in the USA he taught at several universities and colleges, for instance the University of Idaho, the Illinois Institute of Technology and St John's College in Annapolis, Maryland. However he was unable to find a full-time position and Saunders MacLane recently wrote:... most mathematicians fleeing Europe were helped to some sort of position in the United States ... I recall two cases of failures: Max Dehn, noted for work in topology, got only a weak position. The weak position, referred to by MacLane, was at Black Mountain College. This college had no accredited degrees and taught mainly creative arts. There was no trained mathematician on the staff when Dehn was invited to give two lectures there in 1944. He realised that he could not lecture on advanced mathematics so he gave his lectures on Common roots of mathematics and ornamentics and Some moments in the development of mathematical ideas.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dehn.html (1 of 3) [2/16/2002 11:06:49 PM]

Dehn

He was offered a permanent post there at $25 a month. He held out for $40 a month which was agreed. Dehn joined the Faculty in 1945 and remained there until his death. Dehn was the only mathematician ever to teach at the College which closed in 1956. Dehn's outstanding research record is in stark contrast with his low level final post. He was an intuitive geometer, stimulated by Hilbert's axiomatic approach to the subject. Dehn had solved the third of Hilbert's 23 problems on the congruence of polyhedra. In 1907 Dehn wrote one of the first systematic expositions of topology jointly with Heegaard. At that time topology was called 'analysis situs'. Dehn's work in topology had led him into the study of groups, particularly group presentations which arise naturally from topological considerations. Dehn formulated important problems on group presentations, namely the word problem and the isomorphism problem. The word problem asks the fundamental question of whether there is an algorithm to determine whether a word in a group given by a presentation is trivial. It has since been shown that no such algorithm exists in general. Research on questions of this type are still of major importance in combinatorial group theory. Dehn also wrote on statics, projective planes and the history of mathematics. Paul L Chessin has described some interesting personal recollections of Dehn:In 1945, [Dehn] replaced Rudolph Langer, then head of the mathematics department in Madison, Wisconsin, who went on sabbatical to the University of Texas. Prof Dehn came to teach several graduate courses. I attended his course in Non-linear Partial Differential Equations. We were delighted to have a German-speaking instructor since we were to take our German language examinations (required for the doctorate). He, in turn, refused since it was that term that he was to take his examination for US citizenship ! Max would hold forth in the Rathskellar (the only campus beer establishment within the Big Ten Universities). We learned more in the beer stube - such as his personal life. He was the black sheep in his family. So long as he remained in his university, he was supported. Thus he received many doctorates. He would declaim in Greek, some passages from the classics, beer stein in hand. At the time for setting out final grades, he merely called the attendance and one by one asked essentially for the titles of the chapters in the textbook. Clearly we could glance ahead to be assured of the correct answer. I believe he gave something like 18 A's and 3-4 B's. Throughout the class sessions he would interrupt with some voiced concerns about passing his citizenship examination. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles)

A Quotation

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dehn.html (2 of 3) [2/16/2002 11:06:49 PM]

Dehn

A Poster of Max Dehn

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1900 to 1910

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dehn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dehn.html (3 of 3) [2/16/2002 11:06:49 PM]

Delamain

Richard Delamain Born: 1600 in London, England Died: 1644 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Richard Delamain was a joiner by trade. He studied mathematics at Gresham College London. After this he remained in London becoming a private tutor of mathematics. Delamain became mathematics tutor to Charles I, who was king of Great Britain and Ireland (1625-49). Delamain was the same age as the king he tutored, both being born in 1600. He received 40 per year in this position. Delamain became a student of Oughtred and they were great friends at first. Oughtred wrote As I did to Delamain, and to some others ... I freely gave ... my helpe and instruction. ... But Delamain was already corrupted with doring upon instruments, and quite lost from ever being made an artist. They had a bitter dispute over the invention of a circular slide rule. Oughtred described the slide rule in 1622 but the circular slide rule was not described by him until 1632. Delamain described a circular slide rule in a 32 page pamphlet Grammelogia which was sent to the King in 1629 and published the following year. His fame as a mathematician rests on this work. Delamain also published The Making, Description, and Use of . . . a Horizontal Quadrant (1631) which Oughtred claimed consisted of ideas stolen from him. Delamain argued with Oughtred, not only concerning the invention of the circular slide rule but also as regards the use of instruments in teaching mathematics. Oughtred, as an attack on Delamain, claimed that .. the true way of art is not by instruments, but by demonstration: and that it is a preposterous course of vulgar teachers to begin with instruments. Delamain replied ... theory is as the mother that produceth the daughter, the very sinewes and life of practise, the excellencie and highest degree of true mathematical knowledge: but for those that would make but a step as it were into that kind of learning, whose onely desire is expedition and facilitie ... all are best effected with instrument rather than with tedious regular demonstrations... If we think of modern instruments as computers then one would have to say that Delamain's views have a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Delamain.html (1 of 2) [2/16/2002 11:06:50 PM]

Delamain

ring of realism in today's world which are somewhat lacking in Oughtred's high ideals. As well as mathematical instruments Delamain also made sundials. He died in the Civil war some time before 1645 about five years before Charles I was executed. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Delamain.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Delamain.html (2 of 2) [2/16/2002 11:06:50 PM]

Delambre

Jean Baptiste Joseph Delambre Born: 19 Sept 1749 in Amiens, France Died: 19 Aug 1822 in Paris, France

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

In 1771 Jean Baptiste Delambre tutored the son of M d'Assy, Receiver General of Finances, and in 1788 d'Assy built an observatory for Delambre. Delambre worked at his observatory and in 1792 he published Tables du Soleil, de Jupiter, de Saturne, d'Uranus et des satellites de Jupiter. In 1795 he was admitted to the Institute de France and in 1803 became secretary to the mathematical section. In his Rapport historique.. which he read to the Institute in February 1808 he says In almost all branches of Mathematics one is blocked by insurmountable difficulties (but) the spectacle of analysis and mechanics in our time (convinces me that) the generations to come will not see anything impossible in what remains to be done. Delambre served at the Bureau des Longitudes from 1795 and measured the arc of the meridian extending from Dunkirk to Barcelona. His account appears in Base du système métrique. In 1807 Delambre was appointed to the chair of astronomy at the Collège de France in Paris. he was treasurer to the Imperial University from 1808. Delambre also wrote histories of ancient, medieval and 'modern' astronomy. He is honoured by having a large lunar crater named after him. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Delambre.html (1 of 2) [2/16/2002 11:06:52 PM]

Delambre

List of References (9 books/articles)

A Quotation

A Poster of Jean Baptiste Delambre

Mathematicians born in the same country

Cross-references to History Topics

1. Mathematical discovery of planets 2. Greek Astronomy

Honours awarded to Jean Baptiste Delambre (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1791

Lunar features

Crater Delambre

Paris street names

Rue Delambre and Square Delambre (14th Arrondissement)

Commemorated on the Eiffel Tower Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Delambre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Delambre.html (2 of 2) [2/16/2002 11:06:52 PM]

Delaunay

Charles Eugene Delaunay Born: 9 April 1816 in Lusigny-sur-Barse, France Died: 5 Aug 1872 in At sea (near Cherbourg, France)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Delaunay studied under Biot's direction at the Sorbonne, then worked in astronomy and mechanics. He taught mechanics at the Ecole Polytechnique from 1850 and later taught at the Ecole des Mines (where he had been a student). He published, in 1860 and 1867, two volumes on lunar theory La Théorie du mouvement de la lune which were the result of 20 years work. This is an important case of the three body problem. Delaunay found the longitude, latitude and parallax of the Moon as infinite series. These gave results correct to 1 second of arc but were not too practical as the series converged slowly. However this work was important in the beginnings of functional analysis. Delaunay succeeded Le Verrier as director of the Paris Observatory in 1870 but 2 years later he and three companions drowned in a boating accident. Among his works on mechanics were Cours élémentaire de mécanique (1850) and Traité de mécanique rationnelle (1856). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Delaunay.html (1 of 2) [2/16/2002 11:06:53 PM]

Delaunay

Cross-references to History Topics

Orbits and gravitation

Other references in MacTutor

Chronology: 1860 to 1870

Honours awarded to Charles E Delaunay (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1869

Lunar features

Crater Delaunay

Paris street names

Impasse Delaunay, Rue Delaunay and Square Delaunay (11th Arrondissement)

Commemorated on the Eiffel Tower Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Delaunay.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Delaunay.html (2 of 2) [2/16/2002 11:06:53 PM]

Deligne

Pierre René Deligne Born: 3 Oct 1944 in Brussels, Belgium

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Deligne attended the Free University of Brussels receiving his licence in mathematics in 1966. He continued to study for his doctorate which was awarded in 1968. Before the award of his doctorate, Deligne was a junior scientist at the Belgium National Foundation for Scientific Research in 1967-68. In 1968 he went to the Institut des Hautes Etudes Scientifiques at Bures-sur-Yvette in France where he was a visiting member until 1970 when he became a permanent member of the Institute. Deligne remained based at the Institut des Hautes Etudes Scientifiques until 1984 when he went to the Institute for Advanced Study at Princeton where he was appointed a professor. André Weil gave for the first time a theory of varieties defined by equations with coefficients in an arbitrary field, in his Foundations of Algebraic Geometry (1946). This used Zariski's ideas and also made good use of geometric concepts. Weil's work on polynomial equations led to questions on what properties of a geometric object can be determined purely algebraically. Weil's work related questions about integer solutions to polynomial equations to questions in algebraic geometry. He conjectured results about the number of solutions to polynomial equations over the integers using intuition on how algebraic topology should apply in this novel situation. The third of his conjectures was a generalisation of the Riemann hypothesis on the zeta function. These problems quickly became major research challenges to mathematicians. A solution of the three Weil conjectures was given by Deligne. This work brought together algebraic geometry and algebraic number theory and it led to Deligne being awarded a Fields Medal at the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Deligne.html (1 of 2) [2/16/2002 11:06:55 PM]

Deligne

International Congress of Mathematicians in Helsinki in 1978. A solution to these problems required the development of a new kind of algebraic topology. Deligne has worked on many other important problems. The areas on which he has worked, in addition to algebraic geometry, are Hilbert's 21st problem, Hodge theory, theory of moduli, modular forms, Galois representations, L-series and the Langlands conjectures, and representations of algebraic groups. In addition to the Fields Medal, Deligne was awarded the Crafoord Prize of the Royal Swedish Academy of Sciences in 1988:... for his fundamental research in algebraic geometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1970 to 1980

Honours awarded to Pierre Deligne (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1978

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Deligne.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Deligne.html (2 of 2) [2/16/2002 11:06:55 PM]

Delone

Boris Nikolaevich Delone Born: 15 March 1890 in St Petersburg, Russia Died: 1980

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Boris Delone graduated from Kiev University in 1913. At Kiev he was a student of Grave and he followed Grave's work in algebra and number theory. After graduating Delone taught in Kiev where he was a member of the Mathematical Society which had among its members Ch T Bialobzeski, P V Voronets, N B Delone, D A Grave, A A Friedmann, A P Kotelnikov, V P Linnik (I V Linnik's father) and O Yu Schmidt. Following the Revolution of 1917 there was a change in policy towards education which, certainly in the Ukraine, had to become more technology based and more practical. Algebra certainly did not fit into this new educational philosophy and Grave's algebra seminar was forced to close. Some mathematicians, such as Grave himself, changed to study applied mathematical topics. Delone, however, chose to continue to study algebra and so was forced, in the 1920s, to leave the Ukraine. Delone moved to Petrograd in 1922. Petrograd was the name that St Petersburg had been given in 1914 and, two years after Delone began working there, in 1924, it was again renamed, this time to Leningrad. Delone worked at Leningrad University from 1922 until 1935. The Institute of Physics and Mathematics had been established by Steklov in Petrograd (as it was called at the time) in 1921. In 1932 the Institute of Physics and Mathematics was divided into two independent Departments, the Mathematics Department headed by Vinogradov and the Physics Department headed by Vavilov. Vinogradov invited some outstanding mathematicians to join the new Mathematics Department including Delone.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Delone.html (1 of 2) [2/16/2002 11:06:57 PM]

Delone

In this new Mathematics Department, Delone became a colleague of Sergi Bernstein, Luzin, Smirnov, Kuzmin, N S Koshlyakov, Kochin, Sobolev and Faddeev. The St Petersburg Mathematical Society was founded in 1890 but disbanded at the time of the 1917 Revolution. However the Society was reformed in 1921 as the Petrograd Physical and Mathematical Society and Delone joined in the following year. He played an active role in the Society along with other outstanding mathematicians such as Ya V Uspenskii, V I Smirnov, V A Steklov, A A Friedmann, V A Fok, A S Besicovitch, Sergi Bernstein, Ya D Tamarkin, R O Kuzmin and B G Galerkin. In 1934 the Division of Mathematical and Natural Sciences of the Academy of Sciences of the USSR decided to split the Departments of the Steklov Institute of Physics and Mathematics into independent Institutes, the Steklov Mathematical Institute and the Lebedev Physical Institute. Delone became the Head of the Algebra Department of the Steklov Mathematical Institute. However, the Steklov Mathematical Institute moved to Moscow and, in 1935 Delone moved to Moscow. Delone was professor of mathematics at the University of Moscow from 1935 to 1942. The mathematical topics that Delone studied include algebra, the geometry of numbers. He also did important work on the structural analysis of crystals. In addition to his fame as a mathematician, Delone was a famous rock climber. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Delone.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Delone.html (2 of 2) [2/16/2002 11:06:57 PM]

Delsarte

Jean Frédéric Auguste Delsarte Born: 19 Oct 1903 in Fourmies, France Died: 28 Nov 1968 in Nancy, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean Delsarte's father was the head of a textile factory in Fourmies but in 1914 the German armies advanced on the town and Jean left his home town with all the family, except his father, and fled to safety. Jean's father remained in Fourmies trying to save the remnants of his destroyed factory. In 1922 Delsarte entered the Ecole Normale Supérieure in Paris, graduating in 1925. After completing his compulsory military service, he wrote a doctoral thesis during the one year in which he held a research fellowship. In 1928 he was awarded his doctorate and, in the same year, he was appointed to a post at the University of Nancy. In fact Delsarte was to remain on the staff at Nancy for the rest of his career. Among the senior posts he held there was the post of Dean of Science which he held during the years 1945-48. Delsarte worked in analysis extending work on series expansions due to Whittaker and Watson. He was greatly influenced by their text A Course of Modern Analysis and by Watson's Treatise on the Theory of Bessel Functions. Dieudonné, writing in [1], says:These works had convinced him that a good understanding of the formal properties of [series expansions of functions] was necessary to a fruitful study of their domains of definition and their mode of convergence. This was the course he followed with remarkable success, opening up new fields of research that are still far from having been thoroughly explored. One of the most surprising of Delsarte's results was a generalisation of a result due to Gauss. Gauss had shown that if a continuous function f on Rn has at each point x a value equal to its mean value on every

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Delsarte.html (1 of 2) [2/16/2002 11:06:58 PM]

Delsarte

sphere of centre x, then f is harmonic. Delsarte showed that f is harmonic under the weaker condition that f(x) is the mean value on two spheres centre x, radius a and b provided a/b does not take one of a finite set of values. This result is explained in Lectures on Topics in Mean Periodic functions and the Two Radius Theorem published in Bombay in 1961. Delsarte worked independently of other mathematicians and had very few research students. His work is highly original but, because he was rather isolated, had less influence than would otherwise have been the case. Although Delsarte remained at Nancy all his life he did lecture in many different universities from 1950 onwards, particularly ones in India, North America and South America. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Delsarte.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Delsarte.html (2 of 2) [2/16/2002 11:06:58 PM]

Democritus

Democritus of Abdera Born: about 460 BC in Abdera, Thrace, Greece Died: about 370 BC

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Democritus of Abdera is best known for his atomic theory but he was also an excellent geometer. Very little is known of his life but we know that Leucippus was his teacher. Democritus certainly visited Athens when he was a young man, principally to visit Anaxagoras, but Democritus complained how little he was known there. He said, according to Diogenes Laertius writing in the second century AD [5]:I came to Athens and no one knew me. Democritus was disappointed by his trip to Athens because Anaxagoras, then an old man, had refused to see him. As Brumbaugh points out in [3]:How different he would find the trip today, where the main approach to the city from the northeast runs past the impressive "Democritus Nuclear Research Laboratory". Certainly Democritus made many journeys other than the one to Athens. Russell in [9] writes:He travelled widely in southern and eastern lands in search of knowledge, he perhaps spent a considerable time in Egypt, and he certainly visited Persia. He then returned to Abdera, where he remained. Democritus himself wrote (but some historians dispute that the quote is authentic) (see [5]):Of all my contemporaries I have covered the most ground in my travels, making the most exhaustive inquiries the while; I have seen the most climates and countries and listened to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Democritus.html (1 of 5) [2/16/2002 11:07:00 PM]

Democritus

the greatest number of learned men. His travels certainly took him to Egypt and Persia, as Russell suggests, but he almost certainly also travelled to Babylon, and some claim he travelled to India and Ethiopia. Certainly he was a man of great learning. As Heath writes in [7]:... there was no subject to which he did not notably contribute, from mathematics and physics on the one hand to ethics and poetics on the other; he even went by the name of 'wisdom'. Although little is known of his life, quite a lot is known of his physics and philosophy. There are two main sources for our knowledge of his of physical and philosophical theories. Firstly Aristotle discusses Democritus's ideas thoroughly because he strongly disagreed with his ideas of atomism. The second source is in the work of Epicurus but, in contrast to Aristotle, Epicurus is a strong believer in Democritus's atomic theory. This work of Epicurus is preserved by Diogenes Laertius in his second century AD book [5]. Certainly Democritus was not the first to propose an atomic theory. His teacher Leucippus had proposed an atomic system, as had Anaxagoras of Clazomenae. In fact traces of an atomic theory go back further than this, perhaps to the Pythagorean notion of the regular solids playing a fundamental role in the makeup of the universe. However Democritus produced a much more elaborate and systematic view of the physical world than had any of his predecessors. His view is summarised in [2]:Democritus asserted that space, or the Void, had an equal right with reality, or Being, to be considered existent. He conceived of the Void as a vacuum, an infinite space in which moved an infinite number of atoms that made up Being (i.e. the physical world). These atoms are eternal and invisible; absolutely small, so small that their size cannot be diminished (hence the name atomon, or "indivisible"); absolutely full and incompressible, as they are without pores and entirely fill the space they occupy; and homogeneous, differing only in shape, arrangement, position, and magnitude. With this as a basis to the physical world, Democritus could explain all changes in the world as changes in motion of the atoms, or changes in the way that they were packed together. This was a remarkable theory which attempted to explain the whole of physics based on a small number of ideas and also brought mathematics into a fundamental physical role since the whole of the structure proposed by Democritus was quantitative and subject to mathematical laws. Another fundamental idea in Democritus's theory is that nature behaves like a machine, it is nothing more than a highly complex mechanism. There are then questions for Democritus to answer. Where do qualities such as warmth, colour, and taste fit into the atomic theory? To Democritus atoms differ only in quantity, and all qualitative differences are only apparent and result from impressions of an observer caused by differing configurations of atoms. The properties of warmth, colour, taste are only by convention - the only things that actually exist are atoms and the Void. Democritus's philosophy contains an early form of the conservation of energy. In his theory atoms are eternal and so is motion. Democritus explained the origin of the universe through atoms moving randomly and colliding to form larger bodies and worlds. There was no place in his theory for divine intervention. Instead he postulated a world which had always existed, and would always exist, and was filled with atoms moving randomly. Vortex motions occurred due to collisions of the atoms and in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Democritus.html (2 of 5) [2/16/2002 11:07:00 PM]

Democritus

resulting vortex motion created differentiation of the atoms into different levels due only to their differing mass. This was not a world which came about through the design or purpose of some supernatural being, but rather it was a world which came about through necessity, that is from the nature of the atoms themselves. Democritus built an ethical theory on top of his atomist philosophy. His system was purely deterministic so he could not admit freedom of choice to individuals. To Democritus freedom of choice was an illusion since we are unaware of all the causes for a decision. Democritus believed that [3]:... the soul will either be disturbed, so that its motion affects the body in a violent way, or it will be at rest in which case it regulates thoughts and actions harmoniously. Freedom from disturbance is the condition that causes human happiness, and this is the ethical goal. Democritus describes the ultimate good, which he identifies with cheerfulness, as:... a state in which the soul lives peacefully and tranquilly, undisturbed by fear or superstition or any other feeling. He wanted to remove the belief in gods which were, he believed, only introduced to explain phenomena for which no scientific explanation was then available. Very little is known for certainty about Democritus's contributions to mathematics. As stated in the Oxford Classical Dictionary :Little is known (although much is written) about the mathematics of Democritus. We do know that Democritus wrote many mathematical works. Diogenes Laertius (see [5]) lists his works and gives Thrasyllus as the source of this information. He wrote On numbers, On geometry, On tangencies, On mappings, On irrationals but none of these works survive. However we do know a little from other references. Heath [7] writes:In the Method of Archimedes, happily discovered in 1906, we are told that Democritus was the first to state the important propositions that the volume of a cone is one third of that of a cylinder having the same base and equal height, and that the volume of a pyramid is one third of that of a prism having the same base and equal height; that is to say, Democritus enunciated these propositions some fifty years or more before they were first scientifically proved by Eudoxus. There is another intriguing piece of information about Democritus which is given by Plutarch in his Common notions against the Stoics where he reports on a dilemma proposed by Democritus as reported by the Stoic Chrysippus (see [7], [10] or [11]). If a cone were cut by a plane parallel to the base [by which he means a plane indefinitely close to the base], what must we think of the surfaces forming the sections? Are they equal or unequal? For, if they are unequal, they will make the cone irregular as having many indentations, like steps, and unevennesses; but, if they are equal, the sections will be equal, and the cone will appear to have the property of the cylinder and to be made up of equal, not unequal, circles, which is very absurd. There are important ideas in this dilemma. Firstly notice, as Heath points out in [7], that Democritus has the idea of a solid being the sum of infinitely many parallel planes and he may have used this idea to find the volumes of the cone and pyramid as reported by Archimedes. This idea of Democritus may have led Archimedes later to apply the same idea to great effect. This idea would eventually lead to theories of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Democritus.html (3 of 5) [2/16/2002 11:07:00 PM]

Democritus

integration. There is much discussion in [7], [8], [10] and [11] as to whether Democritus distinguished between the geometrical continuum and the physical discrete of his atomic system. Heath points out that if Democritus carried over his atomic theory to geometrical lines then there is no dilemma for him since his cone is indeed stepped with atom sized steps. Heath certainly believed that to Democritus lines were infinitely divisible. Others, see for example [10], have come to the opposite conclusion, believing that Democritus made contributions to problems of applied mathematics but, because of his atomic theory, he could not deal with the infinitesimal questions arising. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles)

Some Quotations (4)

A Poster of Democritus

Mathematicians born in the same country

Cross-references to History Topics

1. Greek Astronomy 2. An overview of the history of mathematics 3. The rise of the calculus

Honours awarded to Democritus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Democritus

Other Web sites

1. Drury College 2. Internet Encyclopedia of Philosophy 3. S M Cohen (Atomism) 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Democritus.html (4 of 5) [2/16/2002 11:07:00 PM]

Democritus

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Democritus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Democritus.html (5 of 5) [2/16/2002 11:07:00 PM]

Denjoy

Arnaud Denjoy Born: 5 Jan 1884 in Auch, Gers, France Died: 21 Jan 1974 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Arnaud Denjoy's father, Jean Denjoy, was a wine merchant in Perpignan, which is in southern France about 30 km north of the border with Spain. We do not know Arnaud's mother's first name, but her surname was Jayez and she came from Catalonia. Arnaud was born and attended a secondary school in Auch which is the capital town of the Gers region in southwestern France. After attending secondary school in Auch, Arnaud completed his schooling in Montpellier which is close to the Mediterranean coast in southern France. He remained associated with these regions throughout his life. In 1902 Denjoy entered the Ecole Normale Supérieure where he studied under Borel, Painlevé and Emile Picard. These great mathematicians gave Denjoy a strong background in complex function theory, continued fractions and differential equations and set him on the road to his great discoveries. Denjoy enjoyed the highest success during his undergraduate years, being the top student in his class. His success was translated into the winning of a prestigious fellowship, the Fondation Thiers fellowship, which supported him for three years during which he worked on his dissertation Sur les produits canoniques d'ordre infini which he submitted in 1909. Denjoy's dissertation, although not considered by him as among his greatest achievements when he looked back on his career in 1934, is now considered to contain some remarkable contributions. It studies the asymptotic behaviour of integral functions of finite order, Weierstrass products of integral functions and the boundary behaviour of conformal representations. From Paris, Denjoy moved back to Montpellier in 1909 when he was appointed "Maitre des conferences" at the University of Montpellier. This post saw him preparing students for the "agregation" examination

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Denjoy.html (1 of 3) [2/16/2002 11:07:02 PM]

Denjoy

and it was the same post in the same university to which Baire had been appointed in 1901. Baire had left Montpellier before Denjoy took up this appointment, but he was an important figure in the development of Denjoy's mathematics. Denjoy was one of a small number of people who appreciated Baire's innovative ideas when he first produced them. For example when Baire was told that what he taught was so difficult that it was beyond human ability to understand it, he wrote:... but look at Denjoy - he understood it, hence it must not be so difficult ... Denjoy taught at Montpellier until the start of World War I in 1914. He suffered from poor eyesight so he was not fit for military service during the war. He was appointed to a professorship at Utrecht in 1917 then at the University of Paris from 1922, a post he held until he retired in 1955. Not long after his appointment at the University of Paris, he married Thérèse-Marie Chevresson in June 1923. They had three children, all sons. Denjoy worked on functions of a real variable in the same areas as Borel, Baire and Lebesgue. He combined topological and metrical methods to attack problems of real analysis. In 1934 he wrote that his greatest achievements had been the integration of derivatives, the computation of the coefficients of a converging trigonometric series, a theorem on quasi-analytic functions, and differential equations on a torus. The second of these topics, computation of the coefficients of a converging trigonometric series, was the subject of a four volume work Lectures on the computation of coefficients in a trigonometric series which appeared between 1941 and 1949. These four volumes were an expanded version of work which had appeared in a series of papers by Denjoy beginning in 1920. Included in these papers was his introduction of the Denjoy index for the points of a perfect set. Also in these papers is his introduction of the second symmetric derivative of a function. This work is studied in detail in [3] where Bullen makes the contents of Denjoy's work, which is not easy to read, more accessible to modern analysts. Although considered by Denjoy as one of his most important pieces of work in his 1934 review, Choquet writes in [1] that these papers may be:... considered more feats of intellectual strength than sources of practical applications. However, Choquet describes the four volume work Lectures on the computation of coefficients in a trigonometric series which contains the famous Denjoy integral, as [1]:... an explosion of beautiful theorems and examples. Choquet, very fairly, suggests that Denjoy's work on differential equations on a torus, not nearly so highly rated by Denjoy himself, is one of his most influential pieces of work and has [1]:... grown into a vast field involving dynamical systems. Similarly Denjoy's theorem on quasi-analytic functions has been the foundation of studies by Mandelbrot and has proved important in the development of large areas of current research. Let us end by saying something of Denjoy's character. He was a quiet man, who liked to carry out his mathematical research in the peace of his own home. For leisure he enjoyed being in the countryside, particularly walking and cycling in wooded country. A relatively poor lecturer he was, nevertheless, a fine writer and an entertaining man with whom to hold a conversation. Denjoy was not a man lacking interests outside mathematics: on the contrary he was fascinated by topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Denjoy.html (2 of 3) [2/16/2002 11:07:02 PM]

Denjoy

such as philosophy, psychology, and social studies. He approached such topics from his position as an atheist and an active participant in socialist politics. In particular he was an active member of the Radical Party which was headed by Edouard Herriot. Although Denjoy did not aspire to the political career of Herriot, who served in nine different cabinets and was premier of France three times, Denjoy's involvement with the Radical Party led to him serving as a town councillor for Montpellier in 1912, and as county councillor for Gers from 1920. He served in this capacity for twenty years. For his outstanding contributions to the theory of functions of a real variable, Denjoy received many honours. As well as election to the Académie des Sciences in 1941, he was also elected to the academies in Amsterdam, Warsaw, and Liege. He was also honoured by being elected as vice-president of the International Mathematical Union in 1954. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1910 to 1920

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Denjoy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Denjoy.html (3 of 3) [2/16/2002 11:07:02 PM]

Deparcieux

Antoine Deparcieux Born: 28 Oct 1703 in Clotet-de-Cessous, France Died: 2 Sept 1768 in Paris, France Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Antoine Deparcieux's father was Jean-Antoine Deparcieux while his mother was Jeanne Donzel. It was a poor family, Antoine's father being a farm worker. He attended the schools of Porte and Saint-Florent where he learned to read and write. Sadly, before his twelfth birthday, Antoine became an orphan and his education and upbringing became the responsibility of his elder brother Pierre Deparcieux. When Antoine was fifteen years old, Pierre sent him to study at the Jesuit College in Alès in southeastern France. This would not have been possible without the financial support of at least one patron. He studied at this College until he was seventeen years old. He showed great promise in scientific subjects at the College and when he left he went to Paris to study mathematics. Again he had financial support, this time from Montcarville who was his patron. In Paris Deparcieux found that the financial support provided by Montcarville was not sufficient to allow him to study without a job. He became a maker of sundials and, although this is not the reason that he appears in this archive, it was an occupation which at that time gave him a reasonable income. He was also interested in hydrodynamics and hydraulics and he moved on from making sundials to invent other machinery, in particular pumps. He is chiefly known, however, for his work on mathematical and statistical tables. In 1741 Deparcieux published Nouveaux traités de trigonométrie rectiligne et sphérique which consisted of tables of sins, tans, secs (calculated to seven decimal places), and log sins and log tans (calculated to eight decimal places). This work also contains interesting trigonometric formulas for tan a/2. In 1746 Deparcieux published a treatise on annuities and mortality: it was one of the first statistical work of its kind and is the main reason for his fame. Deparcieux's interest in mortality tables resulted from his interest in life expectancy which he had investigated in several different contexts. Lorenzo de Tonti from Naples was a financier who had devised the tontine life insurance plan in the seventeenth century. Those taking part in the plan contributed money which eventually went to the one who survives all the others. Deparcieux studied the way that such plans worked, and he also studied life expectancy in individual families and religious communities. The reason for choosing data such groups was that Deparcieux was fully aware that if one looked at births and deaths in a city (as had been done before), then migration made the data unreliable. However, as Johnson and Kotz write in [3]:... the results obtained for restricted populations, although accurately derived - even according to current standards - could not reasonably be ascribed to surrounding areas, a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Deparcieux.html (1 of 2) [2/16/2002 11:07:04 PM]

Deparcieux

point which caused some criticism. The results of Deparcieux's investigations were published in the 1746 treatise Essai sur les probabilités de la durée de la vie humaine. Busard writes in [1]:Deparcieux showed a real progress in his theoretical explanation of the properties of the tables of mortality. However, his tables, which were for a long time the only ones on life expectancies in France, indicated too small a value for the probable life expectancy at every age. In 1746 Deparcieux was elected a member of the Académie des Sciences and it would appear his 1746 publication was the reason for this election. Deparcieux was, as we have indicated above, interested in hydrodynamics. He devised a way of bringing the water of the river Yvette to Paris and, although he did not see his plan put into being during his lifetime, it was carried out after his death. Nicolas, in [4], describes Deparcieux's character:He was modest, not overambitious always keeping in mind his humble origins. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Antoine Deparcieux (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Deparcieux (14th Arrondissement)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Deparcieux.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Deparcieux.html (2 of 2) [2/16/2002 11:07:04 PM]

Desargues

Girard Desargues Born: 21 Feb 1591 in Lyon, France Died: Sept 1661 in Lyon, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Little is known about Girard Desargues' personal life. His family (on both his mother's and his father's side) had been very rich for several generations and had supplied lawyers and judges to the Parlement in Paris as well as to that in Lyon (then the second most important city in France). Desargues seems to have made several extended visits to Paris in connection with a lawsuit for the recovery of a huge debt. Despite this loss, the family still owned several large houses in Lyon, a manor house (and its estate) at the nearby village of Vourles, and a small chateau surrounded by the best vineyards in the vicinity. It is thus clear that Desargues had every opportunity of acquiring a good education, could afford to buy what books he chose, and had leisure to indulge in whatever pursuits he might enjoy. In his later years, these seem to have included designing an elaborate spiral staircase, and an ingenious new form of pump, but the most important of Desargues' interests was Geometry. He invented a new, non-Greek way of doing geometry, now called 'projective' or 'modern' geometry. As a mathematician he was very good indeed: highly original and completely rigorous. He is, however, far from lucid in his mathematical style. When in Paris, Desargues became part of the mathematical circle surrounding Marin Mersenne (1588 1648). This circle included Rene Descartes (1597 -1650), Etienne Pascal (1588 -1651) and his son Blaise Pascal (1623 - 1662). It was probably essentially for this limited readership of friends that Desargues prepared his mathematical works, and had them printed. Some of them were later expanded into more publishable form by Abraham Bosse (1602 -1676), who is now best remembered as an engraver, but was also a teacher of perspective.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Desargues.html (1 of 3) [2/16/2002 11:07:05 PM]

Desargues

Desargues wrote on 'practical' subjects such as perspective (1636), the cutting of stones for use in building (1640) and sundials (1640). His writings are, however, dense in content and theoretical in their approach to the subjects concerned. There is none of the wordy and elementary step-by-step explanation which one finds in texts that are truly addressed to artisans. Desargues' most important work, the one in which he invented his new form of geometry, has the title Rough draft for an essay on the results of taking plane sections of a cone (Brouillon project d'une atteinte aux evenemens des rencontres du Cone avec un Plan). A small number of copies was printed in Paris in 1639. Only one is now known to survive, and until this was rediscovered, in 1951, Desargues' work was known only through a manuscript copy made by Philippe de la Hire (1640 - 1718). The book is short, but very dense. It begins with pencils of lines and ranges of points on a line, considers involutions of six points (Desargues does not use or define a cross ratio), gives a rigorous treatment of cases involving 'infinite' distances, and then moves on to conics, showing that they can be discussed in terms of properties that are invariant under projection. We are given a unified theory of conics. Desargues' famous 'perspective theorem' - that when two triangles are in perspective the meets of corresponding sides are collinear - was first published in 1648, in a work on perspective by Abraham Bosse. It is clear that, despite his determination to explain matters in the vernacular, and without direct reference to the theorems or the vocabulary of Ancient mathematicians, Desargues is well aware of the work of ancient geometers, for instance Apollonius and Pappus. His choosing to explain himself differently may perhaps be due to his recognition that his own work was also deeply indebted to the practical tradition, specifically to the study of perspective (which is a form of conical projection). It seems highly likely that it was in fact from his work on perspective and related matters that Desargues' new ideas arose. When projective geometry was reinvented, by the pupils of Gaspard Monge (1746 -1818), the reinvention was from descriptive geometry, a technique that has much in common with perspective. Article by: J. V. Field, London Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles) A Poster of Girard Desargues

Mathematicians born in the same country

Some pages from publications

A page from Brouillon Projet (1640), a diagram from this and a second diagram showing projection.

Cross-references to Famous Curves

1. cycloid 2. Epicycloid 3. Epitrochoid 4. Hypercycloid 5. Hypertrochoid

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Desargues.html (2 of 3) [2/16/2002 11:07:05 PM]

Desargues

Other references in MacTutor

1. Desargues' theorem 2. Chronology: 1625 to 1650

Honours awarded to Girard Desargues (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Desargues

Paris street names

Rue Desargues (11th Arrondissement)

Other Web sites

1. The Galileo Project 2. Minnesota (Desargues theorem and its relationship to one of Monge's geometry theorems) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR August 1995

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Desargues.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Desargues.html (3 of 3) [2/16/2002 11:07:05 PM]

Descartes

René Descartes Born: 31 March 1596 in La Haye (now Descartes),Touraine, France Died: 11 Feb 1650 in Stockholm, Sweden

Click the picture above to see eight larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

René Descartes was a philosopher whose work, La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry. Descartes was educated at the Jesuit college of La Flèche in Anjou. He entered the college at the age of eight years, just a few months after the opening of the college in January 1604. He studied there until 1612, studying classics, logic and traditional Aristotelian philosophy. He also learnt mathematics from the books of Clavius. While in the school his health was poor and he was granted permission to remain in bed until 11 o'clock in the morning, a custom he maintained until the year of his death. School had made Descartes understand how little he knew, the only subject which was satisfactory in his eyes was mathematics. This idea became the foundation for his way of thinking, and was to form the basis for all his works. Descartes spent a while in Paris, apparently keeping very much to himself, then he studied at the University of Poitiers. He received a law degree from Poitiers in 1616 then enlisted in the military school at Breda. In 1618 he started studying mathematics and mechanics under the Dutch scientist Isaac Beeckman, and began to seek a unified science of nature. After two years in Holland he travelled through Europe. Then in 1619 he joined the Bavarian army. From 1620 to 1628 Descartes travelled through Europe, spending time in Bohemia (1620), Hungary (1621), Germany, Holland and France (1622-23). He spent time in 1623 in Paris where he made contact with Mersenne, an important contact which kept him in touch with the scientific world for many years. From Paris he travelled to Italy where he spent some time in Venice, then he returned to France again http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html (1 of 5) [2/16/2002 11:07:07 PM]

Descartes

(1625). By 1628 Descartes tired of the continual travelling and decided to settle down. He gave much thought to choosing a country suited to his nature and chose Holland. It was a good decision which he did not seem to regret over the next twenty years. Soon after he settled in Holland Descartes began work on his first major treatise on physics, Le Monde, ou Traité de la Lumière. This work was near completion when news that Galileo was condemned to house arrest reached him. He, perhaps wisely, decided not to risk publication and the work was published, only in part, after his death. He explained later his change of direction saying:... in order to express my judgement more freely, without being called upon to assent to, or to refute the opinions of the learned, I resolved to leave all this world to them and to speak solely of what would happen in a new world, if God were now to create ... and allow her to act in accordance with the laws He had established. In Holland Descartes had a number of scientific friends as well as continued contact with Mersenne. His friendship with Beeckman continued and he also had contact with Mydorge, Hortensius, Huygens and Frans van Schooten (the elder). Descartes was pressed by his friends to publish his ideas and, although he was adamant in not publishing Le Monde, he wrote a treatise on science under the title Discours de la méthod pour bien conduire sa raison et chercher la vérité dans les sciences. Three appendices to this work were La Dioptrique, Les Météores, and La Géométrie. The treatise was published at Leiden in 1637 and Descartes wrote to Mersenne saying:I have tried in my Dioptrique and my Météores to show that my Méthod is better than the vulgar, and in my Géométrie to have demonstrated it. The work describes what Descartes considers is a more satisfactory means of acquiring knowledge than that presented by Aristotle's logic. Only mathematics, Descartes feels, is certain, so all must be based on mathematics. La Dioptrique is a work on optics and, although Descartes does not cite previous scientists for the ideas he puts forward, in fact there is little new. However his approach through experiment was an important contribution. Les Météores is a work on meteorology and is important in being the first work which attempts to put the study of weather on a scientific basis. However many of Descartes' claims are not only wrong but could have easily been seen to be wrong if he had done some easy experiments. For example Roger Bacon had demonstrated the error in the commonly held belief that water which has been boiled freezes more quickly. However Descartes claims:... and we see by experience that water which has been kept on a fire for some time freezes more quickly than otherwise, the reason being that those of its parts which can be most easily folded and bent are driven off during the heating, leaving only those which are rigid. Despite its many faults, the subject of meteorology was set on course after publication of Les Météores particularly through the work of Boyle, Hooke and Halley. La Géométrie is by far the most important part of this work. In [17] Scott summarises the importance of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html (2 of 5) [2/16/2002 11:07:07 PM]

Descartes

this work in four points:1. He makes the first step towards a theory of invariants, which at later stages derelativises the system of reference and removes arbitrariness. 2. Algebra makes it possible to recognise the typical problems in geometry and to bring together problems which in geometrical dress would not appear to be related at all. 3. Algebra imports into geometry the most natural principles of division and the most natural hierarchy of method. 4. Not only can questions of solvability and geometrical possibility be decided elegantly, quickly and fully from the parallel algebra, without it they cannot be decided at all. Some ideas in La Géométrie may have come from earlier work of Oresme but in Oresme's work there is no evidence of linking algebra and geometry. Wallis in Algebra (1685) strongly argues the the ideas of La Géométrie were copied from Harriot. Wallis writes:... the Praxis was read by Descartes, and every line of Descartes' analysis bears token of the impression. There seems little to justify Wallis's claim, which was probably made partly through partiotism but also through his just desires to give Harriot more credit for his work. Harriot's work on equations, however, may indeed have influenced Descartes who always claimed, clearly falsely, that nothing in his work was influenced by the work of others. Descartes' Meditations on First Philosophy, was published in 1641, designed for the philosopher and for the theologian. It consists of six meditations, Of the Things that we may doubt, Of the Nature of the Human Mind, Of God: that He exists, Of Truth and Error, Of the Essence of Material Things, Of the Existence of Material Things and of the Real Distinction between the Mind and the Body of Man. However many scientists were opposed to Descartes' ideas including Arnauld, Hobbes and Gassendi. The most comprehensive of Descartes' works, Principia Philosophiae was published in Amsterdam in 1644. In four parts, The Principles of Human Knowledge, The Principles of Material Things, Of the Visible World and The Earth, it attempts to put the whole universe on a mathematical foundation reducing the study to one of mechanics. This is an important point of view and was to point the way forward. Descartes did not believe in action at a distance. Therefore, given this, there could be no vacuum around the Earth otherwise there was way that forces could be transferred. In many ways Descartes's theory, where forces work through contact, is more satisfactory than the mysterious effect of gravity acting at a distance. However Descartes' mechanics leaves much to be desired. He assumes that the universe is filled with matter which, due to some initial motion, has settled down into a system of vortices which carry the sun, the stars, the planets and comets in their paths. Despite the problems with the vortex theory it was championed in France for nearly one hundred years even after Newton showed it was impossible as a dynamical system. As Brewster, one of Newton's 19th century biographers, puts it:Thus entrenched as the Cartesian system was ... it was not to be wondered at that the pure and sublime doctrines of the Principia were distrustfully received ... The uninstructed mind could not readily admit the idea that the great masses of the planets were suspended in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html (3 of 5) [2/16/2002 11:07:07 PM]

Descartes

empty space, and retained their orbits by an invisible influence... Pleasing as Descartes's theory was even the supporters of his natural philosophy, such as the Cambridge metaphysical theologian Henry More, found objections. Certainly More admired Descartes, writing:I should look upon Des-Cartes as a man most truly inspired in the knowledge of Nature, than any that have professed themselves so these sixteen hundred years... However between 1648 and 1649 they exchanged a number of letters in which More made some telling objections, Descartes however in his replies making no concessions to More's points. More went on to ask:Why are not your vortices in the form of columns or cylinders rather than ellipses, since any point of the axis of a vortex is as it were a centre from which the celestial matter recedes with, as far as I can see, a wholly constant impetus? ... Who causes all the planets not to revolve in one plane (the plane of the ecliptic)? ... And the Moon itself, neither in the plane of the Earth's equator nor in a plane parallel to this? In 1644, the year his Meditations were published, Descartes visited France. He returned again in 1647, when he met Pascal and argued with him that a vacuum could not exist, and then again in 1648. In 1649 Queen Christina of Sweden persuaded Descartes to go to Stockholm. However the Queen wanted to draw tangents at 5 a.m. and Descartes broke the habit of his lifetime of getting up at 11 o'clock. After only a few months in the cold northern climate, walking to the palace for 5 o'clock every morning, he died of pneumonia. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (104 books/articles)

Some Quotations (15)

A Poster of René Descartes

Mathematicians born in the same country

Cross-references to History Topics

1. Topology in mathematics 2. Beginnings of set theory 3. Quadratic etc equations 4. Perfect numbers 5. The fundamental theorem of algebra 6. An overview of the history of mathematics 7. The rise of the calculus 8. Abstract linear spaces 9. Matrices and determinants

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html (4 of 5) [2/16/2002 11:07:07 PM]

Descartes

Cross-references to Famous Curves

1. Cartesian ovals 2. Cycloid 3. Equiangular spiral 4. Folium of Descartes 5. Trident of Newton

Other references in MacTutor

1. Descartes' geometric solution of a quadratic equation 2. Chronology: 1625 to 1650

Honours awarded to René Descartes (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Descartes

Paris street names

Rue Descartes ( 5th Arrondissement)

Other Web sites

1. The Galileo Project 2. Rouse Ball 3. The Catholic Encyclopedia 4. Internet Encyclopedia of Philosophy 5. K A Bryson (Rules for the direction of the mind) 6. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Descartes.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html (5 of 5) [2/16/2002 11:07:07 PM]

Dickson

Leonard Eugene Dickson Born: 22 Jan 1874 in Independence, Iowa, USA Died: 17 Jan 1954 in Harlingen, Texas, USA

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leonard Dickson is known for his contributions to number theory and group theory. Dickson's family moved to Texas while he was a young child and he attended both primary and secondary school in his home town of Cleburne. He entered the University of Texas and quickly came under the influence of Halsted who encouraged him to study mathematics. Dickson studied widely within mathematics but specialised in Halsted's own subjects of euclidean and non-euclidean geometry. Dickson received his B.S. in 1893 and his M.S. in 1894, again under Halsted's supervision. Dickson applied for doctoral fellowships at both Harvard and Chicago. He accepted an offer from Harvard but, on receiving a later offer from Chicago, changed his mind. At Chicago he was supervised by Eliakim Moore, but others there influenced him, for example Bolza and Maschke. Dickson received a Ph.D. from the University of Chicago in 1896 for a dissertation entitled The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group. Dickson then spent some time with Lie at Leipzig and later with Jordan in Paris. On returning to the USA he became an instructor at the University of California in Berkeley. He was appointed to the University of Texas at Austin in 1899. However Eliakim Moore and his colleagues in Chicago were keen that Dickson should return there and they offered him a permanent post on the faculty. He accepted immediately and served as professor at the University of Chicago from 1900

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dickson.html (1 of 3) [2/16/2002 11:07:09 PM]

Dickson

to 1939. Dickson worked on finite fields and extended the theory of linear associative algebras initiated by Wedderburn and Cartan. He proved many interesting results in number theory, using results of Vinogradov to deduce the ideal Waring theorem in his investigations of additive number theory. In 1901 his famous book Linear groups with an exposition of the Galois field theory was published. This was a revised and expanded version of his 1896 doctoral thesis. In the proposal for his book, sent to Klein, Dickson wrote:The book here announced proposes to treat of linear congruence groups, or more generally, of linear groups in a Galois field, a subject enriched by the labors of Galois, Betti, Mathieu, Jordan and many recent writers. In his letter to Klein, Dickson also talks of introducing marked simplifications and presenting parts of the theory without the difficult calculations given in the published papers. Parshall in [6] describing the book writes:Dickson presented a unified, complete, and general theory of the classical linear groups not merely over the prime field GF(p) as Jordan had done - but over the general finite field GF (pn), and he did this against the backdrop of a well- developed theory of these underlying fields. ... his book represented the first systematic treatment of finite fields in the mathematical literature. Dickson published 17 books in addition to Linear groups with an exposition of the Galois field theory. The 3-volume History of the Theory of Numbers (1919 - 23) is another famous work still much consulted today. Dickson was awarded many honours. The American Association for the Advancement of Science decided to set up a prize for the most major contribution to the advancement of science. Dickson was the first recipient of the prize. He was also the first recipient of the Cole Prize for algebra awarded by the American Mathematical Society in 1928 for his book Algebren und ihre Zahlentheorie published in Zurich and Leipzig in 1927. In fact Dickson was much involved with the American Mathematical Society, becoming its president in 1917-1918 and having earlier, in 1913, been its Colloquium Lecturer. Princeton and Harvard were among the universities that awarded him an honorary degrees. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles)

A Quotation

A Poster of Leonard E Dickson

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1900 to 1910

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dickson.html (2 of 3) [2/16/2002 11:07:09 PM]

Dickson

Honours awarded to Leonard E Dickson (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1917 - 1918

AMS Colloquium Lecturer

1913

AMS Cole Prize

Awarded 1928

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dickson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dickson.html (3 of 3) [2/16/2002 11:07:09 PM]

Dickstein

Samuel Dickstein Born: 12 May 1851 in Warsaw, Poland Died: 29 Sept 1939 in Warsaw, Poland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Samuel Dickstein was brought up during difficult years for Poles, most of whom aspired to see the country of Poland re-established. Poland did not formally exist at the time of Dickstein's birth and much of the pattern of his life was dominated by the aim of Poles to restore their country. Poland had been partitioned in 1772 with the south was called Galicia and under Austrian control while Russia and Prussia controlled the rest of the country. In 1846, three years before Dickstein was born, there was an attempted revolution by Polish nationalists. The Prussian police had discovered their plans to start an uprising and they put a stop to it in their area. However the uprising spread to Galicia but there it was soon defeated by Austrian troops. During the following years when Dickstein was a young child Poles sought independence but were kept down by armed force. The Crimean War which ended in 1856 had a great influence within the Russian Empire. Some reforms were put in place in the Russian areas of Poland (which included Warsaw where Dickstein lived) but these only seemed to invoke anger among young patriotic Poles. There were political demonstrations and, towards the end of 1862, riots broke out in Warsaw. On 22 January 1863 there was a move to force young Poles into the Russian army and a widespread rebellion took place. For a year Dickstein, a youth aged 12, saw the Polish uprising being crushed. The victorious Russian occupiers then carried out executions, confiscations, and deportations, and one can only imagine how a young man like Dickstein might have felt knowing that Poles could not hope to rule their own country again in the foreseeable future. In a policy implemented between 1869 and 1874, all secondary schooling was in the Russian language. There was no Polish university for Dickstein to attend so, in 1866, he entered one of the only higher http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dickstein.html (1 of 3) [2/16/2002 11:07:11 PM]

Dickstein

education establishments in Warsaw, the teacher's college. He studied there until 1869, the year in which it was converted into the Russian University of Warsaw . From 1870 to 1876 Dickstein attended this Russian university in Warsaw specialising in mathematics. He graduated with a Master's degree in 1876 but all the time he spent at university he held positions in secondary schools teaching mathematics to provide the means to support his studies. With the education system controlled by the Russian rulers, Dickstein decided to do what he could to promote a Polish education and he directed his own private school for ten years beginning in 1878. However during this period he began other ventures to promote Polish science. Dickstein was one of the main instigators of publishing mathematical journals in Poland. In 1884 he was one of the two founders of a series of mathematics and physics textbooks which were written in Polish. A few years later he was one of three scientists who set up the journal Mathematical and Physical Papers editing the journal from 1888. From 1897 he edited Mathematical News another publication which he was involved in setting up. He also continued publication of Circle of Polish Mathematicians which had begun publishing in St Petersburg in 1880. Kuratowski, thinking about the development of Polish mathematics, notes in [2] the importance of the publications:The establishment of "Mathematical and Physical Papers" and "Mathematical News" made possible for Polish mathematicians to publish the results of their research in Poland, and thus it favoured the increase of mathematical activity in our country ... It was not only with his role in publishing that Dickstein made a major contribution to Polish mathematics. In 1903 Dickstein was a founder of the Warsaw Scientific Society and he was important in the development of the Polish Mathematical Society. These played a vital role in the development of Polish mathematics and we have described above the political situation out of which the Warsaw Scientific Society was born. The first meeting of a group trying to set up a Polish institution in Warsaw took place on 21 December 1903. Dickstein was one of two mathematicians in this founding group of fourteen, and was elected as secretary of the group. The work of the Warsaw Scientific Society started properly in November 1907. A Mathematical Study was set up with Dickstein donating a fine library of mathematical texts. World War I brought major changes in Poland. In August 1915 the Russian forces which had held Poland for many years withdrew from Warsaw. Germany and Austria-Hungary took control of most of the country and a German governor general was installed in Warsaw. One of the first moves after the Russian withdrawal was the refounding of the University of Warsaw and it began operating as a Polish university in November 1915. Dickstein taught in the newly established university, giving the first year lectures on algebra. He became a professor of mathematics at the University of Warsaw in 1919 when the university became properly constituted after the end of the war. Dickstein's work was mostly in algebra and the history of mathematics. In particular he had written an important monograph on Wronski in 1896. Kuratowski writes in [2] that Dickstein:... was not a scholar of outstanding creative achievements, and certainly his lectures presented a somewhat outdated algebra; they were, however, excellently shaped lectures by an enthusiast for mathematics, who infected young adepts in mathematics with his own ardour; this was by no means unimportant for the new staff of renascent Polish Science. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dickstein.html (2 of 3) [2/16/2002 11:07:11 PM]

Dickstein

Ulam, at age 23, met Dickstein, who was then in his eighties, at the International Mathematical Congress in Zurich in 1932. He writes that Dickstein was:... wandering around looking for his contemporaries. Dickstein's teacher had been a student of Cauchy in the early nineteenth century, and he still considered Poincaré, who died in 1912, a bright young man. To me this was like going into the prehistory of mathematics and it filled me with a kind of philosophical awe. Dickstein died in the Nazi bombing of Warsaw in 1939 and all his family died during the German occupation of Poland. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dickstein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dickstein.html (3 of 3) [2/16/2002 11:07:11 PM]

Dieudonne

Jean Alexandre Eugène Dieudonné Born: 1 July 1906 in Lille, France Died: 29 Nov 1992

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean Dieudonné was educated in Paris, receiving both his bachelor's degree (1927) and his doctorate (1931) from the Ecole Normale Supérieure. After working at Rennes, Nancy and Sao Paulo in Brazil, he went to the USA in 1952 and was appointed professor of mathematics at the University of Michigan. After teaching at Northwestern University Dieudonné returned, in 1959, to Paris. In 1964 he accepted a chair at Nice. Dieudonné was one of the two main contributors to the Bourbaki series of texts. He began his mathematical career working on the analysis of polynomials. He worked in a wide variety of mathematical areas including general topology, topological vector spaces, algebraic geometry, invariant theory and the classical groups. His best known books are La Géométrie des groupes classiques (1955), Foundations of Modern Analysis (1960), and Algèbre linéaire et géométrie élémentaire (1964). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles)

Some Quotations (5)

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dieudonne.html (1 of 2) [2/16/2002 11:07:13 PM]

Dieudonne

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dieudonne.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dieudonne.html (2 of 2) [2/16/2002 11:07:13 PM]

Digges

Thomas Digges Born: 1546 in Wotton (near Canterbury), Kent, England Died: 24 Aug 1595 in London, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Digges received his early education from his father who himself was a good scientist writing on surveying. Thomas later received advanced mathematical instruction from John Dee. He was to remain a friend of Dee's and undertook joint work with him. Digges wrote on platonic solids and archimedian solids which appear in Pantometria (1571). This work includes contributions by Digges's father. In 1573 Digges published Alae seu scalae mathematicae, a work on the position of the '(Tycho Brahe)'s supernova' of 1572. This work includes observations of the position of the 'new star' and trigonometric theorems which could be used to determine the parallax of the star. The observations are particularly impressive making Digges the ablest observer of his time. Digges's friend Dee published a similar work on the supernova. Digges was the leader of the English Copernicans. He translated part of Copernicus's De revolutionibus and added his own ideas of an infinite universe with the stars at varying distances an infinite space. He published A Perfit Description of the Caelestial Orbes in 1576 which again restates Copernicus's views. As well as having a military career, Digges also wrote and worked on other military matters. His book Stratioticos (1579) is a mathematics book for soldiers and contains the first discussion of ballistics in a work published in England. He also worked on fortifications, being in charge of the fortification of Dover harbour in 1582. A year earlier he had been involved in producing plans for Dover castle. Digges was a member of parliament from 1572 and again in 1584. His military career was with the English forces in the Netherlands from 1586 to 1594. The modern state of the Netherlands came into existence with the Treaty of Utrecht in 1579. This was the year Digges wrote his military work Stratioticos which he dedicated to Robert Dudley, Earl of Leicester. Dudley was named governor-general of the Netherlands in 1586 and Dudley appointed Digges to be master-general of his army to assist him in the campaign. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Digges.html (1 of 2) [2/16/2002 11:07:14 PM]

Digges

List of References (7 books/articles)

A Quotation

Mathematicians born in the same country Other references in MacTutor

1. Platonic solids 2. Semi-regular (or Archimedean) solids

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Digges.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Digges.html (2 of 2) [2/16/2002 11:07:14 PM]

Dilworth

Robert Palmer Dilworth Born: 2 Dec 1914 in Hemet, Califormia, USA Died: 29 Oct 1993 in Califormia, USA

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Dilworth (known as Bob to his friends and colleagues) was brought up on a ranch in California at the foot of the San Jacinto Mountains. This gave him a love of the outdoors which he kept throughout his life. He studied for his Bachelor of Science degree at the California Institute of Technology. Caltech was to play a very important role in Dilworth's life, for it was the Institution that he was associated with for almost the whole of his career. He received his B.S. degree in 1936 and remained at Caltech to undertake postgraduate work for his doctorate. At Caltech, Dilworth's doctoral studies were supervised by Morgan Ward. Ward had himself been a student of E T Bell and Bell was still on the Faculty at Caltech at this time. Like Bell, Ward was someone who valued all aspects of mathematics. He was as interested in teaching elementary mathematics as he was in the latest research problem on which he was working. His ideas of what constituted "a mathematician" rubbed off on Dilworth, as Chase relates in [3]:Professor Ward helped instil in Professor Dilworth his profound respect for the teaching of mathematics, at all levels, even very elementary levels. Dilworth obtained his doctorate in 1939 and was then awarded a Sterling Research fellowship to study at Yale. He held this Fellowship at Yale during the academic year 1939-40 and he was then appointed as an Instructor there. Dilworth married Miriam White on 23 December 1940 after taking up the Instructor appointment. He held this position from 1940 until 1943 when he returned to Caltech as Assistant Professor of Mathematics. At this point Dilworth was back at Caltech and he was to remain there for the rest of his career. Of course 1943 was in the middle of World War II and Dilworth was involved in military service. In July 1944 he became a member of an analysis unit at the 8th Air Force headquarters at Brampton Park in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dilworth.html (1 of 4) [2/16/2002 11:07:16 PM]

Dilworth

England. Dilworth wrote:This unit was to serve as a liaison between the main operational analysis unit located at the headquarters of the 8th Air Force near London and the command of the 1st Air Division. ... In the spring of 1945, in collaboration with the Division navigator, an elaborate experiment was carried out to evaluate the intrinsic accuracy of radar bombing. A special radar target placed in the Wash on the east coast of Britain was used in this exercise. Back at Caltech, Dilworth was promoted to Associate Professor in 1945 and then full Professor in 1950. He held this position for the rest of his career until he retired in 1982. Let us now turn to Dilworth's research contributions. He worked in lattice theory and it would not be an exaggeration to say that he was one of the main factors in the subject moving from being merely a tool of other disciplines to an important subject in its own right. He began his studies in the 1930s by reading the first contributions to lattice theory which were by Dedekind. Dilworth himself remarked that although Dedekind's papers were excellent introductions to the subject it was unclear what his motivation had been. By the time Dilworth began his research, the motivation behind much of lattice theory was to develop methods to attack problems in group theory. This is well explained by Dilworth himself writing in 1959 (see [1]):The theory of groups provided much of the motivation and many of the technical ideas in the early development of lattice theory. Indeed, it was the hope of many of the early researchers that lattice-theoretic methods would lead to the solution of some of the important problems in group theory. Two decades later, it seems to be a fair judgement that, while this hope has not been realised, lattice theory has provided a useful framework for the formulation of certain topics in the theory of groups ... and has produced some interesting and difficult group-theoretic problems ... Dilworth then goes on to explain where the main thrust in developing lattice theory subsequently come from and one has to say that, although he modestly does not say so, he played the major role in this development himself:On the other hand, the fundamental problems of lattice theory have, for the most part, not come from this source but have arisen from attempts to answer the intrinsically natural questions concerning lattices and partially ordered sets; namely, questions concerning the decompositions, representations, imbedding, and free structure of such systems ... As the study of these basic questions has progressed, there has come into being a sizable body of technical ideas and methods which are peculiarly lattice-theoretic in nature. These conceptual tools are intimately related to the underlying order relation and are particularly appropriate for the study of general lattice structure. The main topics in lattice theory to which Dilworth contributed are: Chain partitions in ordered sets, in particular his chain decomposition theorem for partially ordered sets; Uniquely complemented lattices; Lattices with unique irreducible decompositions; Modular and distributive lattices, in particular his covering theorem for modular lattices; Geometric and semimodular lattices; and Multiplicative lattices, where he studied, among other topics, abstract ideal theory, and the representation and embedding theorems for Noether lattices and r-lattices. One important aspect of Dilworth's research was that he always attacked the big problems in lattice theory. He always had a stock of open problems in the subject which he used to direct his research and that of his students. For example in 1959 he writes of about the big problems of the subject [1]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dilworth.html (2 of 4) [2/16/2002 11:07:16 PM]

Dilworth

... the construction of a set of structure invariants for certain classes of Boolean algebras, the characterisation of the lattice of congruence relations of a lattice, the imbedding of finite lattices in finite partition lattices, the word problem for free modular lattices, and a construction of a dimension theory for continuous, non-complemented, modular lattices, have an intrinsic interest independent of the problems associated with other algebraic systems. Furthermore, these and other current problems are sufficiently difficult that imaginative and ingenious methods will be required in their solution. Let us now turn to Dilworth as a teacher. We have already mention the influence of his supervisor Morgan Ward on him. R Freese and J B Nation write (see [1]):When [Dilworth] lectured, he rarely used abbreviations and his handwriting was nearly perfect. Students had to write as fast as they could, using several abbreviations, to keep up with him. When he got stuck he would step back from the blackboard, stare at the problem and whistle "Stars and Stripes Forever". Teaching and examining mathematics played an important part in Dilworth's career. He was appointed to the College Board Advanced Mathematics Committee in 1954. The task of this Committee was to set policy and administer the Advanced Mathematics Examination. Dilworth was Chairman of this Committee from 1957 to 1961. He also became involved with a project to develop mathematics education in African countries. He was Director of the Testing and Evaluation group for this project from 1962 to 1969 and he described its role:The objective was to develop a core of mathematics educators in each of the participating countries who would be able to produce curriculum materials in mathematics which would be appropriate for the needs of each of the countries. During six summer sessions from 1962 to 1968 the representatives of the African countries involved met with mathematics educators from the United States and Britain to develop specimen mathematics texts covering primary and secondary years. It was the responsibility of the Testing and Evaluation group to see that there were African personnel in each of the countries trained in modern testing methods by developing tests and other evaluative materials ... In addition Dilworth served on numerous other bodies concerned with the teaching and examining of mathematics. For example the Board of Examiners in Mathematics, the School Mathematics Study Group Advisory Board, the Miller Mathematics Improvement Program, and several programs set up by the National Science Foundation. Finally we should say a little about Dilworth other than his mathematical interests. As a young man he was an exceptionally good athlete, competing in the decathlon. Later in life he complained that running was damaging his knees and he took up swimming which he did regularly. He kept himself very fit and [3]:He never dawdled, but always walked with a spring in his step, and got wherever he was going very fast. Another of his interests was music and [2]:... he often commented that if he were not accepted to CalTech, he would have made music his life. He loved playing Chopin on the piano late at night in total darkness. He insisted that it improved his mathematical abilities. He played several other instruments as well when necessary for the CalTech Orchestra; however the piano was his relief from the pressures of mathematics. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dilworth.html (3 of 4) [2/16/2002 11:07:16 PM]

Dilworth

Crawley [4] writes:... he was an electrifying teacher and colleague. And apart from his intellectual power as a mathematician, I think this was primarily a product of two traits: Bob Dilworth loved a challenge, and he was tenacious in confronting one; and he had great mathematical taste. Bogart in [2] write that Dilworth:... had a keen sense of humour and was known as a warm and approachable person. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2001 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Dilworth.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dilworth.html (4 of 4) [2/16/2002 11:07:16 PM]

Dinghas

Alexander Dinghas Born: 9 Feb 1908 in Smyrna (now Izmir), Turkey Died: 19 April 1974 in Berlin, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alexander Dinghas's father was a primary school teacher. Alexander attended primary school in Smyrna and began his secondary schooling there. In 1922 his parents moved from Smyrna to Athens and Alexander moved with the family to complete the last three years of his secondary schooling there. In 1925 he entered the Athens Technical University where he studied engineering, graduating in 1930 with a diploma in electrical and mechanical engineering. He married Fanny Grafiadou in 1931. In 1931 Dinghas began his studies at Berlin. His original intention was to study physics and he began taking courses in both physics and mathematics, as well as some philosophy courses. The three professors of mathematics were Schmidt, Schur and Bieberbach. However, many other talented mathematicians and theoretical physicists were also at Berlin and influenced Dinghas. In particular Schrödinger, von Mises, von Laue, von Neumann, Richard Rado, Bernhard Neumann and Wielandt. It was the teaching of Schmidt in particular which convinced Dinghas that mathematics rather than physics was the subject for him to pursue. Right from the time he began his studies in 1931, Dinghas became interested in Nevanlinna theory. He attended lectures on the topic given by Schmidt and it was these lectures which Schmidt gave "with almost religious enthusiasm" which turned Dinghas from an engineer/physicist into a mathematician. He studied for his doctorate under Schmidt and it was awarded in 1936. Two years later he submitted his habilitation thesis and obtained the right to lecture in a university. However, as a non-German his career during the Nazi years was extremely difficult. Despite the award of his habilitation he did not receive a permanent teaching post although he did manage to continue teaching http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dinghas.html (1 of 3) [2/16/2002 11:07:18 PM]

Dinghas

throughout. However, after the end of World War II he became professor at the reopened Humbolt University in 1947. From 1949 until his death he was a professor at the Free University of Berlin and director of the Mathematical Institute there. His work is in many areas of mathematics including differential equations, functions of a complex variable, functions of several complex variables, measure theory and differential geometry. His most important work was in function theory, in particular Nevanlinna theory and the growth of subharmonic functions. Dinghas produced a series of papers on isoperimetric problems in spaces of constant curvature. His work here was much influenced by Schmidt who also produced important results which Dinghas used in his work. The article [5] contains a bibliography of 121 papers by Dinghas, and in addition lists three books and five historical or general articles. Although Dinghas had a wonderful feel for mathematics, he frequently waved his hands somewhat when he gave a proof. His papers were [5]:... not always easy to read and on occasion proofs were only sketched or contained serious gaps. However, the gaps have largely been filled in and the vision of the basic ideas will secure a permanent niche for their author in the theory of functions. His three books are Vorlesungen über Funktionentheorie (1961), Minkowskische Summen und Integrale. Superadditive Mengenfunktionale. Isoperimetrische Ungleichungen (1961), and Einführung in die Cauchy-Weierstrass'sche Funktionentheorie (1968). The first of these is described by a reviewer:This treatise presents an amazing amount of function theory in its modest 400 pages. The presentation is concise and clear. Also each of the nine chapters ends with a section... which presents various interesting topics, frequently in quite abbreviated form. Examples are the formula of Plana-Abel-Cauchy, the theorem of Julia-Wolff-Caratheodory, and the theory of Nevanlinna and of Hallstrom. Each chapter contains a useful section on the history and literature of the chapter's topics. This book will clearly prove valuable as a reference or as a text for any student who already knows a modest amount of elementary function theory. The treatise is in four parts. The final part containing chapters on the maximum principle and the distribution of values, geometric function theory and conformal mapping, and Nevanlinna theory. His 1968 book is described as follows:This little paperback book contains in 107 pages the core material and usual preliminaries of the standard first course in analytic functions of a complex variable. Definitions and theorems are stated precisely in modern terminology, but the underlying attitude is basically traditional and perhaps somewhat innocent topologically. ... Some topics treated which are not always found in the older short elementary texts are the homotopy concept for closed curves, cluster sets of meromorphic functions, removable compact sets of singularities, the monodromy theorem, and the Mittag-Leffler partial fraction expansion of a meromorphic function. Hayman writes of Dinghas's personality in [5]:Dinghas was a complex personality and a German professor of the old school. On the one hand he expected and was awarded the respect due to his position, and his students and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dinghas.html (2 of 3) [2/16/2002 11:07:18 PM]

Dinghas

colleagues were somewhat in awe of him. ... However this was only one side of his nature. he was extremely hospitable and generous and had a puckish sense of humour. ... He felt profound sympathy for those less fortunate. On one occasion he saw a man in a restaurant looking rather forlorn and with a single cup of coffee. Dinghas felt the man's hunger and got the waiter to send him food and drink which Dinghas paid for. On another occasion he felt sorry for a newspaper seller and bought every one of his papers. he always supported students against the teaching staff when he felt they had a good case. Dinghas received many honours for his work. In particular he was elected to membership of the Heidelberg Academy, the Finnish Academy and the Norwegian Academy. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dinghas.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dinghas.html (3 of 3) [2/16/2002 11:07:18 PM]

Dini

Ulisse Dini Born: 14 Nov 1845 in Pisa, Italy Died: 28 Oct 1918 in Pisa, Italy

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ulisse Dini's parents were Pietro Dini and Teresa Marchionneschi Dini. It was certainly not a wealthy family but Dini's parents were able to educate him in his native city of Pisa. He attended the Scuola Normale Superiore in Pisa which was a teacher's college, under the leadership of Enrico Betti, attached to the University of Pisa. Dini grew up at a time when political, and military, events in Italy were intensifying as the country came nearer to unification. There were not only the internal politics of unification but the was problems with Austria and France, both countries having their own agendas. In 1859, when Dini was thirteen years old, there was a war with Austria in which the French at first joined the Italians against the Austrians. However, by 17 March 1861, the Kingdom of Italy was formally created. Rome and Venice were not part of Italy at this stage, however, and there continued high levels of political activity as the govenment structure was discussed. In 1865 Dini entered a competition for a scholarship provide the necessary funds to allow a student to further their studies abroad. He won the scholarship and went to Paris where he studied with Bertrand and Hermite. This was a period of high mathematical activity for Dini and seven publications came out of the research he undertook during his time in Paris. Dini returned to Pisa in 1866, and was appointed to a post in the University of Pisa. There he taught advanced topics in algebra and also the theory of geodesy. Political events in Italy continued, with the Treaty of Vienna bringing Venice into the Italian Kingdom in 1866. Rome was attacked by Italian troops the following year but France defended the city with its troops against the attack. There was widespread

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dini.html (1 of 3) [2/16/2002 11:07:20 PM]

Dini

unrest in Italy due to dissatisfaction with the government and it was far from clear that the newly unified country would not split apart again. In 1870, however, Italian troops captured Rome. Dini progressed quickly in his career at the University of Pisa, being appointed to Betti's chair of analysis and higher geometry in 1871. This was not due to Betti's retiral, but rather because Betti's interests had moved more towards mathematical physics. Dini was not someone who was going to concentrate solely on mathematics and a university career, the political events of the time having a profound effect on someone of Dini's character who was [1]:... an upright, honest, kind man ... devoted to the well-being of his native city and his country ... Despite the workload required in his university career, Dini entered politics in 1871 (although he was only 25 years old at the time) when he was elected to the Pisa City Council. With a period of consolidation for the newly unified Italy, local government became very significant and Dini was keen to do all he could in this important area. His political and academic careers progressed side by side over the following years. In 1877 he was appointed to a second chair in the University of Pisa, from then on holding the chair of infinitesimal analysis in addition to his earlier professorial appointment. Having served many times on the Pisa Council, Dini was elected to the national Italian parliament in 1880 as a representative from Pisa. Not only was Dini highly involved in teaching mathematics and in local and national politics, but in 1888 he reached the highest office in university administration when he became rector of the University of Pisa. He held this post until 1890, then two years later he was elected a senator in the Italian Parliment. He became director of the Scuola Normale Superiore, the teacher's college at which he was himself educated, in 1908 and held this position until his death. Dini's most important work in mathematics was on the theory of functions of a real variable. The paper [4] gives a good survey of the problems which Dini worked on in the 1860s and 1870s, and the most important of the results which he obtained. Bottazzini, the author of [4], puts Dini's work in context showing that it was carried out at a time when those studying real analysis were seeking to determine precisely when the theorems which had earlier been stated and proved in an imprecise way were valid. To achieve this aim mathematicians tried to see how far results could be generalised and they needed to find pathological counterexamples to show the limits to which generalisation was possible. Dini was one of the greatest masters of generalisation and constructing counterexamples. Dini looked at infinite series and generalised results such as a theorem of Kummer and one of Riemann, the ideas for which had first emerged in work of Dirichlet. He discovered a condition, now known as the Dini condition, ensuring the convergence of a Fourier series in terms of the convergence of a definite integral. As well as trigonometric series, Dini studied results on potential theory. He studied surfaces and developed ideas related to those of Liouville and Beltrami. He studied [1]:... surfaces of which the product or the ratio of two principal radii of curvatute remains constant (helicoid surfaces to which Dini's name has been given); [and] ruled surfaces for which one of the principal radii of curvature is a function of the other ... He solved a problem posed by Beltrami of representing one surface on a second surface in such a way that geodesic lines in the first correspond to geodesic lines in the second.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dini.html (2 of 3) [2/16/2002 11:07:20 PM]

Dini

Dini published a number of major texts throughout his career. He published Foundations of the theory of functions of a real variable in 1878; a treatise on Fourier series in 1880; and a two volume work Lessons on infinitesimal analysis with the first volume appearing in 1907 and the second in 1915. In this last work he devoted a chapter to integral equations in which he presented many of his own innovative ideas. Dini's most famous student was Bianchi. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dini.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dini.html (3 of 3) [2/16/2002 11:07:20 PM]

Dinostratus

Dinostratus Born: about 390 BC in Greece Died: about 320 BC Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dinostratus is mentioned by Proclus who says (see for example [1] or [3]):Amyclas of Heraclea, one of the associates of Plato, and Menaechmus, a pupil of Eudoxus who had studied with Plato, and his brother Dinostratus made the whole of geometry still more perfect. It is usually claimed that Dinostratus used the quadratrix, discovered by Hippias, to solve the problem of squaring the circle. Pappus tells us (see for example [1] or [3]):For the squaring of the circle there was used by Dinostratus, Nicomedes and certain other later persons a certain curve which took its name from this property, for it is called by them square-forming [in other words the quadratrix]. It appears from this quote that Hippias discovered the curve but that it was Dinostratus who was the first to use it to find a square equal in area to a given circle. Proclus, who claims to be quoting from Eudemus, writes (see [1]):Nicomedes trisected any rectilinear angle by means of the conchoidal curves, of which he had handed down the origin, order, and properties, being himself the discoverer of their special characteristic. Others have done the same thing by means of the quadratrices of Hippias and Nicomedes. This makes somewhat less convincing the claim that Dinostratus used the quadratrix, discovered by Hippias, to square the circle since Eudemus does not even mention Dinostratus. There is also a suggestion that Hippias wrote a treatise on the quadratrix and if this is the case it seems hard to believe that he did not show how it could be used to square the circle. Whether Dinostratus was indeed the first to square the circle using the quadratrix seems almost irrelevant for, as Bulmer-Thomas writes in [1]:... posterity has firmly associated the name of Dinostratus with the quadrature of the circle by means of the quadratrix.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dinostratus.html (1 of 3) [2/16/2002 11:07:21 PM]

Dinostratus

Pappus relates how the quadratrix was used to square the circle. The construction of the curve is described in the article on Hippias. The quadratrix meets the line AD in the point G and arc BED / AB = AB / AG so the length of the circumference of the circle is expressed in terms of the lengths of straight lines. This leads to the construction of a square equal to the circle.

Pappus reports that Sporus was critical of this construction. He had two objections, the first of which relates to the construction of the quadratrix itself (see the article on Hippias). The second objection relates to Dinostratus's use of the quadratrix to square the circle. Sporus claims that the moving line B'C' never cuts the line AD and so the point G is not determined. The point G can only be found as a limit. There is no doubt that Sporus is quite correct with his objection. Dinostratus probably did much more work on geometry but nothing is known of it. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

Squaring the circle

Cross-references to Famous Curves

the quadratrix

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dinostratus.html (2 of 3) [2/16/2002 11:07:21 PM]

Dinostratus

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dinostratus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dinostratus.html (3 of 3) [2/16/2002 11:07:21 PM]

Diocles

Diocles Born: about 240 BC in Carystus (now Káristos), Euboea (now Evvoia), Greece Died: about 180 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Diocles was a contemporary of Apollonius. Practically all that was knew about him until recently was through fragments of his work preserved by Eutocius in his commentary on the famous text by Archimedes On the sphere and the cylinder. In this work we are told that Diocles studied the cissoid as part of an attempt to duplicate the cube. It is also recorded that he studied the problem of Archimedes to cut a sphere by a plane in such a way that the volumes of the segments shall have a given ratio. The extracts quoted by Eutocius from Diocles' On burning mirrors showed that he was the first to prove the focal property of a parabolic mirror. Although Diocles' text was largely ignored by later Greeks, it had considerable influence on the Arab mathematicians, in particular on al-Haytham. Latin translations from about 1200 of the writings of al-Haytham brought the properties of parabolic mirrors discovered by Diocles to European mathematicians. Recently, however, some more information about Diocles' life has come to us from the Arabic translation of Diocles' On burning mirrors whose discovery is described below. From this work we learn that Zenodorus travelled to Arcadia and entered into discussions with Diocles, so that certainly Diocles was working in Arcadia at the time. This may not seem a very major centre of mathematical importance at the time for such an outstanding scholar as Diocles to be working in, but as Toomer writes in [4]:It would be wrong to conclude from this that Archadia was a cultural centre in this period ... : the whole of the introduction confirms the impression we derive from other contemporary sources, that mathematics during the Hellenistic period was pursued, not in schools established in cultural centres, but by individuals all over the Greek world, who were in lively communication with each other both by correspondence and in their travels. Let us quote from Diocles' introduction to On burning mirrors in the translation by Toomer [4]:Pythian the Thasian geometer wrote a letter to Conon in which he asked him how to find a mirror surface such that when it is placed facing the sun the rays reflected from it meet the circumference of a circle. And when Zenodorus the astronomer came down to Arcadia and was introduced to us, he asked us how to find a mirror surface such that when it is placed facing the sun the rays reflected from it meet a point and thus cause burning.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Diocles.html (1 of 3) [2/16/2002 11:07:23 PM]

Diocles

Toomer notes that his translation of when Zenodorus the astronomer came down to Arcadia and was introduced to us could, perhaps, be translated when Zenodorus the astronomer came down to Arcadia and was appointed to a teaching position there. It is only recently that an Arabic translation of Diocles On burning mirrors has been found in the Shrine Library in Mashhad, Iran. No writing of Diocles was known to Heath in 1921 when he wrote [3], but Toomer translated and published the newly found Arabic translation of the lost treatise On burning mirrors by Diocles in 1976. It has been noticed that On burning mirrors is loosely in three parts, for three separate topics are studied. These three topics are burning mirrors, Archimedes' problem to cut a sphere by a plane, and duplicating the cube. Sesiano (see [4]) has suggested that we may have three short works by Diocles combined into one work and this would have a certain attraction since the title On burning mirrors fails to reflect properly the contents of the whole. If Sesiano's suggestion is correct then we know that the three were combined early on since by the time of Eutocius they formed a single work. On burning mirrors is a collection of sixteen propositions in geometry mostly proving results on conics. It is thought that three of the propositions are later additions to the text, while the remaining ones give a remarkable insight into the theory of conics in the early second century BC. The first of these propositions proves what has long been known to have been first established by Diocles, namely the focal property of the parabola. The next two propositions give properties of spherical mirrors and with Propositions 4 and 5 giving the focus directrix construction of the parabola. These constructions are again properties of the parabola that Diocles was the first to give. The problem of Archimedes to cut a sphere in a given ratio which was known to be in the work through the writing of Eutocius referred to above is studied in Propositions 7 and 8. The duplication of the cube problem, again referred to by Eutocius, is studied by Diocles in Proposition 10. The next two propositions solve the problem of inserting two mean proportions between a pair of magnitudes using the cissoid curve which was invented by Diocles. The final three propositions solve generalisations of the duplication of the cube problem using the cissoid, and another problem of the two mean proportionals type. There are other fascinating deductions that Toomer makes as editor of [4]. A study of the work lead him to claim, contrary to what has long been believed, that the terms "hyperbola", "parabola", and "ellipse" were introduced into the theory of conics before the time of Apollonius. In On burning mirrors Diocles also studies the problem of finding a mirror such that the envelope of reflected rays is a given caustic curve or of finding a mirror such that the focus traces a given curve as the Sun moves across the sky. The solution of this problem would, of course, have interesting consequences for the construction of a sundial. Neugebauer, in an appendix to [4] (see also [6]), shows that this problem cannot be solved exactly while in [5] Hogendijk shows that, using methods available to Diocles, the problem can be solved approximately. Hogendijk in [5] then considers the interesting possibility that Diocles gave arguments of this type in the original text but that later copiers of the text could not understand this part so omitted it.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Diocles.html (2 of 3) [2/16/2002 11:07:23 PM]

Diocles

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Doubling the cube 2. Arabic mathematics : forgotten brilliance? 3. How do we know about Greek mathematicians?

Cross-references to Famous Curves

Cissoid of Diocles

Other references in MacTutor

Chronology: 500BC to 1AD

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Diocles.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Diocles.html (3 of 3) [2/16/2002 11:07:23 PM]

Dionis

Achille Pierre Dionis du Séjour Born: 11 Jan 1734 in Paris, France Died: 22 Aug 1794 in Vernou (near Fontainebleau), France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Achille-Pierre Dionis du Séjour's father was Louis-Achille Dionis du Séjour and his mother was Geneviève-Madeleine Héron. Louis-Achille Dionis du Séjour held a legal position in the Cour des Aides, the board of excise, in Paris. Dionis du Séjour attended the Collège Louis-le-Grand in Paris and the studied at the Faculté de Droit. He published a treatise on the analytic geometry of plane curves in 1756 Traité des courbes algébrique. As a result of this work he was elected associé libre of the Académie des Sciences in the same year as its publication. In 1758 he was appointed as a member of the parliament in Paris. He combined this political career with his research in mathematics and astronomy which, although of high quality, was no more than a hobby for him. However, he wrote extensively on applications of mathematics to astronomy, in particular planetary orbits, and his work was highly regarded by Lagrange, Laplace and Condorcet. Dionis du Séjour applied the latest analytic mathematical methods to the study of problems in astronomy. Over a period of almost 20 years from 1764 he wrote a series of memoirs on eclipses, occultations (when one astronomical body comes in front of another), calculating orbits, and other such topics, and these were brought together in a two volume work Traité analytique des mouvements apparents des corps célestes which he published, volume one in 1786 and volume two in 1789. He published two other volumes on mathematical astronomy. The first was Essai sur les comètes en général; et particulièrement sur celles qui peuvent approacher de l'orbite de la terre (1775) which, as the title suggests considers comets and, in particular, shows that the probability of a collision between a comet and the earth is very low. It is interesting, particularly given the interest in this topic today, to see the subject being explored 225 years ago. The second volume is Essai sur les phémomènes relatifs aux disparitions périodique de l'anneau de Saturne was published in 1776 and, again as suggested by the descriptive title, it explains the variation in the appearance of the rings of Saturn. Dionis du Séjour also worked on the theory of equations, not attaining the depth of results of Bézout or Lagrange. With Condorcet and Laplace he undertook a determination of the population of France [1]:Utilising the list of communes appearing in the Cassini map of France and the most recent information furnished by the civil engineers, this enquiry was based on the empirical http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dionis.html (1 of 2) [2/16/2002 11:07:24 PM]

Dionis

hypothesis that the annual number of births in a given population is approximately one twenty-sixth of the total of that population. The French Revolution caused problems for Dionis du Séjour, both in his political roles and in his scientific work. He was elected as a deputy of the Paris nobility on 10 May 1789. This gave him a seat in the National Assembly. The storming of the Bastille on 14 July 1789 marked the start of the French Revolution and the National Assembly then became the Constituent Assembly. On 30 September 1791 the Constituent Assembly was replaced by Legislative Assembly and Dionis du Séjour's role came to an end at this time. On 31 November 1791 Dionis du Séjour was appointed as a judge of the Paris tribunal but later he resigned and retired to his estate in Argeville. However from 5 September 1793 to 27 July 27 1794 there was the Reign of Terror during which 17,000 were officially executed. Dionis du Séjour was in fear of his life throughout this period and the anxiety he experienced may well have hastened his death which was less than a month after the Reign of Terror ended. As to Dionis du Séjour's character, he is described as [1]:... appreciated for his simplicity, his liberalism, and his humanity. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dionis.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dionis.html (2 of 2) [2/16/2002 11:07:24 PM]

Dionysodorus

Dionysodorus Born: about 250 BC in Caunus, Caria, Asia Minor (now in Turkey) Died: about 190 BC Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

There is certainly more than one mathematician called Dionysodorus and this does make it a little difficult in deciding exactly what was studied by each. Strabo, the Greek geographer and historian (about 64 BC - about 24 AD), describes a mathematician named Dionysodorus who was born in Amisene, Pontus in northeastern Anatolia on the Black Sea. The Dionysodorus we are interested in here is the mathematician Dionysodorus who Eutocius states solved the problem of the cubic equation using the intersection of a parabola and a hyperbola. This was related to a problem of Archimedes given in On the Sphere and Cylinder. It was thought until early this century that the Dionysodorus who Eutocius refers to was Dionysodorus of Amisene described by Strabo. There is a second Dionysodorus who appears in the writings of Pliny. In Natural history Pliny mentions a certain Dionysodorus who measured the earth's radius and gave the value 42000 stades. Strabo distinguishes this Dionysodorus from Dionysodorus of Amisene and it is now thought that the Dionysodorus referred to by Pliny is not the mathematician who solved the problem of the cubic equation. Interestingly Pliny died as a result of the eruption of Vesuvius in 79 AD and it is as a consequence of this eruption that new information regarding a mathematician Dionysodorus was published in 1900. This new information was found by W Cronert in a papyrus found at Herculaneum. When Vesuvius erupted in 79 AD, Herculaneum together with Pompeii and Stabiae, was destroyed. Herculaneum was buried by a compact mass of material about 16 metres deep which preserved the city until excavations began in the 18th century. Special conditions of humidity of the ground conserved wood, cloth, food, and in particular papyri which give us important information. One papyrus states [3]:Philonides was a pupil, first of Eudemus, and afterwards of Dionysodorus, the son of Dionysodorus the Caunian. Eudemus is Eudemus of Pergamum whom Apollonius dedicated two books of his Conics and, in the introduction to Book II, asks Eudemus to show the book to Philonides. We can date Dionysodorus from this information as just a little younger than Apollonius. There is another interesting comment in the papyrus which states that Philonides published some of the lectures by his teacher Dionysodorus. Shortly after Cronert published details of the fragments of papyri relating to Dionysodorus which had

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dionysodorus.html (1 of 3) [2/16/2002 11:07:25 PM]

Dionysodorus

been found at Herculaneum, Schmidt published a commentary on the material in which he argued convincingly that the Dionysodorus who solved the cubic equation using the intersection of a parabola and a hyperbola was the Dionysodorus of Caunus referred to in the Herculaneum papyrus. Caunus is in Caria and is now in Turkey. It is close to Perga in Pamphylia where Apollonius was born. The method which Eutocius describes to cut a sphere in a given ratio, crediting it to Dionysodorus, uses a parabola and a rectangular hyperbola. It is a beautiful construction and in the description that follows we essentially follow the method described by Eutocius (see also [1] and [3]). Let AA' be the diameter of the sphere centre O. We wish to find a plane which divides the sphere in the ratio m : n. Take F on A'A produced so that FA = AO. Let AG be perpendicular to AA' where G is the point with FA : AG = (m+n) : n. Let H be the point on AG with AH2 = FA.AG and draw the parabola with vertex at F through H. Draw the rectangular hyperbola through G with the x and y axes as its asymptotes. Let the hyperbola cut the parabola at P and draw PM perpendicular to AA'. Then Dionysodorus proved that the plane through M with AA' as its normal will cut the sphere in the given ratio m : n. Heron also mentions Dionysodorus as the author of a work On the Tore which, because of the subject matter, must almost certainly be written by the Dionysodorus we are describing here. In this work Dionysodorus calculates the volume of a torus and shows that it is equal to the product of the area of the generating circle with the length of the circle traced by its centre rotating about the axis of revolution. It is clear that Dionysodorus used the methods of Archimedes in proving his result. Dionysodorus is believed to have invented a conical sundial. The report fails to make it clear which Dionysodorus this is, but the fact that the Dionysodorus described here worked on conic sections makes it likely that he is also the person to have studied a conical sundial. In [2] the likely form that Dionysodorus's sundial would take is discussed. A conjectured reconstruction is given in [1]. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dionysodorus.html (2 of 3) [2/16/2002 11:07:25 PM]

Dionysodorus

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dionysodorus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dionysodorus.html (3 of 3) [2/16/2002 11:07:25 PM]

Diophantus

Diophantus of Alexandria Born: about 200 Died: about 284 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Diophantus, often known as the 'father of algebra', is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers. However, essentially nothing is known of his life and there has been much debate regarding the date at which he lived. There are a few limits which can be put on the dates of Diophantus's life. On the one hand Diophantus quotes the definition of a polygonal number from the work of Hypsicles so he must have written this later than 150 BC. On the other hand Theon of Alexandria, the father of Hypatia, quotes one of Diophantus's definitions so this means that Diophantus wrote no later than 350 AD. However this leaves a span of 500 years, so we have not narrowed down Diophantus's dates a great deal by these pieces of information. There is another piece of information which was accepted for many years as giving fairly accurate dates. Heath [3] quotes from a letter by Michael Psellus who lived in the last half of the 11th century. Psellus wrote (Heath's translation in [3]):Diophantus dealt with [Egyptian arithmetic] more accurately, but the very learned Anatolius collected the most essential parts of the doctrine as stated by Diophantus in a different way and in the most succinct form, dedicating his work to Diophantus. Psellus also describes in this letter the fact that Diophantus gave different names to powers of the unknown to those given by the Egyptians. This letter was first published by Paul Tannery in [7] and in that work he comments that he believes that Psellus is quoting from a commentary on Diophantus which is now lost and was probably written by Hypatia. However, the quote given above has been used to date Diophantus using the theory that the Anatolius referred to here is the bishop of Laodicea who was a writer and teacher of mathematics and lived in the third century. From this it was deduced that Diophantus wrote around 250 AD and the dates we have given for him are based on this argument. Knorr in [16] criticises this interpretation, however:But one immediately suspects something is amiss: it seems peculiar that someone would compile an abridgement of another man's work and then dedicate it to him, while the qualification "in a different way", in itself vacuous, ought to be redundant, in view of the terms "most essential" and "most succinct". Knorr gives a different translation of the same passage (showing how difficult the study of Greek mathematics is for anyone who is not an expert in classical Greek) which has a remarkably different meaning:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.html (1 of 6) [2/16/2002 11:07:27 PM]

Diophantus

Diophantus dealt with [Egyptian arithmetic] more accurately, but the very learned Anatolius, having collected the most essential parts of that man's doctrine, to a different Diophantus most succinctly addressed it. The conclusion of Knorr as to Diophantus's dates is [16]:... we must entertain the possibility that Diophantus lived earlier than the third century, possibly even earlier that Heron in the first century. The most details we have of Diophantus's life (and these may be totally fictitious) come from the Greek Anthology, compiled by Metrodorus around 500 AD. This collection of puzzles contain one about Diophantus which says:... his boyhood lasted 1/6th of his life; he married after 1/7th more; his beard grew after 1/ th more, and his son was born 5 years later; the son lived to half his father's age, and 12 the father died 4 years after the son. So he married at the age of 26 and had a son who died at the age of 42, four years before Diophantus himself died aged 84. Based on this information we have given him a life span of 84 years. The Arithmetica is a collection of 130 problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. The method for solving the latter is now known as Diophantine analysis. Only six of the original 13 books were thought to have survived and it was also thought that the others must have been lost quite soon after they were written. There are many Arabic translations, for example by Abu'l-Wafa, but only material from these six books appeared. Heath writes in [4] in 1920:The missing books were evidently lost at a very early date. Paul Tannery suggests that Hypatia's commentary extended only to the first six books, and that she left untouched the remaining seven, which, partly as a consequence, were first forgotten and then lost. However, an Arabic manuscript in the library Astan-i Quds (The Holy Shrine library) in Meshed, Iran has a title claiming it is a translation by Qusta ibn Luqa, who died in 912, of Books IV to VII of Arithmetica by Diophantus of Alexandria. F Sezgin made this remarkable discovery in 1968. In [19] and [20] Rashed compares the four books in this Arabic translation with the known six Greek books and claims that this text is a translation of the lost books of Diophantus. Rozenfeld, in reviewing these two articles is, however, not completely convinced:The reviewer, familiar with the Arabic text of this manuscript, does not doubt that this manuscript is the translation from the Greek text written in Alexandria but the great difference between the Greek books of Diophantus's Arithmetic combining questions of algebra with deep questions of the theory of numbers and these books containing only algebraic material make it very probable that this text was written not by Diophantus but by some one of his commentators (perhaps Hypatia ?). It is time to take a look at this most outstanding work on algebra in Greek mathematics. The work considers the solution of many problems concerning linear and quadratic equations, but considers only positive rational solutions to these problems. Equations which would lead to solutions which are negative or irrational square roots, Diophantus considers as useless. To give one specific example, he calls the equation 4 = 4x + 20 'absurd' because it would lead to a meaningless answer. In other words how could a problem lead to the solution -4 books? There is no evidence to suggest that Diophantus realised that a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.html (2 of 6) [2/16/2002 11:07:27 PM]

Diophantus

quadratic equation could have two solutions. However, the fact that he was always satisfied with a rational solution and did not require a whole number is more sophisticated than we might realise today. Diophantus looked at three types of quadratic equations ax2 + bx = c, ax2 = bx + c and ax2 + c = bx. The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers a, b, c to all be positive in each of the three cases above. There are, however, many other types of problems considered by Diophantus. He solved problems such as pairs of simultaneous quadratic equations. Consider y + z = 10, yz = 9. Diophantus would solve this by creating a single quadratic equation in x. Put 2x = y - z so, adding y + z = 10 and y - z = 2x, we have y = 5 + x, then subtracting them gives z = 5 - x. Now 9 = yz = (5 + x)(5 - x) = 25 - x2, so x2 = 16, x = 4 leading to y = 9, z = 1. In Book III, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares. For example he shows how to find x to make 10x + 9 and 5x + 4 both squares (he finds x = 28). Other problems seek a value for x such that particular types of polynomials in x up to degree 6 are squares. For example he solves the problem of finding x such that x3 - 3x2 + 3x + 1 is a square in Book VI. Again in Book VI he solves problems such as finding x such that simultaneously 4x + 2 is a cube and 2x + 1 is a square (for which he easily finds the answer x = 3/2). Another type of problem which Diophantus studies, this time in Book IV, is to find powers between given limits. For example to find a square between 5/4 and 2 he multiplies both by 64, spots the square 100 between 80 and 128, so obtaining the solution 25/16 to the original problem. In Book V he solves problems such as writing 13 as the sum of two square each greater than 6 (and he gives the solution 66049/10201 and 66564/10201). He also writes 10 as the sum of three squares each greater than 3, finding the three squares 1745041/505521, 1651225/505521, 1658944/505521. Heath looks at number theory results of which Diophantus was clearly aware, yet it is unclear whether he had a proof. Of course these results may have been proved in other books written by Diophantus or he may have felt they were "obviously" true due to his experimental evidence. Among such results are [4]:... no number of the form 4n + 3 or 4n - 1 can be the sum of two squares; ... a number of the form 24n + 7 cannot be the sum of three squares. Diophantus also appears to know that every number can be written as the sum of four squares. If indeed he did know this result it would be truly remarkable for even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Lagrange proved it using results due to Euler. Although Diophantus did not use sophisticated algebraic notation, he did introduce an algebraic symbolism that used an abbreviation for the unknown and for the powers of the unknown. As Vogel writes in [1]:The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.html (3 of 6) [2/16/2002 11:07:27 PM]

Diophantus

Since an abbreviation is also employed for the word "equals", Diophantus took a fundamental step from verbal algebra towards symbolic algebra. One thing will be clear from the examples we have quoted and that is that Diophantus is concerned with particular problems more often than with general methods. The reason for this is that although he made important advances in symbolism, he still lacked the necessary notation to express more general methods. For instance he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked a symbol for a general number n. Where we would write (12 + 6n)/(n2 -3), Diophantus has to write in words:... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three. Despite the improved notation and that Diophantus introduced, algebra had a long way to go before really general problems could be written down and solved succinctly. Fragments of another of Diophantus's books On polygonal numbers, a topic of great interest to Pythagoras and his followers, has survived. In [1] it is stated that this work contains:... little that is original, [and] is immediately differentiated from the Arithmetica by its use of geometric proofs. Diophantus himself refers to another work which consists of a collection of lemmas called The Porisms but this book is entirely lost. We do know three lemmas contained in The Porisms since Diophantus refers to them in the Arithmetica. One such lemma is that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any numbers a, b then there exist numbers c, d such that a3 - b3 = c3 + d3. Another extant work Preliminaries to the geometric elements, which has been attributed to Heron, has been studied recently in [16] where it is suggested that the attribution to Heron is incorrect and that the work is due to Diophantus. The author of the article [14] thinks that he may have identified yet another work by Diophantus. He writes:We conjecture the existence of a lost theoretical treatise of Diophantus, entitled "Teaching of the elements of arithmetic". Our claims are based on a scholium of an anonymous Byzantine commentator. European mathematicians did not learn of the gems in Diophantus's Arithmetica until Regiomontanus wrote in 1463:No one has yet translated from the Greek into Latin the thirteen Books of Diophantus, in which the very flower of the whole of arithmetic lies hid... Bombelli translated much of the work in 1570 but it was never published. Bombelli did borrow many of Diophantus's problems for his own Algebra. The most famous Latin translation of the Diophantus's Arithmetica is due to Bachet in 1621 and it is that edition which Fermat studied. Certainly Fermat was inspired by this work which has become famous in recent years due to its connection with Fermat's Last Theorem. We began this article with the remark that Diophantus is often regarded as the 'father of algebra' but there is no doubt that many of the methods for solving linear and quadratic equations go back to Babylonian

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.html (4 of 6) [2/16/2002 11:07:27 PM]

Diophantus

mathematics. For this reason Vogel writes [1]:... Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (25 books/articles)

A Quotation

Some pages from publications

The title page from the translation by Bachet of Arithmetica (1670) and another page showing the transcription of Fermat's marginal note

Cross-references to History Topics

Mathematicians born in the same country

1. Fermat's last theorem 2. Arabic mathematics : forgotten brilliance? 3. Mathematical games and recreations

Other references in MacTutor

Chronology: 1AD to 500

Honours awarded to Diophantus (Click a link below for the full list of mathematicians honoured in this way) Crater Diophantus and Rima Diophantus

Lunar features

1. Karen H Parshall

Other Web sites

2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.html (5 of 6) [2/16/2002 11:07:27 PM]

Diophantus

Mathematicians of the day JOC/EFR February 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Diophantus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Diophantus.html (6 of 6) [2/16/2002 11:07:27 PM]

Dirac

Paul Adrien Maurice Dirac Born: 8 Aug 1902 in Bristol, Gloucestershire, England Died: 20 Oct 1984 in Tallahassee, Florida, USA

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Dirac's father was Charles Adrien Ladislas Dirac and his mother was Florence Hannah Holten. Charles Dirac was a Swiss citizen born in Monthey, Valais while his mother came from Cornwall in England. Charles had been educated at the University of Geneva, then came to England in around 1888 and taught French in Bristol. There he met Florence, whose father had moved to Bristol as Master Mariner on a Bristol ship, when she was working in the library there. Charles and Florence married in 1899 and they moved into a house in Bishopston, Bristol which they named Monthey after the town of Charles's birth. By this time Charles was teaching French at the secondary school attached to the Merchant Venturers Technical College in Bristol. Paul was one of three children, his older brother being Reginald Charles Felix Dirac and his younger sister being Beatrice Isabelle Marguerite Walla Dirac. Paul had a very strict family upbringing. His father insisted that only French be spoken at the dinner table and, as a result, Paul was the only one to eat with his father in the dining room. Paul's father was so strict with his sons that both were alienated and Paul was brought up in a somewhat unhappy home. The first school which Paul attended was Bishop Primary school and already in this school his exceptional ability in mathematics became clear to his teachers. When he was twelve years old he entered secondary school, attending the secondary school where his father taught which was part of the Merchant Venturers Technical College. At about the time Paul entered the school World War I began and this had a beneficial effect for Paul since the older boys in the school left for military service and the younger boys had more access to the science laboratories and other facilities. Paul himself wrote about his school years in [13]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirac.html (1 of 6) [2/16/2002 11:07:29 PM]

Dirac

The Merchant Venturers was an excellent school for science and modern languages. There was no Latin or Greek, something of which I was rather glad, because I did not appreciate the value of old cultures. I consider myself very lucky in having been able to attend the school. ... I was rushed through the lower forms, and was introduced at an especially early age to the basis of mathematics, physics and chemistry in the higher forms. In mathematics I was studying from books which mostly were ahead of the rest of the class. This rapid advancement was a great help to me in my latter career. He completed his school education in 1918 and then studied electrical engineering at the University of Bristol. By this time the University had combined with the Merchant Venturers Technical College so Dirac remained in the same building as he had studied during his four years at secondary school. Although mathematics was his favourite subject he chose to study an engineering course at university since he thought that the only possible career for a mathematician was school teaching and he certainly wanted to avoid that profession. He obtained his degree in engineering in 1921 but following this, after an undistinguished summer job in an engineering works, he did not find a permanent job. By this time he was developing a real passion for mathematics but his attempts to study at Cambridge failed for rather strange reasons. Taking the Cambridge scholarship examinations in June 1921 he was awarded a scholarship to study mathematics at St John's College Cambridge but it did not provide enough to support him. Additional support would have been expected from his local education authority, but he was refused support on the grounds that his father had not been a British citizen for long enough. Dirac was offered the chance to study mathematics at Bristol without paying fees and he did so being awarded first class honours in 1923. Following this he was awarded a grant to undertake research at Cambridge and he began his studies there in 1923. Dirac had been hoping to have his research supervised by Ebenezer Cunningham, for by this time Dirac had become fascinated in the general theory of relativity and wanted to undertake research on this topic. Cunningham already had as many research students as he was prepared to take on and so Dirac was supervised by Ralph Fowler. The authors of [12] write:Fowler was then the leading theoretician in Cambridge, well versed in the quantum theory of atoms; his own research was mostly on statistical mechanics. He recognised in Dirac a student of unusual ability. Under his influence Dirac worked on some problems in statistical mechanics. Within six months of arriving in Cambridge he wrote two papers on these problems. No doubt Fowler aroused his interest in the quantum theory, and in May 1924 Dirac completed his first paper dealing with quantum problems. Four more papers were completed by November 1925. Despite the obvious academic success Dirac enjoyed as a research student this was no easy time for him. His brother Reginald Dirac committed suicide during this period. No reason for the suicide seems to be known but Dirac's relations with his father, already strained, seemed almost to end completely after this which does suggest that Dirac felt that his father carried at least some responsibility. Already a person who had few friends, this personal tragedy had the effect of making him even more withdrawn. Although he had already made an excellent start to his research career, even more impressive work was to follow. This was as a result of Dirac being given proofs of a paper by Heisenberg to read in the summer of 1925. The significance of the algebraic properties of Heisenberg's commutators struck Dirac

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirac.html (2 of 6) [2/16/2002 11:07:29 PM]

Dirac

when he was out for a walk in the country. He realised that Heisenberg's uncertainty principle was a statement of the noncommutativity of the quantum mechanical observables. He realised the analogy with Poisson brackets in Hamiltonian mechanics. Higgs writes in [13]:This similarity provided the clue which led him to formulate for the first time a mathematically consistent general theory of quantum mechanics in correspondence with Hamiltonian mechanics. The ideas were laid out in Dirac's doctoral thesis Quantum mechanics for which he was awarded a Ph.D. in 1926. It is remarkable that Dirac had eleven papers in print before submitting his doctoral dissertation. Following the award of the degree he went to Copenhagen to work with Niels Bohr, moving on to Göttingen in February 1927 where he interacted with Robert Oppenheimer, Max Born, James Franck and the Russian Igor Tamm. Accepting an invitation from Ehrenfest, he spent a few weeks in Leiden on his way back to Cambridge. He was elected a Fellow of St John's College, Cambridge in 1927. Dirac visited the Soviet Union in 1928. It was the first of many visits for he went again in 1929, 1930, 1932, 1933, 1935, 1936, 1937, 1957, 1965, and 1973. Also in 1928 he found a connection between relativity and quantum mechanics, his famous spin-1/2 Dirac equation. In 1929 he made his first visit to the United States, lecturing at the Universities of Wisconsin and Michigan. After the visit, along with Heisenberg, he crossed the Pacific and lectured in Japan. He returned via the trans-Siberian railway. In 1930 Dirac published The principles of Quantum Mechanics and for this work he was awarded the Nobel Prize for Physics in 1933. De Facio, reviewing [3], says of this book:Dirac was not influenced by the feeding frenzy in experimental phenomenology of the time. This has given Dirac's book ... a lasting quality that few works can match. The authors of [12] comment that the book:... reflects Dirac's very characteristic approach: abstract but simple, always selecting the important points and arguing with unbeatable logic. Also in 1930 Dirac was elected a Fellow of the Royal Society. This honour came on the first occasion that his name was put forward, in itself quite an unusual event which says much about the extremely high opinion that Dirac's fellow scientists had of him. Dirac was appointed Lucasian professor of mathematics at the University of Cambridge in 1932, a post he held for 37 years. In 1933 he published a pioneering paper on Lagrangian quantum mechanics which became the foundation on which Feynman later built his ideas of the path integral. In the same year Dirac received the Nobel prize for physics which he shared with Schrödinger. It is an interesting comment on Dirac's nature that his first thought was to turn down the prize on the grounds that he hated publicity. However when it was pointed out to him that he would receive far more publicity if he turned down the prize, he accepted it. Another comment about this event is that Dirac was told that he could invite his parents to the award ceremony in Stockholm, but he chose to invite only his mother and not his father. The academic year 1934-35 was important for Dirac both for personal and professional reasons. He visited the Institute for Advanced Study at Princeton and there he became friendly with Wigner. While Dirac was there Wigner's sister Margit, who lived in Budapest, visited her brother. This chance meeting led, in January 1937, to Dirac marrying Margit in London. Margit had been married before and had two children Judith and Gabriel Andrew from her first marriage. Both children adopted the name Dirac and

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirac.html (3 of 6) [2/16/2002 11:07:29 PM]

Dirac

Gabriel Andrew Dirac went on the became a famous pure mathematician, particularly contributing to graph theory, becoming professor of pure mathematics at the University of Aarhus in Denmark. In 1937, the same year that he married, Dirac published his first paper on large numbers and cosmological matters. We comment further on his ideas on cosmology below. He published his famous paper on classical electron theory, which included mass renormalisation and radiative reaction in 1938. Dirac worked during World War II on uranium separation and nuclear weapons. In particular he acted as a consultant to a group in Birmingham working on atomic energy. This association led to Dirac being prevented by the British government from visiting the Soviet Union after the end of the war; he was not able to visit again until 1957. We noted above that Dirac was elected a fellow of the Royal Society in 1930. He was awarded the Royal Society's Royal Medal in 1939 and the Society awarded him their Copley Medal in 1952:... in recognition of his remarkable contributions to relativistic dynamics of a particle in quantum mechanics. In 1969 Dirac retired from the Lucasian chair of mathematics at Cambridge and went with his family to Florida in the United States. He held visiting appointments at the University of Miami and at Florida State University. Then, in 1971, Dirac was appointed professor of physics at Florida State University where he continued his research. In 1973 and 1975 Dirac lectured in the Physical Engineering Institute in Leningrad. In these lectures he spoke about the problems of cosmology or, to be more precise, to the problems of non-dimensional combinations of world constants. Although Dirac made vastly important contributions to physics, it is important to realise that he was always motivated by principles of mathematical beauty. Dirac unified the theories of quantum mechanics and relativity theory, but he also is remembered for his outstanding work on the magnetic monopole, fundamental length, antimatter, the d-function, bra-kets, etc. There is a standard folklore of Dirac stories, mostly revolving around Dirac saying exactly what he meant and no more. Once when someone, making polite conversation at dinner, commented that it was windy, Dirac left the table and went to the door, looked out, returned to the table and replied that indeed it was windy. It has been said in jest that his spoken vocabulary consisted of "Yes", "No", and "I don't know". Certainly when Chandrasekhar was explaining his ideas to Dirac he continually interjected "yes" then explained to Chandrasekhar that "yes" did not mean that he agreed with what he was saying, only that he wished him to continue. He once said:I was taught at school never to start a sentence without knowing the end of it. This may explain much about his conversation, and also about his beautifully written sentences in his books and papers. Dirac received many honours for his work, some of which we have mentioned above. He refused to accept honorary degrees but he did accept honorary membership of academies and learned societies. The list of these is long but among them are USSR Academy of Sciences (1931), Indian Academy of Sciences (1939), Chinese Physical Society (1943), Royal Irish Academy (1944), Royal Society of Edinburgh (1946), Institut de France (1946), National Institute of Sciences of India (1947), American Physical Society (1948), National Academy of Sciences (1949), National Academy of Arts and Sciences (1950), Accademia delle Scienze di Torino (1951), Academia das Ciencias de Lisboa (1953), Pontifical http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirac.html (4 of 6) [2/16/2002 11:07:29 PM]

Dirac

Academy of Sciences, Vatican City (1958), Accademia Nazionale dei Lincei, Rome (1960), Royal Danish Academy (1962), and Académie des Sciences Paris (1963). He was appointed to the Order of Merit in 1973. A memorial meeting was held at the University of Cambridge on 19 April 1985 and the papers presented at this meeting were published in Tributes to Paul Dirac, Cambridge, 1985 (Bristol, 1987). The papers [11], [14], [24], [29], [36], [38] and [39] come from this volume. Achuthan, reviewing the volume, writes:... we vividly see everywhere the brilliant imprints of Dirac, unifier of quantum mechanics and relativity theory. Each of the pieces not only is in praise of an exceptionally gifted intellect but also places on record how deeply and abidingly the human mind can delve into the realms of mathematical insight and modelling, keeping intact the spirit of beauty and clarity of a creative genius. Only a few Nobel laureates ever can compare as well with this giant of mathematical sciences in whose demise the world of original thinking certainly has lost one of the most precious souls retaining fortunately still the glory for others to sing and emulate for a long time to come. In November 1995 of a plaque was unveiled in Westminster Abbey commemorating Paul Dirac. The volume [8] consists of lectures presented to the Royal Society on this occasion. The memorial address was presented by Stephen Hawking who was Dirac's successor in the Lucasian chair of mathematics at Cambridge which was also Newton's chair. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (45 books/articles)

Some Quotations (10)

A Poster of Paul Dirac

Mathematicians born in the same country

Honours awarded to Paul Dirac (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1933

Fellow of the Royal Society

Elected 1930

Royal Society Copley Medal

Awarded 1952

Royal Society Royal Medal

Awarded 1939

Royal Society Bakerian lecturer

1941

Fellow of the Royal Society of Edinburgh Lucasian Professor of Mathematics

1932

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirac.html (5 of 6) [2/16/2002 11:07:29 PM]

Dirac

Other Web sites

1. Bob Bruen 2. Nobel prizes site (A biography of Dirac and his Nobel prize presentation speech) 3. West Chester University

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dirac.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirac.html (6 of 6) [2/16/2002 11:07:29 PM]

Dirichlet

Johann Peter Gustav Lejeune Dirichlet Born: 13 Feb 1805 in Düren, French Empire (now Germany) Died: 5 May 1859 in Göttingen, Hanover (now Germany)

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lejeune Dirichlet's family came from the Belgian town of Richelet where Dirichlet's grandfather lived. This explains the origin of his name which comes from "Le jeune de Richelet" meaning "Young from Richelet". Many details of the Dirichlet family are given in [6] where it is shown that the Dirichlets came from the neighbourhood of Liège in Belgium and not, as many had claimed, from France. His father was the postmaster of Düren, the town of his birth situated about halfway between Aachen and Cologne. Even before he entered the Gymnasium in Bonn in 1817, at the age of 12, he had developed a passion for mathematics and spent his pocket-money on buying mathematics books. At the Gymnasium he was a model pupil being [1]:... an unusually attentive and well-behaved pupil who was particularly interested in history as well as mathematics. After two years at the Gymnasium in Bonn his parents decided that they would rather have him attend the Jesuit College in Cologne and there he had the good fortune to be taught by Ohm. By the age of 16 Dirichlet had completed his school qualifications and was ready to enter university. However, the standards in German universities were not high at this time so Dirichlet decided to study in Paris. It is interesting to note that some years later the standards in German universities would become the best in the world and Dirichlet himself would play a hand in the transformation. Dirichlet set off for France carrying with him Gauss's Disquisitiones arithmeticae a work he treasured and kept constantly with him as others might do with the Bible. In Paris by May 1822, Dirichlet soon contracted smallpox. It did not keep him away from his lectures in the Collège de France and the Faculté

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirichlet.html (1 of 5) [2/16/2002 11:07:31 PM]

Dirichlet

des Sciences for long and soon he could return to lectures. He had some of the leading mathematicians as teachers and he was able to profit greatly from the experience of coming in contact with Biot, Fourier, Francoeur, Hachette, Laplace, Lacroix, Legendre, and Poisson. From the summer of 1823 Dirichlet was employed by General Maximilien Sébastien Foy, living in his house in Paris. General Foy had been a major figure in the army during the Napoleonic Wars, retiring after Napoleon's defeat at Waterloo. In 1819 he was elected to the Chamber of Deputies where he was leader of the liberal opposition until his death. Dirichlet was very well treated by General Foy, he was well paid yet treated like a member of the family. In return Dirichlet taught German to General Foy's wife and children. Dirichlet's first paper was to bring him instant fame since it concerned the famous Fermat's Last Theorem. The theorem claimed that for n > 2 there are no non-zero integers x, y, z such that xn + yn = zn. The cases n = 3 and n = 4 had been proved by Euler and Fermat, and Dirichlet attacked the theorem for n = 5. Now if n = 5 then one of x, y, z is even and one is divisible by 5. There are two cases: case 1 is when the number divisible by 5 is even, while case 2 is when the even number and the one divisible by 5 are distinct. Dirichlet proved case 1 and presented his paper to the Paris Academy in July 1825. Legendre was appointed one of the referees and he was able to prove case 2 thus completing the proof for n = 5. The complete proof was published in September 1825. In fact Dirichlet was able to complete his own proof of the n = 5 case with an argument for case 2 which was an extension of his own argument for case 1. It is worth noting that Dirichlet made a later contribution proving the n = 14 case (a near miss for the n = 7 case!). On 28 November 1825 General Foy died and Dirichlet decided to return to Germany. He was encouraged in this by Alexander von Humboldt who made recommendations on his behalf. There was a problem for Dirichlet since in order to teach in a German university he needed an habilitation. Although Dirichlet could easily submit an habilitation thesis, this was not allowed since he did not hold a doctorate, nor could he speak Latin, a requirement in the early nineteenth century. The problem was nicely solved by the University of Cologne giving Dirichlet an honorary doctorate, thus allowing him to submit his habilitation thesis on polynomials with a special class of prime divisors to the University of Breslau. There was, however, much controversy over Dirichlet's appointment and the large correspondence between German professors both for and against his appointment is considered in [15]. From 1827 Dirichlet taught at Breslau but Dirichlet encountered the same problem which made him choose Paris for his own education, namely that the standards at the university were low. Again with von Humboldt's help, he moved to the Berlin in 1828 where he was appointed at the Military College. The Military College was not the attraction, of course, rather it was that Dirichlet had an agreement that he would be able to teach at the University of Berlin. Soon after this he was appointed a professor at the University of Berlin where he remained from 1828 to 1855. He retained his position in the Military College which made his teaching and other administrative duties rather heavier than he would have liked. Dirichlet was appointed to the Berlin Academy of Sciences in 1831 and an improving salary from the university put him in a position to marry, and he married Rebecca Mendelssohn, one of the composer Felix Mendelssohn's two sisters. Dirichlet had a lifelong friend in Jacobi, who taught at Königsberg, and the two exerted considerable influence on each other in their researches in number theory.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirichlet.html (2 of 5) [2/16/2002 11:07:31 PM]

Dirichlet

In the 1843 Jacobi became unwell and diabetes was diagnosed. He was advised by his doctor to spend time in Italy where the climate would help him recover. However, Jacobi was not a wealthy man and Dirichlet, after visiting Jacobi and discovering his plight, wrote to Alexander von Humboldt asking him to help obtain some financial assistance for Jacobi from Friedrich Wilhelm IV. Dirichlet then made a request for assistance from Friedrich Wilhelm IV, supported strongly by Alexander von Humboldt, which was successful. Dirichlet obtained leave of absence from Berlin for eighteen months and in the autumn of 1843 set off for Italy with Jacobi and Borchardt. After stopping in several towns and attending a mathematical meeting in Lucca, they arrived in Rome on 16 November 1843. Schläfli and Steiner were also with them, Schläfli's main task being to act as their interpreter but he studied mathematics with Dirichlet as his tutor. Dirichlet did not remain in Rome for the whole period, but visited Sicily and then spent the winter of 1844/45 in Florence before returning to Berlin in the spring of 1845. Dirichlet had a high teaching load at the University of Berlin, being also required to teach in the Military College and in 1853 he complained in a letter to his pupil Kronecker that he had thirteen lectures a week to give in addition to many other duties. It was therefore something of a relief when, on Gauss's death in 1855, he was offered his chair at Göttingen. Dirichlet did not accept the offer from Göttingen immediately but used it to try to obtain better conditions in Berlin. He requested of the Prussian Ministry of Culture that he be allowed to end lecturing at the Military College. However he received no quick reply to his modest request so he wrote to Göttingen accepting the offer of Gauss's chair. After he had accepted the Göttingen offer the Prussian Ministry of Culture did try to offer him improved conditions and salary but this came too late. The quieter life in Göttingen seemed to suit Dirichlet. He had more time for research and some outstanding research students. However, sadly he was not to enjoy the new life for long. In the summer of 1858 he lectured at a conference in Montreux but while in the Swiss town he suffered a heart attack. He returned to Göttingen, with the greatest difficulty, and while gravely ill had the added sadness that his wife died of a stroke. We should now look at Dirichlet's remarkable contributions to mathematics. We have already commented on his contributions to Fermat's Last Theorem made in 1825. Around this time he also published a paper inspired by Gauss's work on the law of biquadratic reciprocity. Details are given in [13] where Rowe discusses the importance of the intellectual and personal relationship between Gauss and Dirichlet. He proved in 1837 that in any arithmetic progression with first term coprime to the difference there are infinitely many primes. This had been conjectured by Gauss. Davenport wrote in 1980 (see [16]):Analytic number theory may be said to begin with the work of Dirichlet, and in particular with Dirichlet's memoir of 1837 on the existence of primes in a given arithmetic progression. Shortly after publishing this paper Dirichlet published two further papers on analytic number theory, one in 1838 with the next in the following year. These papers introduce Dirichlet series and determine, among other things, the formula for the class number for quadratic forms.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirichlet.html (3 of 5) [2/16/2002 11:07:31 PM]

Dirichlet

His work on units in algebraic number theory Vorlesungen über Zahlentheorie (published 1863) contains important work on ideals. He also proposed in 1837 the modern definition of a function:If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x. In mechanics he investigated the equilibrium of systems and potential theory. These investigations began in 1839 with papers which gave methods to evaluate multiple integrals and he applied this to the problem of the gravitational attraction of an ellipsoid on points both inside and outside. He turned to Laplace's problem of proving the stability of the solar system and produced an analysis which avoided the problem of using series expansion with quadratic and higher terms disregarded. This work led him to the Dirichlet problem concerning harmonic functions with given boundary conditions. Some work on mechanics later in his career is of quite outstanding importance. In 1852 he studied the problem of a sphere placed in an incompressible fluid, in the course of this investigation becoming the first person to integrate the hydrodynamic equations exactly. Dirichlet is also well known for his papers on conditions for the convergence of trigonometric series and the use of the series to represent arbitrary functions. These series had been used previously by Fourier in solving differential equations. Dirichlet's work is published in Crelle's Journal in 1828. Earlier work by Poisson on the convergence of Fourier series was shown to be non-rigorous by Cauchy. Cauchy's work itself was shown to be in error by Dirichlet who wrote of Cauchy's paper:The author of this work himself admits that his proof is defective for certain functions for which the convergence is, however, incontestable. Because of this work Dirichlet is considered the founder of the theory of Fourier series. Riemann, who was a student of Dirichlet, wrote in the introduction to his habilitation thesis on Fourier series that it was Dirichlet [11]:... who wrote the first profound paper about the subject. In [1] Dirichlet's character and teaching qualities are summed up as follows:He was an excellent teacher, always expressing himself with great clarity. His manner was modest; in his later years he was shy and at times reserved. He seldom spoke at meetings and was reluctant to make public appearances. At age 45 Dirichlet was described by Thomas Hirst as follows:He is a rather tall, lanky-looking man, with moustache and beard about to turn grey with a somewhat harsh voice and rather deaf. He was unwashed, with his cup of coffee and cigar. One of his failings is forgetting time, he pulls his watch out, finds it past three, and runs out without even finishing the sentence. Koch, in [11], sums up Dirichlet's contribution writing that:... important parts of mathematics were influenced by Dirichlet. His proofs characteristically started with surprisingly simple observations, followed by extremely sharp analysis of the remaining problem. With Dirichlet began the golden age of mathematics in Berlin.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirichlet.html (4 of 5) [2/16/2002 11:07:31 PM]

Dirichlet

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (16 books/articles) A Poster of Lejeune Dirichlet

Mathematicians born in the same country

Cross-references to History Topics

1. A comment from Thomas Hirst's diary 2. Fermat's last theorem 3. Prime numbers

Other references in MacTutor

Chronology: 1830 to 1840

Honours awarded to Lejeune Dirichlet (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1855

Lunar features

Crater Dirichlet

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dirichlet.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dirichlet.html (5 of 5) [2/16/2002 11:07:31 PM]

Dixon

Alfred Cardew Dixon Born: 22 May 1865 in Northallerton, Yorkshire, England Died: 4 May 1936 in Northwood, Middlesex, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alfred Dixon was the older brother of Arthur Dixon. Alfred was educated at the Quaker School in Kendal. From Kendal he went to Bath where he attended Kingswood School. Whittaker, in [4], describes him as a prodigy:... the school examiner, finding that [Dixon] obtained the maximum marks in papers set at the annual examination, one year deliberately set a problem that could not be done, only to receive from Dixon an elaborate proof of its impossibility. In 1883 he entered Trinity College, Cambridge and he graduated in 1886 as Senior Wrangler (placed first). He had been taught by a number of famous mathematicians at Cambridge, including Glaisher, Rouse Ball, Forsyth and he attended lectures by Cayley. Dixon was appointed a Fellow of Trinity College in 1888 and was awarded a Smith's prize. Dixon was appointed to the Chair of Mathematics at Queen's College, Galway, Ireland in 1893. (This university is now named University College, Galway.) Dixon's appointment was to fill the chair left vacant when George J Allman, a noted historian of mathematics, retired. The appointment to Galway seems to have been good for Dixon since his mathematical output improved in both quantity and quality from this time. In 1901 Dixon was appointed to the chair of mathematics at Queen's College, Belfast. His chair at Galway was filled one year later by another equally outstanding mathematician when Bromwich was appointed.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dixon.html (1 of 3) [2/16/2002 11:07:33 PM]

Dixon

Dixon's main area of research was in differential equations and he did early work on Fredholm integrals independently of Fredholm. He worked both on ordinary differential equations and on partial differential equations studying abelian integrals, automorphic functions and functional equations. Whittaker describes his research as follows:Dixon was never content with formal results, but always took care to investigate carefully the conditions under which his results were valid. In this respect his analysis is comparable in thoroughness with that of Carleman, Hardy, Hilbert and Schmidt. Soon after his appointment at Galway Dixon published his only textbook The Elementary Properties of Elliptic Functions (1894). During his time in Galway, Dixon proved a combinatorial identity which was later generalised by Fjelsted in 1954. Dixon's identity states that:The sum from k = 0 to n of (-1)k(nCk)3 is 0 if n is not 2m and is (-1)m(3m)!/(m!)3 if n = 2m. As J Ward notes in [2]:In general there are few known identities involving sums of products of several binomial coefficients. A spectacular generalisation of Dixon's beautiful identity is given by equation 5.31 on page 171 of [R L Graham, D E Knuth and O Patashnik, Concrete Mathematics (1989)] which must surely be the non plus ultra of the species. Later in his career Dixon worked on the problem of a loaded elastic rectangular plate. The problem was proposed to him first by the professor of engineering at Belfast. In its original form the question was to study the effect produced by placing a weight on a thin uniform rectangular plate which was clamped round the edges. A devout Methodist Dixon was active in the Philharmonic Orchestra. He was elected to the Royal Society in 1904. After he retired from his chair in Belfast in 1930 he served as president of the London Mathematical Society from 1931 until 1933. He did not wish to retire but [4]:... his retirement was made inevitable by eye trouble which culminated ... in an operation involving the loss of one eye. However, he recovered after a time and undertook such a burst of examination work that he complained of being more overworked in retirement than he had ever been in office. ... His death ... came as a shock to his friends: he had a sudden seizure and died from heart failure in an hour. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) A Poster of Alfred Dixon

Mathematicians born in the same country

Honours awarded to Alfred Dixon (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1904

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dixon.html (2 of 3) [2/16/2002 11:07:33 PM]

Dixon

London Maths Society President

1931 - 1933

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dixon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dixon.html (3 of 3) [2/16/2002 11:07:33 PM]

Dixon_Arthur

Arthur Lee Dixon Born: 27 Nov 1867 in Pickering, Yorkshire, England Died: 20 Feb 1955 in Sandgate, Kent, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Arthur Lee Dixon was the younger brother of Alfred Cardew Dixon. Arthur was educated at Kingswood School in Bath which he attended from 1879 to 1885. This school was a Methodist school founded by Wesley, the founder of Methodism. After leaving Kingswood School Arthur Dixon entered Worcester College, Oxford where he studied mathematics, graduating in 1889. Arthur Dixon won a prize fellowship to Merton College, Oxford where he was appointed in 1891. Merton College was one of the Oxford Colleges with a strong historical mathematical connection, since the first school of mathematics there was organised by Thomas Bradwardine in the middle of the 14th century. A further fellowship allowed Dixon to continue at Merton College until he was appointed to the Savilian chair of pure mathematics at Oxford in 1922. He held this chair until he retired in 1945. Arthur Dixon always said that the biggest influences on his study of mathematics were Elliott, who inspired his particular line of research, and C L Dodgson who he once met. His mathematics, very much in the English tradition of Cayley, studied applications of algebra to geometry, elliptic functions and hyperelliptic functions. In 1908 Dixon began a series of publications on algebraic eliminants, carrying the subject forward from the point where Cayley had left it. He also published a number of papers on the cubic surface, studying lines on the surface and other topics such as the Schur quadric. In the latter part of his career, Dixon published a series of around twelve joint paper with W L Ferrar on analytic number theory, summation http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dixon_Arthur.html (1 of 3) [2/16/2002 11:07:35 PM]

Dixon_Arthur

formulas, Bessel functions and other topics in analysis. In 1912 Arthur Dixon was honoured by being elected a Fellow of the Royal Society. He was also a strong supporter of the London Mathematical Society, serving as its President in 1924-26. His older brother, A C Dixon, would hold this same office five years later. Arthur Dixon shared with Elliott, who had inspired him, an old-fashioned approach to mathematics. Chaundy, writing in [1], describes Dixon's feelings on this as follows:He had no great sympathy with much of the mathematics now in vogue. Matrices, he agreed, meant something; but so often in modern writing, when one had mastered a notation and terminology that were unfamiliar (and you suspected, repellent), one discovered it was something one had known all along. Dixon had many talents in addition to his mathematical ones. He was a great sportsman who played hockey, tennis, squash and croquet. Another side of this many talented man was his skills as a linguist and his great musical talents (he played the flute in an orchestra). Chaundy, in [1], describes Dixon's character:He was a man of the sunniest disposition, radiating bonhomie, welcome in every company and with a wide circle of friends. He was gentle in manner, somewhat reserved in speech, with a quiescence that was never to be thought inertia. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Arthur Lee Dixon (Click a link below for the full list of mathematicians honoured in this way) London Maths Society President

1924 - 1926

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dixon_Arthur.html (2 of 3) [2/16/2002 11:07:35 PM]

Dixon_Arthur

JOC/EFR October 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dixon_Arthur.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dixon_Arthur.html (3 of 3) [2/16/2002 11:07:35 PM]

Dodgson

Charles Lutwidge Dodgson Born: 27 Jan 1832 in Daresbury, England Died: 14 Jan 1898 in Guilford, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Dodgson is known especially for Alice's adventures in wonderland (1865) and Through the looking glass (1872), children's books that are also distinguished as satire and as examples of verbal wit. He invented his pen name of Lewis Carroll by anglicizing the translation of his first two names into the Latin 'Carolus Lodovicus'. The son of a clergyman Dodgson, from 1846 to 1850, attended Rugby School and graduated from Christ Church College Oxford in 1854, coming first in the Finals. Dodgson remained there, lecturing on mathematics and writing treatises and guides for students until 1881. Although he took deacon's orders in 1861, Dodgson was never ordained a priest, partly because he was afflicted with a stammer that made preaching difficult and partly, perhaps, because he had discovered other interests. Among Dodgson's hobbies was photography, at which he became proficient. He excelled especially at photographing children. Alice Liddell, one of the three daughters of Henry George Liddell, the dean of Christ Church, was one of his photographic subjects and the model for the fictional Alice. As a mathematician, Dodgson was conservative. He was the author of a fair number of mathematics books, for instance A syllabus of plane algebraical geometry (1860). None of his mathematics books have proved of enduring importance except for Euclid and his modern rivals (1879) which is of historical interest. As a logician, he was more interested in logic as a game than as an instrument for testing reason. He contributed in 'Jabberwocky', the word 'chortle', a word that combines 'snort' and 'chuckle', to the English language. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dodgson.html (1 of 2) [2/16/2002 11:07:37 PM]

Dodgson

Article by: J J O'Connor and E F Robertson List of References (11 books/articles)

Some Quotations (7)

A Poster of Charles Dodgson

Mathematicians born in the same country

Cross-references to History Topics

Mathematical games and recreations

Other Web sites

1. The Lewis Carroll home page 2. Gutenberg Project (Search for Carroll) 3. ThinkQuest 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dodgson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dodgson.html (2 of 2) [2/16/2002 11:07:37 PM]

Doeblin

Wolfang Doeblin Born: 17 March 1915 in Berlin, Germany Died: 20 June 1940 in Housseras, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wolfang Doeblin is arguably one of the four major contributors to probability theory in the first half of the 20th century up to World War II (the other three are Khinchin, Kolmogorov and Paul Lévy). His work contains several profound results, and his importance is also due to his innovativeness and introduction of new methods. He laid many of the cornerstones of the modern theory for Markov chains and processes, to be developed after the war by others. The article in Mathematical Reviews on the historical paper by Paul Lévy [8] from 1955 about Doeblin's work states that:It is a pity that this tribute to Doeblin's genius was written without reference to later developments of his work. After all there can be no grater testimony to a man's work than its influence on others. Fortunately, for Doeblin, this influence has been visible and is still continuing. On limit theorems his work has been complemented and completed by Gnedenko and other Russian authors. On Markov processes it has been carried out mostly in the United States by Doob, T E Harris and the reviewer. Here his mine of ideas and techniques is still being explored. Considering Doeblin's short career, it is remarkable that he published 13 papers and 13 contributions to Comptes Rendus; for bibliographies, cf. Paul Lévy [8] or Lindvall [9]. In his work on the theory of Markov chains and processes, his main field, we notice major contributions to: Markov chains with general state spaces, Jump Markov processes, the coupling method (innovation), and diffusions. The importance of these is to a large extent due to Doeblin's emphasis on path methods rather than analytical ones; much of what is standard approaches today stem from him. The book by Doob [3] from 1953 has been crucial for the development of probability theory; for a large part of its contents on Markov chains http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doeblin.html (1 of 3) [2/16/2002 11:07:39 PM]

Doeblin

and processes, Doeblin's work is the base. Concerning Markov chains with general state space, Orey [5] from 1971 states that:It is essentially Doblin's theory as completed during the quarter of a century following the publication of his papers that is presented here. In Doeblin's mine of ideas, the coupling method was paid attention to by very few until the early 1970's; then the time was ripe to explore it, and the method is now a major tool in probability theory, with applications ranging from elementary theory to front research. The term "diffusion" was not coined until the 1950's. Nevertheless, the first steps were taken in the 1930's. This type of process were the main interest in the last phase of Doeblin's mathematical career, interrupted by World War II; the spare time he had as a soldier was spent on this. The results presented (without proofs) in Comptes Rendus are remarkable, and so are the contents of the file that was sealed in February 1940 and not opened until May 2000. It contains pieces of what we now call stochastic calculus, including a version of Ito's formula. On Doeblin's work concerning sums of independent random variables, Feller [4] writes:The interest in the theory was stimulated by W. Doblin's masterful analysis of the domain of attraction (1939). His criteria were the first to to involve regularly varying functions. The modern theory still carries the imprint... [This concerns weak convergence in arrays]. The last result was obtained by Doblin in a masterful study in 1940, following previous work by Khinchin (1937). [This concerns the so called Doblin's universal law.] Doeblin also contributed to the theory of random chains with complete connection, some of which was used in a paper by him on ergodic properties of continued fractions. Doeblin's life and fate are remarkable and gripping. His father was Alfred Döblin, a medical doctor but best known as author; the novel Berlin Alexanderplatz (1929) stands out among his many books. Wolfgang was born in Berlin, but he spent his first three years behind the German front in World War I, in Saargemünd where his father volunteered as an army doctor. At the peace in 1918, the family moved back to Berlin, where the reputation of the father, as author and left-wing participant in political and cultural debates, started to rise. Warned by friends, Alfred Döblin left Berlin for Zürich the day after the Reichstag-fire in February 1933. He was on the black list of the Nazis: a Jew, controversial for his political views. Wolfgang stayed until May to finish school. The sojourn in Zürich lasted for the summer 1933 only; the family settled in Paris after that. Doeblin immediately made a strong impression in Paris; Fréchet was his adviser, but Doeblin also got in touch with Paul Lévy, with whom he wrote his first note. He received his PhD from the Sorbonne in 1938, but at that time he was far into topics beyond or not related to those of his thesis. He was enrolled in the French army in the autumn 1939 for military training, and called up for front service in September 1939. This means that his mathematical career comprised five years only of concentrated work. In February 1940, the Nazi invasion was expected to come in the spring to follow, and Doeblin decided to file his work on diffusions at l'Académie de Science in Paris. The part of Lorraine where he was commanded fell in June. With the Nazi troops just minutes away from the little village Housseras, he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doeblin.html (2 of 3) [2/16/2002 11:07:39 PM]

Doeblin

decided to take his own life there rather than giving himself up as a prisoner of war. Housseras is located some 100 kilometres from Sarreguemines (French after World War I), the place where Doeblin had spent the first three years of his life. The reader is referred to [1], [6], [7], [9], [10] for biographical texts on Doeblin. Article by: Torgny Lindvall Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Doeblin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doeblin.html (3 of 3) [2/16/2002 11:07:39 PM]

Domninus

Domninus of Larissa Born: about 420 in Larissa, Syria Died: about 480 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Domninus was a Syrian, and by religion a Jew, who was born in the town of Larissa (often identified with Laodicea but probably a separate town) on the Orontes River. He went to Athens where he became a pupil of Syrianus who was the head of Plato's Academy there. Proclus, although slightly older than Domninus, was also a pupil of Syrianus at the Academy at the same time. Marinus, who was later a pupil of Proclus and eventually took over as head of the Academy following Proclus, writes about a rivalry between Domninus and Proclus [1]:[Syrianus] offered to discourse to them on either the Orphic theories or the oracles; but Domninus wanted Orphism, Proclus the oracles, and they had not agreed when Syrianus died... If at first Domninus and Proclus were merely student rivals, certainly it grew into a more serious disagreement centred on how Plato's philosophy should be interpreted. This serious disagreement saw Proclus come out as the victor in the sense that the Academy preferred his views. Proclus succeeded Syrianus as head of the Academy in Athens while, a short while later, Domninus left Athens and returned to his home town of Larissa. The mathematical work of Domninus only came to light after the publication of his Manual of Introductory Arithmetic in 1832, and its importance was not realised until Paul Tannery began publishing a number of works on Domninus in 1884. Although Domninus wrote a number of books the only other known in detail is How to take a ratio out of a ratio which was not published until 1883 in [3]. The Manual of Introductory Arithmetic studies numbers, means, and proportion. The book is in five parts [1]:... an examination of numbers in themselves, an examination of numbers in relation to other numbers, the theory of numbers both in themselves and in relation to others, the theory of means and proportions, and the theory of numbers as figures. At the end of the book Domninus says that he intends to treat some of the subjects more fully in Elements of Arithmetic but it is not known if he ever wrote it! Certainly he would not be the first or last mathematician to refer to a future work which never materialised. Heath [2] writes of the Manual of Introductory Arithmetic :-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Domninus.html (1 of 2) [2/16/2002 11:07:40 PM]

Domninus

It is a sketch of the elements of the theory of numbers, very concise and well arranged, and is interesting because it indicates a serious attempt at a reaction against the Introductio arithmetica of Nicomachus and a return to the doctrine of Euclid. The second book, How to take a ratio out of a ratio, published in translation in 1883, studies manipulation of ratios into other forms. Heath casts some doubt as to whether this book is actually by Domninus. He writes [2]:... if it not by Domninus, it probably belongs to the same period. Bulmer-Thomas in [1] is more certain that it is by Domninus and conjectures that the work was, at least in part, work by Domninus towards the Elements of Arithmetic which he had promised to write. We do have some indications of the character of Domninus, but these may be very unfair since they are related to us by Damascius, the last head of the Academy. Since Domninus's philosophy was considered old-fashioned and out of favour by the Academy the claims made by Damascius may have been aimed at discrediting him. Damascius wrote that when Domninus was an old man he [1]:...loved only the conversation of those who praised his superiority and that he would not admit to his company a young man who argued with him about a point in arithmetic. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Domninus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Domninus.html (2 of 2) [2/16/2002 11:07:40 PM]

Donaldson

Simon Kirwan Donaldson Born: 20 Aug 1957 in Cambridge, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Simon Donaldson's secondary school education was at Sevenoaks School in Kent which he attended from 1970 to 1975. He then entered Pembroke College, Cambridge where he studied until 1980, receiving his B.A. in 1979. One of his tutors at Cambridge described him as a very good student but certainly not the top student in his year. Apparently he would always come to his tutorials carrying a violin case. In 1980 Donaldson began postgraduate work at Worcester College, Oxford, first under Nigel Hitchen's supervision and later under Atiyah's supervision. Atiyah writes in [2]:In 1982, when he was a second-year graduate student, Simon Donaldson proved a result that stunned the mathematical world. This result was published by Donaldson in a paper Self-dual connections and the topology of smooth 4-manifolds which appeared in the Bulletin of the American Mathematical Society in 1983. Atiyah continues his description of Donaldson's work [2]:Together with the important work of Michael Freedman ..., Donaldson's result implied that there are "exotic" 4-spaces, i.e. 4-dimensional differentiable manifolds which are topologically but not differentiably equivalent to the standard Euclidean 4-space R4. What makes this result so surprising is that n = 4 is the only value for which such exotic n-spaces exist. These exotic 4-spaces have the remarkable property that (unlike R4) they contain compact sets which cannot be contained inside any differentiably embedded 3-sphere ! After being awarded his doctorate from Oxford in 1983, Donaldson was appointed a Junior Research Fellow at All Souls College, Oxford. He spent the academic year 1983-84 at the Institute for Advanced

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Donaldson.html (1 of 3) [2/16/2002 11:07:41 PM]

Donaldson

Study at Princeton, After returning to Oxford he was appointed Wallis Professor of Mathematics in 1985, a position he continues to hold. Donaldson has received many honours for his work. He received the Junior Whitehead Prize from the London Mathematical Society in 1985. In the following year he was elected a Fellow of the Royal Society and, also in 1986, he received a Fields Medal at the International Congress at Berkeley. In 1991 Donaldson received the Sir William Hopkins Prize from the Cambridge Philosophical Society. Then, the following year, he received the Royal Medal from the Royal Society. He also received the Crafoord Prize from the Royal Swedish Academy of Sciences in 1994:... for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants ... Atiyah describes the contribution which led to Donaldson's award of a Fields Medal in [2]. He sums up Donaldson's contribution:When Donaldson produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration. Slowly the message has gotten across and now Donaldson's ideas are beginning to be used by others in a variety of ways. ... Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry. The article [3] is very interesting and provides both a collection of reminiscences by Donaldson on how he came to make his major discoveries while a graduate student at Oxford and also a survey of areas which he has worked on in recent years. Donaldson writes in [3] that nearly all his work has all come under the headings:(1) Differential geometry of holomorphic vector bundles. (2) Applications of gauge theory to 4-manifold topology. and he relates his contribution to that of many others in the field. Donaldson's work in summed up by R Stern in [6]:In 1982 Simon Donaldson began a rich geometrical journey that is leading us to an exciting conclusion to this century. He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Donaldson.html (2 of 3) [2/16/2002 11:07:41 PM]

Donaldson

Other references in MacTutor

Chronology: 1980 to 1990

Honours awarded to Simon Donaldson (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1986

Fellow of the Royal Society

Elected 1986

Royal Society Royal Medal winner

1992

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Donaldson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Donaldson.html (3 of 3) [2/16/2002 11:07:41 PM]

Doob

Joseph Leo Doob Born: 27 Feb 1910 in Cincinnati, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Doob studied at Harvard and obtained his doctorate there in 1932 for a thesis entitled Boundary Values of Analytic Functions. He was appointed to the University of Illinois in 1935. He became a full professor there 10 years later. Doob's work was in probability and measure theory, in particular he studied the relations between probability and potential theory. Doob built on work by Paul Lévy and, during the 1940's and 1950's, he developed basic martingale theory and many of its applications. Doob's work has become one of the most powerful tools available to study stochastic processes. In 1953 he published a book which gives a comprehensive treatment of stochastic processes, including much of his own development of martingale theory. This book Stochastic Processes has become a classic and was reissued in 1964. He is also the author of a well known book on measure theory. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Quotation Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doob.html (1 of 2) [2/16/2002 11:07:44 PM]

Doob

Honours awarded to Joseph L Doob (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1963 - 1964

AMS Colloquium Lecturer

1959

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Doob.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doob.html (2 of 2) [2/16/2002 11:07:44 PM]

Doppelmayr

Johann Gabriel Doppelmayr Born: 1677 in Nuremberg, Germany Died: 1 Dec 1750 in Nuremberg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Doppelmayr entered the Aegidien Gymnasium in Nuremberg in 1689. The Aegidien Gymnasium had been founded in the 16th century by the Swiss Protestant reformer Ulrich Zwingli. Doppelmayr was to return to this Gymnasium as a professor later in his career. From Nuremberg Doppelmayr went to the University of Altdorf where he studied law, mathematics and natural philosophy. Four years later he studied at the University of Halle, then spent 2 years travelling in Germany, Holland and England. He studied at Utrecht, Leyden, Oxford and London during his travels. Doppelmayr clearly made a good impression on people during these visits since he was elected to membership of a number of scientific societies including the Berlin Academy, St Petersburg Academy and the Royal Society. Doppelmayr was appointed professor of mathematics at the Aegidien Gymnasium in Nuremberg in 1704 and he remained there for the rest of his life. Doppelmayr wrote on astronomy, spherical trigonometry, sundials and mathematical instruments. He has a book of tremendous value giving biographical details of 360 mathematicians and instrument makers of Nuremberg from the 15th to the 18th Century. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doppelmayr.html (1 of 2) [2/16/2002 11:07:45 PM]

Doppelmayr

List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Johann Doppelmayr (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1733

Lunar features

Crater Doppelmayr and Rimae Doppelmayer

Other Web sites

1. The Galileo Project 2. Linda Hall Library (Star Atlas)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Doppelmayr.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doppelmayr.html (2 of 2) [2/16/2002 11:07:45 PM]

Doppler

Christian Andreas Doppler Born: 29 Nov 1803 in Salzburg, Austria Died: 17 March 1853 in Venice, Italy

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Christian Doppler's family were stonemason's who had a successful business in Salzburg, Austria from 1674. The prospering business led to the building of a fine house in the Hannibal Platz [now named Makart Platz] in Salzburg, near to the river. Christian Doppler was born in this family house and, of course, the family tradition would have had him grow up to take over the stonemason's business. However his health was never very good and he was quite frail so he could not follow in the family tradition. Doppler attended primary school in Salzburg and then attended secondary school in Linz. His parents were unsure of his academic potential and consulted the professor of mathematics at the Salzburg Lyceum who recommended that Doppler should study mathematics at the Vienna Polytechnic Institute. The Polytechnic Institute had only been founded in 1815, so it was still a new establishment when Doppler began his studies there in 1822. He excelled in his mathematical and other studies and graduated in 1825. After this he returned to Salzburg, attended philosophy lectures at the Salzburg Lyceum, then went to the University of Vienna where he studied higher mathematics, mechanics and astronomy. At the end of his studies at the University of Vienna in 1829, Doppler was appointed as assistant to the professor of higher mathematics and mechanics at the University, Professor A Burg. He published four mathematics papers during his four years as Burg's assistant, his first being A contribution to the theory of parallels. This assistantship was only a temporary post and Doppler, rather older than most others, began to seek a permanent post at the age of 30. In [11] Seidlerova explains how applications worked at that time in Austria:From 1825 all vacant professorships at Austrian universities and polytechnics were filled by http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doppler.html (1 of 6) [2/16/2002 11:07:47 PM]

Doppler

public competition. It actually meant admission examination, where the questions were determined ... The applicants at various schools of the monarchy had to answer them in written form, which could take up to twelve hours. Part of the examination was also a short probationary lecture on an arbitrary topic in front of the appointed commission. The sealed answers, together with an evaluation of the lecture, were then sent to the school where the competition had been announced. The final decisions were taken by the commission in Vienna but the applicants were only selected on their teaching ability, any sign of higher levels of knowledge would be treated as telling against the candidate. Doppler submitted himself to a number of these competitions, both for school and university places. He applied to schools in Linz, Salzburg, Gorizia and Ljubljana and for the chair of higher mathematics at Vienna Polytechnic and on 23 March 1833 for the professorship of arithmetic, algebra, theoretical geometry and accountancy at the Technical Secondary School in Prague. While this was going on Doppler had to earn his living and he spent 18 months as a bookkeeper at a cotton spinning factory. This was a period of sadness and great difficulty for Doppler and it is not surprising that he decided to give up the unequal struggle and emigrate to America. He began to sell his possessions and visited the American Consul in Munich to make the necessary arrangements. However, when he was close to making the final decision he received an offer of the post at the Technical Secondary School in Prague. It had taken a long time for the process of appointing to reach its conclusion and Doppler took up his post in March 1835, almost exactly two years after entering the competition. Doppler was ambitious and teaching elementary mathematics at the Technical School was not greatly to his liking. He tried for a post of professor of higher mathematics at the Polytechnic in Prague but without success. However, during 1836-38 he was able to teach higher mathematics for four hours a week at the Polytechnic. This brought in extra money which he certainly needed since he married in 1836. Doppler did get another chance of a post at the Polytechnic, however, and at the end of 1837 the professorship in practical geometry and elementary mathematics became vacant. Doppler assumed the duties of the post but things were not that straightforward. Despite the fact that he was carrying out the duties, a competition for the post was held on 3 October 1839. Doppler did not have to take part in the competition but was hurt by the fact that it was held at all. He was formally appointed to the post in March 1841. Doppler did not have an easy time teaching at the Polytechnic. Seidlerova writes in [11]:The examinations were very stressful. The terms of both oral and written exams had to be reported in advance to the Land Committee which also nominated the examination commissioner. For example, in January and February 1843 Doppler had to examine 256 students in 17 days, both in writing and orally, in arithmetic and algebra. The examinations took a minimum of six hours daily. The same number of students sat for the examination in "theoretical geometry" in June and July of the same year in a twelve day examination. Additionally in July and August it was necessary to examine 145 students in geodesy in eight days. ... In July 1847 Doppler orally examined 526 students in mathematics and 289 in geodesy. By 1844 Doppler's health, always less than good, failed under the strain. He had to give up teaching and requested sick leave. He had support from Bolzano who wrote [13]:It is hard to believe how fruitful a genius Austria has in this man. I have written to ... many http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doppler.html (2 of 6) [2/16/2002 11:07:47 PM]

Doppler

people who can save Doppler for science and not let him die under the yoke. Unfortunately I fear the worst. The situation was made worse by Doppler's students complaining that he was too harsh in his examining. Doppler was investigated and reprimanded while the students were allowed to retake their examinations. Doppler considered himself totally innocent and demanded that the reprimand be withdrawn. Eventually the reprimand was reluctantly withdrawn by the end of 1844 but Doppler was not well enough to return to his duties until 1846. With such a difficult time in Prague, it is no surprise that Doppler wanted to move and he was offered the professorship of mathematics, physics and mechanics at the Academy of Mines and Forests in Banska Stiavnica. Stoll writes [13]:When Doppler left Prague for Banska Stiavnica, he did not suspect that his stay in this city would be so short. The stormy year 1848 shook all parts of the monarchy and revolution broke out in Prague, Vienna and Budapest. Due to political unrest the situation in Banska Stiavnica became complicated and Doppler was once again seeking refuge. He was now a figure of some importance so a move was now able to be made with less difficulty. He was appointed to Vienna Polytechnic, then on 17 January 1850 he was appointed as the first director of the new Institute of Physics at Vienna University. Doppler had reached the high point of his career. What qualities had carried Doppler through the struggles of his early experiences to this important position? It was not, it seems, his great mathematical abilities, for despite his career as a mathematician he was always short of the topmost level when it came to mathematical research. In fact his grasp of mathematics may have been even less good than this for he wrote an elementary text Arithmetic and algebra published in Prague in 1843. Seidlerova, describing this work in [11] writes:Doppler's explanations were conducted in a very unfortunate way and demonstrated that in the basic questions of mathematics he groped more than his eminent contemporaries. However, despite this Doppler did have genius within him. It was a genius that Bolzano saw from the very first. Bolzano reviewed the first paper which Doppler submitted to the Royal Bohemian Society of Sciences in 1837. After recommending Doppler's paper on applied analysis for publication, Bolzano commented about Doppler himself. Dated Prague 25 September 1839, the report reads [13]:Mr Doppler has already demonstrated his very promising abilities to the scientific community through his numerous published works in mathematics and physics. The expectations raised by his hitherto published works would multiply when one enters into personal acquaintance with him. You are not only struck by how many highly interesting and fruitful ideas, in many areas of knowledge, that so young a scientist is able to produce, but you also convince yourself with the greatest pleasure that this exceptional spiritual power combines with an amiable character, genuine unaffected determination and with that pure love of science and truth ... So Bolzano, himself a great mathematical innovator, could see the genius in Doppler. Not everyone could see it however. Kulik, who was professor of mathematics at the Charles University of Prague while Doppler worked at the Polytechnic, [13]:... did not have much understanding of Doppler's originality or of his intuitive ways of thinking.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doppler.html (3 of 6) [2/16/2002 11:07:47 PM]

Doppler

Bolzano moved to Prague in 1842 and became secretary to the mathematical section of the Royal Bohemian Society of Sciences. He was then in closer contact with Doppler and Bolzano wrote in 1844 [13]:Professor Doppler over several weeks has excited me with his ideas, the one more brilliant than the other. I must think about them day and night. However two years before Bolzano wrote this, Doppler had presented his most famous brilliant idea to the Royal Bohemian Society. On the 25 May 1842 Doppler presented the paper On the coloured light of the double stars and certain other stars of the heavens. The minutes of the meeting reported on Doppler's lecture [10]:Mr Doppler talked about a wonderful phenomenon of the coloured light of the double stars and some other stars in the heavens. He sought the explanation of this striking phenomenon in formulating a new general theory, which included in itself as an integral part the theory of Bradley. The paper presented for the first time the Doppler principle which relates the frequency of a source to its velocity relative to an observer. Doppler derived the principle in a few lines treating both light and sound as longitudinal waves in the ether and matter, respectively. Doppler was incorrect regarding light being a longitudinal wave. In fact Fresnel had already published his theory that light was a transverse wave but, although Doppler had read Fresnel's work, he did not accept it. However the error does not really affect the result of Doppler's principle. Doppler also was wrong when he tried to illustrate his theory with an application to the colours of double stars. Although Doppler was correct in saying that his principle would change the colours of double stars, depending on which star was approaching or receding from the Earth, the effect is too small to be significant. Doppler does, however, make a remarkable prediction in his paper:It is almost to be accepted with certainty that this will in the not too distant future offer astronomers a welcome means to determine the movements and distances of such stars which, because of their unmeasurable distances from us and the consequent smallness of the parallactic angles, until this moment hardly presented the hope of such measurements and determinations. Although changes in colours were impossible to observe with the instruments of the time, the situation with sound was rather different. As early as 1845 experiments were conducted with musicians on railway trains playing instruments and other trained musicians writing down the apparent note as the train approached them and receded from them. In 1846 Doppler published a better version of his principle where he considered both the motion of the source and the motion of the observer. Not everyone of course was immedately convinced by Doppler's theory. His most vigorous opponent was Petzval, by this time professor of mathematics at the University of Vienna. Their dispute was based on a misunderstanding, in some sense both were correct but they could not see that they were arguing about different things. No other work by Doppler came anywhere close in matching the importance of his publications on the Doppler principle. He did publish on electricity and magnetism, the variation of magnetic declination with time as well as several publications on optics and astronomical topics. His mind would continually come up with new ideas and so he was led to invent many instruments, particularly optical instruments,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doppler.html (4 of 6) [2/16/2002 11:07:47 PM]

Doppler

and improve existing ones. Most of his ideas are quite revolutionary, he was certainly a very original thinker, but on the down side most would just not work in practice. However one can often see the germ of some important future discovery there, even though the idea as presented by Doppler is basically incorrect. Doppler had some difficulty becoming a member of the Royal Bohemian Society despite very strong support from Bolzano and his good relations with the Society. In 1837, when he reviewed the first paper that Doppler submitted to the Society, Bolzano requested in his report that Doppler be elected to the Society. This was not acted on but, in the following year, Doppler was proposed again and not elected in a ballot. On 28 June 1840 Doppler was eleced an associate member of the Royal Bohemian Society after a close ballot of 7 for and 5 against. It does appear that after his 1842 paper he gained more favour for he was elected as an ordinary member of the Society on 5 November 1843 with 9 votes in his favour and only one against. In 1847 he was elected deputy secretary of the Society and became one of the leaders of the Society who showed, according to Bolzano's words [10]:... pure love of science and truth which rises high above narrowminded party-spirit as well as conceited inflexibility. Other honours which came Doppler's way in 1848 were election to ordinary membership of the Imperial Academy of Sciences in Vienna and an honorary doctorate from the University of Prague. Doppler's time as the first Director of the Institute of Physics at Vienna University was a short one. He was appointed by Imperial Decree on 17 January 1850. His health continued to deteriorate with severe chest problems and, in November 1852, he travelled to Venice in the hope that the warmer climate would bring about some improvement. It was not to be, however, and by March 1853 it was clear that he was sinking fast. Doppler's wife, who had given him staunch support throughout their marriage, had remained in Vienna with their three sons and two daughters awaiting his return but, on realising that his end was near, she made the journey to Venice and was with Doppler when he died. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) Mathematicians born in the same country Cross-references to History Topics

Special relativity

Honours awarded to Christian Doppler (Click a link below for the full list of mathematicians honoured in this way) Lunar features Other Web sites

Crater Doppler 1. West Chester University 2. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doppler.html (5 of 6) [2/16/2002 11:07:47 PM]

Doppler

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Doppler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Doppler.html (6 of 6) [2/16/2002 11:07:47 PM]

Douglas

Jesse Douglas Born: 3 July 1897 in New York, USA Died: 7 Oct 1965 in New York, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jesse Douglas developed a love of geometry studying Plateau Problems. He studied differential geometry at Columbia College from 1920 to 1926. Then from 1926 to 1930 he visited Princeton, Harvard, Chicago, Paris and Göttingen. He gave a complete solution to the Plateau problem which was first posed by Lagrange in 1760. It was studied by Riemann, Weierstrass and Schwarz. The problem is proving the existence of a surface of minimal area bounded by a contour. Before Douglas's solution only special cases had been solved. Then he went on to study generalisations of the problem. In 1943 Douglas was awarded the Bôcher Prize by the American Mathematical Society for his memoirs on the Plateau Problem. In particular the award was for three papers all published in 1939: Green's function and the problem of Plateau published in the American Journal of Mathematics, The most general form of the problem of Plateau published in the American Journal of Mathematics and Solution of the inverse problem of the calculus of variations published in the Proceedings of the National Academy of Sciences. Douglas also worked on the calculus of variations and, in 1951, studied groups with two generators x, y such that every element can be expressed in the form xmyn with m, n integers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Douglas.html (1 of 2) [2/16/2002 11:07:49 PM]

Douglas

List of References (8 books/articles) A Poster of Jesse Douglas

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1930 to 1940

Honours awarded to Jesse Douglas (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1936

AMS Bôcher Prize

Awarded 1943

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Douglas.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Douglas.html (2 of 2) [2/16/2002 11:07:49 PM]

Dowker

Clifford Hugh Dowker Born: 2 March 1912 in Parkhill, Western Ontario, Canada Died: 14 Oct 1982 in London, England

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hugh Dowker's parents owned a small farm in Western Ontario in Canada. his father's ancestors came from Yorkshire in England while his mother's came from Scotland. It was not a family with any previous record of academic achievement and, in fact, Hugh was the only one of his parent's three sons to have a university education. Now Hugh certainly did not have an easy time with his school education for in primary school he had to walk two miles every day to reach the school. When he progressed to secondary school in Parkhill things became even more difficult for he found himself with a mathematics teacher who appeared to have no understanding of the subject. Most of the great mathematicians in this archive were inspired to become mathematicians by fine teaching at school but in Dowker's case he received payment to stay behind after school to show the mathematics teacher how to do the problems. Despite these difficulties with his schooling Dowker won a scholarship to enable him to study at the University of Western Ontario. There he showed his outstanding ability in mathematics but he studied other subjects as well such as physics and economics for his B.A. which was awarded in 1933. Now this went far further than someone from Dowker's background could ever expect to go in academic pursuits and it was natural for him to wish to end his education at this point. His aim had been to become a school teacher, after all he had shown at secondary school that he could teach the teacher mathematics, and he was now well on his way to achieving this. Others, realising that he had extraordinary talents for mathematics, worked hard to persuade him to continue. they were successful and in the following year he studied for his Master's degree at the University of Toronto. Again Dowker was surprised to find that his lecturers at Toronto advised him to go to Princeton to study for a doctorate under Lefschetz. Strauss writes in [3] (or see [2]):-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dowker.html (1 of 3) [2/16/2002 11:07:51 PM]

Dowker

It was at Princeton that Dowker became fully aware of the power and beauty of mathematics, and he became an active topologist, running one of Lefschetz's seminars. He obtained his Ph.D. there in 1938. Apart from Lefschetz, the mathematicians who were to have an important influence on Dowker included Aleksandrov, Fox, Hurewicz and Steenrod. Dowker's doctoral thesis extended basic results in homotopy theory from compact metric spaces to normal and parametric spaces. After the award of his doctorate in 1938 Dowker was appointed as an instructor to the University of Western Ontario for the year 1938-39. Following this he was an assistant to von Neumann at the Institute for Advanced Study at Princeton in 1939-40. He then spent three years as an instructor at Johns Hopkins university in Baltimore. It was there that he met Yael Naim, a young mathematics student from Israel. They married in 1944 but the year before this both began working at the Massachusetts Institute of Technology Radiation Laboratory. Dowker also did war work with the United States Air Force, working on the trajectory of projectiles. In carrying out this work he visited Libya and Egypt. When World war II ended, Dowker was appointed as an associate professor at Tufts. however both Dowker and his wife became increasingly tense as McCarthy's hunt for communists became increasingly menacing. Senator Joseph R McCarthy whipped up strong feelings against communism in the United States and several of Dowker's friends began to come under suspicion from authorities who saw imaginary problems everywhere. He decided that he disliked living in such an atmosphere and looked for positions in England for himself and his wife. He was appointed as a Reader in Applied Mathematics at Birkbeck College in London while Yael was appointed to the University of Manchester. Although most of Dowker's work was in topology, his war work set him up well for the applied mathematics post where he continued his research on projectiles. Dowker was appointed to a personal chair at Birkbeck College in 1962, a post which he held until his retirement in 1970. James in [1] summarises his topological work:While he is best known for his work in point set topology, he also made contributions to category theory, sheaf theory and the theory of knots. He had a long-standing interest in homology theory for general spaces. Among many other important results he showed that the Cech and Vietoris homology groups coincide, for general spaces, as do the Cech and Alexander cohomology groups. In 1956 Dowker published Lectures on sheaf theory which followed the approach which had been adopted by Henri Cartan. In his review of the work Atiyah writes:Unlike the Cartan seminars, however, the exposition does not aim at conciseness. Many parts of the theory are treated in much greater detail, most calculations are given in full, and much standard algebra is developed ab initio. In this respect the work may serve a very useful purpose. Perhaps the most characteristic feature of the lectures is the way each new notion is analysed very carefully, its peculiarities examined on "bad spaces", and counter-examples given whenever possible. For example, both fine and locally fine sheaves are defined, and it is shown that they coincide on paracompact normal spaces but not otherwise. There is also a very careful treatment of presheaves as opposed to sheaves. The lectures on coherent sheaves are dealt with in the same analytical spirit, and there is no attempt to go far into the applications to algebraic geometry or complex manifolds.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dowker.html (2 of 3) [2/16/2002 11:07:51 PM]

Dowker

In 1983, the year in which Dowker died after a long illness, he published a joint paper Classification of knot projections with Morwen B Thistlethwaite. The authors describe their results as follows:The first step in tabulating the noncomposite knots with n crossings is the tabulation of the nonsingular plane projections of such knots, where two (piecewise linear) projections are regarded as equivalent, or in the same class, if they agree up to homeomorphism of the extended plane, i.e., the two-sphere. This first step is here reduced to a simple algorithm suitable for computer use. In [3] (or see [2]) Strauss describes Dowker's character:In manner Dowker was reserved and gentle, with an innate dignity and a penetrating wit. he possessed a high degree of integrity and moral strength which enabled him to endure seven years of illness uncomplainingly. Although supremely tolerant towards others, he had only the highest standards of behaviour for himself. He was totally without ostentation or pretention and totally disinterested in wealth, honours or managerial power. There is another side to Dowker and his wife which we should mention before completing this biography. this is their work with children, particularly with gifted children who were having difficulties at school. They undertook this work for the National association for Gifted Children. Dowker was able to give these children what any mathematician will recognise as an incredibly wonderful experience, namely the experience of discovering new mathematical results. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Dowker.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dowker.html (3 of 3) [2/16/2002 11:07:51 PM]

Drach

Jules Joseph Drach Born: 13 March 1871 in Sainte-Marie-aux-Mines, France Died: 8 March 1941 in Cavalaire, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jules Drach worked as an architect in his youth to help his family. He went to the Lycée in Nancy at the age of 18 then to Ecole Normale Supérieure in Paris. Drach was encouraged by Jules Tannery to undertake mathematical research and he obtained a doctorate from Ecole Normale Supérieure in 1898. He taught at many universities including Lille, Poitiers, Toulouse and Paris. Drach viewed Emile Picard's application, in 1887, of Galois theory to linear differential equations as a model of perfection and he tried to extend Galois theory to differential equations in general building on the work of Lie and Vessiot in addition to that of Emile Picard. Drach was a friend of Borel, and together they published lectures by Poincaré and by Jules Tannery. Drach helped to prepare Poincaré's works for publication. These appeared in 11 volumes between 1916 and 1956. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Drach.html (1 of 2) [2/16/2002 11:07:52 PM]

Drach

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Drach.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Drach.html (2 of 2) [2/16/2002 11:07:52 PM]

Drinfeld

Vladimir Gershonovich Drinfeld Born: 1954 in Kharkov, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vladimir Drinfeld studied at Moscow University from 1969 until 1974. He graduated in 1974 and remained at Moscow University to undertake research under Yuri Ivanovich Manin's supervision. Drinfeld completed his postgraduate studies in 1977 and he defended his "candidate" thesis in 1978 at Moscow University. The "candidate" thesis is the Russian equivalent of the British or American Ph.D. Since 1981 Drinfeld has been working at B Verkin Institute for Low Temperature Physics and Engineering of the Academy of Sciences of the Ukraine. Drinfeld defended his "doctor" thesis in 1988 at Steklov Institute, Moscow. The "doctor" thesis is the Russian equivalent of German habilitation. On 21 August 1990 Drinfeld was awarded a Fields Medal at the International Congress of Mathematicians in Kyoto, Japan:... for his work on quantum groups and for his work in number theory. A Jaffe and B Mazur, write in [2] about Drinfeld's work which led to the award of the Fields Medal:Drinfeld's interests can only be described as "broad". Not only do the span work in algebraic geometry and number theory, but his most recent ideas have taken a strikingly different direction: he has been doing significant work on mathematical questions motivated by physics, including the relatively new theory of quantum groups. Drinfeld defies any easy classification ... His breakthroughs have the magic that one would expect of a revolutionary mathematical discovery: they have seemingly inexhaustible consequences. On the other hand, they seem deeply personal pieces of mathematics: "only Drinfeld could have thought of them!" But contradictorily they seem transparently natural; once understood, "everyone should have thought of them!" http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Drinfeld.html (1 of 2) [2/16/2002 11:07:54 PM]

Drinfeld

Drinfeld's main achievements are his proof of the Langlands conjecture for GL(2) over a functional field; and his work in quantum group theory. Although he only proved a special case of the Langlands conjecture, Drinfeld has introduced important new ideas in his solution and made a real breakthrough. He introduced the idea of an elliptic module in his proof and this notion is leading to a whole new topic within number theory. The interactions between mathematics and mathematical physics studied by Atiyah led to the introduction of instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory. Drinfeld and Manin worked on the construction of instantons using ideas from algebraic geometry. In 1992 Drinfeld was elected a member of the Academy of Sciences of the Ukraine. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1990 to 2000

Honours awarded to Vladimir Drinfeld (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1990

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Drinfeld.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Drinfeld.html (2 of 2) [2/16/2002 11:07:54 PM]

Du_Bois-Reymond

Paul David Gustav Du Bois-Reymond Born: 2 Dec 1831 in Berlin, Germany Died: 7 April 1889 in Freiburg, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul du Bois-Reymond was a brother of the famous physiologist Emil du Bois-Reymond and studied at Berlin and then medicine at Zurich. Moving to Königsberg he was influenced by Franz Neumann to change to mathematics. After a doctorate from Berlin he held chairs in Heidelberg, Freiburg and Tübingen where he succeeded Hankel. Finally he was appointed to a chair in Berlin. His work is almost exclusively on calculus, in particular differential equations and functions of a real variable. He generalised Monge's idea of the characteristic of a partial differential equation from 2nd order equations to nth order equations. This work formed a basis of what Lie was to generalise later. In 1873 he gave a continuous function with divergent Fourier series at any point solving a major problem. The term 'integral equation' is due to Du Bois-Reymond.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Paul du Bois-Reymond

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Du_Bois-Reymond.html (1 of 2) [2/16/2002 11:07:55 PM]

Du_Bois-Reymond

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Du_Bois-Reymond.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Du_Bois-Reymond.html (2 of 2) [2/16/2002 11:07:55 PM]

Du_Val

Patrick du Val Born: 26 March 1903 in Cheadle Hulme (near Manchester), Cheshire, England Died: 22 Jan 1987 in Cambridge, Cambridgeshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Patrick du Val suffered from bad health as a child and was educated mostly by his mother. He took a correspondence course and was awarded First Class Honours from the University of London in 1926, having been an external student. Talented in many different subjects such as mathematics, languages and history, perhaps the most significant step in his life came when he and his mother moved to a village near Cambridge. They got to know Baker and he persuaded Patrick to research into algebraic geometry at Cambridge. Baker supervised his research and he received a Ph.D. in 1930. During his period as a research student he had many famous geometers as fellow research students and he formed a particular friendship with Coxeter and Semple. Du Val visited Rome working with Enriques, then in 1934 he visited Princeton and attended lectures by Alexander, Eisenhart, Lefschetz, Veblen, Wedderburn and Weyl. He held posts in Manchester, where he stayed for five years, Istanbul, where he learnt then wrote in Turkish and the USA where he spent three years. Rather unhappy in the USA, he returned to England, first taking up a post in Bristol, then London in 1954 where he remained until he retired in 1970. Together with Semple he led the London Geometry Seminar during the time he spent in London. After he retired Du Val returned to Istanbul and for three years he held the same post as he had held 30 years before. Then he returned to England and lived in Cambridge in his retirement.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Du_Val.html (1 of 2) [2/16/2002 11:08:00 PM]

Du_Val

Du Val's early work before he became a research student at Cambridge was on on relativity. He published on the De Sitter model of the universe and Grassmann's tensor calculus. His doctorate was on algebraic geometry and in his thesis he generalised a result of Schoute. He worked on algebraic surfaces, especially during his time in Rome, and later in his career du Val became interested in elliptic functions. Du Val was always interested in teaching as well as research. His style as a lecturer is described in these terms in [1]:... his raucous asthmatic delivery gave him a somewhat forbidding manner, yet he was most kind and sympathetic to his students and always willing to spend time coaching the weaker ones among them. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Du_Val.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Du_Val.html (2 of 2) [2/16/2002 11:08:00 PM]

Dubreil

Paul Dubreil Born: 1 March 1904 in Le Mans, Maine, France Died: 9 March 1994 in Soisy sur Ecole (near Paris), France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Dubreil attended the Lycée in Le Mans where his father was the professor of mathematics. From there he went to Paris where he studied at the Lycée St Louis, preparing for the entrance examinations to the Ecole Normale Supérieure and the Ecole Polytechnique. He entered the Ecole Normale Supérieure, then went to the Sorbonne to complete his License es Sciences. In the national examinations of 1926 Dubreil was placed first in the whole of France and was appointed a lecturer at the Ecole Normale Supérieure in 1927. There he worked for his doctorate. In 1929 he won the prestigious Rockefeller scholarship which enabled him to visit Hamburg to study with Artin. He returned to Paris briefly to defend his doctoral thesis in October 1930, returning to Hamburg. Emmy Noether visited Hamburg in 1931 and Dubreil found discussions with her extremely useful. He visited Göttingen to work with van der Waerden, then went to Frankfurt because by that time Emmy Noether was in Frankfurt and Dubreil wanted to continue exchanging ideas with her. Certainly Dubreil was going to get the most out of his Rockefeller scholarship and he next went to Rome where he discussed problems with the geometers Castelnuovo, Enriques and Severi. Dubreil's main interest at that time was algebraic varieties and he believed that he had learnt most from Emmy Noether so, before returning to France, his final visit was again to Göttingen to visit Emmy Noether. The exciting travels of his fellowship over, Dubreil took up his first permanent post at the University of Lille. Two years later, in 1933, he moved to Nancy where he was to spend the period of World War II. In 1946 Dubreil returned to the Sorbonne and there, in 1954, he was appointed to the chair of arithmetic and number theory. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dubreil.html (1 of 2) [2/16/2002 11:08:02 PM]

Dubreil

Dubreil began to work in more general algebraic structures around 1936 when he became interested in generalising the familiar elementary properties to groups into more general settings. He studied the lattice of equivalence relations on sets and from there was led to study semigroups. Thue had published on semigroups as early as 1914 when he had posed word problem for semigroups. However it was not until the late 1930s and early 1940s that the study of semigroups became a major topic. Malcev, Clifford and Dubreil became major figures in the new subject, Dubreil's first major work being Contribution à la théorie des demi-groups. Lallement who was a student of Dubreil's, describes his lectures in [1]:Each of his lectures was a brilliant piece of exposition, clear, precise, polished, and at the same time inspiring by his ability to relate the topics in hand to past and current research. I remember vividly the keen competition this course [Algebra and Number Theory] generated among students who were all eager to solve the most challenging weekly problems. When I became one of his doctoral students in 1961, I realised how deep Paul Dubreil's influence on young algebraists was. He was directing a large number of theses, and the weekly meetings of the Dubreil-Pisot seminar were rallying points of researchers, coming even from distant universities to listen to a very wide variety of guest lectures. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dubreil.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dubreil.html (2 of 2) [2/16/2002 11:08:02 PM]

Dudeney

Henry Ernest Dudeney Born: 10 April 1857 in Mayfield, Sussex, England Died: 24 April 1930 in Lewes, Sussex, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henry Dudeney learnt to play chess at a young age and became interested in chess problems. From the age of nine he was composing problems and puzzles which he published in a local paper. Although he only had a basic education, he had a particular interest in mathematics and studied mathematics and its history in his spare time. Dudeney worked as a clerk in the Civil Service from the age of 13 but continued to study mathematics and chess. He began to write articles for magazines and joined a group of authors which included Arthur Conan Doyle. He was doing well publishing mathematical puzzles under the pseudonym 'Sphinx'. In 1884 Dudeney married and his wife, who wrote full length novels which proved popular helped made the family very well off financially. Sam Loyd started sending his puzzles to England in 1893 and a correspondence started between him and Dudeney. The two were the main creators of mathematical puzzles and recreations of their day and it was natural that they should exchange ideas. Of the two Dudeney showed the more subtle mathematical skills. He sent a large number of his puzzles to Loyd and became very upset when Loyd began to publish them under his own name. In [4] Newing describes how one of Dudeney's daughters:... recalled her father raging and seething with anger to such an extent that she was very frightened and, thereafter, equated Sam Loyd with the devil. Dudeney contributed to the Strand Magazine for over 30 years and his very popular collection of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dudeney.html (1 of 2) [2/16/2002 11:08:03 PM]

Dudeney

mathematical puzzles Modern Puzzles was published in 1926. After Dudeney's death his wife helped edit a collection of his puzzles Puzzles and Curious Problems (1931) and later on she again helped edit a second collection A Puzzle Mine. References [1] and [2] show that Dudeney's puzzles are still of interest to many mathematicians. In [1] a generalisation of Problem 229 in H E Dudeney's 536 puzzles and curious problems is discussed. In [2] the following problem of Dudeney posed first in 1905 and appearing in Amusements in mathematics (1915) is discussed. Is it possible to seat n people at a round table on (n - 1)(n - 2)/2 occasions so that each person has the same pair of neighbours exactly once. A proof is given in [2] for n even, but the case of n odd still appears to be open. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) A Poster of Henry E Dudeney

Mathematicians born in the same country

Cross-references to History Topics

Mathematical games and recreations

Other Web sites

Thinks.com (Some puzzles)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dudeney.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dudeney.html (2 of 2) [2/16/2002 11:08:03 PM]

Duhamel

Jean Marie Constant Duhamel Born: 5 Feb 1797 in St Malo, France Died: 29 April 1872 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean-Marie Duhamel was a student at the Ecole Polytechnique and then he became professor there in 1830. He was highly thought of as a teacher of mathematics and is reported to have given very fine lectures. During the period 1848 until 1851 Duhamel was Director of Studies at the Ecole Polytechnique. From 1851, he again filled the analysis chair at the Ecole Polytechnique. Also from 1851 he was professor at the Faculté des Sciences in Paris. Duhamel worked on partial differential equations and applied his methods to the theory of heat, to rational mechanics and to acoustics. His acoustical studies involved vibrating strings and the vibration of air in cylindrical and conical pipes. His techniques in the theory of heat were mathematically similar to Fresnel's work in optics. His theory of the transmission of heat in crystal structures was based on work of Fourier and Poisson. 'Duhamel's principle' in partial differential equations arose from his work on the distribution of heat in a solid with a variable boundary temperature. Duhamel was elected to the Académie des Sciences in 1840. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Duhamel.html (1 of 2) [2/16/2002 11:08:05 PM]

Duhamel

List of References (4 books/articles) A Poster of Jean-Marie Duhamel

Mathematicians born in the same country

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Duhamel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Duhamel.html (2 of 2) [2/16/2002 11:08:05 PM]

Duhem

Pierre Maurice Marie Duhem Born: 10 June 1861 in Paris, France Died: 14 Sept 1916 in Cabrespine, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Duhem's father was Pierre-Joseph Duhem, a commercial traveller, and his mother was Alexandrine Fabre. It was a Flemish family. Pierre, the eldest of his parents four children, was born in the Rue des Jeûneurs in Paris after his parents moved there. When he was eleven years old he entered the Collège Stanislas where he proved a brilliant student. Leaving the Collège Stanislas with outstanding achievements in Latin, Greek, science, mathematics and other subjects, he had to choose between studying at the Ecole Polytechnique which, in principle, prepared one to be an engineer, and the Ecole Normale, the more academic of the two. Duhem's father wanted him to study science at the Ecole Polytechnique since he wanted his son to follow a technical career. Duhem's mother, on the other hand, wanted him to study Latin and Greek at the Ecole Normale, principally because she feared that a study of science would turn him away from the Roman Catholic beliefs that she had instilled in her children. Duhem was ranked first in the entrance examinations of both institutions but he chose to please neither of his parents by studying pure scientific at the Ecole Normale. He began his studies on 2 August 1882. When Hadamard arrived at the Ecole Normale Duhem was beginning his third year of study there. However the two quickly became firm friends. Hadamard wrote in [17]:In these long and precious conversations during which, from the moment of my entry to the Ecole, our friendship grew, how I felt him being thrilled by the genius of Hermite, or that of Poincaré, whose works he followed better than most of us could (I mean the most specialised in mathematics)! But in a general way, all the great mathematical ideas, all the one which were truly fruitful, were familiar to him. From this time I owe him revelations, insights (how http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Duhem.html (1 of 5) [2/16/2002 11:08:07 PM]

Duhem

broad, how disdainful of details to the profit of that which was really essential!) which for me, effortlessly and as if unconsciously, replaced long months of study. In 1884, while still at the Ecole Normale, Duhem published his first paper which was on electrochemical cells. Even before receiving his licence in mathematics, Duhem submitted his doctoral thesis in 1884. Suddenly his brilliant career shuddered to a halt. The thesis was on thermodynamic potential in physics and chemistry and in it he defined the criterion for chemical reactions in terms of free energy. In this he was replacing the incorrect criterion which Marcellin Berthelot had put forward twenty years earlier. Quite correctly Duhem criticised Berthelot's theory putting forward a correct alternative. Sadly being right is not always good enough and a scientist as influential as Berthelot was able to arrange for Duhem's thesis to be rejected. Duhem knew he was right and boldly published the rejected thesis in 1886. This certainly did not help his relations with Berthelot, as one might imagine, and the bad news for Duhem was that Berthelot became French Minister of Education in 1886. Duhem meanwhile worked on a second thesis, this time wisely choosing a mathematical topic which was less likely to be affected by the fate of his first thesis. His mathematical work on magnetism was accepted in 1888 but he suffered all his life because of Berthelot. Before his second thesis was submitted Duhem was already teaching at Lille. He worked there from the time he took up the appointment on 13 October 1887 until 1893. In Lille he lectured on hydrodynamics, elasticity, and accoustics, publishing these lectures in 1891. While in Lille he married Adèle Chayet in October 1890. She died two years later during the birth of their second daughter, who also died. This personal tragedy may have made it harder for him to get on with his superiors in Lille, something he always found hard despite having many good personal friendships. It was after a dispute with the Dean, M. Demartres, that Duhem requested a move from Lille and was appointed maître de conférence at Rennes in October 1893. Arriving in Rennes he found that it was not well equipped for his work and he at once requested another post. He became professor of theoretical physics at the University of Bordeaux on 13 October 1894 but a move to Paris, which a scientist of his outstanding ability would naturally expect, was blocked. Jaki writes in [4]:In spite of having grown aware of Duhem's scientific triumph over him, Berthelot could not bring himself to acknowledge this to the extent of letting him obtain a chair in Paris. At stake was the renown of the theoretical interpretation which Berthelot gave to his vast and most valuable experimental researches. It was all too human of Berthelot to protect that interpretation from Duhem's devastating criticism which, if delivered from a chair in Paris, would have forced Berthelot into the open. Herein lies the clue to the slighting which affected Duhem for thirty years, from his first doctoral dissertation to his very death, that is his whole academic career. Without a careful look at it a presentation of Duhem's life would not appear that poignant drama which it actually was. Hadamard had been in Bordeaux for a year when Duhem took up the chair there. He writes in [17]:For my part, our meeting again at the Faculté des Sciences of Bordeaux gave me the good fortune of supplementing my reading with invaluable and constant exchanges of views. It is to this reading, to these exchanges of views, that I owe the greater part of my later works, almost all of which deal with the calculus of variations, the theory of Hugoniot, hyperbolic partial differential equations, Huygens' principle. Duhem himself returned to almost all

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Duhem.html (2 of 5) [2/16/2002 11:08:07 PM]

Duhem

these questions in the continuation of his immense work, and most of the theories which he had so happily and so clearly explained, suggested to him sometimes some observations on details, sometimes some additions of fundamental importance. One would have to add that Duhem's thesis was not the only reason that he did not achieve the appointment in Paris. As well as the scientific dispute, Duhem was at odds with Berthelot on religious issues too. In addition as Miller writes in [1] he:... was of a contentious and acrimonious disposition, with a talent for making personal enemies over scientific matters. After becoming a corresponding member of the Académie des Sciences on 30 July 1900, in the following year he again requested a move from Bordeaux but again it was refused. Few scientists have contributed in works of leading importance, as Duhem did, to the philosophy of science, the historiography of science, and science itself. One has to see much of his writings, however, strongly influenced by his ultra-Catholic views which prevented him approaching subjects with an open mind. His mother had not wanted him to study science in case science would diminish his religious beliefs. Perhaps in the event the opposite happened and his religious beliefs played too large a part in formulating his scientific beliefs. These comments, however, should not be taken as in any way lessening the importance of the views that Duhem put forward in all three areas of his interests. His interests in science itself were mainly in the area of mathematical physics, and in particular thermodynamics, hydrodynamics, elasticity, mathematical chemistry, and mechanics. He viewed mechanics as a special case of a more general theory of space and he considered that a generalised version of thermodynamics would provide a theory to explain all of physics and chemistry. The paper [27] considers Duhem's:... unification of theoretical physics: the thermodynamical potentials, the Lagrangian analytical formalism and Duhem's philosophical conception of theoretical physics. The mathematical aspects of this unification and Duhem's priority in the axiomatization of thermodynamics are emphasized. In many way Duhem can be seen as very modern in his approach. He would begin by setting up axioms which the physical system that he was studying satisfied. He then studied in depth the consequences of the initial axioms deducing properties of the physical system from mathematical theorems developed from the axioms alone. He was opposed, however, to studying mathematical problems which did not arise from physical situations. His contributions to thermodynamics are of major importance and he also studied magnetism following the work of Gibbs and Helmholtz. Some of his most important papers on these topics are Etude sur les travaux thermodynamiques de J Willard Gibbs (1887) and Commentaire aux principes de la theormodynamique (1892). Three major treatises are Thermodynamique et chimie (1902), the two volume work Recherches sur l'hydrodynamique (1903-4), and Recherches sur l'élasticité (1906). As well as his major contributions to science he was led to write articles of major importance on the philosophy of science. His approach was very much in line with the way that he went about his scientific studies as we have outlined above. He wrote:A physical theory ... is a system of mathematical propositions, deduced from a small number of principles, which has the object of representing a set of experimental laws as simply, as

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Duhem.html (3 of 5) [2/16/2002 11:08:07 PM]

Duhem

completely, and as exactly as possible. As one might imagine, Duhem disagreed with Poincaré on many aspects of the philosophy of science and the two engaged in a vigorous debate. One of his most important works on philosophy of science was La Théorie physique, son objet et sa structure (1906). If scientific work itself led Duhem towards the philosophy of science, then in turn the philosophy of science led him towards the history of science. His paper L'évolution de la méchanique in 1902 is really an article on the philosophy of science but it is based heavily on using historical examples. Earlier important work on the history of science was Les théories de la chaleur published in 1895. His most important work on the history of science was, however, research which showed that the period from 1200 onwards was not a period when science had been ignored. Of course he was very keen to show that this was the case since the Catholic Church had been blamed by many for preventing scientific work during this period. While working on Les origines de la statique in the late autumn of 1903, Duhem came across the scientist Jordanus Nemorarius who worked before Leonardo da Vinci. Until that time Duhem had accepted the commonly held view that there had been no scientific work in the Middle Ages. It was this surprise which led Duhem to look for other scientists who worked before the development of Renaissance mechanics. His most famous works in this area include Etudes sur Léonard de Vinci (1906-13). In 1913 he began publication of Le Système du monde, Histoire des doctrines cosmologiques, de Platon à Copernic (1913-17) but only 5 of the intended 10 volumes were written before his death. Duhem saw different national characteristics lead to different approaches to science. He disliked British science, in particular the work of Maxwell, and he described it as broad and shallow while he said that French science was narrow and deep. German sciences he claimed were highly geometrical, which for Duhem was a criticism for he considered an approach using an analytical style of mathematics to be far superior to a geometrical one. Hadamard writes (see for example [7]):At the beginning of the first World War one of our greatest scientists and historians of sciences, the physicist Duhem, was misled by [nationalistic passions]. ... In a rather detailed article, he depicts German scientists, especially mathematicians, as lacking intuition or even deliberately setting it aside ... Late in his career Duhem was offered a professorship in Paris as a historian of science and not as a mathematical physicist. Duhem refused the chance to work in Paris that he had always longed for saying that he was a mathematical physicist and did not want to get to Paris through the back door. He died while on a walking holiday in Cabrespine. Some reports say it was a heart attack, others that he died of a chest infection. Article by: J J O'Connor and E F Robertson List of References (31 books/articles)

A Quotation

Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Duhem.html (4 of 5) [2/16/2002 11:08:07 PM]

Duhem

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Duhem.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Duhem.html (5 of 5) [2/16/2002 11:08:07 PM]

Dupin

Pierre Charles François Dupin Born: 6 Oct 1784 in Varzy, France Died: 18 Jan 1873 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Dupin's father was Charles-André-Dupin and his mother was Cathérine Agnès Dupin. In fact Dupin was both her married and maiden name. Charles' father was a lawyer and Charles himself was brought up in Nivernais, the region of his birth. Nivernais was not part of the French crown when he was born there, being owned from 1659 by Cardinal Mazarin and his descendants until 1790 when it became the département of Nièvre. Charles was the middle of the three sons of his parents; his older brother André also achieved fame in his profession as a lawyer. Dupin was educated at the Ecole Polytechnique in Paris, where he learnt geometry from Monge. While an undergraduate he made his famous discovery of what are called today 'Dupin's cyclides' guided in this work by Monge. He graduated in 1803 and then became a naval engineer. He often went on long sea voyages which resulted in his publications being much delayed. After being assigned to duties in Antwerp, Genoa and then Toulon, he was sent to Corfu in 1807 to take charge of the damaged naval arsenal there. While in Corfu he carried out his naval engineer's duties of repairing the port, but he also carried out tasks relating to his scientific interests. He was appointed as secretary to the Ionian Academy which had been founded only a short time before and he undertook deep research on mathematical topic, in particular studying the differential geometry of surfaces, and applied mechanics where he investigated the resistance of materials. After three years in Corfu he set out to return to France but, while passing through Pisa, he was taken ill. It took him a while to recover from the illness, but Dupin was not idle while recovering in Pisa. Rather he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dupin.html (1 of 3) [2/16/2002 11:08:09 PM]

Dupin

worked on this occasion on preparing for publication the writings of a friend who had died. In 1813 he was in Toulon, and there he set up a maritime museum which was highly influential in the way that maritime museums were organised. Also in 1813 Dupin published his Développments de géométrie which [1]:... contains many contributions to differential geometry, notably the introduction of conjugate and asymptotic lines on a surface ... Other contributions to differential geometry which occur in this work include his invention of the 'Dupin indicatrix' which gives an indication of the local behaviour of a surface up to the terms of degree two. The year 1813 saw Dupin elected to the Institut de France, the new organisation set up to replace the Académie des Sciences after the French Revolution. With the reestablishment of the Académie des Sciences, Dupin was elected to that body in 1818. Dupin was appointed professor at Conservatoire des Arts et Métiers in Paris in 1819. He held this post until 1854 and he gave many public lectures on the applications of mathematics and mechanics to industry. These lectures proved extremely popular, mainly since Dupin was an exceptional lecturer. But he not only had an academic life, publishing further important works on the applications of differential geometry to industry and the arts, but he also took a major part in politics from 1828. In that year he was elected as a deputy for Tarn, a département in southern France. In 1830, at the age of 45, Dupin married Rosalie Anne Joubert. His political career continued and his expertise as a naval engineer clearly stood him in good stead when he was appointed as minister of maritime affairs in 1834. Four years later he became a peer and in 1852 he was appointed to the Senate. Struik writes in [1] that Dupin:... tirelessly encouraged the establishment of schools and libraries, the founding of savings banks, the construction of roads and canals, and the use of steam power. As well as the honour of election to the Institut de France and the Académie des Sciences which we referred to above, he was also honoured with election to the Académie des Sciences Morales et Politique in 1832. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of Charles Dupin

Mathematicians born in the same country

Honours awarded to Charles Dupin (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Dupin (6th Arrondissement)

Other Web sites

The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dupin.html (2 of 3) [2/16/2002 11:08:09 PM]

Dupin

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dupin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dupin.html (3 of 3) [2/16/2002 11:08:09 PM]

Dupre

Athanase Louis Victoire Dupré Born: 28 Dec 1808 in Cerisiers, France Died: 10 Aug 1869 in Rennes, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Athanase Dupré was educated at the Collège in Auxerre. He entered the Ecole Normale Supérieure in Paris in 1826. He was placed first in his class when he graduated in 1829. After taking his degree, Dupré was appointed to the Collège Royal in Rennes. He taught mathematics and physics there until 1847. In 1847 Dupré was appointed to the chair of mathematics in the Faculty of Science at Rennes. In 1866 he was appointed dean of the Faculty. Dupré received relatively little in the way of honours for his mathematical work in view of its quality. For example he was never elected to the Academy of Sciences. However he did receive the Legion of Honour in 1863. For the first part of his career, from 1826 until 1859, Dupré contributed to a number of areas in mathematics and physics. He won an honourable mention for the 1858 Grand Prix of the Academy of Sciences with a paper on Legendre's theory of numbers. He was awarded one half of the 3000 franc prize. It looks as if Dupré was somewhat unhappy with his treatment. He certainly changed research topic following the prize announcement. For the last ten years of his career he studied the mechanical theory of heat. He published 40 papers on this topic in the Academy of Sciences and made the concepts of thermodynamics well known in France. Dupré wrote a successful advanced textbook Théorie méchanique de la chaleur (1869). He entered for the Academy of Science's Bordin prize in 1866 with the same half success as his entry on number theory. Again he was awarded one half of the prize with an honourable mention. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dupre.html (1 of 2) [2/16/2002 11:08:10 PM]

Dupre

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dupre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dupre.html (2 of 2) [2/16/2002 11:08:10 PM]

Durer

Albrecht Dürer Born: 21 May 1471 in Imperial Free City of Nürnberg (now in Germany) Died: 6 April 1528 in Imperial Free City of Nürnberg (now in Germany)

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Albrecht Dürer was the third son of Albrecht Dürer and Barbara Holfer. He was one of their eighteen children. The Dürer family came from Hungary, Albrecht Dürer senior being born there, and at this time the family name was Ajtos. The name Ajtos means "door" in Hungarian and when Dürer senior and his brothers came to Germany they chose the name Türer which sounds like the German "Tür" meaning door. The name changed to Dürer but Albrecht Dürer senior always signed himself Türer rather than Dürer. Here are portraits of his father and mother. Albrecht Dürer senior was a jeweller who had served his apprenticeship with Hieronymus Holfer, and then married Holfer's daughter. Albrecht Dürer junior wrote about his father and his upbringing (see for example [3]):My father suffered much and toiled painfully all his life, for he had no resources other than the proceeds of his trade from which to support himself and his wife and family. He led an honest, God-fearing life. His character was gentle and patient. He was friendly towards all and full of gratitude to his Maker. He cared little for society and nothing for worldly amusements. A man of very few words and deeply pious, he paid great attention to the religious education of his children. His most earnest hope was that the high principles he instilled into their minds would render them ever more worthy of divine protection and the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Durer.html (1 of 6) [2/16/2002 11:08:12 PM]

Durer

sympathy of mankind. He told us every day that we must love God and be honourable in our dealings with our neighbours. As a young boy Dürer was educated at the Lateinschule in St Lorenz and he also worked in his father's workshop learning the trade of a goldsmith and jeweller. By the age of 13 he was already a skilled painter as seen from a self portrait which he painted at that time. This was the first of many self-portraits which Dürer painted and they provide a wonderful record. Here is our collection of such self-portraits. In 1486 Dürer became an apprentice painter and woodcut designer to Michael Wolgemut, the leading producer of altarpieces. After an apprenticeship of four years, Dürer had learnt all he could from Wolgemut and had reached a level of artistic quality exceeding that of his famous teacher. Wolgemut advised Dürer to travel to widen his experience and meet other artists. Following Wolgemut's advice, Dürer delayed visiting Italy (which Wolgemut himself never visited), where there were very different artistic styles, until he had fully developed his own style and learnt more techniques from other German artists. Here is a portrait of Wolgemut. Dürer travelled first to Nördlingen, where he met artists of the Swabian school. The Swabian style had been influenced by Dutch artistic design which Dürer had not met before. His next visit was to Ulm where he met more artists of the Swabian school. Dürer [3]:... participated with keen enjoyment in the discussions among artists of his own age, in the low-ceilinged taverns, over foaming mugs of beer. These youthful enthusiasts, in common with those of all nations throughout history, were bent on rejuvenation of the art of the world. They were delighted with Dürer's drawings, with his first engravings and the small pictures he had already painted, independently of Wolgemut's directions or opinions. Leaving Ulm, Dürer made his way to Constance which charmed him with its fairyland appearance. Basel was the next town which Dürer visited, and he found it quite similar to his home town of Nürnberg. Finally Dürer returned home, making visits to Colmar and Strasbourg on the way. It had been a long journey of great importance to Dürer which had taken nearly four years, but after he returned to Nürnberg in 1494 he felt disappointed that he had not visited Italy. He had also become convinced that [1]:... the new art must be based upon science - in particular, upon mathematics, as the most exact, logical, and graphically constructive of the sciences. Italy was not only a country with new ideas to offer Dürer in art, but it was also leading the world at this time in the revival of mathematics. Before setting out for Italy, however, Dürer married Agnes Frey, the daughter of a learned man Hans Frey who had made quite a lot of money through making jewellery, musical instruments, and mechanical devices. Here are portraits of Agnes. The marriage seems to have been more the idea of the parents of Agnes and Albrecht, and the pair were married on 7 July 1494. It was a marriage which helped raise Dürer's status in Nürnberg, as well as provide him with money which helped him set up his own studio. Before the end of 1494, Dürer was on his travels again, leaving Agnes behind in Nürnberg. First he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Durer.html (2 of 6) [2/16/2002 11:08:12 PM]

Durer

visited Augsburg where he met strong Italian artistic influences for the first time. Travelling through the Tyrol, he reached Trento and his first view of Italy. Here is one of his paintings of Trento. He travelled on to Verona before reaching Venice which was his main objective. In Venice, Dürer, as he had done throughout his journeys, sketched scenes, visited galleries and churches, and met with the local artists. One of the artists that he met in Venice, Giovanni Bellini, had an important influence on Dürer for [3]:... everything that [Venice] could teach him was to be found in Giovanni's paintings. He cultivated the artist's society, therefore, with a devotion both impassioned and deferential, retaining throughout his life, with his whole heart and soul, unbounded feelings of gratitude to the man whose pictures had unveiled so wonderful a world to him. Dürer returned to Nürnberg in 1495, and although he does not seem to have met with any of the major Italian mathematicians on his journeys, he did meet Jacopo de Barbari who told him of the mathematical work of Pacioli and its importance to the theory of beauty and art. Nor did Dürer meet with Leonardo da Vinci while in Italy, but he learnt of the importance which that artist placed in mathematics. Back in Nürnberg, Dürer began a serious study of mathematics. He read Euclid's Elements and the important treatise De architectura (On Architecture) by Vitruvius (1st century BC), the famous Roman architect and engineer. He also became familiar with the work of Alberti and Pacioli on mathematics and art, in particular work on proportion. It was not only this scientific approach to art that influenced Dürer as he began his artistic career in Nürnberg, but he also benefited from seeing different artistic styles and the different scenery which he had viewed [3]:The variety of regions through which Dürer had passed in the course of his travels and the care he had taken with the drawings and water-colours he had made of the most attractive or unfamiliar of them had provided him with a great range of pictorial motives emanating from the most diverse sources. In 1495 Dürer was still not well known as an artist in the highest circles but news of his skill reached Frederick the Wise, Elector of Saxony, and Dürer was commissioned to paint his portrait. Frederick liked his portrait which Dürer painted in April 1496 when Frederick had visited Nürnberg. Despite Frederick's attempts to persuade Dürer to move to Weimar and become Court painter, the artist did not wish to leave Nürnberg. He was deeply attached to Nürnberg, painting these views of the city in 1497. From about 1500 Dürer's art showed the influence of the mathematical theory of proportion which he continued to spend so much time studying. It is claimed that his self-portrait in a wig made in 1500 has the dimensions of the head constructed proportionally. For the engraving Adam and Eve made in 1504, Dürer described the intricate ruler and compass constructions which he made to construct the figures. It was not only the mathematical theory of proportion which influenced Dürer's art at this period, but also his mastery of perspective through his study of geometry. This is most clearly seen in his woodcuts Life of the Virgin made between 1502 and 1505. During the ten years after 1496 Dürer went from a relatively unknown artist to someone with a wide reputation as both an artist and a mathematician. His personal circumstances had changed greatly. His http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Durer.html (3 of 6) [2/16/2002 11:08:12 PM]

Durer

father had died in 1502 and Dürer was left to care for his invalid, and nearly blind, mother. He had set up his own printing press while he, or often his wife, sold his works to buyers at local fairs. It was a difficult life and one in which Dürer's health began to suffer. In fact he would never regain full health during the rest of his life. From 1505 to 1507 Dürer made a second visit to Italy, spending much time again in Venice. It was a very different visit from his first, with Dürer now more interested in his international fame than in learning about art. He was so conscious of his fame, and the threat he perceived that he might hold to the local artists, that [3]:... he refused invitations to dinner in case someone should try to poison him. It was not about art that Dürer now wished to learn from the Italians, but rather about mathematics. He visited Bologna to meet with Pacioli whom he considered held the mathematical secrets of art. He also visited Jacopo de Barbari and the great efforts which Dürer made to meet de Barbari shows the importance which Dürer more and more attached to mathematical knowledge. Dürer returned to Nürnberg from this second visit to Italy feeling that he must delve yet more deeply into the study of mathematics. In about 1508 Dürer began to collect material for a major work on mathematics and its applications to the arts. This work would never be finished but Dürer did use parts of the material in later published work. He continued to produce art of outstanding quality, and he produced one of his most famous engravings Melancholia in 1514. It contains the first magic square to be seen in Europe, cleverly including the date 1514 as two entries in the middle of the bottom row. Also of mathematical interest in Melancholia is the polyhedron in the picture. The faces of the polyhedron appear to consist of two equilateral triangles and six somewhat irregular pentagons. An interesting reconstruction of the polyhedron is given in [19], see also [18] for further details. Dürer worked for Maximilian I, the Holy Roman emperor, from about 1512. Maximilian, however, had little in the way of wealth to pay for Dürer's work and he asked the councillors of Nürnberg to exempt Dürer from taxes as compensation. He then asked the councillors to pay Dürer a pension on his behalf, which certainly did not please them. From about 1515 the councillors tried to avoid paying this pension. Dürer met Maximilian personally for the first time in 1518 and, probably from one sitting in Augsburg, painted Maximilian's portrait. The following year Maximilian died and this was the final excuse for the councillors to refuse to make any further payment, saying that the new emperor Charles would have to agree to the pension. Although Dürer was fairly well off by this time and the pension was not necessary for him, it was more a matter of prestige to have his pension restored. He set off for Antwerp on 15 July 1520 with his wife and their maid to visit the Emperor Charles V. Passing through Aachen, Dürer sketched the cathedral at Aachen. Dürer had a second reason for this visit to the Netherlands, for he believed that Maximilian's daughter had a book by Jacopo de Barbari on applications of mathematics to art, and Dürer had long sought the truths which he believed this work contained. On meeting Maximilian's daughter he offered her the portrait of her father which he had painted, but was distressed to find that she did not want the portrait. She had already given the book by Jacopo de Barbari to another artist so Dürer's quest was in vain. He did persuade Charles V to restore his pension, however, which was formally agreed on 12 November

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Durer.html (4 of 6) [2/16/2002 11:08:12 PM]

Durer

1520. After returning to Nürnberg, Dürer's health became still worse. He did not slacken his work on either mathematics or painting but most of his effort went into his work Treatise on proportion. Although it was completed in 1523, Dürer realised that it required mathematical knowledge which went well beyond what any reader could be expected to have, so he decided to write a more elementary text. He published this more elementary treatise, in four books, in 1525 publishing the work through his own publishing company. This treatise, Unterweisung der Messung mit dem Zirkel und Richtscheit, is the first mathematics book published in German (if one discounts an earlier commercial arithmetic book) and places Dürer as one of the most important of the Renaissance mathematicians. Dürer's sources for this work are discussed in [21] where three main sources are suggested (i) the practical recipes of craftsmen, (ii) classical mathematics from printed works and manuscripts, and (iii) the manuals of Italian artists. The article [16] gives many details of the mathematics contained in the treatise. The first of the four books describes the construction of a large number of curves, including the Spiral of Archimedes, the Equiangular or Logarithmic Spiral, the Conchoid, Dürer's Shell Curves, the Epicycloid, the Epitrochoid, the Hypocycloid, the Hypotrochoid, and the Limaçon of Pascal (although of course Dürer did not use that name!). Details about Dürer's descriptions of the curves, in particular one he calls a "muschellini", is given in [9]. In the second book he gave exact and approximate methods to construct regular polygons. Dürer's constructions of regular polygons with 5, 7, 9, 11 and 13 sides is discussed in [12]. Dürer also gave approximate methods to square the circle using ruler and compass constructions in this book. A method to obtain a good approximation to the trisector of an angle by Euclidean construction is also given. Book three considers pyramids, cylinders and other solid bodies. The second part of this book studies sundials and other astronomical instruments. The final book studies the five Platonic solids as well as the semi-regular Archimedean solids. Also in this book is Dürer's theory of shadows and an introduction to the theory of perspective. In 1527 Dürer published another work, this time on fortifications. There were strong reasons why he produced a work on fortifications at this time, for the people of Germany were in fear of an invasion by the Turks. Many cities, including Nürnberg, would improve their fortifications using the methods set out by Dürer in this book. Dürer's final masterpiece was his Treatise on proportion which was at the proof stage at the time of his death. Descriptive geometry originated with Dürer in this work although it was only put on a sound mathematical basis in later work of Monge. One of the methods of overcoming the problems of projection, and describing the movement of bodies in space, is descriptive geometry. Dürer's remarkable achievement was through applying mathematics to art, he developed such fundamentally new and important ideas within mathematics itself. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Durer.html (5 of 6) [2/16/2002 11:08:12 PM]

Durer

List of References (23 books/articles)

Some Quotations (3)

A Poster of Albrecht Dürer

Mathematicians born in the same country

Cross-references to History Topics

Mathematical games and recreations

Other references in MacTutor

1. 2. 3. 4.

Melancholia Dürer's magic square Dürer's work on facial proportions Chronology: 1500 to 1600

Honours awarded to Albrecht Dürer (Click a link below for the full list of mathematicians honoured in this way) Planetary features

Crater Dürer on Mercury

Other Web sites

1. The Galileo Project 2. The Catholic Encyclopedia 3. Mark Harden's Artchive (More self portraits and other works) 4. George W Hart (Dürer's polyhedra) 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Durer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Durer.html (6 of 6) [2/16/2002 11:08:12 PM]

Dynkin

Eugene Borisovich Dynkin Born: 11 May 1924

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Evgenii Dynkin was born into a family of Jewish origins at a time when Russia was suffering extreme unrest and repression. He lived with his family in Leningrad until 1935 when they were exiled to Kasakhstan and his father was designated one of the 'people's enemies'. Although he was totally innocent, his father disappeared in the Gulag two years later and became one of the millions to perish under Stalin. Things looked particularly bleak for Dynkin at this stage. Being of Jewish origin and the son of a 'people's enemy' should have prevented Dynkin from succeeding in the system. Yet, as Dynkin recalls in [3]:It was almost a miracle that I was admitted (at the age of sixteen) to Moscow University. Every step in my professional career was difficult because the fate of my father, in combination with my Jewish origin, made me permanently undesirable for the party authorities at the university. Only special efforts by A N Kolmogorov, who put, more than once, his influence at stake, made it possible for me to progress through the graduate school to a teaching position at Moscow University. Admitted to Moscow University in 1940, he was saved from military service through poor eyesight and he was able to continue his studies throughout World War II, graduating with an M.S. from the Mechanics and Mathematics Faculty in 1945. His work at this time was partly in algebra and partly in probability. He attended the seminars of Gelfand on Lie groups and of Kolmogorov on Markov chains. At this time he discovered the 'Dynkin diagram' approach to the classification of the semisimple Lie algebras. This work came out of Dynkin trying to understand the papers by Weyl and by van der Waerden on semisimple Lie groups. Dynkin was not the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dynkin.html (1 of 3) [2/16/2002 11:08:13 PM]

Dynkin

only person to introduce graph of this type. Coxeter had independently introduced them in his work on crystallographic groups. After graduating, Dynkin remained at Moscow University where he became a research student of Kolmogorov. For ten years he worked both on the theory of Lie algebras and on probability theory although his main work during this period was in algebra. In 1945 he solved a problem on Markov chains suggested by Kolmogorov and his first publication in probability resulted. In 1948 Dynkin was awarded his Ph.D. and he became an assistant professor of Kolmogorov's who held the Probability Chair. Dynkin became Doctor of Physics and Mathematics in 1951 and Kolmogorov pressed for Dynkin to be awarded a chair. However there was no way that the Communist Party leaders of Moscow University would allow a person of Dynkin's background to hold a chair at this time. In 1953 Stalin died and the situation in Russia eased. The following year, with Kolmogorov's strong support, Dynkin was appointed to a chair at the University of Moscow and he held this chair until 1968. From the time he was appointed to the chair, Dynkin's work became more and more devoted to probability theory. His work from this period is contained in two major books Foundations of the Theory of Markov Processes (1959) and Markov Processes (1963) which have become classics of probability theory. This work on Markov processes is described in [4] and is introduced as follows:Following Kolmogorov, Feller, Doob , and Ito, Dynkin opened a new chapter in the theory of Markov processes. He created the fundamental concept of a Markov process as a family of measures corresponding to various initial times and states and he defined time homogeneous processes in terms of the shift operators ... . Dynkin's work at Moscow University ended in 1968 as described in [2]:In 1968 Dynkin's work at Moscow University was compulsorily interrupted and from 1968 to 1976 he was a senior scientific worker at the Central Economics and Mathematics Institute at the USSR Academy of Sciences. During his short spell of work there he organized a group of young workers together with whom he obtained important results in the theory of economic growth and economic equilibrium that culminated in the first Soviet report on this topic at the International Mathematics Congress in Vancouver (to which, incidentally, in the usual way, he was not allowed to go). In 1976 Dynkin emigrated to the United States but, as explained in [4], this was a brave move:At the end of 1976, Dynkin left the USSR. The decision to leave was very hard: pupils, friends, and youth were left behind. To apply for emigration was a great risk, especially for an outstanding scientist: many such applicants have been denied exit visas, they have lost their jobs and lived for years as outcasts of Soviet society. Dynkin took the risk because life in the USSR had became more and more unbearable, and the Dynkins' only daughter had already left for Israel. In 1977 Dynkin was appointed to Cornell University in Ithaca, New York. His work there gained a new lease of life as described in [3]:Around 1980 Dynkin interpreted and vastly generalized an identity which had first come up in the context of quantum field theory. In his hands it became a remarkable relation between occupation times of a Markov process and a related Gaussian random field. This identity

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dynkin.html (2 of 3) [2/16/2002 11:08:13 PM]

Dynkin

has led to many deep studies, by Dynkin himself as well as a host of others ... In the last few years Dynkin has obtained exciting results in the theory of "superprocesses" ... a class of measure-valued Markov processes [which] can be used to give probabilistic solutions to certain nonlinear PDE's in a way which is analogous to the classical solution of the Dirichlet problem by means of Brownian motion. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Dynkin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Dynkin.html (3 of 3) [2/16/2002 11:08:13 PM]

Eckert_John

John Presper Eckert Born: 9 April 1919 in Philadelphia, Pennsylvania , USA Died: 3 June 1995 in Bryn Mawr, Pennsylvania, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

J Presper Eckert Jr. attended the William Penn Carter School in Germanstown. In 1937, after graduating from school, he entered the Moore School of Electrical Engineering at the University of Pennsylvania from where he graduated in 1941. Eckert, an outstanding electrical engineering student, was given a post as an instructor at the Moore School soon after his graduation. The Moore School was by this time heavily involved with research specifically directed towards the war effort. Eckert taught a defence course at the Moore School and one of his students on the course was John Mauchly. It might seem strange that Mauchly, who was twelve years older than Eckert, should be his student. Mauchly was already an established academic teaching physics but he became involved in defence training as part of his contribution to the war effort. Eckert quickly became interested in Mauchly's ideas for the development of computers and for a while the two discussed these ideas frequently. Soon, however, Eckert moved on to undertake other military work at the School. Ashurst [2] relates how Eckert was:.... eventually involved with work on ultraviolet light and the development of the means to measure metal fatigue. Later, he went on to develop a method for measuring small magnetic fields to be used in detecting marine mines. He then went on to work on the electronics of radar and target locating and following equipment; these devices played a decisive part in weaponry, and their development and construction was considered to be of the very highest priority. After Mauchly's report on the construction of a computer was accepted, Eckert collaborated with him in the construction of the Electronic Integrator and Computer (ENIAC). At this time, May 1943, Eckert had http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eckert_John.html (1 of 3) [2/16/2002 11:08:15 PM]

Eckert_John

almost completed the work for his masters degree and he was appointed as chief engineer on the project with the specific task of designing the electronic circuits. One of the major problems which had to be solved was how a machine with 18000 valves could function when the valves were relatively unreliable. This was one of the many problems which Eckert had to solve and he eventually achieved a lifetime of 2500 hours for each individual valve which made the operation of the computer viable. The ENIAC computer was intended to be a general purpose one, but it was also designed for a very specific task, namely compiling tables for the trajectories of bombs and shells. ENIAC is described in [1]:Completed by February 1946, the ENIAC was the first general-purpose electronic digital computer. It contained roughly 18000 vacuum tubes and measured about 2.5 metres in height and 24 metres in length. The machine was more than 1000 times faster than its electromechanical predecessors and could execute up to 5000 additions per second. Its operation was controlled by a program that was set up externally by wires on plugboards. The ENIAC was the most complex and influential electronic computer of its time ... Electronic computers are today machines based on binary arithmetic but this was not so for the ENIAC computer. Eckert designed electronic calculators which worked to base 10 for the ENIAC, reducing the number of components over a binary machine, but of course at the price of greater complexity. Of course by 1946 when the ENIAC was completed World War II was over but the computer was used intensively, particularly on top secret problems associated with the development of nuclear weapons. Von Neumann was working on this project and became involved with the ENIAC computer and used it to solve systems of partial differential equations which were crucial in the work on atomic weapons at Los Almos. Eckert left the Moore School at the University of Pennsylvania in October 1946, as did Mauchly. They started up the Electronic Control Company which they received an order from Northrop Aircraft Company to build the Binary Automatic Computer (BINAC). One of the major advances of this machine, which was used from August 1950, was that data was stored on magnetic tape rather than on punched cards. The Electronic Control Company become the Eckert-Mauchly Computer Corporation and it received an order from the National Bureau of Standards to build the Universal Automatic Computer (UNIVAC). This was the first computer to be produced commercially in the United States with 46 UNIVACs being built. One of the major advances which the UNIVAC introduced was an ability to handle both numerical and alphabetical information with equal success. Eckert and Mauchly were better at computer design than they were at the economics of running a company. The problem really lay in the fact that this was such a new area that costs of production were extremely hard to estimate. As a consequence the Eckert-Mauchly Computer Corporation soon hit financial difficulties. In 1950 the Remington Rand Corporation acquired the Eckert-Mauchly Computer Corporation and changed its name to the Univac Division of Remington Rand. Eckert remained with Remington Rand and became an executive of the corporation. He continued with the company as it merged with the Burroughs Corporation to become Unisys. In 1989 Eckert retired from Unisys but continued to act as a consultant for the company. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eckert_John.html (2 of 3) [2/16/2002 11:08:15 PM]

Eckert_John

Between 1948 and 1966 Eckert took out patents on 85 inventions, almost all electronic in nature. He received many awards for his pioneering work on computers. In 1966 he received the Harry M Goode Memorial Award, a medal and $2,000 awarded by the Computer Society:For his pioneering contributions to automatic computing by participating in the design and construction of the ENIAC, the world's first all-electronic computer, and of the BINAC and the UNIVAC, and for his continuing work in the design of electronic computing systems. Eckert received many awards for his pioneering work on computers. He was elected to the National Academy of Engineering in 1967 but perhaps the most prestigious honour was being awarded the US National Medal of Science in 1969. Both Eckert and Mauchly received the IEEE Computer Society Pioneer Award in 1980. Eckert died from complications relating to leukaemia. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other references in MacTutor

1. A picture of ENIAC 2. Chronology: 1940 to 1950

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eckert_John.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eckert_John.html (3 of 3) [2/16/2002 11:08:15 PM]

Eckert_Wallace

Wallace John Eckert Born: 19 June 1902 in Pittsburgh, Pennsylvania, USA Died: 24 Aug 1971 in Englewood, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wallace J Eckert earned his PhD was from Yale in 1931 in astronomy. At that time Ernest Brown was a member of the astronomy department and Brown's work on the Moon was an important ingredient of Eckert's later work. Eckert had joined the Faculty at Columbia University in 1926 and later he became professor there. Eckert was an early user of IBM punch card equipment to reduce astronomical data and solve numerically planetary orbits. In 1937 Columbia University and IBM established the Thomas J Watson Astronomical Computing Bureau as a result of the collaboration with Eckert. In fact the work which led to this development was published by Eckert in Punched card methods in scientific computation (1940). In 1940 Eckert became director of the US Nautical Almanac Office and produced work vital to navigation during World War II. In this post he introduced machine methods to compute and print tables and he began publication of the Air Almanac in 1940. In 1945 Eckert became director of the Watson Scientific Computing Laboratory at Columbia University. As stated in [3]:During the more than 20 years he was in charge of the laboratory, it was a major training center for scientific computation, where more than 1,000 astronomers, physicists, crystallographers, statisticians, and other scientists studied. Eckert directed the construction of a number of innovative computers. In 1949 the Selective Sequence Electronic Calculator (SSEC) was built. Later the Naval Ordnance Research Calculator (NORC) was built. Completed in 1954 it was for many years the most powerful computer in the world. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eckert_Wallace.html (1 of 3) [2/16/2002 11:08:17 PM]

Eckert_Wallace

Eckert applied computers, in particular the SSEC and NORC, to compute precise planetary positions and contribute to the theory of the orbit of the Moon. In particular he used the SSEC to compute the positions of Jupiter, Saturn, Uranus, Neptune and Pluto, publishing the results in 1951 in Coordinates of the five outer planets. The NORC was used by Eckert to work on the problem of the position of the Moon. Writing in 1954 Eckert explained the how Brown had calculated the Moon's position:Since 1923 the work of E W Brown has constituted the basis for the published ephemerides of the moon. His monumental calculation, which occupied most of his lifetime, consists of two distinct steps. The first is the development of the theory or the solution of the differential equations of motion expressing the coordinates of the moon as explicit functions of time. Secondly, in order to reduce the necessary labor involved in computing the coordinates of the moon for any given date from these formulae, Brown computed from his theory a set of Tables which, including the necessary explanations, comprise over 650 large quarto pages. ... In order to bring the Tables within even their present length, various parts of the basic equations were curtailed whenever permissible in the light of observational requirements (as then visualised). However by the 1950s it was realised that the Tables were not accurate enough. Eckert therefore decided not to recompute new tables but to compute the ephemerides directly from Brown's equations. The task was immense for, see [3]:... Brown's formulae involved some 1,650 trigonometric terms, many of them with variable coefficients. The accuracy of Eckert's calculations of the Moon's orbit was so good that in 1965 he was able to correctly show that there was a concentration of mass near the lunar surface. In 1967 he produced theoretical work which improved on Brown's theory of the Moon. Eckert's work is summed up in [3]:Eckert retired in 1967 from IBM and in 1970 from his professorship at Columbia, greatly honored by his fellow astronomers but because of his modest nature little known to the public. Hardly any other astronomer of his generation influenced our science more profoundly. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Wallace J Eckert (Click a link below for the full list of mathematicians honoured in this way) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eckert_Wallace.html (2 of 3) [2/16/2002 11:08:17 PM]

Eckert_Wallace

Lunar features

Crater Eckert

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eckert_Wallace.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eckert_Wallace.html (3 of 3) [2/16/2002 11:08:17 PM]

Eckmann

Beno Eckmann Born: 31 March 1917 in Bern, Switzerland

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Beno Eckmann was brought up in Bern where he attended the Gymnasium obtaining his Diploma in 1935. He then entered the Eidgenössische Technische Hochschule Zürich where he was taught mathematics by a number of outstanding mathematicians. Eckmann fully appreciated the importance of this aspect in his mathematical development writing [1]:I feel that the best thing that can happen to a University mathematician is to have good teacher and/or good students; I was fortunate enough to have both. With regard to good teachers it certainly suffices if I mention names: Heinz Hopf, Plancherel, Pólya, Bernays. As a student I attended their courses and seminars and was introduced in an unusually personal way to the world of mathematics. Eckmann graduated with his Master's Degree from the Eidgenössische Technische Hochschule Zürich (ETH) in 1939. He was the appointed as an Assistant to Plancherel while he worked for his doctorate under Hopf. Eckmann explains how these two provided a remarkable foundation for his career in [1]:As an assistant to Plancherel I had the opportunity to work, and even lecture, in different fields, including analysis. Under the wonderful guidance of Heinz Hopf I then got into my doctoral thesis work. It was characteristic of Hopf's views on our science that this meant not only learning algebraic topology - then a very young field - but also getting acquainted with group theory, differential geometry, and algebra in the "abstract" sense of the Emmy Noether school. The combination of these fields, considered at that time to be largely separated from each other, remained a constant challenge during all my later work ... In 1941 Eckmann was awarded his Dr. sc. math. (equivalent to a Ph.D.) from ETH Zürich. He submitted a thesis which was judge quite outstanding, even relative to the high standards they had, and Eckmann was awarded the Kern Prize and silver medal for his work. After the award of the doctorate he remained at Zürich as a Privatdozent until 1942 when he appointed as a lecturer at the University of Lausanne. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eckmann.html (1 of 3) [2/16/2002 11:08:19 PM]

Eckmann

The most important person to influence Eckmann during his years at the University of Lausanne was Georges de Rham who was an extraordinary professor there when Eckmann arrived and became full professor in 1943. Eckmann writes:My first University appointment was in Lausanne 1942-48 where close contacts with Georges de Rham provided friendship and stimulation. While holding the position in Lausanne, Eckmann visited the Institute for Advanced Study at Princeton. He arrived there in January 1947 and remained there for most of the year, returning to Lausanne in September. This gave him:... the possibility of concentrated research and meeting so many mathematicians whose names had been familiar before, when contacts were not possible because of the war. In 1948 Eckmann, who had already been promoted to Extraordinary Professor at Lausanne, was offered a full Professorship back at the Eidgenössische Technische Hochschule Zürich. It was an opportunity that he certainly was not going to turn down and, pleased to be back, he remained as Professor there until he retired in 1984. He did, however, make many visits as a visiting professor. For example he was a visiting Professor at the University of Michigan during the summer of 1950, the University of Illinois during the winter of 1952, the University of California at Berkeley during the summer of 1955 and the Scuola Normale Superiore in Pisa in the spring of 1958. An important change in Eckmann's duties in Zürich occurred in 1964. In that year the Forschungsinstitut für Mathematik (mathematics Research Institute) was founded in Zürich. Eckmann was appointed as Head of the Institute and he held this position until he retired in 1984. He writes that the Institute was:... designed especially for the purpose of inviting visitors to cooperate with the department members and their students. The Institute rapidly developed into a well-known international centre of mathematics research; needless to say, it was of great benefit to me ... In April 1977 a Colloquium was held in Zürich to celebrate Eckmann's 60th birthday. Saunders MacLane spoke at the Colloquium about Eckmann's contributions to the founding of homological algebra and category theory. Peter Hilton, who had been a personal friend of Eckmann's for many years spoke in detail of Eckmann's research in topology: continuous solutions of systems of linear equations, a group-theoretical proof of the Hurwitz-Radon theorem, complexes with operators, spaces with means, simple homotopy type. Details of these research contributions are given in [2]. Hilton described Eckmann's contributions to mathematics as being described by the names:... unification, clarification and penetration. Eckmann, [1], expresses his thanks to Peter Hilton for their many years of collaborations:I am deeply grateful, not only for stimulation and sharing so many ideas with me, but also for his friendship ... It was not only with Peter Hilton that Eckmann collaborated. Rather collaboration was what he liked to do and many of his research papers arose from these collaborations, many of which were with his students. Finally we should comment on other roles which Eckmann held and honours which he has received. He was Secretary to the International Mathematical Union from 1956 to 1961, President of the Swiss mathematical Society in 1961-62, and a member of the research Council of the Swiss National Science

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eckmann.html (2 of 3) [2/16/2002 11:08:19 PM]

Eckmann

Foundation from 1964 to 1984. He received honorary degrees from Fribourg in 1964, Lausanne Polytechnique in 1969, and the Israel Institute of Technology in 1983. He was also awarded the Prix Mondial Nessim Habif from the University of Geneva in 1967. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eckmann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eckmann.html (3 of 3) [2/16/2002 11:08:19 PM]

Eddington

Arthur Stanley Eddington Born: 28 Dec 1882 in Kendal, Westmorland, England Died: 22 Nov 1944 in Cambridge, Cambridgeshire, England

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Arthur Eddington's father, Arthur Henry Eddington, taught at a Quaker training college in Lancashire before moving to Kendal to become headmaster of Stramongate School. He died of typhoid in an epidemic which swept the country in 1884 before his son was two years old. Arthur Eddington's mother, Sarah Ann Stout, came from Darlington and, like her husband, was also from a Quaker family. On Arthur Henry Eddington's death she was left to bring up Arthur and his older sister with relatively little income. The family moved to Weston-super-Mare where at first Arthur was educated at home before spending three years at a preparatory school. In 1893 Arthur entered Brymelyn School in Weston which was mainly for boarders but he did not board at the school, being one of a small number of day pupils. The school provided a good education within the limited resources available to it and allowed Arthur to excel in mathematics and English literature. His progress through the school was rapid and he earned high distinction in mathematics. The level to which the school was able to take Arthur was, however, not very advanced and his good grounding in mathematics stopped short of reaching the differential and integral calculus. In 1898 he was awarded a scholarship of 60 a year for three years by Somerset County (Weston-super-Mare is now in Avon but it was at that time in Somerset). Eddington had not reached sixteen years of age at the time, and so officially he was too young to enter university. This was a problem which was solved quickly, however, and did not cause him to delay his entry to Owens College, Manchester which he attended from 1898 to 1902. In his first year of study Eddington took general subjects before spending the next three years studying mainly physics. Although on a physics course, Eddington attended the mathematics lectures, being greatly influenced by one of his mathematics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eddington.html (1 of 5) [2/16/2002 11:08:21 PM]

Eddington

teachers, Horace Lamb. Of course the financial position of his family meant that they were not able to provide him with financial support but his outstanding academic work allowed him to win a number of highly competitive scholarships to provide enough money to let him complete his B.Sc. course with First Class Honours in 1902. He was awarded a Natural Science scholarship of 75 a year to study at Trinity College, Cambridge near the end of 1901. Entering Trinity in 1902 he received, in March 1903, a Mathematics Scholarship of 100 a year instead of the Natural Science scholarship. At Trinity he was taught by E T Whittaker, A N Whitehead and E W Barnes. He became Senior Wrangler in the Mathematical Tripos in 1904 and graduated with a M.A. in the following year. After graduating, he began a research project in the Cavendish Laboratory on thermionic emission but it appears not to have gone too well and he gave up the project. He began research in mathematics, also in 1905, but this was no more successful than his work in physics although he was to make use of the ideas many years later when he applied these early research ideas in mathematics to an astronomy problem. Before the end of 1905 Eddington had made the move to astronomy with his appointment to a post at the Royal Observatory at Greenwich. Astronomy had been a topic of interest to him from an early age and he had been given a loan of a 3 inch telescope when less than 10 years old which had heightened his interest. On being appointed to fill a vacancy at the Royal Observatory he was immediately involved with a research project which had been underway since 1900 when photographic plates of Eros had been taken over the period of a year. Eddington's first task was to complete the reduction of these photographic observations to determine an accurate value for the solar parallax. Plummer writes in [16]:He had introduced his method of analysis of two star-drifts, and his prevailing interest in statistical stellar astronomy was concentrated on the systematic motions and distribution of the stars throughout his Greenwich years. Eddington was a Smith's prize winner for an essay on the proper motions of stars in 1907, and he was awarded a Trinity College Fellowship. George Darwin, a son of Charles Darwin and Plumian professor of astronomy at Cambridge, died in December 1912. In 1913 Eddington was appointed to fill the vacant position of Plumian Professor of Astronomy. There were in fact two chairs of astronomy at Cambridge, the other being the Lowndean chair. Originally the Plumian chair covered the experimental side of the subject while the Lowndean chair covered the theoretical side. Although this distinction had become somewhat blurred over the years the appointment of Eddington was certainly seen as an appointment in experimental astronomy. However, the holder of the Lowndean chair died towards the end of 1913 and, in 1914, Eddington became director of the Cambridge Observatory. In doing so he effectively took over responsibility for both theoretical and experimental astronomy at Cambridge. Shortly after his appointment as director of the Cambridge Observatory he was elected a Fellow of the Royal Society. Shortly after taking up his role of leading astronomy research at Cambridge, World War I broke out. As we noted above Eddington came from a Quaker tradition and, as a conscientious objector, he avoided active war service and was able to continue his research at Cambridge during the war years of 1914-18. This was, however, not an easy time for him giving him a highly stressful time right at the beginning of his tenure of the Cambridge chair. Eddington made important contributions to the theory of general relativity. His interest in this topic started in 1915 when he received papers by Einstein and by de Sitter which came to him via the Royal Astronomical Society. He became interested in this theory, particularly since it provided an explanation http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eddington.html (2 of 5) [2/16/2002 11:08:21 PM]

Eddington

for the previously noticed, but unexplained, advance of the perihelion of Mercury. He lectured on relativity at the British Association meeting in 1916 and produced a major report on the topic for the Physical Society in 1918. In the following year Eddington led an eclipse expedition to Principe Island in West Africa. Its aim was to verify the bending of light passing close to the sun which was predicted by relativity theory. At that time such observations of stars close to the sun in the sky could only be made during a total eclipse. He sailed from England in March 1919 and by mid-May had his instruments set up on Principe Island. The eclipse was due to occur at two o'clock in the afternoon of 29 May but that morning there was a storm with heavy rain. Eddington wrote (see for example [5]):The rain stopped about noon and about 1.30 ... we began to get a glimpse of the sun. We had to carry out our photographs in faith. I did not see the eclipse, being too busy changing plates, except for one glance to make sure that it had begun and another half-way through to see how much cloud there was. We took sixteen photographs. They are all good of the sun, showing a very remarkable prominence; but the cloud has interfered with the star images. The last few photographs show a few images which I hope will give us what we need ... He remained on Principe Island to develop the photographs and to try to measure the deviation in the stellar positions. The cloud made the plates of poor quality and hard to measure. On 3 June he recorded in his notebook:... one plate I measured gave a result agreeing with Einstein. The results from the Africa expedition provided the first confirmation of Einstein's theory that gravity will bend the path of light when it passes near a massive star. Eddington wrote, in a parody of the Rubaiyat of Omar Khayyam (see for example [5]):Oh leave the Wise our measures to collate One thing at least is certain, light has weight One thing is certain and the rest debate Light rays, when near the Sun, do not go straight. Eddington lectured on relativity at Cambridge, giving a beautiful mathematical treatment of the topic. He used these lectures as a basis for his book Mathematical Theory of Relativity which was published in 1923. Einstein said that this work was:... the finest presentation of the subject in any language. In addition to his work in relativity theory Eddington also did important work on the internal structure of stars. He discovered the mass-luminosity relationship for stars, he calculated the abundance of hydrogen and he produced a theory to explain the pulsation of Cepheid variable stars. His early research on this is contained in the important work The Internal Constitution of Stars (1926). Eddington had a long running argument with James Jeans over the mechanism by which energy was created in stars. He wrote, correctly of course, that as to the process of generating energy:... probably the simplest hypothesis ... is that there may be a slow process of annihilation of matter. Jeans, however, favoured the theory that the energy was the result of contraction. Of course this is not entirely wrong since a star when it forms will initially heat up under the energy generated by contraction before nuclear reactions begin and then provide the energy source for most of the star's life. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eddington.html (3 of 5) [2/16/2002 11:08:21 PM]

Eddington

Among Eddington's many books were philosophical works such as The Nature of the Physical World (1928), New Pathways of Science (1935) and The Philosophy of Physical Science (1939). Eddington's rather unusual view of the importance of the history of a subject comes over in these works. He believed that familiarity with the history of a subject was a hindrance to creative research in that subject. The authors of this archive would have to register their strong disagreement with Eddington on this issue! Eddington had a fascination with the fundamental constants of nature and produced some surprising numerical coincidences most of which were published after his death in Fundamental Theory (1946), a book prepared for publication by Whittaker. He writes in that book that his aim was to determine the relation between the sizes of different physical systems. Ronan writes [17]:Eddington, hard-headed mathematician and down-to-earth astronomer though he might be, possessed a mystical side to his nature and the last years of his life were spent in an attempt to construct a huge relativistic synthesis of the physical universe, an edifice in which the bricks would be subatomic and astronomical evidence of the observer and the mortar the underlying mathematical relationships between them. In [8] Kilmister delves deeply into the ideas which led Eddington to the theories he put forward in Fundamental Theory in attempting to unite quantum mechanics and general relativity. Kilmister explains how Eddington considered that epistemology is at the basis of physics, that physical laws and physical constants are the consequences of the condition of observation. It was Dirac's 1928 paper on the wave equation of the electron which had first set Eddington on the path of seeking ways to unify quantum mechanics and general relativity. Kilmister explains how Dirac's use of spinors had surprised Eddington and led him to study a generalisation of the Dirac algebra. His work on algebras which would give a symmetrical description of nature is also examined in [18]. Eddington was knighted in 1930 and received the Order of Merit in 1938. He received many other honours including gold medals from the Astronomical Society of the Pacific (1923), the Royal Astronomical Society (1924), The National Academy of Washington (1924), the French Astronomical Society (1928), and the Royal Society (1928). In addition to election to the Royal Society, he was elected to the Royal Society of Edinburgh, the Royal Irish Academy, the National Academy of Sciences, the Russian Academy of Sciences, the Prussian Academy of Sciences and many others. He was invited to give the Bakerian Lecture of the Royal Society of London in 1926 when he lectured on Diffuse matter in interstellar space. Plummer writes in [16]:To his splendid equipment as a mathematical physicist he owed much ... A bold imagination was coupled with an exceptional knowledge of those features which are accessible to observation. ... To launch out into unknown seas, to be venturesome even at the risk of error, Eddington felt himself called, and the reward of the pioneer came to him. ... Simplicity and modesty were his outstanding characteristics ... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (19 books/articles)

Some Quotations (16)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eddington.html (4 of 5) [2/16/2002 11:08:21 PM]

Eddington

A Poster of Arthur Eddington

Mathematicians born in the same country

Honours awarded to Arthur Eddington (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1914

Royal Society Royal Medal

Awarded 1928

Royal Society Bakerian lecturer

1926

ASP Bruce Medallist

1924

Lunar features

Crater Eddington

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Eddington.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eddington.html (5 of 5) [2/16/2002 11:08:21 PM]

Edge

William Leonard Edge Born: 8 Nov 1904 in Stockport, England Died: 27 Sept 1997 in Bonnyrigg, Scotland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Edge's parents were both schoolteachers. He was educated at his local school, Stockport Grammar School, and from there he went to Cambridge where he studied mathematics at Trinity College. After graduating, he continued working for his doctorate at Trinity on projective geometry. Cambridge was at that time a centre for geometry research with Baker's school flourishing there. Edge's fellow students included P du Val and J G Semple but other famous geometers joined the group while Edge was at Cambridge including the slightly younger men H S M Coxeter and J A Todd. After holding a fellowship at Trinity he was offered a lectureship at the University of Edinburgh by E T Whittaker which he accepted and took up the post in 1932. Edge was to spend the rest of his career at Edinburgh and David Monk, writing in [1], suggests that the reason that Edge never moved to a chair in another university was because:... the Scottish hills and mountains, which he loved, kept him in Edinburgh. Edge played a major role in the success of the Mathematics Department at Edinburgh, first under Whittaker's and then under Aitken's leadership. He formed a close friendship with both these men and supported their work with his high international reputation for research and his lecture courses packed with gems. He did not, however, find administration to his liking and he preferred to avoid this whenever possible. After studying classical geometry Edge moved towards the topic which is most associated with him, finite geometry. He had an amazing geometrical feel for complex situations as well as a skill at handling intricate combinatorial arguments which were characteristic of his work. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Edge.html (1 of 4) [2/16/2002 11:08:23 PM]

Edge

Edge wrote nearly 100 papers and his mastery of the area ranks him with Coxeter as one of the leading geometers of this century. His work was a continuation of work started by the great geometers of the late 19th Century and early 20th Century, in particular Castelnuovo, Cayley, Clebsch, Cremona, Fano, Fricke, Georges Humbert, Klein, Plücker and Schläfli. Humbert discovered a plane sextic curve of genus 5 having five cusps for its singular points. These have interesting geometrical properties and Edge investigated them in a series of papers spanning 40 years. In 1890 Castelnuovo studied and classified algebraic surfaces with hyperelliptic prime sections. Edge continued and completed Castelnuovo's investigations. Castelnuovo proved that a non-ruled surface whose prime sections have genus 2 is the projection of a nonsingular rational surface of order 12 in projective 11-space. Edge explicitly examined one such projection in a paper on Castelnuovo's normal surface. The equation of the scroll of tangents of the common curve of two quadrics is due to Cayley in 1850. Salmon, in his famous text, gave an equation in covariant form. Edge gave a procedure for finding this equation in 1979. Bring's curve was first studied in Klein's 1884 book in connection with the transformation to reduce the general quintic equation to the form x5 + ex + f = 0. Some of Edge's work on Bring's curve extends work due to Clebsch. Edge investigated a pencil of canonical curves of genus 6 on a del Pezzo quintic surface in a 5-dimensional projective space. He investigated the group of self-projectivities of the space, which is isomorphic to the symmetric group S5. He also used geometrical configurations to investigate groups and, although his work was out of fashion at a time when group theorists were moving towards the classification of finite simple groups, his work did provide a deeper understanding of some of these groups, for example Conway's simple groups. Edge was not someone uninterested in modern techniques, however, and in a 1991 paper he included computer-drawn pictures. Other topics Edge worked on, all of which exhibit his mastery of the subject, include nets of quadric surfaces, the geometry of the Veronese surface, Klein's quartic, Maschke's quartic surfaces, Kummer's quartic, the Kummer surface, Weddle surfaces, Fricke's octavic curve, the geometry of certain groups, finite planes and permutation representations of groups arising from geometry. His papers are almost all written as single author papers but he did collaborate with his friends Coxeter and Du Val. In fact when he attended the celebrations for Coxeter in Toronto in 1979 it was the first time Edge had crossed the Atlantic and he said only his great friendship with Coxeter had made him overcome his reluctance to travel. I [EFR] asked Edge a few years ago if he would come to St Andrews and give a talk on the history of mathematics. He said he knew nothing of the history of mathematics. I did not give up that easily and asked him if he would not speak on Cayley's mathematics. "Never met Cayley" replied Edge. He paused for a second before adding "Knew his landlady though". For many years Edge was someone I [EFR] expected to see whenever I went to the Edinburgh University staff club. For some reason I never quite understood, there was frequently a note on the blackboard at the entrance to the club saying there was a message for W L Edge. He was a tall straight man with an http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Edge.html (2 of 4) [2/16/2002 11:08:23 PM]

Edge

imposing figure, certainly someone who one noticed. Typically he wore a green corduroy jacket and his hair blew about in an uncontrolled manner. A colleague now at St Andrews, C M Campbell, attended Edge's courses in the 1960s. He described them as difficult lectures which required a lot of work to appreciate their content but, once this work had been put in, the quality and insight in Edge's lectures became apparent. Edge taught the algebra courses at Edinburgh at this time but he taught algebra with a strong geometric flavour reflecting his deep knowledge, feel and love for geometry. Edge had a deep concern for his students, both while they were studying at Edinburgh and after they had graduated. He kept in touch with these students in many different ways including sending his best wishes when he saw a notice of marriage in the press. Monk [1] describes Edge's lifestyle and interests outside mathematics as follows:Edge never married. He lived in a succession of lodgings, carefully chosen for the quality of the cooking and space for a piano. Music was an abiding interest and he had a fine singing as well as a sonorous speaking voice. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article)

A Quotation

Mathematicians born in the same country Honours awarded to William Edge (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh Honorary Fellow of the Edinburgh Maths Society

Elected 1983

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Edge.html (3 of 4) [2/16/2002 11:08:23 PM]

Edge

JOC/EFR October 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Edge.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Edge.html (4 of 4) [2/16/2002 11:08:23 PM]

Edgeworth

Francis Ysidro Edgeworth Born: 8 Feb 1845 in Edgeworthstown, County Longford, Ireland Died: 13 Feb 1926 in Oxford, Oxfordshire, England

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Francis Edgeworth came to study statistics after an education in ancient and modern languages. He entered Trinity College, Dublin at the age of 17 and studied French, German, Spanish and Italian. After graduating, he was awarded a scholarship to study at Oxford and he entered Exeter College in January 1867. At Oxford he spent some time at Magdalen and at Balliol, graduating in 1869. Exactly what Edgeworth did in the years after leaving Oxford is unclear. He must have studied law at some time since he was called to the Bar in 1877. Three years later, however, he was lecturing in logic at King's College, London. In 1888 he was appointed Professor of Political Economics at King's College, London and, two years later, he was appointed to the Tooke chair of Economic Science. The surprising part is that somewhere in this varied career Edgeworth studied mathematics. We have to assume that he was self-taught in mathematics and this might explain why he seemed to believe that advanced mathematics was understood by all. For example his first serious publication New and old methods of ethics (1877) is described by Kendall in [6] as follows:None of his writings, at any time in his life, consisted of the kind of prose, or orderly presentation of ideas, which give pleasure for their own sake, and in this particular work he actually writes down variational integrals, which must have put it beyond the understanding of most of those who were interested in ethical problems at that time. In 1881 he published Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. This work, really on economics, looks at the Economical Calculus and the Utilitarian Calculus. He formulated mathematically a capacity for happiness and a capacity for work. His conclusions that women have less capacity for pleasure and for work than do men would not be popular in the 1990's. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Edgeworth.html (1 of 3) [2/16/2002 11:08:24 PM]

Edgeworth

Edgeworth published Methods of Statistics in 1885 which presented an exposition of the application and interpretation of significance tests for the comparison of means. In 1891 Edgeworth left London to take up the Drummond Chair of Political Economy at Oxford. He obtained a fellowship at All Souls College and he held both the chair and the fellowship until he retired in 1922. Another event of significance in 1891 was that the Economic Journal began publication with Edgeworth as its first editor. He continued to be editor until 1926 when Keynes took over the editorship. In 1892 Edgeworth examined correlation and methods of estimating correlation coefficients in a series of papers. The first of these papers was Correlated Averages. Edgeworth's work was to influence Pearson although bad feeling developed between the two and later Pearson was to deny Edgeworth's influence. At the Galton dinner in February 1926 Pearson spoke of Edgeworth's death a few days earlier:... him we can almost call a biometrician for he contributed to Biometrika ... Only last December he came and spoke as he had always spoken ... and his criticism failed as it had always failed, because he spoke not the language of the people. ... I should like to reckon him among the biometricians if he ploughed always right across the line of our furrows. Besides we owe him something, like a good German he knew that the Greek k is not a modern c, and, if any of you at any time wonder where the k in Biometrika comes from, I will frankly confess that I stole it from Edgeworth. Whenever you see that k call to mind dear old Edgeworth. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) A Poster of Francis Edgeworth

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1880 to 1890

Other Web sites

1. Paul Walker (A history of Game Theory) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Edgeworth.html (2 of 3) [2/16/2002 11:08:24 PM]

Edgeworth

JOC/EFR February 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Edgeworth.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Edgeworth.html (3 of 3) [2/16/2002 11:08:24 PM]

Egorov

Dimitri Feddrovich Egorov Born: 22 Dec 1869 in Moscow, Russia Died: 10 Sept 1931 in Kazan, USSR

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dimitri Egorov attended school in Moscow then entered Moscow University to study mathematics and physics, enrolling in 1887. The teacher to influence him most at this time was Bugaev. Egorov wrote his first paper in 1892 on numerical integrals and derivatives clearly influenced by Bugaev's work in this area. Egorov taught at Moscow University from 1894, obtaining a doctorate in 1901. He spent a year abroad, then in 1903 he returned to become a professor at Moscow University. Egorov worked on triply orthogonal systems and potential surfaces, making a major contribution to differential geometry. Some of Egorov's work was presented by Darboux in his famous four volume work Leçons sur la théorie général des surfaces et les applications géométriques du calcul infinitésimal. Egorov also worked on integral equations and a theorem in the theory of functions of a real variable is named after him. Luzin was Egorov's first student and became a member of the school Egorov created in Moscow dealing with functions of a real variable. In 1917 Egorov became secretary of the Moscow Mathematical Society. Then in 1921 he was elected vice-president, becoming president the following year. In 1923 Egorov became director of the Institute for Mechanics and Mathematics at Moscow State University. However Egorov was a deeply religious man and when the Church was repressed after the revolution, Egorov defended them. In 1922-23 there were mass execution of clergy and in 1928 the attack was renewed. Egorov was in a position of power in the Moscow Mathematical Society and he tried to shelter http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Egorov.html (1 of 2) [2/16/2002 11:08:26 PM]

Egorov

academics who had been dismissed from their posts. He tried to prevent the attempt to impose Marxist methodology on scientists. In 1929 Egorov was dismissed as director of the Institute for Mechanics and Mathematics and given a public rebuke. Some time later he was arrested as a "religious sectarian" and put in prison. The Moscow Mathematical Society continued to support Egorov, refusing to expel him, and those who presented papers at the next meeting, including Kurosh, were to be expelled by an "Initiative group" took over the Society in November 1930. They expelled Egorov denouncing him as a reactionary and a churchman. Egorov went on a hunger strike in prison and eventually, by this time close to death, he was taken to the prison hospital in Kazan. Chebotaryov's wife was working as a doctor in the prison hospital and, although it sounds rather unlikely, it is reported that Egorov died at Chebotaryov's home. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Egorov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Egorov.html (2 of 2) [2/16/2002 11:08:26 PM]

Ehrenfest

Paul Ehrenfest Born: 18 Jan 1880 in Vienna, Austria Died: 25 Sept 1933 in Leiden, Netherlands

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Ehrenfest's father, Sigmund Ehrenfest, came from a poor Jewish family. He was working in a weaving mill in the Jewish village of Loschwitz in Moravia when he married Johanna Jellinek. After the marriage they moved to Vienna where they set up a grocery business which fared rather well. They had five children who survived birth, and Paul was the youngest, having four older brothers Arthur, Emil, Hugo, and Otto. Johanna Ehrenfest worked long hours in their shop and Paul was looked after at home by a nursemaid. As a child Paul's health was poor. He was sickly, had dizzy spells, and suffered frequent nosebleeds. He suffered from anti-Semitic comments from other children in the neighbourhood but his brothers supported him strongly and played an important role in his childhood. His oldest brother was twenty-two years old by the time Paul was five and it was through his brothers that Paul became interested in education. They gave him puzzles which he enjoyed solving. By the time he was six years old Paul could read, write and count. Mostly he had taught himself these things, helped a little by his mother, and encouraged by his brothers. At this age, in 1886, he began to attend primary school, moving to a different primary school in 1888. He was introduced to science and mathematics by his brothers, rather than from his school, and their attitude was one which would have quite an influence on him [2]:This early impression of science as something to be learned with joy, something to be discussed and argued about, was absorbed into Ehrenfest's innermost being. He completed his primary schooling in 1890 and his school reports show that it had been a very successful time academically with top marks in all his subjects, but already his life was becoming http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ehrenfest.html (1 of 6) [2/16/2002 11:08:28 PM]

Ehrenfest

unhappy. His mother who had been ill for some time died of breast cancer in 1890. Paul's father was also in poor health, suffering from stomach ulcers. Soon after Johanna 's death Paul's father married again, his second wife being Josephine Jellinek, his first wife's younger sister. Josephine was about the same age as Paul's oldest brother. In 1890 Paul began his secondary education at the Akademisches Gymnasium. Perhaps not surprisingly given his problems at home his performance at school deteriorated greatly, both his marks and his behaviour. The only subject he continued to excel in was mathematics. Clearly he was an unhappy child [2]:He was often miserable, deeply depressed and at odds with himself and the world. Life did not get better, for when he was sixteen years old his father died from the stomach trouble which had got progressively worse for many years. Arthur, his oldest brother, became his guardian and managed to persuade Paul not to leave school as he wished to do. Things got somewhat better [2]:Paul was apparently able to work himself out of his depression, which had sometimes been deep enough to make him contemplate suicide. His intellectual interests grew stronger, perhaps as a form of self-protection. In the summer of 1899 Ehrenfest successfully took his school examinations, but his experiences of school had been decidedly negative as he showed later in life when he insisted that his own children were educated at home. Ehrenfest became a student at the Technische Hochschule in Vienna in October 1999. There he formed a close friendship with three other students of mathematics, Heinrich Tietze, Hans Hahn and Herglotz. They called themselves the 'inseparable four'. Ehrenfest attended Boltzmann's lectures on the mechanical theory of heat during 1899-1900. Suddenly, thanks mainly to Boltzmann, the negative thoughts about education which he had at school were replaced with a great love for mathematics and physics. As was the custom at the time, students in Austria and Germany did not usually stay at a single university for their whole undergraduate course. In 1901 Ehrenfest moved to Göttingen where he studied under Klein and Hilbert. There he took Max Abraham's course on the electromagnetic theory of light and also attended courses by Stark, Walther Nernst, Schwarzschild and Zermelo. While attending courses by Klein and Hilbert, Ehrenfest saw a young Russian student of mathematics Tatyana Alexeyevna Afnassjewa. He wondered why she did not come to meetings of the mathematics club but then discovered that the reason was that women were not allowed to attend. Ehrenfest challenged this rule and, after quite a battle, was able to get the rule changed. It was the beginning of their friendship which led eventually to their marriage. Ehrenfest returned to Vienna after spending eighteen months in Göttingen. He obtained his doctorate from Vienna in 1904, under Boltzmann's supervision, on a topic in classical mechanics The motion of rigid bodies in fluids and the mechanics of Hertz. It was considered a good piece of work but Ehrenfest himself never rated it very highly and chose not to publish it after receiving his doctorate on 23 June 1904. After this Tatyana came from Göttingen to join Ehrenfest in Vienna and they married after overcoming the severe problem of having different faiths. Each had to renounce their religion before the marriage was allowed - it took place on 21 December 1904. In 1905 Ehrenfest published a paper on Planck's theory of black-body radiation. It [2]:... shows Ehrenfest's real interests and particular talents and the beginning of his personal http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ehrenfest.html (2 of 6) [2/16/2002 11:08:28 PM]

Ehrenfest

style. He remained at Vienna but without a post. He returned to Göttingen in September 1906, hoping there might be a position available but there was not. He was shocked to learn, however, that Boltzmann had committed suicide on 6 September. Ehrenfest took on the task of writing his obituary. Klein, Hilbert, Minkowski and Carathéodory were all working in Göttingen at this time and it was an important period for Ehrenfest's research. Klein asked him to write, jointly with his wife if he wished, an article on statistical mechanics. The two Ehrenfests began working on the article which would not appear in print until 1911. In 1907 Ehrenfest went to St Petersburg. It was not that he had a post there but his wife was Russian and the move was probably aimed at finding somewhere where they could feel at home. Certainly Ehrenfest had mixed feelings about his own country given the anti-Semitic attitudes he had encountered. Once in St Petersburg he made contact with Tamarkin, Friedmann, Steklov and other mathematicians and physicists. The Ehrenfests spent five years in St Petersburg. It was a time when Ehrenfest was deeply engrossed in research problems. Together with his wife he worked on the review article on statistical mechanics which took longer to complete than expected. He corresponded with Klein who told him that what was required was a survey, not a complete solution of all the problems of the subject by Ehrenfest himself. In the hope that this might lead to an academic post Ehrenfest, despite holding a doctorate, took the degree of Master of Physics at St Petersburg. He was successful in obtaining the degree but not the academic post for which he hoped. He tried to find a position by writing to many institutions, including some in North America, but nothing came of any of his enquiries. An important paper was published by Ehrenfest in 1911 in Annalen der Physik on the essential features of quantum theory. In January 1912 Ehrenfest set out on a tour of universities in the German speaking world in the hope of a position. He visited Berlin where he saw Planck, Leipzig where he saw his old friend Herglotz, Munich where he met Sommerfeld, then Zurich, Vienna, and Prague where he met Einstein for the first time. On his travels he learnt that Poincaré had written a paper on quantum theory which gave similar results to those in his Annalen der Physik paper. Poincaré had not known of Ehrenfest's contribution and therefore had not referred to his work. Ehrenfest returned to St Petersburg saddened that Poincaré's paper had been published before he could point out his own contribution to him - he was paying the price for being isolated from mainstream research in St Petersburg. However, his fortunes were about to change. Lorentz was looking for someone to succeed to his chair at Leiden. Sommerfeld recommended Ehrenfest, writing (see [2]):He lectures like a master. I have hardly ever heard a man speak with such fascination and brilliance. Significant phrases, witty points and dialectic are all at his disposal in an extraordinary manner ... He knows how to make the most difficult things concrete and intuitively clear. Mathematical arguments are translated by him into easily comprehensible pictures. On 29 September 1912 Ehrenfest received a telegram saying that he had been name professor at Leiden. He remained at Leiden for the rest of his career. We examine now some of the contributions which he made while working there. In 1917 and 1920 Ehrenfest published papers investigating the problem of the extent to which the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ehrenfest.html (3 of 6) [2/16/2002 11:08:28 PM]

Ehrenfest

three-dimensional nature of physical space is determined by the structure of basic physical equations or is reflected in these basic equations. Ehrenfest's arguments were based both on Newton's celestial mechanics and also on Einstein's relativity theory. Among Ehrenfest's contribution to quantum statistics was an understanding of the nature of photons, and their properties which were implied by Planck's radiation law. He worked on quantum theory applying it to rotating bodies. He recognised that Ampère's molecular currents are incompatible with classical statistical mechanics. He proposed a model of diffusion in order to illuminate the statistical interpretation of the second law of thermodynamics, that the entropy of a closed system can only increase. The modern theory of nonequilibrium thermodynamics brings together the molecular, collisional ideas of Boltzmann with the statistical ideas of Ehrenfest's to give a nonlinear, statistical theory. In 1933 Ehrenfest presented a classification of phase transitions based on the discontinuity in derivatives of the free energy function. Uhlenbeck was a student of Ehrenfest who began research for his Master's degree in 1920. He spoke of Ehrenfest's teaching style [9]:First the assertion, then the proof ... His famous clarity [should] not to be confused with rigour. ... He never gave of made problems; he did not believe in them; in his opinion the only problems worth considering were those you proposed yourself. ... He worked with essentially one student at a time, and that practically every afternoon during the week. As to Ehrenfest's mathematical skills, Uhlenbeck wrote [9]:Although he knew mathematics it was not simple for him. He was not a computer. He could not compute. Pais writes about Ehrenfest's lectures in the early 1920s [10]:Ehrenfest's graduate lectures consisted of a two-year course: Maxwell theory, ending with the theory of electrons and some relativity, one year; and statistical mechanics, ending with atomic structure and quantum theory the other. In 1925 when quantum mechanics began to dominate work in theoretical physics, Ehrenfest felt he had problems [9]:I think he always hated it. All these youngsters who had, with great facility, made these calculations ... and you didn't have to understand much. You just computed and you did this and you did that and everything came out ... There was, therefore, a mathematical apparatus built with Hilbert's basis, with operators, which had a sort of abstractness. It was so against his creed that I'm sure he suffered from it. ... he understood it all alright. He said he was just too old. It was against his creed to really take part in it. Niels Bohr and Ehrenfest began to correspond in 1918. When they met in Leiden [7]:[Ehrenfest] and Bohr had much to talk about together -- from the current problems of quantum theory to the Icelandic sagas, from the stages of a child's development to the difference between genuine physicists and the other. Their exchanges ranged over heaven and earth as Ehrenfest showed his new friend the treasures of the Dutch museums and the brilliant colours of the bulb fields. Ehrenfest presented Bohr's results to the third Solvay conference in 1921. Bohr did not attend through http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ehrenfest.html (4 of 6) [2/16/2002 11:08:28 PM]

Ehrenfest

overwork. Later in 1921 Bohr invited Ehrenfest to Copenhagen. He replied:Dear, dear Bohr, I would like so terribly much to be with you again. Ehrenfest was unhappy at the disagreement between Bohr and Einstein over quantum theory. He brought them together at his home in Leiden in December 1925 in an attempt to have them reach an agreed position. They did not and Ehrenfest was very unhappy that he was forced to take sides with one of his two close friends. He said in 1927, while in tears, that forced to make a choice he would have to agree with Bohr. May 1931 Ehrenfest wrote to Bohr [3]:I have completely lost contact with theoretical physics. I cannot read anything any more and feel myself incompetent to have even the most modest grasp about what makes sense in the flood of articles and books. Perhaps I cannot at all be helped any more. All through his life Ehrenfest had suffered from low self esteem, which was in marked contrast to the high esteem in which he was held by his fellow scientists. He was also greatly saddened by his son Wassik being a mongol and having severe problems both physically and mentally. His last letter (which was never sent) is a sad document (see for example [3]):My dear friends: Bohr, Einstein, Franck, Herglotz, Joffé, Kohnstamm, and Tolman! I absolutely do not know any more how to carry further during the next few months the burden of my life which has become unbearable. I cannot stand it any longer to let my professorship in Leiden go down the drain. I must vacate my position here. Perhaps it may happen that I can use up the rest of my strength in Russia. .. If, however, it will not become clear rather soon that I can do that, then it is as good as certain that I shall kill myself. And if that will happen some time then I should like to know that I have written, calmly and without rush, to you whose friendship has played such a great role in my life. ... In recent years it has become ever more difficult for me to follow the developments in physics with understanding. After trying, ever more enervated and torn, I have finally given up in desperation. This made me completely weary of life .. I did feel condemned to live on mainly because of the economic cares for the children. I tried other things but that helps only briefly. Therefore I concentrate more and more on the precise details of suicide. I have no other practical possibility than suicide, and that after having first killed Wassik. Forgive me ... May you and those dear to you stay well. This letter, and a similar letter which he wrote to his students, was never sent. Ehrenfest shot Wassik in the waiting room of the Professor Watering Institute in Amsterdam where Wassik was being treated. Then he shot himself. The Dutch papers only reported his sudden death and gave lengthy accounts of his achievements. Einstein said of Ehrenfest [2] that he was:He was not merely the best teacher in our profession whom I have ever known; he was also passionately preoccupied with the development and destiny of men, especially his students. To understand others, to gain their friendship and trust, to aid anyone embroiled in outer or inner struggles, to encourage youthful talent - all this was his real element, almost more than his emersion in scientific problems. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ehrenfest.html (5 of 6) [2/16/2002 11:08:28 PM]

Ehrenfest

Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country Cross-references to History Topics

General relativity

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ehrenfest.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ehrenfest.html (6 of 6) [2/16/2002 11:08:28 PM]

Ehresmann

Charles Ehresmann Born: 19 April 1905 in Strasbourg, France Died: 22 Sept 1979 in Amiens France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Ehresmann came from a poor family in Alsace. His father was a gardener and the family spoke Alsatian which is related to the Germanic languages. Alsace, which was originally French, had came under German rule in 1871 but by 1902 had effective self-government. After 1911 it had its own constitution and progress was made toward Germanisation in the region. Certainly Charles began his schooling entirely in German up to the end of World War I in 1918. He attended the Lycée Kléber in Strasbourg but, after 1919 Alsace was returned to France and French language schools were set up. This met with resistance from the German speaking population but their attempts at autonomy within the French Republic were unsuccessful. Charles, however, was from that time on taught in French. In 1924 Ehresmann entered the Ecole Normale Supérieure in Paris. He graduated in 1927, spent a year doing military service, then taught mathematics at the French speaking Lycée in Rabat, the national capital of Morocco. After spending the years 1928-29 in Rabat, Ehresmann continued his education going to Göttingen to undertake research. During his time in Göttingen in the years 1930 and 1931 it was the leading centre for mathematical research in the world although, of course, shortly after this the rise to power of the Nazis would change the mathematical world significantly. From 1932 to 1934 Ehresmann studied at Princeton in the United States. Having left the leading centre of Göttingen, he had gone to the place which in many ways would replace it as the leading mathematical centre as the Jewish mathematicians left Germany after the Nazis passed their anti-Jewish legislation in 1933. Ehresmann's doctorate was awarded by Paris in 1934. In his doctoral dissertation, and during the time from 1934 to 1939 when he was carrying out research in the Centre Nationale de la Recherche http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ehresmann.html (1 of 3) [2/16/2002 11:08:30 PM]

Ehresmann

Scientifique, he studied topological properties of differential manifolds. In particular he described [1]:... the homology of classical types of homogeneous manifolds, such as Grassmannians, flag manifolds, Stiefel manifolds, and classical groups. He became a lecturer in the University of Strasbourg in 1939 but shortly after this he was back in the middle of the France/Germany conflict of his youth but this time on a quite different scale as the Germans invaded Alsace in 1940. During the German occupation of World War II, the University of Strasbourg's faculties were moved to Clermont Ferrand University in central France, then back to Strasbourg in 1945. Ehresmann followed the moves of the university then, in 1955, a chair of topology was specially created for him in the University of Paris. He held this chair until he retired in 1975. Although he was 70 years old when he retired, Ehresmann did not give up lecturing for at this time he moved to Amiens, where his second wife was a professor of mathematics, and he taught there. Ehresmann was one of the creators of differential topology. Beginning in 1941, Ehresmann made major contributions toward establishing the current view of fibre spaces, manifolds, foliations and jets. His work in the creation and development of fibre spaces followed on from the study of a special case made earlier by Seifert and Whitney. After 1957 Ehresmann became a leader in category theory and he worked in this area for 20 years. His principal achievements in this area concern local categories and structures defined by atlases, and germs of categories. The article [3] contains a list of 139 articles written by Ehresmann during his productive career as well as listing several volumes which he edited. Between 1980 and 1983 Andrée Charles Ehresmann, his wife, edited his complete works. These appeared in seven volumes: Charles Ehresmann: Oeuvres complètes et commentées as supplements to the Journal Cahiers de Topologie et Geometrie Differentielle Categoriques which Charles Ehresmann created in 1957. In [1] Dieudonné describes Ehresmann's personality as:... distinguished by forthrightness, simplicity, and total absence of conceit or careerism. As a teacher he was outstanding, not so much for the brilliance of his lectures as for the inspiration and tireless guidance he generously gave to his research students ... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ehresmann.html (2 of 3) [2/16/2002 11:08:30 PM]

Ehresmann

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ehresmann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ehresmann.html (3 of 3) [2/16/2002 11:08:30 PM]

Eilenberg

Samuel Eilenberg Born: 30 Sept 1913 in Warsaw, Poland Died: 30 Jan 1998 in New York, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Samuel Eilenberg's father was educated at a Jewish school but became a brewer was he married into a family of brewers. Sammy, as Eilenberg was always called, studied at the University of Warsaw. It is not surprising that Eilenberg's interests quickly turned towards point set topology which, of course, was an area which flourished at the University of Warsaw at that time. A remarkable collection of mathematicians were on the staff at the University of Warsaw while Eilenberg studied there. For example Mazurkiewicz, Kuratowski, Sierpinski, Saks and Borsuk taught there. Eilenberg was awarded his MA from the University of Warsaw in 1934. Then in 1936 he received his doctorate after studying under Borsuk. MacLane writes in [1]:His thesis, concerned with the topology of the plane, was published in Fundamenta Mathematica in 1936. Its results were well received both in Poland and the USA. The second mathematical centre in Poland at this time was Lvov. It was there that Eilenberg met Banach, who led the Lvov mathematicians. He joined the community of mathematicians working and drinking in the Scottish Café and he contributed problems to the Scottish Book, the famous book in which the mathematicians working in the Café entered unsolved problems. You can see a picture of the Scottish Café. Most of Eilenberg's publications from this period were on point-set topology but there were signs, even at this early stage of his career, that he was moving towards more algebraic topics. MacLane writes [1]:In 1938 he published in [Fundamenta Mathematica] another influential paper on the action http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eilenberg.html (1 of 5) [2/16/2002 11:08:32 PM]

Eilenberg

of the fundamental group on the higher homotopy groups of a space. Algebra was not foreign to his topology! This paper was an early sign that Eilenberg was moving into the area for which he has become famous. It was one of a truly remarkable collection of papers published by Eilenberg for, from his days as an undergraduate up until 1939 when he left Poland for the United States, he published 37 papers. In 1939 Eilenberg's father convinced him that the right course of action was to emigrate to the United States. Once there he went to Princeton where Veblen and Lefschetz helped him to find a university post. This was not too long in coming and, in 1940, he was appointed as an instructor at the University of Michigan. This was an excellent place for Eilenberg to begin his teaching career in the United States for there he could interact with leading topologists. Wilder was on the staff at Ann Arbor and Steenrod, who had studied there earlier, continued to have close links and returned onto the staff at Ann Arbor in 1942. In 1940 there was an important topology conference organised at Michigan. World War II was by this time dominating the international scene so the number of participants at the conference from outside the United States was much less that one would have otherwise expected. Eilenberg lectured at the conference on Extension and classification of continuous mappings. Eilenberg was only an instructor for one year, then in 1941 he was promoted at assistant professor at the University of Michigan. In 1945 he was promoted again, this time to associate professor. He spent the year 1945-46 as a visiting lecturer at Princeton before being appointed as a full professor at the University of Indiana in 1946. After one year he moved to Columbia University in New York where he remained for the rest of his career. In 1948, the year after he took up the post at Columbia, Eilenberg became a US citizen. He married Natasa Chterenzon in 1960. Perhaps the most obvious feature of Eilenberg's work was the amount which was done in collaboration with other mathematicians. One major collaboration was his work with Bourbaki. In 1949 André Weil was working at the University of Chicago and he contacted Eilenberg to ask him to collaborate on writing about homotopy groups and fibre spaces as part of the Bourbaki project. Eilenberg became a member of the Bourbaki team spending 1950-51 as a visiting professor in Paris and participating in the two week summer meetings until 1966. He had been awarded Fulbright and Guggenheim scholarships to fund his year in Paris. One of the first collaborations with which Eilenberg entered was with MacLane. The two first met in 1940 in Ann Arbor and from that time until about 1954 the pair produced fifteen papers on a whole range of topics including category theory, cohomology of groups, the relation between homology and homotopy, Eilenberg-MacLane spaces, and generic cycles. In 1942 they published a paper in which they introduced Hom and Ext for the first time. They introduced the terms functor and natural isomorphism and, in 1945, added the terms category and natural transformation. Ann Arbor again provided the means to bring Eilenberg and Steenrod together. In 1945 they set out the axioms for homology and cohomology theory but they did not give proofs in their paper, leaving these to appear in their famous text Foundations of algebraic topology in 1952. MacLane writes in [1]:At that time there were many different and confusing versions of homology theory, some singular some cellular. The book used categories to show that they all could be described conceptually as presenting homology functors from the category of pairs of spaces to groups http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eilenberg.html (2 of 5) [2/16/2002 11:08:32 PM]

Eilenberg

or to rings, satisfying suitable axioms such as "excision". Thanks to Sammy's insight and his enthusiasm, this text drastically changed the teaching of topology. In fact Eilenberg had written a definitive treatment of singular homology and cohomology in a paper in the Annals of Mathematics in 1944. He had written this paper since he found the treatment of the topic by Lefschetz in his 1942 book unsatisfactory. In 1948 Eilenberg, in a joint paper with Chevalley, gave an algebraic approach to the cohomology of Lie groups, using the Lie algebra as a basic object. They showed that in characteristic zero the cohomology of a compact Lie group is isomorphic as an algebra to the cohomology of the corresponding Lie algebra. Another collaboration of major importance was between Eilenberg and Henri Cartan. The two first met in 1947 and began to exchange ideas by letter in the following years. However as we mentioned above Eilenberg spent 1950-51 in Paris and it was during this time that they made remarkable progress. Henri Cartan writes in [1]:We went from discovery to discovery, Sammy having an extraordinary gift for formulating at each moment the conclusions that would emerge from the discussion. And it was always he who wrote everything up as we went along in precise and concise English. ... Of course, this work together took several years. Sammy made several trips to my country houses (in Die and in Dolomieu). Outside of our work hours he participated in our family life. The outcome of this collaboration was the book Homological algebra the title being a term which the two mathematicians invented. Although they had completed the manuscript by 1953 it was not published until 1956. Hochschild reviewing the book wrote:The title "Homological Algebra" is intended to designate a part of pure algebra which is the result of making algebraic homology theory independent of its original habitat in topology and building it up to a general theory of modules over associative rings. ... The conceptual flavour of homological algebra derives less specifically from topology than from the general "naturalistic" trend of mathematics as a whole to supplement the study of the anatomy of any mathematical entity with an analysis of its behaviour under the maps belonging to the larger mathematical system with which it is associated. In particular, homological algebra is concerned not so much with the intrinsic structure of modules but primarily with the pattern of compositions of homomorphisms between modules and their interplay with the various constructions by which new modules may be obtained from given ones. He also notes:The appearance of this book must mean that the experimental phase of homological algebra is now surpassed. The diverse original homological constructions in various algebraic systems which were frequently of an ad hoc and artificial nature have been absorbed in a general theory whose significance goes far beyond its sources. The basic principles of homological algebra, and in particular the full functorial control over the manipulation of tensor products and modules of operator homomorphisms, will undoubtedly become standard algebraic technique already on the elementary level. We should mention another major two volume text which Eilenberg published in 1974 and 1976. This text was Automata, languages, and machines which was described by a reviewer as:... one of the most important events in the mathematical study of the foundations of computer science and in applied mathematics. The work includes a unifying mathematical http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eilenberg.html (3 of 5) [2/16/2002 11:08:32 PM]

Eilenberg

presentation of almost all major topics of automata and formal language theory. It was a topic which Eilenberg had been interested in from 1966 onwards and it is worth noting that it is one of the few major works by Eilenberg which he worked on alone. The book examines rational structures, that is those that can be recognised by a finite state automaton. So far we have only talked about Sammy the mathematician. There was another side to Eilenberg however, for he was a dealer in the art world in which he was known as "Professor". He dealt in Indian art and he was a leading expert on the subject. Hyman Bass writes in [1]:Over the years Sammy gathered on of the world's most important collections of Southeast Asian art. His fame among certain art collectors overshadows his mathematical reputation. In a gesture characteristically marked by its generosity and elegance, Sammy in 1987 donated much of his collection to the Metropolitan Museum of art in New York, which in turn was thus motivated to contribute substantially to the endowment of the Eilenberg Visiting Professorship in Mathematics at Columbia university. Eilenberg received many honours for his work. In particular we should mention the Wolf Prize which he shared with Selberg in 1986 and his election to the National Academy of Sciences. Finally let us give a quote regarding Eilenberg's personality. Bass writes in [1]:Though his mathematical ideas may seem to have a kind of crystalline austerity, Sammy was a warm, robust, and very animated human being. For him mathematics was a social activity, whence his many collaborations. He liked to do mathematics on his feet, often prancing while he explained his thoughts. When something connected, one could read it in his impish smile and the sparkle in his eyes. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Other references in MacTutor

1. Chronology: 1940 to 1950 2. Chronology: 1950 to 1960

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eilenberg.html (4 of 5) [2/16/2002 11:08:32 PM]

Eilenberg

JOC/EFR September 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eilenberg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eilenberg.html (5 of 5) [2/16/2002 11:08:32 PM]

Einstein

Albert Einstein Born: 14 March 1879 in Ulm, Württemberg, Germany Died: 18 April 1955 in Princeton, New Jersey, USA

Click the picture above to see fifteen larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Around 1886 Albert Einstein began his school career in Munich. As well as his violin lessons, which he had from age six to age thirteen, he also had religious education at home where he was taught Judaism. Two years later he entered the Luitpold Gymnasium and after this his religious education was given at school. He studied mathematics, in particular the calculus, beginning around 1891. In 1894 Einstein's family moved to Milan but Einstein remained in Munich. In 1895 Einstein failed an examination that would have allowed him to study for a diploma as an electrical engineer at the Eidgenössische Technische Hochschule in Zurich. Einstein renounced German citizenship in 1896 and was to be stateless for a number of years. He did not even apply for Swiss citizenship until 1899, citizenship being granted in 1901. Following the failing of the entrance exam to the ETH, Einstein attended secondary school at Aarau planning to use this route to enter the ETH in Zurich. While at Aarau he wrote an essay (for which was only given a little above half marks!) in which he wrote of his plans for the future, see [13]:If I were to have the good fortune to pass my examinations, I would go to Zurich. I would stay there for four years in order to study mathematics and physics. I imagine myself becoming a teacher in those branches of the natural sciences, choosing the theoretical part of them. Here are the reasons which lead me to this plan. Above all, it is my disposition for abstract and mathematical thought, and my lack of imagination and practical ability. Indeed Einstein succeeded with his plan graduating in 1900 as a teacher of mathematics and physics. One of his friends at ETH was Marcel Grossmann who was in the same class as Einstein. Einstein tried to

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html (1 of 6) [2/16/2002 11:08:34 PM]

Einstein

obtain a post, writing to Hurwitz who held out some hope of a position but nothing came of it. Three of Einstein's fellow students, including Grossmann, were appointed assistants at ETH in Zurich but clearly Einstein had not impressed enough and still in 1901 he was writing round universities in the hope of obtaining a job, but without success. He did manage to avoid Swiss military service on the grounds that he had flat feet and varicose veins. By mid 1901 he had a temporary job as a teacher, teaching mathematics at the Technical High School in Winterthur. Around this time he wrote:I have given up the ambition to get to a university ... Another temporary position teaching in a private school in Schaffhausen followed. Then Grossmann's father tried to help Einstein get a job by recommending him to the director of the patent office in Bern. Einstein was appointed as a technical expert third class. Einstein worked in this patent office from 1902 to 1909, holding a temporary post when he was first appointed, but by 1904 the position was made permanent and in 1906 he was promoted to technical expert second class. While in the Bern patent office he completed an astonishing range of theoretical physics publications, written in his spare time without the benefit of close contact with scientific literature or colleagues. Einstein earned a doctorate from the University of Zurich in 1905 for a thesis On a new determination of molecular dimensions. He dedicated the thesis to Grossmann. In the first of three papers, all written in 1905, Einstein examined the phenomenon discovered by Max Planck, according to which electromagnetic energy seemed to be emitted from radiating objects in discrete quantities. The energy of these quanta was directly proportional to the frequency of the radiation. This seemed to contradict classical electromagnetic theory, based on Maxwell's equations and the laws of thermodynamics which assumed that electromagnetic energy consisted of waves which could contain any small amount of energy. Einstein used Planck's quantum hypothesis to describe the electromagnetic radiation of light. Einstein's second 1905 paper proposed what is today called the special theory of relativity. He based his new theory on a reinterpretation of the classical principle of relativity, namely that the laws of physics had to have the same form in any frame of reference. As a second fundamental hypothesis, Einstein assumed that the speed of light remained constant in all frames of reference, as required by Maxwell's theory. Later in 1905 Einstein showed how mass and energy were equivalent. Einstein was not the first to propose all the components of special theory of relativity. His contribution is unifying important parts of classical mechanics and Maxwell's electrodynamics. The third of Einstein's papers of 1905 concerned statistical mechanics, a field of that had been studied by Ludwig Boltzmann and Josiah Gibbs. After 1905 Einstein continued working in the areas described above. He made important contributions to quantum theory, but he sought to extend the special theory of relativity to phenomena involving acceleration. The key appeared in 1907 with the principle of equivalence, in which gravitational acceleration was held to be indistinguishable from acceleration caused by mechanical forces. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html (2 of 6) [2/16/2002 11:08:34 PM]

Einstein

Gravitational mass was therefore identical with inertial mass. In 1908 Einstein became a lecturer at the University of Bern after submitting his Habilitation thesis Consequences for the constitution of radiation following from the energy distribution law of black bodies. The following year he become professor of physics at the University of Zurich, having resigned his lectureship at Bern and his job in the patent office in Bern. By 1909 Einstein was recognised as a leading scientific thinker and in that year he resigned from the patent office. He was appointed a full professor at the Karl-Ferdinand University in Prague in 1911. In fact 1911 was a very significant year for Einstein since he was able to make preliminary predictions about how a ray of light from a distant star, passing near the Sun, would appear to be bent slightly, in the direction of the Sun. This would be highly significant as it would lead to the first experimental evidence in favour of Einstein's theory. About 1912, Einstein began a new phase of his gravitational research, with the help of his mathematician friend Marcel Grossmann, by expressing his work in terms of the tensor calculus of Tullio Levi-Civita and Gregorio Ricci-Curbastro. Einstein called his new work the general theory of relativity. He moved from Prague to Zurich in 1912 to take up a chair at the Eidgenössische Technische Hochschule in Zurich. Einstein returned to Germany in 1914 but did not reapply for German citizenship. What he accepted was an impressive offer. It was a research position in the Prussian Academy of Sciences together with a chair (but no teaching duties) at the University of Berlin. He was also offered the directorship of the Kaiser Wilhelm Institute of Physics in Berlin which was about to be established. After a number of false starts Einstein published, late in 1915, the definitive version of general theory. Just before publishing this work he lectured on general relativity at Göttingen and he wrote:To my great joy, I completely succeeded in convincing Hilbert and Klein. In fact Hilbert submitted for publication, a week before Einstein completed his work, a paper which contains the correct field equations of general relativity. When British eclipse expeditions in 1919 confirmed his predictions, Einstein was idolised by the popular press. The London Times ran the headline on 7 November 1919:Revolution in science - New theory of the Universe - Newtonian ideas overthrown. In 1920 Einstein's lectures in Berlin were disrupted by demonstrations which, although officially denied, were almost certainly anti-Jewish. Certainly there were strong feelings expressed against his works during this period which Einstein replied to in the press quoting Lorentz, Planck and Eddington as supporting his theories and stating that certain Germans would have attacked them if he had been:... a German national with or without swastika instead of a Jew with liberal international convictions... During 1921 Einstein made his first visit to the United States. His main reason was to raise funds for the planned Hebrew University of Jerusalem. However he received the Barnard Medal during his visit and lectured several times on relativity. He is reported to have commented to the chairman at the lecture he gave in a large hall at Princeton which was overflowing with people:I never realised that so many Americans were interested in tensor analysis. Einstein received the Nobel Prize in 1921 but not for relativity rather for his 1905 work on the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html (3 of 6) [2/16/2002 11:08:34 PM]

Einstein

photoelectric effect. In fact he was not present in December 1922 to receive the prize being on a voyage to Japan. Around this time he made many international visits. He had visited Paris earlier in 1922 and during 1923 he visited Palestine. After making his last major scientific discovery on the association of waves with matter in 1924 he made further visits in 1925, this time to South America. Among further honours which Einstein received were the Copley Medal of the Royal Society in 1925 and the Gold Medal of the Royal Astronomical Society in 1926. Niels Bohr and Einstein were to carry on a debate on quantum theory which began at the Solvay Conference in 1927. Planck, Niels Bohr, de Broglie, Heisenberg, Schrödinger and Dirac were at this conference, in addition to Einstein. Einstein had declined to give a paper at the conference and:... said hardly anything beyond presenting a very simple objection to the probability interpretation .... Then he fell back into silence ... Indeed Einstein's life had been hectic and he was to pay the price in 1928 with a physical collapse brought on through overwork. However he made a full recovery despite having to take things easy throughout 1928. By 1930 he was making international visits again, back to the United States. A third visit to the United States in 1932 was followed by the offer of a post at Princeton. The idea was that Einstein would spend seven months a year in Berlin, five months at Princeton. Einstein accepted and left Germany in December 1932 for the United States. The following month the Nazis came to power in Germany and Einstein was never to return there. During 1933 Einstein travelled in Europe visiting Oxford, Glasgow, Brussels and Zurich. Offers of academic posts which he had found it so hard to get in 1901, were plentiful. He received offers from Jerusalem, Leiden, Oxford, Madrid and Paris. What was intended only as a visit became a permanent arrangement by 1935 when he applied and was granted permanent residency in the United States. At Princeton his work attempted to unify the laws of physics. However he was attempting problems of great depth and he wrote:I have locked myself into quite hopeless scientific problems - the more so since, as an elderly man, I have remained estranged from the society here... In 1940 Einstein became a citizen of the United States, but chose to retain his Swiss citizenship. He made many contributions to peace during his life. In 1944 he made a contribution to the war effort by hand writing his 1905 paper on special relativity and putting it up for auction. It raised six million dollars, the manuscript today being in the Library of Congress. By 1949 Einstein was unwell. A spell in hospital helped him recover but he began to prepare for death by drawing up his will in 1950. He left his scientific papers to the Hebrew University in Jerusalem, a university which he had raised funds for on his first visit to the USA, served as a governor of the university from 1925 to 1928 but he had turned down the offer of a post in 1933 as he was very critical of its administration. One more major event was to take place in his life. After the death of the first president of Israel in 1952, the Israeli government decided to offer the post of second president to Einstein. He refused but found the offer an embarrassment since it was hard for him to refuse without causing offence.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html (4 of 6) [2/16/2002 11:08:34 PM]

Einstein

One week before his death Einstein signed his last letter. It was a letter to Bertrand Russell in which he agreed that his name should go on a manifesto urging all nations to give up nuclear weapons. It is fitting that one of his last acts was to argue, as he had done all his life, for international peace. Einstein was cremated at Trenton, New Jersey at 4 pm on 18 April 1955 (the day of his death). His ashes were scattered at an undisclosed place. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (365 books/articles)

Some Quotations (43)

A Poster of Albert Einstein

Mathematicians born in the same country

Cross-references to History Topics

1. The quantum age begins 2. A visit to James Clerk Maxwell's house 3. A brief history of cosmology 4. General relativity 5. Special relativity

Other references in MacTutor

1. A meeting with Einstein. 2. Chronology: 1900 to 1910 3. Chronology: 1910 to 1920

Honours awarded to Albert Einstein (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1921

Fellow of the Royal Society

Elected 1921

Royal Society Copley Medal

Awarded 1925

AMS Gibbs Lecturer

1934

Lunar features

Crater Einstein

Other Web sites

1. Albert Einstein Online 2. American Institute of Physics 3. Institute of Physics 4. University of Glasgow 5. Nobel prizes site (A biography of Einstein and his Nobel prize presentation speech) 6. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html (5 of 6) [2/16/2002 11:08:34 PM]

Einstein

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Einstein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Einstein.html (6 of 6) [2/16/2002 11:08:34 PM]

Eisenhart

Luther Pfahler Eisenhart Born: 13 Jan 1876 in York, Pennsylvania, USA Died: 28 Oct 1965 in Princeton, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Luther Eisenhart was a student at Gettysburg College from 1892 until 1896, receiving his A.B. in 1826. After teaching in a school for a year he undertook graduate study at Johns Hopkins University. He obtained a doctorate in 1900 for a thesis entitled Infinitesimal deformations of surfaces. This work was heavily influenced by Darboux's treatise on the subject and he received little supervision for his doctorate. Eisenhart spent most of his career at Princeton where he became an instructor in mathematics in 1900. He was promoted to professor in 1909 and worked there until he retired in 1945. He served as Dean of the Faculty from 1925 to 1933 when he became Dean of the Graduate School. There are two stages in Eisenhart's work although it is all in differential geometry. The first stage continued his doctoral work studying deformations of surfaces. His first book A Treatise in the Differential Geometry of Curves and Surfaces, published in 1909, was on this topic and was a development of courses he had given at Princeton for several years. In [2] this book is described as:... in textbook form, with numerous problems, introducing the student to classical and modern methods. One of the most interesting novelties of the volume was the so-called 'moving trihedrals' for twisted curves as well as surfaces so freely used in writings of Darboux and others. From the first, methods of the theory of functions of a real variable are employed. The work was of great value in introducing the American student to an important field by the most modern method of the time. The second stage started after 1921 when Eisenhart, prompted by Einstein's general theory of relativity

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eisenhart.html (1 of 3) [2/16/2002 11:08:36 PM]

Eisenhart

and the related geometries, studied generalisations of Riemannian geometry. He published Riemannian Geometry in 1926 and Non-Riemannian Geometry in 1927. The scene is set for the first of these works in [5]:Riemann proposed the generalisation of the theory of surfaces as developed by Gauss, to spaces of any order, and introduced certain fundamental ideas in this general theory. Important contributions to it were made by Bianchi, Beltrami, Christoffel, Schur, Voss, and others, and Ricci-Curbastro coordinated and extended the theory with the use of tensor analysis and his absolute calculus. The book gave a presentation of the existing theory of Riemannian geometry after a period of considerable study and development of the subject by Levi-Civita, Eisenhart, and many others. In 1933 Eisenhart published Continuous Groups of Transformations which continues the work of his earlier books looking at Lie's theory using the methods of the tensor calculus and differential geometry. Again quoting [5]:The study of continuous groups of transformations inaugurated by Lie resulted in the developments by Engel, Killing , Scheffers, Schur, Cartan, Bianchi and Fubini, a chapter which closed about the turn of the century. The new chapter began about 1920 with the extended studies of tensor analysis, Riemannian geometry and its generalizations, and the application of the theory of continuous groups to the new physical theories. Eisenhart has thus developed a remarkable body of original material and has notably served his colleagues by frequent surveys of fields in which he had become a specialist. Eisenhart had a long association with the American Mathematical Society being Vice-President in 1914, Colloquium lecturer in 1925 when he lectured on non-Riemannian geometry, he edited the Transactions of the American Mathematical Society from 1917 to 1923, being managing editor in 1920-23, and was President from 1931 to 1932. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Luther Eisenhart (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1931 - 1932

AMS Colloquium Lecturer

1925

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eisenhart.html (2 of 3) [2/16/2002 11:08:36 PM]

Eisenhart

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eisenhart.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eisenhart.html (3 of 3) [2/16/2002 11:08:36 PM]

Eisenstein

Ferdinand Gotthold Max Eisenstein Born: 16 April 1823 in Berlin, Germany Died: 11 Oct 1852 in Berlin, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gotthold Eisenstein's father was Johan Konstantin Eisenstein and his mother was Helene Pollack. The family were Jewish but before Gotthold, who was their first child, was born they had converted from Judaism to become Protestants. They family were not well off for Johan Eisenstein, after serving in the Prussian army for eight years, found it hard to adjust to a steady job in civilian life. Despite trying a variety of jobs he did not find a successful occupation for most of his life, although towards the end of his life things did go right for him. Eisenstein suffered all his life from bad health but at least he survived childhood which none of his five brothers and sisters succeeded in doing. All of them died of meningitis, and Gotthold himself also contracted the disease but he survived it. This disease and the many others which he suffered from as a child certainly had a psychological as well as a physical effect on him and he was a hypochondriac all his life. His mother, Helene Eisenstein, had a major role in her son's early education. He wrote an autobiography when about two years old and in it he describes the way that his mother taught him the alphabet when he was a child, associating objects with each letter to suggest their shape, like a door for O and a key for K. He also describes his early talent for mathematics in these autobiographical writings (see for example [1]):As a boy of six I could understand the proof of a mathematical theorem more readily than that meat had to be cut with one's knife, not one's fork. He also showed a considerable talent for music from a young age and he played the piano and composed music throughout his life.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eisenstein.html (1 of 6) [2/16/2002 11:08:38 PM]

Eisenstein

While he was at elementary school he had health problems but these may have had a lot to do with the schools which he attended. When he was about ten years old his parents tried to find a solution to his continual health problems by sending him to Cauer Academy in Charlottenburg, a district of Berlin which was not incorporated into the city until 1920. This school adopted an almost military style of discipline and a strict formal approach to education which did nothing for Eisenstein's creative nature. Rather than improve his health problem, it had the opposite effect and in addition to continuing physical illnesses he suffered from depression. In 1837, when he was fourteen years old, Eisenstein entered the Friedrich Wilhelm Gymnasium then moved to the Friedrich Werder Gymnasium in Berlin to complete his schooling. His mathematical talents were recognised by his teachers as soon as he entered the Friedrich Wilhelm Gymnasium and his teachers gave him every encouragement. However, he soon went well beyond the school syllabus in mathematics and from the age of fifteen he was buying mathematics books to study on his own. He began by learning the differential and integral calculus from the works of Euler and Lagrange. By the time he was seventeen, although he was still at school, he began to attend lectures by Dirichlet and other mathematicians at the University of Berlin. It was around this time that his father, having failed to find satisfactory employment in Germany, went to England to try to find a better life. Eisenstein remained at school in Berlin becoming more and more devoted to mathematics. He wrote in his autobiography about the reasons that he was so attracted to mathematics:What attracted me so strongly and exclusively to mathematics, apart from the actual content, was particularly the specific nature of the mental processes by which mathematical concepts are handled. This way of deducing and discovering new truths from old ones, and the extraordinary clarity and self-evidence of the theorems, the ingeniousness of the ideas ... had an irresistible fascination for me. Beginning from the individual theorems, I grew accustomed to delve more deeply into their relationships and to grasp whole theories as a single entity. That is how I conceived the idea of mathematical beauty ... In 1842 he bought a French translation of Gauss's Disquisitiones arithmeticae and, like Dirichlet, he became fascinated by the number theory which he read there. In the summer of 1842, before taking his final school examinations, he travelled with his mother to England where they joined his father who was searching for a better life. In [12] Warnecke argues that during this visit to England Eisenstein became familiar with applied technology and science which aroused his interest in mathematics generally and in particular contributed to desire to become a mathematician. The family tried spending time in Wales and Ireland but Eisenstein's father could not find the right job to give him satisfaction and financial security. As they moved from place to place Eisenstein read Disquisitiones arithmeticae and played the piano whenever it was possible. While in Ireland in 1843 Eisenstein met Hamilton in Dublin, a city he would have dearly liked to have settled in, and Hamilton gave him a copy of a paper that he had written on Abel's work on the impossibility of solving quintic equations. This further stimulated Eisenstein to begin research in mathematics. In June 1843 Eisenstein returned to Germany with his mother who separated from his father at this time. Eisenstein applied to take his final school examinations and was allowed to do so in August/September. He graduated with a glowing report from his mathematics teacher [1]:His knowledge of mathematics goes far beyond the scope of the secondary school

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eisenstein.html (2 of 6) [2/16/2002 11:08:38 PM]

Eisenstein

curriculum. His talent and zeal lead one to expect that some day he will make an important contribution to the development and expansion of science. His teacher, Schellbach, was right and it would not be long before his expectations were fulfilled. Eisenstein enrolled at the University of Berlin in the autumn of 1843 and in January 1844 he delivered Hamilton's paper to the Berlin Academy. At the same time as he submitted to the Berlin Academy his own paper on cubic forms with two variables. He was working on a variety of topics at this time including quadratic forms and cubic forms, the reciprocity theorem for cubic residues, quadratic partition of prime numbers and reciprocity laws. Crelle was appointed as referee for Eisenstein's paper and, with his usual intuition for spotting young mathematical talent, Crelle immediately realised that here was a potential genius. Crelle communicated with Alexander von Humboldt who also took immediate note of the extraordinarily talented youngster. Eisenstein met von Humboldt in March 1844. Eisenstein's financial position was poor and von Humboldt went out of his way to obtain grants from the King, the Prussian government, and the Berlin Academy. These were given somewhat grudgingly, always for a short period, arriving late and rather lacking generosity. Had it not been for von Humboldt's personal generosity, Eisenstein would have had a harder time than in fact he had. But Eisenstein was a sensitive person and he was not happy to receive the grants, particularly when he felt that the official ones were given grudgingly. The authorities should certainly have been pleased with the return for their money since Eisenstein published 23 papers and two problems in Crelle's Journal in 1844. In June 1844 Eisenstein went to Göttingen for two weeks to visit Gauss. Gauss had a reputation for being extremely hard to impress, but Eisenstein had sent some of his papers to Gauss before the visit and Gauss was full of praise. At this time Eisenstein was working on a variety of topics including quadratic and cubic forms and the reciprocity theorem for cubic residues. It was a highly successful visit and Eisenstein made a friend at Göttingen, namely Moritz Stern. Despite the instant international fame that Göttingen achieved while still in his first year at university, he was depressed and this depression would only grow worse through his short life. Kummer arranged that the University of Breslaw award Eisenstein an honorary doctorate in February 1845. Jacobi had also been involved in arranging this honour, but Eisenstein and Jacobi were not always on the best of terms having a very up and down relationship. From 1846 to 1847 Eisenstein worked on elliptic functions and in the first of these years he was involved in a priority dispute with Jacobi. He wrote to Stern explaining the situation (see for example [1]):... the whole trouble is that, when I learned of [Jacobi's] work on cyclotomy, I did not immediately and publicly acknowledge him as the originator, while I frequently have done this in the case of Gauss. That I omitted to do so in this instance is merely the fault of my naive innocence. In 1847 Eisenstein received his habilitation from the University of Berlin and began to lecture. Riemann attended lectures that he gave on elliptic functions in that year and we comment below on possible interaction between Riemann and Eisenstein at this time. By 1848 conditions were bad in the German Confederation. Unemployment and crop failures had led to discontent and disturbances. The news that Louis-Philippe had been overthrown by an uprising in Paris http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eisenstein.html (3 of 6) [2/16/2002 11:08:38 PM]

Eisenstein

in February 1848 led to revolutions in many states and there was fighting in Berlin. Republican and socialist feelings meant that the monarchy was in trouble. Eisenstein attended some pro-democracy meetings but did not play any active political role. However, on 19 March 1848, during street fighting in Berlin shots were fired on the King's troops from a house which Eisenstein was in (although it was not his own house) and he was arrested. He was released on the following day but the severe treatment which he had received caused a sharp deterioration in his already delicate health. The arrest had another bad side effect for it convinced those funding him that he had republican sympathies and it became much harder for him to obtain money although von Humboldt continued to strenuously support him. Writing of his mathematical works written during this period Weil writes in [3]:As any reader of Eisenstein must realise, he felt hard pressed for time during the whole of his short mathematical career. ... His papers, although brilliantly conceived, must have been written by fits and starts, with the details worked out only as the occasion arose; sometimes a development is cut short, only to be taken up again at a later stage. Occasionally Crelle let him send part of a paper to the press before the whole was finished. One is frequently reminded of Galois' tragic remark 'Je n'ai pas le temps'. Despite his health problems Eisenstein published one treatise after another on quadratic partition of prime numbers and reciprocity laws. He was receiving many honours, for example Gauss proposed Eisenstein for election to the Göttingen Academy and he was elected in 1851. Early in 1852, at Dirichlet's request, Eisenstein was elected to the Berlin Academy. Eisenstein died of pulmonary tuberculosis at the age of 29. His great supporter Alexander von Humboldt, by that time 83 years of age, followed Eisenstein's coffin at the cemetery. He had successfully obtained funds to allow Eisenstein to spend time in Sicily in order to recover his health, but it was too late. There are three major areas of mathematics to which Eisenstein contributed and we have already mentioned them above. He worked on the theory of forms with the aim of generalising the results obtained by Gauss in Disquisitiones arithmeticae for the theory of quadratic forms. He examined the higher reciprocity laws, with the aim of generalising Gauss's results on quadratic reciprocity, again contained in Disquisitiones arithmeticae. In his work on this topic Eisenstein used Kummer's theory of ideals. The work of both Kummer and Eisenstein, and the rivalry which existed between the two in their work published in 1850 on the higher reciprocity laws, is discussed in [7]. These two topics on which Eisenstein worked were both strongly motivated by Gauss's Disquisitiones arithmeticae and the paper [13] discusses the copy of this work which Eisenstein owned from his days at school which is now in the mathematical library in Giessen. In the paper [13] Weil examines the annotations in the book made by Eisenstein and conjectures that Riemann received ideas in conversations with Eisenstein which led to his famous paper on the zeta function. The third topic to which Eisenstein made a major contribution was the theory of elliptic functions. Weil writes in [3]:Eisenstein, having laid the foundations for a theory of elliptic functions, was able to carry out much of his design for the building itself, and to indicate how he wished it completed. Although the topic was pushed forward greatly by Abel and Jacobi, Eisenstein's paper on the topic in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eisenstein.html (4 of 6) [2/16/2002 11:08:38 PM]

Eisenstein

1847 [10]:... developed his own independent analytic theory of elliptic functions, based on the technique of summing certain conditionally convergent series. Kronecker wrote (see for example [3]):Essentially new points of view ... particularly concerning the transformation theory of theta-functions ...were introduced by Eisenstein in the fundamental but seldom quoted "Beitrage zur Theorie der elliptischen Funktionen" published in Crelle's Journal in 1847, which are based upon entirely original ideas ... In fact the book [3], the first edition of which appeared in 1976 and was the result of a course given at the Institute for Advanced Study at Princeton in 1974, is devoted to this approach. Kronecker took up these themes [3]:Eisenstein's major themes, properly modulated, lend themselves to a large number of interesting variations; ... much of Kronecker's best work consists of such variations ... This book by Weil shows that Eisenstein's approach is of major importance to the mathematics which is being developed today, a great tribute to a genius who died 150 years ago. Often the power of an approach is illustrated by insight that it adds to simpler well understood cases and indeed this is well illustrated by Weil:As Eisenstein shows, his method for constructing elliptic functions applies beautifully to the simpler case of trigonometric functions. Moreover, this case provides not merely an illuminating introduction to his theory, but also the simplest proofs for a series of results, originally discussed by Euler ... Finally we quote from [10] on the same theme of the relevance of Eisenstein's work today:Looking back from today's vantage, Eisenstein's mathematics appear to us more up to date than ever. It is not so much the harvest of theorems, nor the creation of full-fledged theories, but the way of looking at things which amazes us ... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) A Poster of Gotthold Eisenstein

Mathematicians born in the same country

Cross-references to History Topics

Matrices and determinants

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eisenstein.html (5 of 6) [2/16/2002 11:08:38 PM]

Eisenstein

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eisenstein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eisenstein.html (6 of 6) [2/16/2002 11:08:38 PM]

Elliott

Edwin Bailey Elliott Born: 1 June 1851 in Oxford, England Died: 21 July 1937 in Oxford, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edwin Elliott was educated at Magdalen School in Oxford then, in 1869, he entered Magdalen College of the University of Oxford to study mathematics. After outstanding achievements at university, Elliott became a Fellow and Mathematical Tutor of Queen's College, Oxford in 1874. In addition to his Fellowship at Queen's College, Elliott was appointed a lecturer in mathematics at Corpus Christi College in Oxford in 1884. These appointments came to an end in 1892 when Elliott became the first Waynflete professor of Pure Mathematics. This chair was named after William of Waynflete, the English lord chancellor and bishop of Winchester who founded Magdalen College in the 15th century. The Waynflete chair came with a Fellowship at Magdalen College so Elliott was again attached to his old College. Elliott held the Waynflete chair for 29 years until his retirement in 1921. During this time he was much involved with the London Mathematical Society, being President of the Society from 1896 to 1898. A few years before this, in 1891, he had been honoured by being elected a Fellow of the Royal Society. As Chaundy writes in [1]:Elliott's mathematical life circulated round the twin foci of Oxford and London. Besides his work in formal teaching and lecturing at Oxford, he was one of the founders (1888) of the Oxford Mathematical Society, its first secretary, and later its president. His mathematical work included algebra, algebraic geometry, synthetic geometry, elliptic functions and the theory of convergence. However his most important contribution was the book An introduction to the algebra of quantics which was first published in 1895. This work was a major contribution to invariant

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Elliott.html (1 of 3) [2/16/2002 11:08:39 PM]

Elliott

theory. In the book Elliott presented a wealth of material on invariant theory which had been developed on the Continent of Europe, but he presented it in a style which was more familiar to English mathematicians familiar with the work of Cayley and Sylvester. It was a popular book and, in 1913, Elliott published a second revised edition. All ageing mathematicians should be particularly pleased to learn that a second piece of work by Elliott, which was again of major importance, was his contribution to the theory of integral equations which he made after he retired. In fact he was aged 77 when he produced these important results. In [1] his work is described as:... like all good mathematics .. distinguished by simplicity and naturalness, surprising results being often achieved by the exploitation, with real insight, of ideas in themselves elementary. Elliott wrote well and he always produced clear rigorous arguments. He could not abide sloppy mathematics and anyone producing hand-waving proofs was likely to be severely criticised by him. He was not only critical of others, in many ways he was his own hardest critic. Outside mathematics, Elliott had many interests. Music, natural history and literature were all high on his list of hobbies. However these were often more than mere hobbies. For example he founded the oldest literary society in Queen's College. Chaundy, in [1], describes him as:... modest and retiring, hesitant in speech, unfailing in helpfulness to others, a much loved man. It is fair to say, however, that his impact in the development of mathematics was less than might be expected of someone of his great mathematical gifts. This is almost certainly because he was rather an old-fashioned mathematician whose work:... looked back to a closing epoch. It was typically English, it could be called Victorian; it lacked sympathy with more recent developments, but it had honesty and dignity... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Elliott.html (2 of 3) [2/16/2002 11:08:39 PM]

Elliott

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Elliott.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Elliott.html (3 of 3) [2/16/2002 11:08:39 PM]

Empedocles

Empedocles of Acragas Born: about 492 BC in Acragas (now Agrigento, Sicily,Italy) Died: about 432 BC in Peloponnese, Greece Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Empedocles was born in Acragas on the south coast of Sicily. The name Acragas is Greek, while the Latin name for the town was Agrigentum. Later the town was called Girgenti and more recently it became known by its present name of Agrigento. It was one of the most beautiful cities of the ancient world up to the time it was destroyed by the Carthaginians in 406 BC. It was, in Empedocles time, a rich city containing the finest Greek culture. Some of the Pythagoreans had come there after being attacked in their centre at Croton. Empedocles was born into a rich aristocratic family. He travelled throughout the Greek world participating fully in the extraordinary desire for learning and understanding which gripped that part of the world. He is described as follows by Sarton [5]:He was not only a philosopher but a poet, a seer, a physicist, a social reformer, a man of so much enthusiasm that he would easily be considered a charlatan by some people, or become a legendary hero in the eyes of others. There are many legends regarding Empedocles life. He wrote poetry and 450 lines of such had been preserved by later writers such as Simplicius, Aristotle, Plutarch and others. It is not difficult to see the source of most of the legends about Empedocles for these are built on the poems that he wrote himself. In these he claims god-like powers, but whether this was simply a poetic style or whether he really did believe that he had such powers it is hard to say. Certainly his poems were much appreciated, for example Lucretius admired his hexametric poetry. If we are the gather anything about the character of the man then it will come from the lines of poetry which have been preserved: 400 lines from his poem Peri physeos (On Nature) and the remainder from his poem Katharmoi (Purifications). These [1]:... reveal a man of fervid imagination, versatility, and eloquence, with a touch of theatricality. Some details of his travels appear accurate. He went to Italy and was in the town of Thurii, Lucania shortly after 445 BC. From there he went to the Peloponnese and he was in Olympia in 440 BC. His songs were sung at the Olympic games in that year. He had a young friend, Pausanias the son of Anchitos, who went with him on his travels. Of the many legends regarding his death, the most likely would appear to be that he died following a feast in the Peloponnese. Sarton writes [5]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Empedocles.html (1 of 3) [2/16/2002 11:08:41 PM]

Empedocles

Empedocles was so great and rare a man that he left no school; none of his admirers or disciples, not even the faithful Pausanias, was able to continue the master's work. Certainly Empedocles was attributed with many "firsts". Aristotle is said to have considered him the inventor of rhetoric while Galen regarded him as the founder of the science of medicine in Italy. He is best known, however, for his belief that all matter was composed of four elements: fire, air, water, and earth. The reason for his four element theory was to argue a modification of the belief of the Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, which had been founded by Parmenides in Elea in southern Italy. The philosophy of this school, which included Zeno of Elea, was the claim that the many things which appear to exist are merely a single eternal reality. Empedocles did not go for the "all is one" version, but his "all is composed of the four elements" is extremely important in the development of science since it was adopted by Plato and Aristotle. As Sarton writes [5]:In spite of its arbitrariness, that hypothesis had a singular fortune, for it dominated Western thought in one form or another almost until the eighteenth century. We should also note an important feature of the hypothesis. It, like the ideas of Pythagoras, tried to explain the multitude of complexity seen in the world as being the consequence of a small number of simple underlying properties. Although we no longer believe in Empedocles' four element theory, we do still look for simple mathematics which will explain the complex phenomena that surround us. Empedocles did not base his four element hypothesis on any experimental evidence. He did base some other scientific ideas on experiment, however, and he showed by experiment that air existed and was not empty space. He did this with a clepsydra, a vessel with a hole in the bottom and one in the top. Placing the bottom hole of the vessel under water, Empedocles observed that the vessel filled up with water. If, however, he put his finger over the top hole, then the water did not enter the hole at the bottom but it did once he removed his finger. Empedocles correctly deduced that the air in the container prevented the water entering. Empedocles believed that light travelled with a finite velocity, not through any experimental evidence, of course, but simply through reasoning. Aristotle writes in De sensu :Empedocles says that the light from the Sun arrives first in the intervening space before it comes to the eye, or reaches the Earth. This might plausibly seem to be the case. For whatever is moved through space, is moved from one place to another; hence, there must be a corresponding interval of time also in which it is moved from the one place to the other. But any given time is divisible into parts; so that we should assume a time when the sun's ray was not as yet seen, but was still travelling in the middle space. It is remarkable how many of Empedocles' ideas have turned out to be correct. In addition to his belief in the finite velocity of light he also developed a crude evolutionary theory based on the survival of the fittest. He also had a form of the law of conservation of energy and had a theory of constant proportions in chemical reactions. His ideas, although they had little influence on the development of science, can be seen in the light of our current scientific knowledge to be quite incredible. If we have to explain how such prophetically correct ideas could have such little influence we have to agree with the philosopher Hans Reichenbach who, writing in 1957, said (see [1]):... a good idea stated within an insufficient theoretical frame loses its explanatory power

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Empedocles.html (2 of 3) [2/16/2002 11:08:41 PM]

Empedocles

and is forgotten. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Other Web sites

1. Internet Encyclopedia of Philosophy 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Empedocles.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Empedocles.html (3 of 3) [2/16/2002 11:08:41 PM]

Engel

Friedrich Engel Born: 26 Dec 1861 in Lugau (near Chemnitz), Germany Died: 29 Sept 1941 in Giessen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Friedrich Engel was the son of a Lutheran pastor. He attended the Gymnasium at Greiz, one of the chief towns of Reuss Oberland, from 1872 to 1879. His university studies began in 1879 and he attended the University of Leipzig and the University of Berlin. He received his doctorate from Leipzig in 1883 having studied under Adolph Mayer working on contact transformations. At Leipzig, Engel was taught by Klein who recognised that he was the right man to assist Lie. At Klein's suggestion Engel went to work with Lie in Christiania (now Oslo) from 1884 until 1885. In 1885 Engel's Habilitation thesis was accepted by Leipzig and he became a lecturer there. The year after Engel returned to Leipzig from Christiania, Lie was appointed to succeed Klein and the collaboration of Lie and Engel continued. In 1889 Engel was promoted to assistant professor and, ten years later he was promoted to associate professor. In 1904 he accepted the chair of mathematics at Greifswald when his friend Eduard Study resigned the chair. Engel's final post was the chair of mathematics at Giessen which he accepted in 1913 and he remained there for the rest of his life. In 1931 he retired from the university but continued to work in Giessen. The collaboration between Engel and Lie led to Theorie der Transformationsgruppen a work on three volumes published between 1888 and 1893. This work was:... prepared by S Lie with the cooperation of F Engel... In many ways it was Engel who put Lie's ideas into a coherent form and made them widely accessible. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Engel.html (1 of 2) [2/16/2002 11:08:42 PM]

Engel

From 1922 to 1937 Engel published Lie's collected works in six volumes and prepared a seventh (which in fact was not published until 1960). In [1] Engel's efforts in producing Lie's collected works are described as:... an exceptional service to mathematics in particular, and scholarship in general. Lie's peculiar nature made it necessary for his works to be elucidated by one who knew them intimately and thus Engel's 'Annotations' completed in scope with the text itself. Engel also edited Hermann Grassmann's complete works and really only after this was published did Grassmann get the fame which his work deserved. Engel collaborated with Stäckel in studying the history of non-euclidean geometry. He also wrote on continuous groups and partial differential equations, translated works of Lobachevsky from Russian to German, wrote on discrete groups, Pfaffian equations and other topics. He received many honours for his work including the Lobachevsky Gold Medal and the Norwegian Order of St Olaf. In addition he was awarded an honorary doctorate from the University of Oslo. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) A Poster of Friedrich Engel

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1880 to 1890

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Engel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Engel.html (2 of 2) [2/16/2002 11:08:42 PM]

Enriques

Federigo Enriques Born: 5 Jan 1871 in Leghorn (now Livorno), Tuscany, Italy Died: 14 June 1946 in Rome, Italy

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Federigo Enriques's family moved from Livorno to Pisa where Federigo was educated. He entered the University of Pisa, also studying at the Scuola Normale in Pisa and he was awarded his degree in 1891. He was fortunate to have been taught by Betti in Pisa. In 1892 Enriques asked Castelnuovo, who was in Rome, for advice on which direction his research should take. He took Castelnuovo's advice and worked on algebraic surfaces, sometimes collaborating with Castelnuovo. After his degree, Enriques continued to study at Pisa for a year before moving to Rome to work with Castelnuovo. Again he spent a year studying before moving again, this time to Turin where he worked with Corrado Segre. As well as Betti and Corrado Segre, Enriques had been taught by Dini, Bianchi and Volterra. Enriques was appointed to the University of Bologna where he taught projective geometry and descriptive geometry. Appointed a professor at Bologna in 1896 he remained there until 1923 when he accepted the chair of higher geometry at the University of Rome. While the Fascist regime was in power Enriques felt that he could not work with them and he retired from teaching from 1938 until 1944. Enriques made important contributions to geometry and to the history and philosophy of mathematics. He produced a series of papers over a period of 20 years which, together with Castelnuovo, finally produced a classification of algebraic surfaces. His work on algebraic surfaces gained world-wide recognition when it was highlighted by H F Baker in his presidential address to the International http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Enriques.html (1 of 2) [2/16/2002 11:08:44 PM]

Enriques

Congress in Cambridge in 1912. Another topic which Enriques worked on was differential geometry. In this area he also won fame with the joint award of the Bordin prize to him and Severi in 1907 for work on hyperelliptic surfaces. The foundations of mathematics had always interested Enriques, and, at Klein's request, he wrote an article on the foundations of geometry. His interest also extended to psychology when he asked questions such as:What leads a mathematician to make a conjecture? His book Problemi della scienza written in 1906, stressed the unifying aspect of scientific theories, the association of ideas and of scientific representation. He writes:It is plainly seen that scientific questions include something essential, apart from the special way in which they are conceived in a particular epoch by the scholars who study such problems. ... In the formulation of concepts, we shall see not only economy of thought ... but also a somewhat determinate mental process ... . In addition to his research work, Enriques also wrote textbooks for schools. He was president of the Italian Philosophical Society from 1907 until 1913 and during that time he organised the fourth international congress of philosophy in Bologna in 1911. Enriques was awarded an honorary degree by the University of St Andrews. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (32 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Enriques.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Enriques.html (2 of 2) [2/16/2002 11:08:44 PM]

Enskog

David Enskog Born: 22 April 1884 in Västra Ämtervik, Värmland, Sweden Died: 1 June 1947 in Stockholm, Sweden

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

After completing his secondary schooling, David Enskog entered Uppsala University. He continued to study there for his doctorate after taking his first degree. In 1917 he was awarded his Ph.D. and then he spent several years teaching in secondary schools and colleges. In 1930 Enskog was appointed professor of mathematics and mechanics at the Royal Institute of Technology in Stockholm. Enskog worked on the Maxwell- Boltzmann equations. These had first been formulated by Maxwell in 1867 to describe the flow of molecules, momentum and energy of a gas. This was reformulated by Boltzmann in 1872 in terms of a velocity distribution function. Enskog began to work on this equation for his master's degree at Uppsala and made a remarkable prediction. If a mixture of two gases is subjected to a temperature difference, the gas with the larger molecules concentrates at the lower temperature. A simple theory does not predict this behaviour. However Enskog predicted it in a paper written in 1911. In 1917 Chapman independently predicted it, but their theory was questioned until Chapman persuaded a chemist F W Dootson to conduct experiments; the theory was verified. Hilbert published a new approach to the Maxwell- Boltzmann equations in 1912. Enskog used Hilbert's methods to work out a series expansion of the velocity distribution function and wrote this up for his doctoral dissertation at Uppsala in 1917. How to extend the Maxwell- Boltzmann equation to include collisions of more than two bodies was not http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Enskog.html (1 of 2) [2/16/2002 11:08:46 PM]

Enskog

clear. However Enskog made an important advance in 1921, although it described the rather artificial situation treating molecules as hard spheres. Chapman, who was still working on the Maxwell- Boltzmann equations, saw the importance of Enskog's methods and developed them further. The book S Chapman and T G Cowling, The Mathematical Theory of Non-uniform Gases is the classic text on the modern kinetic theory of gases based on the approach by Enskog and Chapman. Although the advent of quantum theory was to lessen the impact of this theory, it was later seen to be still important in the new context. Chapman recommended Enskog for the chair at the Royal Institute of Technology. However Chapman wrote later:His transfer to a university chair seemed rather to bring him new duties than increased leisure, and this, with renewed ill-health, reduced his productivity in later years. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Enskog.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Enskog.html (2 of 2) [2/16/2002 11:08:46 PM]

Eotvos

Lóránd Baron von Eötvös Born: 27 July 1848 in Pest (now part of Budapest), Hungary Died: 8 April 1919 in Budapest, Hungary

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lóránd Eötvös studied at Heidelberg where he was taught by Kirchhoff, Helmholtz and Bunsen. Then he went to Königsberg and studied under Franz Neumann and Friedrich Richelot. He returned to Hungary and received a doctorate with a thesis which studied problems of Fizeau on the relative motion of a light source. This was one of the first steps towards relativity theory. Eötvös went back to Hungary in 1871 and taught at the University of Budapest where he became professor of experimental physics in 1878. He published on capillarity between 1876 and1886, then he published on gravitation for the rest of his life. He invented the Eötvös balance and showed that, to a high degree of accuracy, gravitational mass and inertial mass are equivalent. Eötvös founded the Hungarian Society for Mathematics in 1885 and he was important in improving educational standards in Hungary. What was once the Péter Pázmány University in Budapest is now known as the Lóránd Eötvös University. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eotvos.html (1 of 2) [2/16/2002 11:08:48 PM]

Eotvos

Honours awarded to Lóránd Eötvös (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Eotvos

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eotvos.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eotvos.html (2 of 2) [2/16/2002 11:08:48 PM]

Epstein

Paul Epstein Born: 24 July 1871 in Frankfurt, Germany Died: 11 Aug 1939 in Dornbusch, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Epstein was brought up in a Jewish family in Frankfurt where his father was a professor at the Philanthropin Academy. After submitting a thesis on abelian functions, he received his doctorate in 1895 from the University of Strasbourg. The city was German at this time (and called Strassburg) and it had been since it was annexed by Germany during the Franco-German War of 1870-71. From 1895 to 1918 he remained in Strasbourg, teaching at the Technical School and also at the University where he had been appointed a Privatdozent. During World War I he did military service. At the end of the war in 1918, however, the city of Strasbourg reverted to France, and Epstein, being German, was forced to leave Alsace. He returned to his native city of Frankfurt. Epstein was appointed to a non-tenured post at the university and he lectured in Frankfurt from 1919. Later he was appointed professor at Frankfurt. On 30 January 1933, however, Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. However, there was an exemption clause which exempted non-Aryans who had fought for Germany in World War I. Epstein certainly qualified under this clause and this allowed him to keep his lecturing post in Frankfurt in 1933. Decisions at the Nuremberg party congress in the autumn of 1935 made it clear that non-Aryans would no longer be able to keep their posts even if they had served in World War I. Siegel writes in [2]:Epstein voluntarily relinquished his teaching position before the Nuremberg laws went into effect. As he explained to me, he had wanted to save the German authorities the trouble of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Epstein.html (1 of 3) [2/16/2002 11:08:50 PM]

Epstein

doing to him what the French had done back in 1918. Epstein did not attempt to emigrate. He was 64 years old and had he emigrated he would have lost all his money except 10 Marks. There was no prospect of a 64 year old obtaining a post. On the Kristallnacht (so called because of the broken glass in the streets on the following morning), the 9-10 November 1938, 91 Jews were murdered, hundreds were seriously injured, and thousands were subjected to horrifying experiences. Thousands of Jewish businesses were burnt down together with over 150 synagogues. The Gestapo arrested 30,000 well-off Jews and a condition of their release was that they emigrate. The Gestapo broke into Epstein's house but found that he was seriously ill and could not be moved. At this point Epstein must have known that his only chance was to leave Germany. It would have been posssible for [2]:... one of his sisters had emigrated earlier and could have supported him. But despite the possibility of escape, he hesitated leaving his books and his native city. He moved to Dornbusch and was visited there by Siegel [2]:... we sat in the sunny garden of the house he was living in then. ... he pointed to the trees and flowers in the garden and said "Isn't it lovely here". About a week after Siegel's visit, Epstein received a summons from the Gestapo. He knew what had happened to others who had received such a summons, many had been tortured and killed. He wanted to avoid the suffering so he took a lethal dose of Veronal. The Gestapo later claimed that they had only summoned him to get him to sign a document to fix a date on which he would emigrate. His work was in number theory, in particular the zeta function. He also worked on the history of mathematics. Perhaps we should mention one other of Epstein's talents which was music, and he took part energetically in the cultural life of Frankfurt. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Epstein.html (2 of 3) [2/16/2002 11:08:50 PM]

Epstein

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Epstein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Epstein.html (3 of 3) [2/16/2002 11:08:50 PM]

Eratosthenes

Eratosthenes of Cyrene Born: 276 BC in Cyrene, North Africa (now Shahhat, Libya) Died: 194 BC in Alexandria, Egypt

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eratosthenes was born in Cyrene which is now in Libya in North Africa. His teachers included the scholar Lysanias of Cyrene and the philosopher Ariston of Chios who had studied under Zeno, the founder of the Stoic school of philosophy. Eratosthenes also studied under the poet and scholar Callimachus who had also been born in Cyrene. Eratosthenes then spent some years studying in Athens. The library at Alexandria was planned by Ptolemy I Soter and the project came to fruition under his son Ptolemy II Philadelphus. The library was based on copies of the works in the library of Aristotle. Ptolemy II Philadelphus appointed one of Eratosthenes' teachers Callimachus as the second librarian. When Ptolemy III Euergetes succeeded his father in 245 BC and he persuaded Eratosthenes to go to Alexandria as the tutor of his son Philopator. On the death of Callimachus in about 240 BC, Eratosthenes became the third librarian at Alexandria, in the library in a temple of the Muses called the Mouseion. The library is said to have contained hundreds of thousands of papyrus and vellum scrolls. Despite being a leading all-round scholar, Eratosthenes was considered to fall short of the highest rank. Heath writes [4]:[Eratosthenes] was, indeed, recognised by his contemporaries as a man of great distinction in all branches of knowledge, though in each subject he just fell short of the highest place. On the latter ground he was called Beta, and another nickname applied to him, Pentathlos, has the same implication, representing as it does an all-round athlete who was not the first runner or wrestler but took the second prize in these contests as well as others. Certainly this is a harsh nickname to give to a man whose accomplishments in many different areas are remembered today not only as historically important but, remarkably in many cases, still providing a basis for modern scientific methods.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eratosthenes.html (1 of 5) [2/16/2002 11:08:52 PM]

Eratosthenes

One of the important works of Eratosthenes was Platonicus which dealt with the mathematics which underlie Plato's philosophy. This work was heavily used by Theon of Smyrna when he wrote Expositio rerum mathematicarum and, although Platonicus is now lost, Theon of Smyrna tells us that Eratosthenes' work studied the basic definitions of geometry and arithmetic, as well as covering such topics as music. One rather surprising source of information concerning Eratosthenes is from a forged letter. In his commentary on Proposition 1 of Archimedes' Sphere and cylinder Book II, Eutocius reproduces a letter reputed to have been written by Eratosthenes to Ptolemy III Euergetes. The letter describes the history of the problem of the duplication of the cube and, in particular, it describes a mechanical device invented by Eratosthenes to find line segments x and y so that, for given segments a and b, a : x = x : y = y : b. By the famous result of Hippocrates it was known that solving the problem of finding two mean proportionals between a number and its double was equivalent to solving the problem of duplicating the cube. Although the letter is a forgery, parts of it are taken from Eratosthenes' own writing. The letter, which occupies an important place in the history of mathematics, is discussed in detail in [14]. An original Arabic text of this letter was once kept in the library of the St Joseph University in Beirut. However it has now vanished and the details given in [14] come from photographs taken of the letter before its disappearance. Other details of what Eratosthenes wrote in Platonicus are given by Theon of Smyrna. In particular he described there the history of the problem of duplicating the cube (see Heath [4]):... when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an alter double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an alter of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry. Eratosthenes erected a column at Alexandria with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [4]:If, good friend, thou mindest to obtain from any small cube a cube the double of it, and duly to change any solid figure into another, this is in thy power; thou canst find the measure of a fold, a pit, or the broad basin of a hollow well, by this method, that is, if thou thus catch between two rulers two means with their extreme ends converging. Do not thou seek to do the difficult business of Archytas's cylinders, or to cut the cone in the triads of Menaechmus, or to compass such a curved form of lines as is described by the god-fearing Eudoxus. Nay thou couldst, on these tablets, easily find a myriad of means, beginning from a small base. Happy art thou, Ptolemy, in that, as a father the equal of his son in youthful vigour, thou hast thyself given him all that is dear to muses and Kings, and may be in the future, O Zeus, god of heaven, also receive the sceptre at thy hands. Thus may it be, and let any one who sees this offering say "This is the gift of Eratosthenes of Cyrene". Eratosthenes also worked on prime numbers. He is remembered for his prime number sieve, the 'Sieve of Eratosthenes' which, in modified form, is still an important tool in number theory research. The sieve appears in the Introduction to arithmetic by Nicomedes.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eratosthenes.html (2 of 5) [2/16/2002 11:08:52 PM]

Eratosthenes

Another book written by Eratosthenes was On means and, although it is now lost, it is mentioned by Pappus as one of the great books of geometry. In the field of geodesy, however, Eratosthenes will always be remembered for his measurements of the Earth. Eratosthenes made a surprisingly accurate measurement of the circumference of the Earth. Details were given in his treatise On the measurement of the Earth which is now lost. However, some details of these calculations appear in works by other authors such as Cleomedes, Theon of Smyrna and Strabo. Eratosthenes compared the noon shadow at midsummer between Syene (now Aswan on the Nile in Egypt) and Alexandria. He assumed that the sun was so far away that its rays were essentially parallel, and then with a knowledge of the distance between Syene and Alexandria, he gave the length of the circumference of the Earth as 250,000 stadia. Of course how accurate this value is depends on the length of the stadium and scholars have argued over this for a long time. The article [11] discusses the various values scholars have given for the stadium. It is certainly true that Eratosthenes obtained a good result, even a remarkable result if one takes 157.2 metres for the stadium as some have deduced from values given by Pliny. It is less good if 166.7 metres was the value used by Eratosthenes as Gulbekian suggests in [11]. Several of the papers referenced, for example [10], [15] and [16], discuss the accuracy of Eratosthenes' result. The paper [15] is particularly interesting. In it Rawlins argues convincingly that the only measurement which Eratosthenes made himself in his calculations was the zenith distance on the summer solstice at Alexandria, and that he obtained the value of 7 12'. Rawlins argues that this is in error by 16' while other data which Eratosthenes used, from unknown sources, was considerably more accurate. Eratosthenes also measured the distance to the sun as 804,000,000 stadia and the distance to the Moon as 780,000 stadia. He computed these distances using data obtained during lunar eclipses. Ptolemy tells us that Eratosthenes measured the tilt of the Earth's axis with great accuracy obtaining the value of 11/83 of 180 , namely 23 51' 15". The value 11/83 has fascinated historians of mathematics, for example the papers [9] and [17] are written just to examine the source of this value. Perhaps the most commonly held view is that the value 11/83 is due to Ptolemy and not to Eratosthenes. Heath [4] argues that Eratosthenes used 24 and that 11/83 of 180 was a refinement due to Ptolemy. Taisbak [17] agrees with attributing 11/83 to Ptolemy although he believes that Eratosthenes used the value 2/15 of 180 . However Rawlins [15] believes that a continued fraction method was used to calculate the value 11/83 while Fowler [9] proposes that the anthyphairesis (or Euclidean algorithm) method was used (see also [3]). Eratosthenes made many other major contributions to the progress of science. He worked out a calendar that included leap years, and he laid the foundations of a systematic chronography of the world when he tried to give the dates of literary and political events from the time of the siege of Troy. He is also said to have compiled a star catalogue containing 675 stars. Eratosthenes made major contributions to geography. He sketched, quite accurately, the route of the Nile to Khartoum, showing the two Ethiopian tributaries. He also suggested that lakes were the source of the river. A study of the Nile had been made by many scholars before Eratosthenes and they had attempted http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eratosthenes.html (3 of 5) [2/16/2002 11:08:52 PM]

Eratosthenes

to explain the rather strange behaviour of the river, but most like Thales were quite wrong in their explanations. Eratosthenes was the first to give what is essentially the correct answer when he suggested that heavy rains sometimes fell in regions near the source of the river and that these would explain the flooding lower down the river. Another contribution that Eratosthenes made to geography was his description of the region "Eudaimon Arabia", now the Yemen, as inhabited by four different races. The situation was somewhat more complicated than that proposed by Eratosthenes, but today the names for the races proposed by Eratosthenes, namely Minaeans, Sabaeans, Qatabanians, and Hadramites, are still used. Eratosthenes writings include the poem Hermes, inspired by astronomy, as well as literary works on the theatre and on ethics which was a favourite topic of the Greeks. Eratosthenes is said to have became blind in old age and it has been claimed that he committed suicide by starvation.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (17 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Longitude and the Académie Royale 2. Greek Astronomy 3. Doubling the cube 4. Prime numbers

Other references in MacTutor

Chronology: 500BC to 1AD

Honours awarded to Eratosthenes (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Eratosthenes

Other Web sites

1. Peter Alfeld 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eratosthenes.html (4 of 5) [2/16/2002 11:08:52 PM]

Eratosthenes

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eratosthenes.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eratosthenes.html (5 of 5) [2/16/2002 11:08:52 PM]

Erdelyi

Arthur Erdélyi Born: 2 Oct 1908 in Budapest, Hungary Died: 12 Dec 1977 in Edinburgh, Scotland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Arthur Erdélyi attended primary and secondary schools in Budapest from 1914 to 1926. His interest in mathematics goes back to this period. As a Jew it was difficult for him to receive a university education in Hungary so he went to Brno, Czechoslovakia to study electrical engineering. However after winning prizes in a mathematics competition in his first year he was persuaded to study mathematics. He began research in mathematics and his first paper was published in 1930. By the end of 1936 Arthur had 18 papers in print, another 11 appearing in 1937. Arthur wrote no doctoral thesis, he merely matriculated at the University of Prague, and submitted his papers instead of a thesis. He was awarded a doctorate in 1938 but because of the Nazi invasion of Czechoslovakia he was told he had to leave the country by the end of 1938 or be sent to a concentration camp. Many of Erdélyi's papers were on the hypergeometric function so it was natural for him to write to Whittaker for help. Whittaker found great difficulties: he had to obtain 400 to support Erdélyi before a visa could be obtained. On 26 January 1939 Erdélyi wrote to Whittaker:Necessity and danger compel me to trouble you once more. ... You know, perhaps, what it means today if a Jew is to be put on the German or Hungarian frontier. However within a few days of writing the letter Erdélyi was in Edinburgh. For two years he held a research grant from Edinburgh then he became a lecturer there. In 1946 Bateman died and Whittaker was asked to recommend someone who could undertake the project of publishing Bateman's manuscripts. Whittaker advised that Erdélyi was the person to undertake the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erdelyi.html (1 of 3) [2/16/2002 11:08:54 PM]

Erdelyi

project and in 1947 Erdélyi arrived in Caltech as Visiting Professor. After returning to Edinburgh for session 1948/49 he resigned in 1949 to take up a chair in California. At Caltech Erdélyi headed a team which produced 3 volumes of Higher Transcendental Functions and 2 volumes of Tables of Integral Transforms. In [2] these works are described as the most widely cited mathematical works of all time and a basic reference source for generations of applied mathematicians and physicists throughout the world. In 1964 he returned to Edinburgh where a second chair of mathematics had been created to provide leadership since Aitken was in very poor health. He remained in Edinburgh until his death, continuing a steady stream of high quality papers up to the time of his death. A list of 178 mathematical papers and articles is given in [2]. Erdélyi was a leading expert on special functions, in particular hypergeometric functions, orthogonal polynomials and Lamé functions. He also worked on asymptotic analysis, fractional integration and singular partial differential equations. In addition to the five volumes which arose from the Bateman project mentioned above, Erdélyi wrote two other texts of major importance Asymptotic expansions (1955) and Operational calculus and generalised functions (1962). Colton in [2] sums up Erdélyi's contribution saying:His reputation was based on much more than his published papers, although this alone would have sufficed to make him one of the leading analysts of his day. It was rather a combination of his mathematical scholarship, his interest and enthusiasm for mathematics, his concern for younger workers, and his willingness to devote his time in aid of the mathematical community that won Erdélyi the admiration and respect of an entire generation of mathematicians. Erdélyi received many honours, the most prestigious being elected a Fellow of the Royal Society in 1975. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Arthur Erdélyi

Mathematicians born in the same country

Honours awarded to Arthur Erdélyi (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1975

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erdelyi.html (2 of 3) [2/16/2002 11:08:54 PM]

Erdelyi

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Erdelyi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erdelyi.html (3 of 3) [2/16/2002 11:08:54 PM]

Erdos

Paul Erdös Born: 26 March 1913 in Budapest, Hungary Died: 20 Sept 1996 in Warsaw, Poland

Click the picture above to see nine larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Erdös came from a Jewish family (the original family name being Engländer) although neither of his parents observed the Jewish religion. Paul's father Lajos and his mother Anna had two daughters, aged three and five, who died of scarlet fever just days before Paul was born. This naturally had the effect making Lajos and Anna extremely protective of Paul. He would be introduced to mathematics by his parents, themselves both teachers of mathematics. Paul was not much over a year old when World War I broke out. Paul's father Lajos was captured by the Russian army as it attacked the Austro-Hungarian troops. He spent six years in captivity in Siberia. As soon as Lajos was captured, with Paul's mother Anna teaching during the day, a German governess was employed to look after Paul. Anna, excessively protective after the loss of her two daughters, kept Paul away from school for much of his early years and a tutor was provided to teach him at home. The situation in Hungary was chaotic at the end of World War I. After a short while as a democratic republic, a communist Béla Kun took over, and Hungary became a left wing Soviet Republic. Anna was at this time made head teacher of her school but when the Communists called for strike action against Kun's regime she continued working, not for political reasons but simply because she did not wish to see children's education suffer. After four months in control of Hungary, Kun fled to Vienna when Romanian troops advanced on Budapest in July 1919. Miklós Horthy, a right-wing nationalist, took over control of the country. He quickly moved against those perceived as Communists and Anna Erdös fell into that category due to her failing to obey the Communist strike call when Kun was in power. She was dismissed from her post and she was left in fear of her life as Horthy's men roamed the streets killing Jews and Communists. By 1920 http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erdos.html (1 of 6) [2/16/2002 11:08:56 PM]

Erdos

Horthy had introduced anti-Jewish laws similar to those Hitler would introduce in Germany thirteen years later. The year 1920 was not all bad for Paul, for his father Lajos returned home from Siberia. He had learnt English to pass the long hours in captivity but, having no English teacher, did not know how to pronounce the words. He now set about teaching Paul to speak English, but the strange English accent which this gave Paul remained one of his characteristics throughout his life. Despite the restrictions on Jews entering universities in Hungary, Erdös, as the winner of a national examination, was allowed to enter in 1930. He studied for his doctorate at the University Pázmány Péter in Budapest. Awarded a doctorate in 1934, he took up a post-doctoral fellowship at Manchester, essentially being forced to leave Hungary because he was Jewish. During his tenure of the fellowship, Erdös travelled widely in the UK. He met Hardy in Cambridge in 1934 and Ulam, also in Cambridge, in 1935. His friendship with Ulam was to prove important later when Erdös was in the United States. The situation in Hungary by the late 1930s clearly made it impossible for someone of Jewish origins to return. However he did visit Budapest three times a year during his tenure of the Manchester fellowship. In March 1938 Hitler took control of Austria and Erdös had to cancel his intended spring visit to Budapest. He did visit during the summer vacation but the Czech crisis on 3 September 1938 made him decide to return hurriedly to England. Within weeks Erdös was on his way to the USA where he took up a fellowship at Princeton. He hoped for his fellowship to be renewed but Erdös did not conform to Princeton's standards so he was offered only a six month extension rather than the expected year. Princeton found him [3]:... uncouth and unconventional... and Ulam invited Erdös to visit Madison to help out. We shall return later to give further details of the strange life which Erdös lived from this time on, devoted exclusively to seeking out and solving good mathematical problems. First we make some comments about his mathematics. The contributions which Erdös made to mathematics were numerous and broad. However, basically Erdös was a solver of problems, not a builder of theories. The problems which attracted him most were problems in combinatorics, graph theory, and number theory. He did not just want to solve problems, however, he wanted to solve them in an elegant and elementary way. To Erdös the proof had to provide insight into why the result was true, not just provide a complicated sequence of steps which would constitute a formal proof yet somehow fail to provide any understanding. Some results with which Erdös is most closely associated had been first proved before Erdös was born. In 1845 Bertrand conjectured that there was always at least one prime between n and 2n for n 2. Chebyshev proved Bertrand's conjecture in 1850 but when Erdös was only an eighteen year old student in Budapest he found an elegant elementary proof of this result. Another result on prime numbers associated with Erdös is the Prime Number Theorem, namely:.. the number of primes n tends to

as n/loge n.

The theorem was conjectured in the 18th century, Chebyshev himself came close to a proof, but it was not proved until 1896, when Hadamard and de la Vallée Poussin independently proved it using complex analysis. In 1949 Erdös and Atle Selberg found an elementary proof. Subsequent events are described in [15]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erdos.html (2 of 6) [2/16/2002 11:08:56 PM]

Erdos

Selberg and Erdös agreed to publish their work in back-to-back papers in the same journal, explaining the work each had done and sharing the credit. But at the last minute Selberg ... raced ahead with his proof and published first. The following year Selberg won the Fields Medal for this work. Erdös was not much concerned with the competitive aspect of mathematics and was philosophical about the episode. This result was typical of the type of mathematics Erdös worked on. He posed and solved problems that were beautiful, simple to understand, but notoriously difficult to solve. Erdös did receive the Cole Prize of the American Mathematical Society in 1951 for his many papers on the theory of numbers, and in particular for the paper On a new method in elementary number theory which leads to an elementary proof of the prime number theorem published in the Proceedings of the National Academy of Sciences in 1949. Whether a rather silly event which took place in August 1941 was to have any real effect on Erdös's life, or whether it was simply used as an excuse, is hard to tell. Erdös and two fellow mathematicians were picked up by the police near a military radio transmitter on Long Island. It was quite an innocent event with the three mathematicians being too absorbed in discussion of mathematics to notice a NO TRESPASSING sign. After a friendly session with the police it was realised that no harm had been intended. However, it gave Erdös an FBI record which was later used against him. Ulam left Madison in 1943 to join other mathematicians and physicists at Los Alamos in New Mexico working on the atomic bomb project. He asked Erdös to join the project but, although he was interested enough to be interviewed, Erdös gave answers to those interviewing him which he must have known were not what they wanted to hear. Erdös was simply too honest in saying that he would wish to return to Budapest at the end of the war. This episode does give the feeling that Erdös never wanted to work at Los Alamos, but was simply amusing himself. In 1943 Erdös worked at Purdue University, taking a part-time appointment. Although it was a difficult time with great uncertainty about the fate of his family in Hungary, yet mathematically Erdös flourished. He had heard nothing from his family between 1941 and the time when Budapest was liberated in 1945. The Jews in Hungary had suffered incredible hardship from 1944 with many being murdered, and others deported to Auschwitz. It is unlikely that the full extent of the horror was understood by Erdös in the United States at the time. However, in August 1945, Erdös received a telegram giving details of his family. His father had died of a heart attack in 1942. His mother had survived while, quite remarkably, a cousin Magda Fredro had been sent to Auschwitz but had survived. The family had suffered terribly through the Nazi campaign against the Jews, however, and four of Erdös's uncles and aunts had been murdered. Near the end of 1948 Erdös was able to return to Hungary for a visit and there he was reunited with his surviving family and friends. For the next three years he travelled frequently between England and the United States before accepting a temporary post at the University of Notre Dame in 1952. It was an inspired offer which gave Erdös complete freedom to rush off to do some joint research whenever he wanted. Erdös could not bring himself to accept the same generous offer on a permanent basis, which both the University of Notre Dame and Erdös's friends tried hard to encourage him to accept. During the early 1950s senator Joseph R McCarthy whipped up strong feelings against communism in the United States. Erdös began to come under suspicion from authorities who saw imaginary problems http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erdos.html (3 of 6) [2/16/2002 11:08:56 PM]

Erdos

everywhere. When asked by US immigration, as he returned after a conference in Amsterdam in 1954, what he thought of Marx, Erdös made the ill judged reply:I'm not competent to judge, but no doubt he was a great man. This was followed by a line of questioning about whether he would ever return to Hungary. Erdös said:I'm not planning to visit Hungary now because I don't know whether they would let me back out. I'm planning to go only to England and Holland. So, was it only the fear of not being let out of Hungary that stopped him going there. Erdös replied innocently:Of course, my mother is there and I have many friends there. Erdös was not allowed back to the United States but no reason was given. The files indicate that the official reasons were not the answers Erdös gave to the above questions, but the fact that he had corresponded with a Chinese mathematician who had subsequently returned from the United States to China and also Erdös's 1941 FBI record. He spent much of the next ten years in Israel. During the early 1960s he made numerous requests to be allowed to return to the United States and a visa was finally granted in November 1963. By this time, however, Erdös had become a traveller moving from one university to another, and from the home of one mathematician to another. However, he did have a home of sorts with his friend Ronald Graham. Erdös and Graham met at a number theory conference in 1963 and soon began a mathematical collaboration. It was Graham who provided a room in his house where Erdös could live when he wanted, he also srored Erdös's papers there and, in many ways, acted as a secretary to Erdös. Although somewhat over the top, the following quote from [12] shows the high regard in which Erdös was held by his fellow mathematicians:Never, mathematicians say, has there been an individual like Paul Erdös. He was one of the century's greatest mathematicians, who posed and solved thorny problems in number theory and other areas and founded the field of discrete mathematics, which is the foundation of computer science. He was also one of the most prolific mathematicians in history, with more than 1,500 papers to his name. And, his friends say, he was also one of the most unusual. Erdös won many prizes including the Wolf Prize of 50 000 dollars in 1983. However he had a lifestyle that needed little money and he gave away:... most of the money he earned from lecturing at mathematics conferences, donating it to help students or as prizes for solving problems he had posed. In 1976 Ulam gave this description of Erdös:He had been a true child prodigy, publishing his first results at the age of eighteen in number theory and in combinatorial analysis. Being Jewish he had to leave Hungary, and as it turned out, this saved his live. In 1941 he was twenty-seven years old, homesick, unhappy, and constantly worried about the fate of his mother who remained in Hungary. ... Erdös is somewhat below medium height, an extremely nervous and agitated person. ... His eyes indicated he was always thinking about mathematics, a process interrupted only by his rather pessimistic statements on world affairs, politics, or human affairs in general, which he viewed darkly. ... His peculiarities are so numerous it is impossible to describe them all. ... Now over sixty, he has more than seven hundred papers to his credit.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erdos.html (4 of 6) [2/16/2002 11:08:56 PM]

Erdos

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (18 books/articles)

Some Quotations (9)

A Poster of Paul Erdös

Mathematicians born in the same country

Other references in MacTutor

1. Prime Number Theorem 2. Chronology: 1940 to 1950

Honours awarded to Paul Erdös (Click a link below for the full list of mathematicians honoured in this way) AMS Cole Prize

Awarded 1951

Other Web sites

1. Odense, Denmark 2. Oakland (The Erdos Number Web site) 3. Erdös's Reciprocal Sum Constants 4. Daily Telegraph (Obituary) 5. Daily Telegraph (Another article) 6. New York Times (Obituary) 7. Washington Post (Obituary) 8. AMS 9. AMS (Erdös's work) 10. AMS 11. MAA (Obituary in Mathland) 12. Paul Hoffman 13. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erdos.html (5 of 6) [2/16/2002 11:08:56 PM]

Erdos

JOC/EFR January 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Erdos.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erdos.html (6 of 6) [2/16/2002 11:08:56 PM]

Erlang

Agner Krarup Erlang Born: 1 Jan 1878 in Lonborg (near Tarm), Jutland, Denmark Died: 3 Feb 1929 in Copenhagen, Denmark

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Agner Erlang was descended on his mother's side from Thomas Fincke. His father was a schoolmaster and Erlang was educated at his father's school when he was young. He took his examinations in Copenhagen at the age of 14 and passed with special distinction after having to obtain special permission to take the examinations because he was below the minimum age. He returned to Lonberg and taught at his father's school for two years. In 1896 he passed the entrance examination to the University of Copenhagen with distinction and, since his parents were poor, he was given free board and lodgings in a College of the University of Copenhagen. His studies at Copenhagen were in mathematics and natural science. He attended the mathematics lectures of Zeuthen and Juel and these gave him an interest in geometrical problems which were to remain with him all his life. After graduating in 1901 with mathematics as his major subject and physics, astronomy and chemistry as secondary subjects, he taught in schools for several years. During this time he kept up his interest in mathematics, and he received an award for an essay on Huygens' solution of infinitesimal problems which he submitted to the University of Copenhagen. His interests turned towards the theory of probability and he kept up his mathematical interests by joining the Mathematical Association. At meetings of the Mathematical Association he met Jensen who was then the chief engineer at the Copenhagen Telephone Company. He persuaded Erlang to apply his skills to the solution of problems which arose from a study of waiting times for telephone calls.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erlang.html (1 of 2) [2/16/2002 11:08:57 PM]

Erlang

In 1908 Erlang joined the Copenhagen Telephone Company and began applying probability to various problems arising in the context of telephone calls. He published his first paper on these problems The theory of probability and telephone conversations in 1909. In 1917 he gave a formula for loss and waiting time which was soon used by telephone companies in many countries including the British Post Office. In addition to his work on probability Erlang was also interested in mathematical tables. This interest is described in [1]:A subject that interested Erlang very much was the calculation and arrangement of numerical tables of mathematical functions, and he had an uncommonly thorough knowledge of the history of mathematical tables from ancient times right up to the present. Erlang set forth a new principle for the calculation of certain forms of mathematical tables, especially tables of logarithms... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other Web sites

Pass Magazine (An article on Erlang and his work)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Erlang.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Erlang.html (2 of 2) [2/16/2002 11:08:57 PM]

Escher

Maurits Cornelius Escher Born: 17 June 1898 in Leeuwarden, Netherlands Died: 27 March 1972 in Laren, Netherlands

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Maurits Escher was always referred to by his parents as Mauk. He was brought up by his father, George Escher, who was a civil engineer, and his second wife Sarah who was the daughter of a government minister. He lived with his four older brothers, Arnold, Johan, Berend, and Edmond. Maurits attended both elementary and secondary school in Arnhem between 1912 and 1918, where he failed to shine in many of his subjects, but exhibited an early interest in both music and carpentry. People expressed the opinion that he possessed a mathematical brain but he never excelled in the subject at any stage during his schooling and treated the subject with some considerable unease. He wrote [7]:At high school in Arnhem, I was extremely poor at arithmetic and algebra because I had, and still have, great difficulty with the abstractions of numbers and letters. When, later, in stereometry [solid geometry], an appeal was made to my imagination, it went a bit better, but in school I never excelled in that subject. But our path through life can take strange turns. Early reports detailed his methodological approach to life which was taken to be an unconscious reaction to his engineering family upbringing. As a child, Maurits always had an intensely creative side and an 'acute sense of wonder'. He often claimed to see shapes that he could relate to in the clouds. Here is the picture Puddle. Maurits, and his good friend Bas Kist both developed a deep interest in printing techniques as a consequence of receiving good reports from their respective art departments who had encouraged their student to experiment. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Escher.html (1 of 8) [2/16/2002 11:09:00 PM]

Escher

Family aspirations that Maurits would train as an architect were disappointed when he failed his final exams in history, constitutional organisations, political economies and book keeping, and as a result he never officially graduated. His family moved to Oosterbeek where a loophole in Dutch law allowed Maurits to enrol at the Higher Technical School in Delft (1918-1919) and thus allowed him to repeat some of the subjects he had failed. Unable and unwilling to catch up following poor health, Maurits decided to concentrate on his drawing and his woodcut techniques. He was influenced and initially trained by R N Roland Holst:He strongly advised me to do some woodcuts, and I immediately followed his advice ... It is wonderful work but far more difficult than working with linoleum. In September 1920 Maurits moved to Haarlem in a final attempt to try follow his father's wish that he study architecture and he enrolled at the School for Architecture and Decorative Arts. A chance meeting with Samuel Jesserum de Mesquita, a graphic arts teacher, proved a landmark event in Escher's life and he became convinced that a graphic arts programme would be better suited to his skills. De Mesquita taught the eager Escher all he knew of woodcut printing techniques, gave him space to experiment, and encouraged him to experiment widely in order to develop his skills. Escher was regularly heard to complain about his lack of natural drawing ability and as a result most of his pieces took a long time to complete, and required numerous attempts before he was completely happy. In his youth he concentrated on landscapes, many of which were drawn from unusual perspectives. He also made numerous sketches of plants and even insects, all of which regularly appear in his later work. These can be seen in, for example in the picture of St Peter's. Travelling took up a large part of Escher's life from this point on. He made a trip with two friends to Florence in April 1922 and spent the whole time sketching and drinking. Escher then spent a further month travelling alone around Italy gathering material to use in his experimental woodcuts. During his early drawing career Escher touched only briefly on the subject of 'filling the plane', signs of which had been visible from an early age. Many years later a lady [7]:... remembered the care with which this little boy [Escher] had selected the shape, quantity and size of his slices of cheese, so that, fitted one against the other, they would cover as exactly as possible the entire slice of bread. This particular trait never left him ... His first work featuring regular division of the plane was named Eight Heads, and was completed in 1922. Here is the picture of Eight Heads. Escher visited Spain in June 1922, making the voyage on a sea freighter, and there his interest in regular division was briefly revitalised. He travelled widely and visited many palaces and was inspired by a great number of both buildings and landscapes. One building which was to have an immense influence on his life was the Alhambra Palace in Grenada. Here are modern pictures of the Lions' Court and the pond in the Alhambra. Escher was overwhelmed by the beauty of the 14th century Moorish palace and in particular, by the decorative majolica tilings which decorated many of the surfaces of the building. Unlike the Moors, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Escher.html (2 of 8) [2/16/2002 11:09:00 PM]

Escher

Escher was both keen and permitted to use recognisable objects in his ad-hoc versions of the tilings. He made a number of attempts at using this style of artwork over the next couple of years but was unhappy about both the length of time this passion was taking (due to its trial and error nature) and the poor quality of his final work, and he left aside regular division for a number of years. We wrote that, in about 1924, [7]:... for the first time I printed on a cloth a single animal motif cut out of wood which repeats itself according to a certain system, thereby adhering to the principle that no blank spaces may occur. I needed at least three colours; with each in turn I rolled my stamping block in order to contrast one motif with its adjoining congruent repetitions. I exhibited this cloth together with my other work, but I did not have any success with it. Following his return from Spain, Escher went to live in Italy. Again he travelled widely and in 1923, whilst staying in the town of Ravello, he met his future wife Jetta Umiker. They married on 12 June 1924 and made their home in Frascati, just outside Rome. They would have three children, George (born 23 June 1926 in Frascati), then Arthur (born 8 December 1928) and Jan (born 6 March 1938). Escher with his family took frequent holidays around Italy during the next decade. Years of sketching Italian landscapes, usually with impossible perspectives, followed before the family were forced to leave Italy as a result of the Fascist political uprising which developed in Italy during the summer months of 1935. They moved to the mountain village of Chateau-d'Oex in Switzerland but Jetta missed Italy and the high Swiss prices forced Escher to sell more prints. The family was unhappy at first in their new surroundings and, lacking inspiration for his work, Maurits and Jetta set out on a Mediterranean excursion. Escher managed to negotiate a deal with the Adria Shipping Company which gave free passage and meals for himself and also a one way ticket for Jetta. He made payment with prints which he completed using sketches made on the journey. The trip began on the 26 April 1936, and during the next two months the pair made volumes of sketches from which to work from in the future. Here is an Alhambra sketch from 1936. Escher's fascination with order and symmetry took over his life after this Mediterranean journey in 1936 after he made his second visit to the Alhambra. Escher remarked that it was [7]:...the richest source of inspiration I have ever tapped. Escher and his wife spent days on end working at the Alhambra Palace, where they sketched as much as they could, much to the amusement of the numerous tourists who visited each day. These sketches were to become a fundamental source for much of Escher's future work. After this trip Escher became obsessed with the concept of regular division of the plane. He wrote [7]:It remains an extremely absorbing activity, a real mania to which I have become addicted, and from which I sometimes find it hard to tear myself away. Escher felt that he could improve upon the work of the Moorish artists and used his sketches as a geometric grid from which to design his own characters to fill the plane. He experimented with many different motifs such as birds, weightlifters and lions, all of which appear in many of his early designs. All of his work during this time period relied heavily on his own imagination along with his Spanish sketches, and was immensely time consuming.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Escher.html (3 of 8) [2/16/2002 11:09:00 PM]

Escher

Here is one of his designs. In October 1937 Escher showed some of his new work to his brother Berend, by then a professor of geology at Leiden University, when both were visiting their parents home in The Hague. Recognising the connection between his brother's woodcuts and crystallography, Berend sent his brother a list of articles that he felt would be of assistance. This was Escher's first contact with mathematics. Escher read Pólya's 1924 paper on plane symmetry groups. Although he did not understand the abstract concept of groups discussed in Pólya's paper he did understand the 17 plane symmetry groups described there. He subsequently taught himself the principles by which each of the 17 groups operated. Between 1937 and 1941 Escher worked on possible periodic tilings producing 43 coloured drawings with a wide variety of symmetry types. He adopted a highly mathematical approach with a systematic study using notation which he invented himself. Escher also studied an article written by F Haag in 1923 and he eventually challenged some of the views expressed in the literature following further research into the topic. Near the end of 1937 the Escher family moved to Belgium which became their home until the 20 February 1941 when the invading German army forced them to flee to Baarn in Holland. World War II was a deeply emotional time for Escher and prevented him from concentrating on his work. Over the years that followed Escher made numerous woodcuts utilising each of the 17 symmetry groups. With practice his skills naturally improved and as a result he could design and complete each piece far quicker than in his earlier years. His art formed an integral part of family life, and Escher would work in his study between 8 am and 4 pm every day. New concepts could take months or even years to come to fruition before the finished work was discussed and explained to the family. One of his children wrote [7]:The end of the cycle, making the first print, gave father a mixture of joy and sadness. It was exciting and satisfying to lift the paper from the inked wood for the first time, to see the finished print, crisp and immaculate, gradually appearing around the edge of the paper as it was carefully raised. But father had always a feeling of disappointment, of not having been able to depict adequately his thoughts. After all his efforts, how far short of the originally so lucid and misleading simple idea did this result fall! Extensive research and investigation culminated in 1941 with his first notebook Regular Division of the plane with Asymmetric congruent Polygons. This notebook was extended and improved over the course of the following year, when the results obtained from extensive colour based division investigations were included. These books were never meant for publication - only for background information to allow him to continue as a visionary artist. The notebooks were evidence of the fact that Escher had become a research mathematician of the highest order, regardless of his personal feeling of mathematical insecurity. He had developed his own categorisation system which covered all the possible combinations of shape, colour and symmetrical properties. As such he had unknowingly studied areas of crystallography years in advance of any professional mathematician working in this field. He wrote [7]:A long time ago, I chanced upon this domain [of regular division of the plane] in one of my wanderings... However, on the other side I landed in a wilderness.... I came to the open gate of mathematics. Sometimes I think I have covered the whole area ... and then I suddenly

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Escher.html (4 of 8) [2/16/2002 11:09:00 PM]

Escher

discover a new path and experience fresh delights. Escher was inundated with requests to give lectures all over the world. In a lecture in 1953 Escher said [3]:... I have often felt closer to people who work scientifically (though I certainly do not do so myself) than to my fellow artists. By around 1956 Escher's interests changed again taking regular division of the plane to the next level by representing infinity on a fixed 2-dimensional plane. Earlier in his career he had used the concept of a closed loop to try to express infinity as demonstrated in Horseman. Here is the Horsemen picture and another version of it. He had put his designs on to a variety of three-dimensional objects such as columns and spheres during the 1940s, again in an attempt to impart an endless perspective to his work. He later tried working with the concept of similarities, using identical motifs of diminishing size, arranged in a series of concentric circles, but as with much of his work, he was unhappy about the final quality. In 1958 Escher met Coxeter and they became life-long friends. Escher came across an article written by Coxeter, and again whilst unable to understand the text, he was able to determine the rules regarding hyperbolic tessellations using only the diagrams in the paper. Escher paid thanks to Coxeter by sending him a copy of one of his new works Circle Limit I. Escher continued to develop and enhance this field and produced many more prints using both circles and squares as the frames for his works. Here is the picture Circle limit I. This style of artwork required enormous dedication because of the careful planning and trial sketches required, coupled with the necessary hand and carving skill, but was an enormous source of satisfaction to Escher. He wrote [7]:I discovered once again that the human hand is capable of executing small and yet completely controlled movements, on the condition that the eye sees sufficiently clearly what the hand is doing. In 1995 Coxeter published a paper which proved that Escher had achieved mathematical perfection in one of his etchings. Circle Limit III was created using only simple drawing instruments and Escher's great intuition, but Coxeter proved that [8]:... [Escher] got it absolutely right to the millimetre, absolutely to the millimetre .... Unfortunately he didn't live long enough to see my mathematical vindication. Here is the picture Circle limit III. This proof serves to highlight Escher's amazing natural ability of being able to combine both his artistic skills and the techniques that he learned from others, into mathematically perfect designs. Here is the picture Circle limit IV (Heaven and Hell). By 1958 Escher had achieved remarkable fame. He continued to give lectures and correspond with people who were eager to learn from him. He had given his first important exhibition of his works and had also been featured in Time magazine. Escher received numerous awards over his career including the Knighthood of the Oranje Nassau (1955) and was regularly commissioned to design art for dignitaries http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Escher.html (5 of 8) [2/16/2002 11:09:00 PM]

Escher

around the world. In 1958 he published Regular Division of the Plane and in this work he says:At first I had no idea at all of the possibility of systematically building up my figures. I did not know ... this was possible for someone untrained in mathematics, and especially as a result of my putting forward my own layman's theory, which forced me to think through the possibilities. Again in Regular Division of the Plane Escher writes:In mathematical quarters, the regular division of the plane has been considered theoretically. ... [Mathematicians] have opened the gate leading to an extensive domain, but they have not entered this domain themselves. By their very nature they are more interested in the way in which the gate is opened than in the garden lying behind it. Escher's work covered a variety of subjects throughout his life. His early love of portraits, Roman and Italian landscapes and of nature, eventually gave way to regular division of the plane. Many of his pieces were drawn from unusual perspectives thus creating enigmatic spatial effects. He was skilled in the art of a number of different printing techniques such as woodcuts, lithographs and mezzotints. Over 150 colourful and recognisable works testify to Escher's ingenuity and interest in regular division of the plane. He managed to capture the notion of hyperbolic space on a fixed 2-dimensional plane as well as translating the principles of regular division onto a number of 3-dimensional objects such as spheres, columns and cubes. A number of his prints combine both 2 and 3-dimensional images with startling effect as demonstrated for example in Reptiles. Here is the picture Reptiles. He wrote [7]:When an element of plane division suggests to me the form of an animal, I immediately think of a volume. The "flat shape" irritates me - I feel as if I were shouting to my figures, "You are too fictitious for me; you just lie there static and frozen together; do something, come out of there and show me what you are capable of!" So I make them come out of the plane. But do they really do that? On the contrary, I am deliberately inconsistent, suggesting plasticity in the plane by means of light and shadow. He was fascinated by topology, which only began to be studied during his lifetime, as illustrated by the Möbius strip. In his later years he learned much from the British mathematician Roger Penrose and used this knowledge to design many "impossible" etchings such as Waterfall or Up and down. Here are the pictures Waterfall and Up and down. Escher used pictures to tell a story in his Metamorphosis series of designs. These designs brought together many of Escher's skills and show the transformation from one distinct object to another, by means of a series of slight changes to a regular pattern in the plane. Metamorphosis 1 in particular, printed in 1933, yields an insight into the change of artistic style which occurred in Escher's life at this time. An Italian coastline is transformed through a series of convex polygons into a regular pattern in the plane until finally a distinct, coloured, human motif emerges, signifying his change of perspective from landscape work to regular division of the plane.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Escher.html (6 of 8) [2/16/2002 11:09:00 PM]

Escher

Here is the picture Metamorphosis I. Escher fell ill initially in 1964 whilst delivering a series of lectures in North America. As a result he was forced to cut down his schedule substantially, later devoting most of his time to correspondence with friends. In [11] his last years are described as follows:When Escher's view of the world turned inward he produced his best known puzzling prints. which, art aside, were truly intellectually playful, yet he was not. His life turned inward, he cut himself off and he had few friends. ... He died after a protracted illness... His final graphic work, a woodcut, Snakes took six months to complete and was finally unveiled in July 1969. This exceptional etching heads off to infinity at both the centre and the edges of the picture. Following further operations Escher moved to the Rosa Spier house in Laren and later died in hospital. Here is the picture Snakes. Here are a pair of unusual self-portraits Reflection in a glass ball and Rind. Click here for a list of the pictures available from this page. [All M C Escher works © 2001 Cordon Art - Baarn - Holland. All rights reserved. Used by permission.]

Article by: J J O'Connor and E F Robertson based on a project by Malcolm Raven. Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles)

Some Quotations (2)

A Poster of Maurits Escher

Mathematicians born in the same country

Other Web sites

1. The official M C Escher Web site 2. Escher's World 3. Yezha (Some of Escher's periodic pictures) 4. Escher Web Sketch (Interactive periodic pictures) 5. National Gallery of Art, Washington, USA 6. ThinkQuest 7. SUNet (Some pictures) 8. Access Indiana (Escher links) 9. Berkeley (A list of Escher Web sites) 10. George W Hart (Escher's polyhedra) 11. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Escher.html (7 of 8) [2/16/2002 11:09:00 PM]

Escher

Previous

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Escher.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Escher.html (8 of 8) [2/16/2002 11:09:00 PM]

Esclangon

Ernest Benjamin Esclangon Born: 17 March 1876 in Misson (N of Sisteron), France Died: 28 Jan 1954 in Eyrenville, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernest Esclangon studied mathematics at Ecole Normale Supérieure. He entered the Ecole in 1895 and graduated with an agregé in mathematics three years later. He became an astronomer in 1899 by accepting, out of curiosity, a position offered to him at Bordeaux. It was only three years earlier, in 1896, that Bordeaux was reconstituted as a university and given autonomy and state financing. Esclangon accepted a post at the university observatory but continued to work on mathematical topics for his doctorate. His doctoral thesis, published in the Annales de l'Observatoire de Bordeaux in 1904 was on quasi-periodic functions. Lévy writes in [1]:Quasi-periodic functions, newly introduced, constitute a remarkable class among the almost periodic functions. Esclangon elaborated a theory for these functions, studied their differentiation and integration, and examined the differential equations which allow them as coefficients. His doctoral thesis established a basis for their employment at a time when their role in mathematical physics was only beginning to be developed. In [4] Gaiduk shows how Esclangon anticipated many of Harald Bohr's results on almost periodic functions. As well as holding the position in the Observatory at Bordeaux, Esclangon taught mathematics in the Faculty of Science there from 1902. He became professor of astronomy at Strasbourg in 1919, having been appointed director of the Observatory there in the previous year. Then, from 1929 to 1944 he was director of the Observatory at Paris, again holding the position of professor of astronomy from 1930 to 1946.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Esclangon.html (1 of 3) [2/16/2002 11:09:02 PM]

Esclangon

The range of topics on which Esclangon worked during his career is quite remarkable. These include [1]:... pure mathematics, applied celestial mechanics, relativity, observational astronomy, instrumental astronomy, astronomical chronometry, aerodynamics, interior and exterior ballistics, and aerial and underwater acoustic detection. In astronomy he worked on making observations more precise by improving astronomical instruments. He is particularly famous in this area for his brilliant ideas on improving the design of a transit instrument. From 1914 he worked on ballistics as part of the war effort. His work in this area was aimed at locating the position of enemy guns. When a missile was fired there were two shock waves emitted, one being a spherical wave at the instant of firing which was centred at the position of the gun, while the other was a conical shock wave from the projectile. By 1916 he had devised a method to calculate the position from which the projectile was fired very accurately allowing enemy gun locations to be targeted. In 1933, using an astronomical calculation of time, he started the 'talking clock' telephone service in Paris. This work was carried out in his role as Director of the Bureau International de l'Heure which he held from 1929 to 1944. Esclangon was honoured with election to the Académie des Sciences in 1939 and the Bureau des Longitudes in 1932. Lévy writes in [1]:He assumed his official functions with simplicity and amiability; he was affable and loved to joke, and did not deny himself leisure time. It would almost seem that he accomplished his body of important work without effort. This is even more remarkable when one realises that during his career he wrote 247 scientific papers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Honours awarded to Ernest Esclangon (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Esclangon

Paris street names

Rue Esclangon (18th Arrondissement)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Esclangon.html (2 of 3) [2/16/2002 11:09:02 PM]

Esclangon

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Esclangon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Esclangon.html (3 of 3) [2/16/2002 11:09:02 PM]

Euclid

Euclid of Alexandria Born: about 325 BC Died: about 265 BC in Alexandria, Egypt

Click the picture above to see six larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements. The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. However little is known of Euclid's life except that he taught at Alexandria in Egypt. Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [1] or [9] or many other sources):Not much younger than these [pupils of Plato] is Euclid, who put together the "Elements", arranging in order many of Eudoxus's theorems, perfecting many of Theaetetus's, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger than Plato's circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says. In his aim he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole "Elements" the construction of the so-called Platonic figures. There is other information about Euclid given by certain authors but it is not thought to be reliable. Two different types of this extra information exists. The first type of extra information is that given by Arabian authors who state that Euclid was the son of Naucrates and that he was born in Tyre. It is believed by historians of mathematics that this is entirely fictitious and was merely invented by the authors.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html (1 of 8) [2/16/2002 11:09:04 PM]

Euclid

The second type of information is that Euclid was born at Megara. This is due to an error on the part of the authors who first gave this information. In fact there was a Euclid of Megara, who was a philosopher who lived about 100 years before the mathematician Euclid of Alexandria. It is not quite the coincidence that it might seem that there were two learned men called Euclid. In fact Euclid was a very common name around this period and this is one further complication that makes it difficult to discover information concerning Euclid of Alexandria since there are references to numerous men called Euclid in the literature of this period. Returning to the quotation from Proclus given above, the first point to make is that there is nothing inconsistent in the dating given. However, although we do not know for certain exactly what reference to Euclid in Archimedes' work Proclus is referring to, in what has come down to us there is only one reference to Euclid and this occurs in On the sphere and the cylinder. The obvious conclusion, therefore, is that all is well with the argument of Proclus and this was assumed until challenged by Hjelmslev in [48]. He argued that the reference to Euclid was added to Archimedes book at a later stage, and indeed it is a rather surprising reference. It was not the tradition of the time to give such references, moreover there are many other places in Archimedes where it would be appropriate to refer to Euclid and there is no such reference. Despite Hjelmslev's claims that the passage has been added later, Bulmer-Thomas writes in [1]:Although it is no longer possible to rely on this reference, a general consideration of Euclid's works ... still shows that he must have written after such pupils of Plato as Eudoxus and before Archimedes. For further discussion on dating Euclid, see for example [8]. This is far from an end to the arguments about Euclid the mathematician. The situation is best summed up by Itard [11] who gives three possible hypotheses. (i) Euclid was an historical character who wrote the Elements and the other works attributed to him. (ii) Euclid was the leader of a team of mathematicians working at Alexandria. They all contributed to writing the 'complete works of Euclid', even continuing to write books under Euclid's name after his death. (iii) Euclid was not an historical character. The 'complete works of Euclid' were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about 100 years earlier. It is worth remarking that Itard, who accepts Hjelmslev's claims that the passage about Euclid was added to Archimedes, favours the second of the three possibilities that we listed above. We should, however, make some comments on the three possibilities which, it is fair to say, sum up pretty well all possible current theories. There is some strong evidence to accept (i). It was accepted without question by everyone for over 2000 years and there is little evidence which is inconsistent with this hypothesis. It is true that there are differences in style between some of the books of the Elements yet many authors vary their style. Again the fact that Euclid undoubtedly based the Elements on previous works means that it would be rather remarkable if no trace of the style of the original author remained.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html (2 of 8) [2/16/2002 11:09:04 PM]

Euclid

Even if we accept (i) then there is little doubt that Euclid built up a vigorous school of mathematics at Alexandria. He therefore would have had some able pupils who may have helped out in writing the books. However hypothesis (ii) goes much further than this and would suggest that different books were written by different mathematicians. Other than the differences in style referred to above, there is little direct evidence of this. Although on the face of it (iii) might seem the most fanciful of the three suggestions, nevertheless the 20th century example of Bourbaki shows that it is far from impossible. Henri Cartan, André Weil, Jean Dieudonné, Claude Chevalley, and Alexander Grothendieck wrote collectively under the name of Bourbaki and Bourbaki's Eléments de mathématique contains more than 30 volumes. Of course if (iii) were the correct hypothesis then Apollonius, who studied with the pupils of Euclid in Alexandria, must have known there was no person 'Euclid' but the fact that he wrote:.... Euclid did not work out the syntheses of the locus with respect to three and four lines, but only a chance portion of it ... certainly does not prove that Euclid was an historical character since there are many similar references to Bourbaki by mathematicians who knew perfectly well that Bourbaki was fictitious. Nevertheless the mathematicians who made up the Bourbaki team are all well known in their own right and this may be the greatest argument against hypothesis (iii) in that the 'Euclid team' would have to have consisted of outstanding mathematicians. So who were they? We shall assume in this article that hypothesis (i) is true but, having no knowledge of Euclid, we must concentrate on his works after making a few comments on possible historical events. Euclid must have studied in Plato's Academy in Athens to have learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar. None of Euclid's works have a preface, at least none has come down to us so it is highly unlikely that any ever existed, so we cannot see any of his character, as we can of some other Greek mathematicians, from the nature of their prefaces. Pappus writes (see for example [1]) that Euclid was:... most fair and well disposed towards all who were able in any measure to advance mathematics, careful in no way to give offence, and although an exact scholar not vaunting himself. Some claim these words have been added to Pappus, and certainly the point of the passage (in a continuation which we have not quoted) is to speak harshly (and almost certainly unfairly) of Apollonius. The picture of Euclid drawn by Pappus is, however, certainly in line with the evidence from his mathematical texts. Another story told by Stobaeus [9] is the following:... someone who had begun to learn geometry with Euclid, when he had learnt the first theorem, asked Euclid "What shall I get by learning these things?" Euclid called his slave and said "Give him threepence since he must make gain out of what he learns". Euclid's most famous work is his treatise on mathematics The Elements. The book was a compilation of knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were first proved by Euclid but the organisation of the material and its exposition are certainly due to him. In fact there is ample evidence that Euclid is using earlier textbooks as he writes the Elements since he introduces quite a number of definitions which are never used such as that of an oblong, a rhombus, and a rhomboid. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html (3 of 8) [2/16/2002 11:09:04 PM]

Euclid

The Elements begins with definitions and five postulates. The first three postulates are postulates of construction, for example the first postulate states that it is possible to draw a straight line between any two points. These postulates also implicitly assume the existence of points, lines and circles and then the existence of other geometric objects are deduced from the fact that these exist. There are other assumptions in the postulates which are not explicit. For example it is assumed that there is a unique line joining any two points. Similarly postulates two and three, on producing straight lines and drawing circles, respectively, assume the uniqueness of the objects the possibility of whose construction is being postulated. The fourth and fifth postulates are of a different nature. Postulate four states that all right angles are equal. This may seem "obvious" but it actually assumes that space in homogeneous - by this we mean that a figure will be independent of the position in space in which it is placed. The famous fifth, or parallel, postulate states that one and only one line can be drawn through a point parallel to a given line. Euclid's decision to make this a postulate led to Euclidean geometry. It was not until the 19th century that this postulate was dropped and non-euclidean geometries were studied. There are also axioms which Euclid calls 'common notions'. These are not specific geometrical properties but rather general assumptions which allow mathematics to proceed as a deductive science. For example:Things which are equal to the same thing are equal to each other. Zeno of Sidon, about 250 years after Euclid wrote the Elements, seems to have been the first to show that Euclid's propositions were not deduced from the postulates and axioms alone, and Euclid does make other subtle assumptions. The Elements is divided into 13 books. Books one to six deal with plane geometry. In particular books one and two set out basic properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties of the circle while book four deals with problems about circles and is thought largely to set out work of the followers of Pythagoras. Book five lays out the work of Eudoxus on proportion applied to commensurable and incommensurable magnitudes. Heath says [9]:Greek mathematics can boast no finer discovery than this theory, which put on a sound footing so much of geometry as depended on the use of proportion. Book six looks at applications of the results of book five to plane geometry. Books seven to nine deal with number theory. In particular book seven is a self-contained introduction to number theory and contains the Euclidean algorithm for finding the greatest common divisor of two numbers. Book eight looks at numbers in geometrical progression but van der Waerden writes in [2] that it contains:... cumbersome enunciations, needless repetitions, and even logical fallacies. Apparently Euclid's exposition excelled only in those parts in which he had excellent sources at his disposal. Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus. Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html (4 of 8) [2/16/2002 11:09:04 PM]

Euclid

Books eleven to thirteen deal with three-dimensional geometry. In book thirteen the basic definitions needed for the three books together are given. The theorems then follow a fairly similar pattern to the two-dimensional analogues previously given in books one and four. The main results of book twelve are that circles are to one another as the squares of their diameters and that spheres are to each other as the cubes of their diameters. These results are certainly due to Eudoxus. Euclid proves these theorems using the "method of exhaustion" as invented by Eudoxus. The Elements ends with book thirteen which discusses the properties of the five regular polyhedra and gives a proof that there are precisely five. This book appears to be based largely on an earlier treatise by Theaetetus. Euclid's Elements is remarkable for the clarity with which the theorems are stated and proved. The standard of rigour was to become a goal for the inventors of the calculus centuries later. As Heath writes in [9]:This wonderful book, with all its imperfections, which are indeed slight enough when account is taken of the date it appeared, is and will doubtless remain the greatest mathematical textbook of all time. ... Even in Greek times the most accomplished mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, altering the language here and there, mostly with a view to greater clearness and consistency... It is a fascinating story how the Elements has survived from Euclid's time and this is told well by Fowler in [7]. He describes the earliest material relating to the Elements which has survived:Our earliest glimpse of Euclidean material will be the most remarkable for a thousand years, six fragmentary ostraca containing text and a figure ... found on Elephantine Island in 1906/07 and 1907/08... These texts are early, though still more than 100 years after the death of Plato (they are dated on palaeographic grounds to the third quarter of the third century BC); advanced (they deal with the results found in the "Elements" [book thirteen] ... on the pentagon, hexagon, decagon, and icosahedron); and they do not follow the text of the Elements. ... So they give evidence of someone in the third century BC, located more than 500 miles south of Alexandria, working through this difficult material... this may be an attempt to understand the mathematics, and not a slavish copying ... The next fragment that we have dates from 75 - 125 AD and again appears to be notes by someone trying to understand the material of the Elements. More than one thousand editions of The Elements have been published since it was first printed in 1482. Heath [9] discusses many of the editions and describes the likely changes to the text over the years. B L van der Waerden assesses the importance of the Elements in [2]:Almost from the time of its writing and lasting almost to the present, the Elements has exerted a continuous and major influence on human affairs. It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century. It is sometimes said that, next to the Bible, the "Elements" may be the most translated, published, and studied of all the books produced in the Western world. Euclid also wrote the following books which have survived: Data (with 94 propositions), which looks at what properties of figures can be deduced when other properties are given; On Divisions which looks at http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html (5 of 8) [2/16/2002 11:09:04 PM]

Euclid

constructions to divide a figure into two parts with areas of given ratio; Optics which is the first Greek work on perspective; and Phaenomena which is an elementary introduction to mathematical astronomy and gives results on the times stars in certain positions will rise and set. Euclid's following books have all been lost: Surface Loci (two books), Porisms (a three book work with, according to Pappus, 171 theorems and 38 lemmas), Conics (four books), Book of Fallacies and Elements of Music. The Book of Fallacies is described by Proclus [1]:Since many things seem to conform with the truth and to follow from scientific principles, but lead astray from the principles and deceive the more superficial, [Euclid] has handed down methods for the clear-sighted understanding of these matters also ... The treatise in which he gave this machinery to us is entitled Fallacies, enumerating in order the various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of the error with practical illustration. Elements of Music is a work which is attributed to Euclid by Proclus. We have two treatises on music which have survived, and have by some authors attributed to Euclid, but it is now thought that they are not the work on music referred to by Proclus. Euclid may not have been a first class mathematician but the long lasting nature of The Elements must make him the leading mathematics teacher of antiquity or perhaps of all time. As a final personal note let me add that my [EFR] own introduction to mathematics at school in the 1950s was from an edition of part of Euclid's Elements and the work provided a logical basis for mathematics and the concept of proof which seem to be lacking in school mathematics today. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (97 books/articles)

Some Quotations (3)

A Poster of Euclid

Mathematicians born in the same country

Some pages from publications

The first page of The Elements in the translation by Campanus. This edition was published in 1482. The five postulates taken from this edition. Another page from the edition of The Elements, Book V published in 1482 The first page of The Elements published in 1505. (This was the first Latin translation directly from the Greek.) Propositions 1 and 2 from The Elements, Book I (1536) Propositions 2 and 3 from The Elements, Book I (1536) Propositions 6 and 7 from The Elements, Book I (The Campanus translation 1482) Propositions 27 and 28 from The Elements, Book I (The Campanus translation 1482) A page from Optics (an edition of 1557)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html (6 of 8) [2/16/2002 11:09:04 PM]

Euclid

Proposition 2 of Elements (Campani edition of 1482) and its continuation Cross-references to History Topics

1. Non-Euclidean geometry 2. Greek Astronomy 3. Perfect numbers 4. Doubling the cube 5. Arabic mathematics : forgotten brilliance? 6. How do we know about Greek mathematicians? 7. How do we know about Greek mathematics? 8. Quadratic, cubic and quartic equations 9. Prime numbers 10. A history of Zero 11. A chronology of pi

Other references in MacTutor

1. Euclid's geometric solution of a quadratic equation 2. The five regular polyhedra 3. Chronology: 500BC to 1AD

Honours awarded to Euclid (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Euclides

Other Web sites

1. David Joyce (The Elements) 2. Vatican exhibition 3. Math Forum 4. R Page (The 5th postulate) 5. Interactive Real Analysis 6. Simon Fraser University 7. G Don Allen (An introduction to The Elements ) 8. Kevin Brown (A discussion of which propositions depend on which axioms) 9. Tufts University (An on-line version of The Elements) 10. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html (7 of 8) [2/16/2002 11:09:04 PM]

Euclid

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Euclid.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euclid.html (8 of 8) [2/16/2002 11:09:04 PM]

Eudemus

Eudemus of Rhodes Born: about 350 BC in Rhodes, Greece Died: about 290 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

We should certainly credit Eudemus of Rhodes for his achievements in this archive since Eudemus seems to have been the first major historian of mathematics. Simplicius informs us that a biography of Eudemus was written by Damas, who is unknown but for this reference, but sadly no trace of this biography has been found. As exciting aspect of the history of mathematics is that the discovery of this text (and other lost texts) in the future, although highly unlikely, always remains a possibility. Eudemus was born on Rhodes and we know that he had a brother called Boethus. Of his parents and early life we know nothing, but we do know that he studied with Aristotle. Aristotle spent time in Athens, Assos and other places and it would certainly be good to understand when Eudemus studied with him. Unfortunately there is no record either of time or of place which would let us answer these questions with any degree of certainty. W Jaeger, however, in his discussion of Aristotle [4] (see also [5]) has argued strongly that Eudemus studied with Aristotle during his period in Assos. Aristotle had two followers, Eudemus and Theophrastus of Lesbos, who were known as his "companions". We should make it clear, however, that there was another philosopher called Eudemus associated with Aristotle, namely Eudemus of Cyprus and it was this other Eudemus after whom Aristotle named his famous text Eudemus. When Aristotle realised that he had only a short time left to live he chose his successor between his two companions, Eudemus and Theophrastus. He chose Theophrastus and it appears that Eudemus, although not unhappy with the decision, left Athens and set up his own school, probably back on his native Rhodes. To say that Eudemus was not an original mathematician may be fair but just a little harsh, for we do know through Proclus that he wrote an original mathematical work called On the Angle. This work is lost so we are unable to judge its importance but it does seem likely to have been considerably less important than his works on the history of mathematics. We know of three works on the history of mathematics by Eudemus, namely History of Arithmetic (two or more books), History of Geometry (two or more books), and History of Astronomy (two or more books). The History of Arithmetic is known to us from only one reference to it in the writing of Porphyry. This reference tell us that the first book dealt with the Pythagorean idea of number and its interrelations with http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eudemus.html (1 of 3) [2/16/2002 11:09:05 PM]

Eudemus

music. The History of Geometry is the most important of the three mathematical histories of Eudemus. Although the work has not survived, it was available to many later writers who made heavy use of it. We are fortunate therefore that much of the knowledge that Eudemus had of the history of Greek mathematics before Euclid (it had to be before Euclid given the dates when Eudemus was writing) has reached us despite the fact that he book has not. In many of the articles in this archive we have quoted from accounts based on Eudemus. To illustrate with one example, the work of Hippocrates on the quadrature of lunes is only known to us through Eudemus's History of Geometry. It is unclear exactly when the History of Geometry was lost. Paul Tannery (see for example [7]) believed that it was lost before the time of Pappus while others such as J L Heiberg have argued that Pappus and Eutocius both wrote with an open copy of Eudemus's History of Geometry in front of them. The History of Astronomy again was heavily used by later writers and in exactly the same way as his geometry text, much information has survived in the works of others despite the loss of the original text. In particular Thales' eclipse prediction was described in Eudemus's work and we believe that Eudoxus's system of concentric spheres was first described there and later transmitted to us through the writing of Simplicius in the second century AD. Other topics in this book included [1]:... the cycle of the great year after which all the heavenly bodies are found in the same relative positions; the realisation by Anaximander that the earth is a heavenly body moving about the middle of the universe; the discovery by Anaximenes that the moon reflects the light of the sun and the explanation of lunar eclipses; and the inequality of the times between the solstices and the equinoxes. We have described above the important contributions of Eudemus to mathematics. However he is even better known for his contribution to saving the work of Aristotle for posterity. But for Eudemus we might not have had access to the works of Aristotle for he used his own lecture notes, Aristotle's lecture notes and recollections from memory to produce volumes of Aristotle's work fit for publication. One further work is definitely due to Eudemus, namely a work on Physics which was a treatise in four books following the work by Aristotle of the same title fairly closely. Simplicius had a copy of this work which he found very helpful in understanding Aristotle's Physics and perhaps this was precisely the role the Eudemus intended for the work. Another work by Eudemus was on logic, in fact he may well have written two logic books and he also wrote On Discourse. Some works by Eudemus are harder to identify with Eudemus of Rhodes and may have been written by others with the same name. Certainly there are many references to a work on animals written by a certain Eudemus and one of the references certainly does refer to Eudemus of Rhodes. Since the work seems to have been a collection of fables about animals the subject matter seems too far removed from the serious, scientific and scholarly works which he certainly wrote. Perhaps more likely is a work on the poet Lindos. Since Lindos had connections with Rhodes the link makes this a more likely possibility. Again there is a reference which seems to imply that Eudemus wrote a history of theology and again this seems highly probable. Many authors refer to Eudemus as the 'pious Eudemus' due to his belief in the 'contemplation of God'. This however may be due to editing by a later Christian who would have seen

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eudemus.html (2 of 3) [2/16/2002 11:09:05 PM]

Eudemus

that clearly Eudemus meant 'contemplation of God' rather than what is much more likely what he wrote 'contemplation of Mind' and "corrected" the text accordingly! Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Cross-references to History Topics

Doubling the cube

Other references in MacTutor

Chronology: 500BC to 1AD

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eudemus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eudemus.html (3 of 3) [2/16/2002 11:09:05 PM]

Eudoxus

Eudoxus of Cnidus Born: 408 BC in Cnidus (on Resadiye peninsula), Asia Minor (now Turkey) Died: 355 BC in Cnidus, Asia Minor (now Turkey) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eudoxus of Cnidus was the son of Aischines. As to his teachers, we know that he travelled to Tarentum, now in Italy, where he studied with Archytas who was a follower of Pythagoras. The problem of duplicating the cube was one which interested Archytas and it would be reasonable to suppose that Eudoxus's interest in that problem was stimulated by his teacher. Other topics that it is probable that he learnt about from Archytas include number theory and the theory of music. Eudoxus also visited Sicily, where he studied medicine with Philiston, before making his first visit to Athens in the company of the physician Theomedon. Eudoxus spent two months in Athens on this visit and he certainly attended lectures on philosophy by Plato and other philosophers at the Academy which had only been established a short time before. Heath [3] writes of Eudoxus as a student in Athens:... so poor was he that he took up his abode at the Piraeus and trudged to Athens and back on foot each day. After leaving Athens, he spent over a year in Egypt where he studied astronomy with the priests at Heliopolis. At this time Eudoxus made astronomical observations from an observatory which was situated between Heliopolis and Cercesura. From Egypt Eudoxus travelled to Cyzicus in northwestern Asia Minor on the south shore of the sea of Marmara. There he established a School which proved very popular and he had many followers. In around 368 BC Eudoxus made a second visit to Athens accompanied by a number of his followers. It is hard to work out exactly what his relationship with Plato and the Academy were at this time. There is some evidence to suggest that Eudoxus had little respect for Plato's analytic ability and it is easy to see why that might be, since as a mathematician his abilities went far beyond those of Plato. It is also suggested that Plato was not entirely pleased to see how successful Eudoxus's School had become. Certainly there is no reason to believe that the two philosophers had much influence on each others ideas. Eudoxus returned to his native Cnidus and there was acclaimed by the people who put him into an important role in the legislature. However he continued his scholarly work, writing books and lecturing on theology, astronomy and meteorology.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eudoxus.html (1 of 6) [2/16/2002 11:09:08 PM]

Eudoxus

He had built an observatory on Cnidus and we know that from there he observed the star Canopus. The observations made at his observatory in Cnidus, as well as those made at the observatory near Heliopolis, formed the basis of two books referred to by Hipparchus. These works were the Mirror and the Phaenomena which are thought by some scholars to be revisions of the same work. Hipparchus tells us that the works concerned the rising and setting of the constellations but unfortunately these books, as all the works of Eudoxus, have been lost. Eudoxus made important contributions to the theory of proportion, where he made a definition allowing possibly irrational lengths to be compared in a similar way to the method of cross multiplying used today. A major difficulty had arisen in mathematics by the time of Eudoxus, namely the fact that certain lengths were not comparable. The method of comparing two lengths x and y by finding a length t so that x = m.t and y = n.t for whole numbers m and n failed to work for lines of lengths 1 and 2 as the Pythagoreans had shown. The theory developed by Eudoxus is set out in Euclid's Elements Book V. Definition 4 in that Book is called the Axiom of Eudoxus and was attributed to him by Archimedes. The definition states (in Heath's translation [3]):Magnitudes are said to have a ratio to one another which is capable, when a multiple of either may exceed the other. By this Eudoxus meant that a length and an area do not have a capable ratio. But a line of length 2 and one of length 1 do have a capable ratio since 1. 2 > 1 and 2.1 > 2. Hence the problem of irrational lengths was solved in the sense that one could compare lines of any lengths, either rational or irrational. Eudoxus then went on to say when two ratios are equal. This appears as Euclid's Elements Book V Definition 5 which is, in Heath's translation [3]:Magnitudes are said to be of the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and the third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or are alike less than the latter equimultiples taken in corresponding order. In modern notation, this says that a : b and c : d are equal (where a, b, c, d are possibly irrational) if for every possible pair of integers m, n (i) if ma < nb then mc < nd, (ii) if ma = nb then mc = nd, (iii) if ma > nb then mc > nd. Huxley writes in [1]:It is difficult to exaggerate the significance of the theory, for it amounts to a rigorous definition of real number. Number theory was allowed to advance again, after the paralysis imposed on it by the Pythagorean discovery of irrationals, to the inestimable benefit of all subsequent mathematics. A number of authors have discussed the ideas of real numbers in the work of Eudoxus and compared his ideas with those of Dedekind, in particular the definition involving 'Dedekind cuts' given in 1872. Dedekind himself emphasised that his work was inspired by the ideas of Eudoxus. Heath [3] writes that Eudoxus's definition of equal ratios:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eudoxus.html (2 of 6) [2/16/2002 11:09:08 PM]

Eudoxus

... corresponds exactly to the modern theory of irrationals due to Dedekind, and that it is word for word the same as Weierstrass's definition of equal numbers. However, some historians take a rather different view. For example, the article [15] (quoting from the author's summary):... analyses, first, the historical significance of the theory of proportions contained in Book V of Euclid's "Elements" and attributed to Eudoxus. It then demonstrates the radical originality, relative to this theory, of the definition of real numbers on the basis of the set of rationals proposed by Dedekind. Two conclusions: (1) there are not in Book V of the "Elements" the gaps perceived by Dedekind; (2) one cannot properly speak of an 'influence' of Eudoxus's ideas on Dedekind's theory. Another remarkable contribution to mathematics made by Eudoxus was his early work on integration using his method of exhaustion. This work developed directly out of his work on the theory of proportion since he was now able to compare irrational numbers. It was also based on earlier ideas of approximating the area of a circle by Antiphon where Antiphon took inscribed regular polygons with increasing numbers of sides. Eudoxus was able to make Antiphon's theory into a rigorous one, applying his methods to give rigorous proofs of the theorems, first stated by Democritus, that (i) the volume of a pyramid is one-third the volume of the prism having the same base and equal height; and (ii) the volume of a cone is one-third the volume of the cylinder having the same base and height. The proofs of these results are attributed to Eudoxus by Archimedes in his work On the sphere and cylinder and of course Archimedes went on to use Eudoxus's method of exhaustion to prove a remarkable collection of theorems. We know that Eudoxus studied the classical problem of the duplication of the cube. Eratosthenes, who wrote a history of the problem, says that Eudoxus solved the problem by means of curved lines. Eutocius wrote about Eudoxus's solution but it appears that he had in front of him a document which, although claiming to give Eudoxus's solution, must have been written by someone who had failed to understand it. Paul Tannery tried to reconstruct Eudoxus's proof from very little evidence, so it must remain no more than a guess. Tannery's ingenious suggestion was that Eudoxus had used the kampyle curve in his solution and, as a consequence, the curve is now known as the kampyle of Eudoxus. Heath, however, doubts Tannery's suggestions [3]:To my mind the objection to it is that it is too close an adaptation of Archytas's ideas ... Eudoxus was, I think, too original a mathematician to content himself with a mere adaptation of Archytas's method of solution. We have still to discuss Eudoxus's planetary theory, perhaps the work for which he is most famous, which he published in the book On velocities which is now lost. Perhaps the first comment that is worth making is that Eudoxus was greatly influenced by the philosophy of the Pythagoreans through his teacher Archytas. Therefore it is not surprising that he developed a system based on spheres following Pythagoras's belief that the sphere was the most perfect shape. The homocentric sphere system proposed by Eudoxus consisted of a number of rotating spheres, each sphere rotating about an axis through the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eudoxus.html (3 of 6) [2/16/2002 11:09:08 PM]

Eudoxus

centre of the Earth. The axis of rotation of each sphere was not fixed in space but, for most spheres, this axis was itself rotating as it was determined by points fixed on another rotating sphere. As in the diagram on the right, suppose we have two spheres S1 and S2, the axis XY of S1 being a diameter of the sphere S2. As S2 rotates about an axis AB, then the axis XY of S1 rotates with it. If the two spheres rotate with constant, but opposite, angular velocity then a point P on the equator of S1 describes a figure of eight curve. This curve was called a hippopede (meaning a horse-fetter). Eudoxus used this construction of the hippopede with two spheres and then considered a planet as the point P traversing the curve. He introduced a third sphere to correspond to the general motion of the planet against the background stars while the motion round the hippopede produced the observed periodic retrograde motion. The three sphere subsystem was set into a fourth sphere which gave the daily rotation of the stars.

The planetary system of Eudoxus is described by Aristotle in Metaphysics and the complete system contains 27 spheres. Simplicius, writing a commentary on Aristotle in about 540 AD, also describes the spheres of Eudoxus. They represent a magnificent geometrical achievement. As Heath writes [3]:... to produce the retrogradations in this theoretical way by superimposed axial rotations of spheres was a remarkable stroke of genius. It was no slight geometrical achievement, for those days, to demonstrate the effect of the hypothesis; but this is nothing in comparison with the speculative power which enabled the man to invent the hypothesis which could produce the effect. There is no doubting this incredible mathematical achievement. But there remain many questions which one must then ask. Did Eudoxus believe that the spheres actually existed? Did he invent them as a geometrical model which was purely a computational device? Did the model accurately represent the way the planets are observed to behave? Did Eudoxus test his model with observational evidence? One argument in favour of thinking that Eudoxus believed in the spheres only as a computational device is the fact that he appears to have made no comment on the substance of the spheres nor on their mode of interconnection. One has to distinguish between Eudoxus's views and those of Aristotle for as Huxley writes in [1]:Eudoxus may have regarded his system simply as an abstract geometrical model, but Aristotle took it to be a description of the physical world... The question of whether Eudoxus thought of his spheres as geometry or a physical reality is studied in the interesting paper [29] which argues that Eudoxus was more interested in actually representing the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eudoxus.html (4 of 6) [2/16/2002 11:09:08 PM]

Eudoxus

paths of the planets than in predicting astronomical phenomena. Certainly the model does not represent, and perhaps more significantly could not represent, the actual paths of the planets with a degree of accuracy which would pass even the simplest of observational tests. As to the question of how much Eudoxus relied on observational data in verifying his hypothesis, Neugebauer writes in [7]:... not only do we not have evidence for numerical data in the construction of Eudoxus's homocentric spheres but it would also be difficult how his theory could have survived a comparison with observational parameters. Perhaps it is just too modern a way of thinking to wonder how Eudoxus could have developed such an intricate theory without testing it out with observational data. Many of the early commentators believed that Plato was the inspiration for Eudoxus's representation of planetary motion by his system of homocentric spheres. These view are still quite widely held but the article [19] argues convincingly that this is not so and that the ideas which influenced Eudoxus to come up with his masterpiece of 3-dimensional geometry were Pythagorean and not from Plato. As a final comment we should note that Eudoxus also wrote a book on geography called Tour of the Earth which, although lost, is fairly well known through around 100 quotes in various sources. The work consisted of seven books and studied the peoples of the Earth known to Eudoxus, in particular examining their political systems, their history and background. Eudoxus wrote about Egypt and the religion of that country with particular authority and it is clear that he learnt much about that country in the year he spent there. In the seventh book Eudoxus wrote at length on the Pythagorean Society in Italy again about which he was clearly extremely knowledgeable. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (32 books/articles)

A Quotation

Mathematicians born in the same country Cross-references to History Topics

1. Pi through the ages 2. Greek Astronomy 3. Doubling the cube 4. The rise of the calculus

Cross-references to Famous Curves

Kampyle curve

Other references in MacTutor

Chronology: 500BC to 1AD

Honours awarded to Eudoxus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Eudoxus

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eudoxus.html (5 of 6) [2/16/2002 11:09:08 PM]

Eudoxus

Other Web sites

1. Eric's treasure troves 2. G Don Allen 3. Simon Fraser University 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eudoxus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eudoxus.html (6 of 6) [2/16/2002 11:09:08 PM]

Euler

Leonhard Euler Born: 15 April 1707 in Basel, Switzerland Died: 18 Sept 1783 in St Petersburg, Russia

Click the picture above to see thirteen larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leonhard Euler's father was Paul Euler. Paul Euler had studied theology at the University of Basel and had attended Jacob Bernoulli's lectures there. In fact Paul Euler and Johann Bernoulli had both lived in Jacob Bernoulli's house while undergraduates at Basel. Paul Euler became a Protestant minister and married Margaret Brucker, the daughter of another Protestant minister. Their son Leonhard Euler was born in Basel, but the family moved to Riehen when he was one year old and it was in Riehen, not far from Basel, that Leonard was brought up. Paul Euler had, as we have mentioned, some mathematical training and he was able to teach his son elementary mathematics along with other subjects. Leonhard was sent to school in Basel and during this time he lived with his grandmother on his mother's side. This school was a rather poor one, by all accounts, and Euler learnt no mathematics at all from the school. However his interest in mathematics had certainly been sparked by his father's teaching, and he read mathematics texts on his own and took some private lessons. Euler's father wanted his son to follow him into the church and sent him to the University of Basel to prepare for the ministry. He entered the University in 1720, at the age of 14, first to obtain a general education before going on to more advanced studies. Johann Bernoulli soon discovered Euler's great potential for mathematics in private tuition that Euler himself engineered. Euler's own account given in his unpublished autobiographical writings, see [1], is as follows:... I soon found an opportunity to be introduced to a famous professor Johann Bernoulli. ... True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (1 of 11) [2/16/2002 11:09:12 PM]

Euler

given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand ... In 1723 Euler completed his Master's degree in philosophy having compared and contrasted the philosophical ideas of Descartes and Newton. He began his study of theology in the autumn of 1723, following his father's wishes, but, although he was to be a devout Christian all his life, he could not find the enthusiasm for the study of theology, Greek and Hebrew that he found in mathematics. Euler obtained his father's consent to change to mathematics after Johann Bernoulli had used his persuasion. The fact that Euler's father had been a friend of Johann Bernoulli's in their undergraduate days undoubtedly made the task of persuasion much easier. Euler completed his studies at the University of Basel in 1726. He had studied many mathematical works during his time in Basel, and Calinger [23] has reconstructed many of the works that Euler read with the advice of Johann Bernoulli. They include works by Varignon, Descartes, Newton, Galileo, von Schooten, Jacob Bernoulli, Hermann, Taylor and Wallis. By 1726 Euler had already a paper in print, a short article on isochronous curves in a resisting medium. In 1727 he published another article on reciprocal trajectories and submitted an entry for the 1727 Grand Prize of the Paris Academy on the best arrangement of masts on a ship. The Prize of 1727 went to Bouguer, an expert on mathematics relating to ships, but Euler's essay won him second place which was a fine achievement for the young graduate. However, Euler now had to find himself an academic appointment and when Nicolaus(II) Bernoulli died in St Petersburg in July 1726 creating a vacancy there, Euler was offered the post which would involve him in teaching applications of mathematics and mechanics to physiology. He accepted the post in November 1726 but stated that he did not want to travel to Russia until the spring of the following year. He had two reasons to delay. He wanted time to study the topics relating to his new post but also he had a chance of a post at the University of Basel since the professor of physics there had died. Euler wrote an article on acoustics, which went on to become a classic, in his bid for selection to the post but he was nor chosen to go forward to the stage where lots were drawn to make the final decision on who would fill the chair. Almost certainly his youth (he was 19 at the time) was against him. However Calinger [23] suggests:This decision ultimately benefited Euler, because it forced him to move from a small republic into a setting more adequate for his brilliant research and technological work. As soon as he knew he would not be appointed to the chair of physics, Euler left Basel on 5 April 1727. He travelled down the Rhine by boat, crossed the German states by post wagon, then by boat from Lübeck arriving in St Petersburg on 17 May 1727. He had joined the St. Petersburg Academy of Science two years after it had been founded by Catherine I the wife of Peter the Great. Through the requests of Daniel Bernoulli and Jakob Hermann, Euler was appointed to the mathematical-physical division of the Academy rather than to the physiology post he had originally been offered. At St Petersburg Euler had many colleagues who would provide an exceptional environment for him [1]:Nowhere else could he have been surrounded by such a group of eminent scientists, including the analyst, geometer Jakob Hermann, a relative; Daniel Bernoulli, with whom Euler was connected not only by personal friendship but also by common interests in the field of applied mathematics; the versatile scholar Christian Goldbach, with whom Euler discussed numerous problems of analysis and the theory of numbers; F Maier, working in trigonometry; and the astronomer and geographer J-N Delisle. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (2 of 11) [2/16/2002 11:09:12 PM]

Euler

Euler served as a medical lieutenant in the Russian navy from 1727 to 1730. In St Petersburg he lived with Daniel Bernoulli who, already unhappy in Russia, had requested that Euler bring him tea, coffee, brandy and other delicacies from Switzerland. Euler became professor of physics at the academy in 1730 and, since this allowed him to became a full member of the Academy, he was able to give up his Russian navy post. Daniel Bernoulli held the senior chair in mathematics at the Academy but when he left St Petersburg to return to Basel in 1733 it was Euler who was appointed to this senior chair of mathematics. The financial improvement which came from this appointment allowed Euler to marry which he did on 7 January 1734, marrying Katharina Gsell, the daughter of a painter from the St Petersburg Gymnasium. Katharina, like Euler, was from a Swiss family. They had 13 children altogether although only five survived their infancy. Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet. We will examine Euler's mathematical achievements later in this article but at this stage it is worth summarising Euler's work in this period of his career. This is done in [23] as follows:... after 1730 he carried out state projects dealing with cartography, science education, magnetism, fire engines, machines, and ship building. ... The core of his research program was now set in place: number theory; infinitary analysis including its emerging branches, differential equations and the calculus of variations; and rational mechanics. He viewed these three fields as intimately interconnected. Studies of number theory were vital to the foundations of calculus, and special functions and differential equations were essential to rational mechanics, which supplied concrete problems. The publication of many articles and his book Mechanica (1736-37), which extensively presented Newtonian dynamics in the form of mathematical analysis for the first time, started Euler on the way to major mathematical work. Euler's health problems began in 1735 when he had a severe fever and almost lost his life. However, he kept this news from his parents and members of the Bernoulli family back in Basel until he had recovered. In his autobiographical writings Euler says that his eyesight problems began in 1738 with overstrain due to his cartographic work and that by 1740 he had [23]:... lost an eye and [the other] currently may be in the same danger. However, Calinger in [23] argues that Euler's eyesight problems almost certainly started earlier and that the severe fever of 1735 was a symptom of the eyestrain. He also argues that a portrait of Euler from 1753 suggests that by that stage the sight of his left eye was still good while that of his right eye was poor but not completely blind. Calinger suggests that Euler's left eye became blind from a later cataract rather than eyestrain. By 1740 Euler had a very high reputation, having won the Grand Prize of the Paris Academy in 1738 and 1740. On both occasions he shared the first prize with others. Euler's reputation was to bring an offer to go to Berlin, but at first he preferred to remain in St Petersburg. However political turmoil in Russia made the position of foreigners particularly difficult and contributed to Euler changing his mind. Accepting an improved offer Euler, at the invitation of Frederick the Great, went to Berlin where an Academy of Science was planned to replace the Society of Sciences. He left St Petersburg on 19 June 1741, arriving in Berlin on 25 July. In a letter to a friend Euler wrote:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (3 of 11) [2/16/2002 11:09:12 PM]

Euler

I can do just what I wish [in my research] ... The king calls me his professor, and I think I am the happiest man in the world. Even while in Berlin Euler continued to receive part of his salary from Russia. For this remuneration he bought books and instruments for the St Petersburg Academy, he continued to write scientific reports for them, and he educated young Russians. Maupertuis was the president of the Berlin Academy when it was founded in 1744 with Euler as director of mathematics. He deputised for Maupertuis in his absence and the two became great friends. Euler undertook an unbelievable amount of work for the Academy [1]:... he supervised the observatory and the botanical gardens; selected the personnel; oversaw various financial matters; and, in particular, managed the publication of various calendars and geographical maps, the sale of which was a source of income for the Academy. The king also charged Euler with practical problems, such as the project in 1749 of correcting the level of the Finow Canal ... At that time he also supervised the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence. This was not the limit of his duties by any means. He served on the committee of the Academy dealing with the library and of scientific publications. He served as an advisor to the government on state lotteries, insurance, annuities and pensions and artillery. On top of this his scientific output during this period was phenomenal. During the twenty-five years spent in Berlin, Euler wrote around 380 articles. He wrote books on the calculus of variations; on the calculation of planetary orbits; on artillery and ballistics (extending the book by Robins); on analysis; on shipbuilding and navigation; on the motion of the moon; lectures on the differential calculus; and a popular scientific publication Letters to a Princess of Germany (3 vols., 1768-72). In 1759 Maupertuis died and Euler assumed the leadership of the Berlin Academy, although not the title of President. The king was in overall charge and Euler was not now on good terms with Frederick despite the early good favour. Euler, who had argued with d'Alembert on scientific matters, was disturbed when Frederick offered d'Alembert the presidency of the Academy in 1763. However d'Alembert refused to move to Berlin but Frederick's continued interference with the running of the Academy made Euler decide that the time had come to leave. In 1766 Euler returned to St Petersburg and Frederick was greatly angered at his departure. Soon after his return to Russia, Euler became almost entirely blind after an illness. In 1771 his home was destroyed by fire and he was able to save only himself and his mathematical manuscripts. A cataract operation shortly after the fire, still in 1771, restored his sight for a few days but Euler seems to have failed to take the necessary care of himself and he became totally blind. Because of his remarkable memory was able to continue with his work on optics, algebra, and lunar motion. Amazingly after his return to St Petersburg (when Euler was 59) he produced almost half his total works despite the total blindness. Euler of course did not achieve this remarkable level of output without help. He was helped by his sons, Johann Albrecht Euler who was appointed to the chair of physics at the Academy in St Petersburg in 1766 (becoming its secretary in 1769) and Christoph Euler who had a military career. Euler was also helped by two other members of the Academy, W L Krafft and A J Lexell, and the young mathematician N Fuss who was invited to the Academy from Switzerland in 1772. Fuss, who was Euler's http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (4 of 11) [2/16/2002 11:09:12 PM]

Euler

grandson-in-law, became his assistant in 1776. Yushkevich writes in [1]:.. the scientists assisting Euler were not mere secretaries; he discussed the general scheme of the works with them, and they developed his ideas, calculating tables, and sometimes compiled examples. For example Euler credits Albrecht, Krafft and Lexell for their help with his 775 page work on the motion of the moon, published in 1772. Fuss helped Euler prepare over 250 articles for publication over a period on about seven years in which he acted as Euler's assistant, including an important work on insurance which was published in 1776. Yushkevich describes the day of Euler's death in [1]:On 18 September 1783 Euler spent the first half of the day as usual. He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motion of balloons; then discussed with Lexell and Fuss the recently discovered planet Uranus. About five o'clock in the afternoon he suffered a brain haemorrhage and uttered only "I am dying" before he lost consciousness. He died about eleven o'clock in the evening. After his death in 1783 the St Petersburg Academy continued to publish Euler's unpublished work for nearly 50 more years. Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficial account of it. He was the most prolific writer of mathematics of all time. He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc. as functions rather than as chords as Ptolemy had done. He made decisive and formative contributions to geometry, calculus and number theory. He integrated Leibniz's differential calculus and Newton's method of fluxions into mathematical analysis. He introduced beta and gamma functions, and integrating factors for differential equations. He studied continuum mechanics, lunar theory with Clairaut, the three body problem, elasticity, acoustics, the wave theory of light, hydraulics, and music. He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies (1765). We owe to Euler the notation f(x) for a function (1734), e for the base of natural logs (1727), i for the square root of -1 (1777), for pi, for summation (1755), the notation for finite differences y and and many others.

2y

Let us examine in a little more detail some of Euler's work. Firstly his work in number theory seems to have been stimulated by Goldbach but probably originally came from the interest that the Bernoullis had in that topic. Goldbach asked Euler, in 1729, if he knew of Fermat's conjecture that the numbers 2n + 1 were always prime if n is a power of 2. Euler verified this for n = 1, 2, 4, 8 and 16 and, by 1732 at the latest, showed that the next case 232 + 1 = 4294967297 is divisible by 641 and so is not prime. Euler also studied other unproved results of Fermat and in so doing introduced the Euler phi function

(n), the

number of integers k with 1 k n and k coprime to n. He proved another of Fermat's assertions, namely that if a and b are coprime then a2 + b2 has no divisor of the form 4n - 1, in 1749. Perhaps the result that brought Euler the most fame in his young days was his solution of what had http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (5 of 11) [2/16/2002 11:09:12 PM]

Euler

become known as the Basel problem. This was to find a closed form for the sum of the infinite series (2) = (1/n2), a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that (2) = 2/6 but he went on to prove much more, namely that (4) = 4/90, (6) = 6/945, (8) = 8/9450, (10) = 10/93555 and (12) = 691 12/638512875. In 1737 he proved the connection of the zeta function with the series of prime numbers giving the famous relation (s) = (1/ns ) = (1 - p-s)-1 Here the sum is over all natural numbers n while the product is over all prime numbers. By 1739 Euler had found the rational coefficients C in (2n) = C

2n

in terms of the Bernoulli numbers.

Other work done by Euler on infinite series included the introduction of his famous Euler's constant 1735, which he showed to be the limit of 1/1 + 1/2 + 1/3 + ... + 1/n - loge n

, in

as n tends to infinity. He calculated the constant to 16 decimal places. Euler also studied Fourier series and in 1744 he was the first to express an algebraic function by such a series when he gave the result /2 - x/2 = sin x + (sin 2x)/2 + (sin 3x)/3 + ... in a letter to Goldbach. Like most of Euler's work there was a fair time delay before the results were published; this result was not published until 1755. Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals of powers, the harmonic series and Euler's constant and other results on series. In particular he wrote [59]:Concerning the summation of very slowly converging series, in the past year I have lectured to our Academy on a special method of which I have given the sums of very many series sufficiently accurately and with very little effort. He then goes on to describe what is now called the Euler-Maclaurin summation formula. Two years later Stirling replied telling Euler that Maclaurin:... will be publishing a book on fluxions. ... he has two theorems for summing series by means of derivatives of the terms, one of which is the self-same result that you sent me. Euler replied:... I have very little desire for anything to be detracted from the fame of the celebrated Mr Maclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer. For I found that theorem about four years ago, at which time I also described its proof and application in greater detail to our Academy. Some of Euler's number theory results have been mentioned above. Further important results in number theory by Euler included his proof of Fermat's Last Theorem for the case of n = 3. Perhaps more significant than the result here was the fact that he introduced a proof involving numbers of the form a + b -3 for integers a and b. Although there were problems with his approach this eventually led to Kummer's major work on Fermats Last Theorem and to the introduction of the concept of a ring. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (6 of 11) [2/16/2002 11:09:12 PM]

Euler

One could claim that mathematical analysis began with Euler. In 1748 in Introductio in analysin infinitorum Euler made ideas of Johann Bernoulli more precise in defining a function, and he stated that mathematical analysis was the study of functions. This work bases the calculus on the theory elementary functions rather than on geometric curves, as had been done previously. Also in this work Euler gave the formula eix = cos x + i sin x. In Introductio in analysin infinitorum Euler dealt with logarithms of a variable taking only positive values although he had discovered the formula ln(-1) = i in 1727. He published his full theory of logarithms of complex numbers in 1751. Analytic functions of a complex variable were investigated by Euler in a number of different contexts, including the study of orthogonal trajectories and cartography. He discovered the Cauchy-Riemann equations in 1777, although d'Alembert had discovered them in 1752 while investigating hydrodynamics. In 1755 Euler published Institutiones calculi differentialis which begins with a study of the calculus of finite differences. The work makes a thorough investigation of how differentiation behaves under substitutions. In Institutiones calculi integralis (1768-70) Euler made a thorough investigation of integrals which can be expressed in terms of elementary functions. He also studied beta and gamma functions, which he had introduced first in 1729. Legendre called these 'Eulerian integrals of the first and second kind' respectively while they were given the names beta function and gamma function by Binet and Gauss respectively. As well as investigating double integrals, Euler considered ordinary and partial differential equations in this work. The calculus of variations is another area in which Euler made fundamental discoveries. His work Methodus inveniendi lineas curvas ... published in 1740 began the proper study of the calculus of variations. In [11] it is noted that Carathéodory considered this as:... one of the most beautiful mathematical works ever written. Problems in mathematical physics had led Euler to a wide study of differential equations. He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others. When considering vibrating membranes, Euler was led to the Bessel equation which he solved by introducing Bessel functions. Euler made substantial contributions to differential geometry, investigating the theory of surfaces and curvature of surfaces. Many unpublished results by Euler in this area were rediscovered by Gauss. Other geometric investigations led him to fundamental ideas in topology such as the Euler characteristic of a polyhedron.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (7 of 11) [2/16/2002 11:09:12 PM]

Euler

In 1736 Euler published Mechanica which provided a major advance in mechanics. As Yushkevich writes in [1]:The distinguishing feature of Euler's investigations in mechanics as compared to those of his predecessors is the systematic and successful application of analysis. Previously the methods of mechanics had been mostly synthetic and geometrical; they demanded too individual an approach to separate problems. Euler was the first to appreciate the importance of introducing uniform analytic methods into mechanics, thus enabling its problems to be solved in a clear and direct way. In Mechanica Euler considered the motion of a point mass both in a vacuum and in a resisting medium. He analysed the motion of a point mass under a central force and also considered the motion of a point mass on a surface. In this latter topic he had to solve various problems of differential geometry and geodesics. Mechanica was followed by another important work in rational mechanics, this time Euler's two volume work on naval science. It is described in [23] as:Outstanding in both theoretical and applied mechanics, it addresses Euler's intense occupation with the problem of ship propulsion. It applies variational principles to determine the optimal ship design and first establish the principles of hydrostatics ... Euler here also begins developing the kinematics and dynamics of rigid bodies, introducing in part the differential equations for their motion. Of course hydrostatics had been studied since Archimedes, but Euler gave a definitive version. In 1765 Euler published another major work on mechanics Theoria motus corporum solidorum in which he decomposed the motion of a solid into a rectilinear motion and a rotational motion. He considered the Euler angles and studied rotational problems which were motivated by the problem of the precession of the equinoxes. Euler's work on fluid mechanics is also quite remarkable. He published a number of major pieces of work through the 1750s setting up the main formulas for the topic, the continuity equation, the Laplace velocity potential equation, and the Euler equations for the motion of an inviscid incompressible fluid. In 1752 he wrote:However sublime are the researches on fluids which we owe to Messrs Bernoulli, Clairaut and d'Alembert, they flow so naturally from my two general formulae that one cannot sufficiently admire this accord of their profound meditations with the simplicity of the principles from which I have drawn my two equations ... Euler contributed to knowledge in many other areas, and in all of them he employed his mathematical knowledge and skill. He did important work in astronomy including [1]:... determination of the orbits of comets and planets by a few observations, methods of calculation of the parallax of the sun, the theory of refraction, consideration of the physical nature of comets, .... His most outstanding works, for which he won many prizes from the Paris Académie des Sciences, are concerned with celestial mechanics, which especially attracted scientists at that time. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (8 of 11) [2/16/2002 11:09:12 PM]

Euler

In fact Euler's lunar theory was used by Tobias Mayer in constructing his tables of the moon. In 1765 Tobias Mayer's widow received 3000 from Britain for the contribution the tables made to the problem of the determination of the longitude, while Euler received 300 from the British government for his theoretical contribution to the work. Euler also published on the theory of music, in particular he published Tentamen novae theoriae musicae in 1739 in which he tried to make music:... part of mathematics and deduce in an orderly manner, from correct principles, everything which can make a fitting together and mingling of tones pleasing. However, according to [7] the work was:... for musicians too advanced in its mathematics and for mathematicians too musical. Cartography was another area that Euler became involved in when he was appointed director of the St Petersburg Academy's geography section in 1735. He had the specific task of helping Delisle prepare a map of the whole of the Russian Empire. The Russian Atlas was the result of this collaboration and it appeared in 1745, consisting of 20 maps. Euler, in Berlin by the time of its publication, proudly remarked that this work put the Russians well ahead of the Germans in the art of cartography. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (86 books/articles)

Some Quotations (5)

A Poster of Leonhard Euler

Mathematicians born in the same country

Some pages from publications

The title page of Introductio in analysin infinitorum (1784) and another page from this work

Cross-references to History Topics

1. Fermat's last theorem 2. The development of group theory 3. Topology enters mathematics 4. Mathematical games and recreations 5. The fundamental theorem of algebra 6. Orbits and gravitation 7. Arabic mathematics : forgotten brilliance? 8. General relativity 9. An overview of the history of mathematics 10. The trigonometric functions 11. Pi through the ages 12. Quadratic, cubic and quartic equations

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (9 of 11) [2/16/2002 11:09:12 PM]

Euler

13. The rise of the calculus 14. Prime numbers Cross-references to Famous Curves

1. Catenary 2. Epicycloid 3. Hypocycloid 4. Tricuspoid 5. Trident of Newton

Other references in MacTutor

1. Bridges in Königsberg 2. Chronology: 1720 to 1740 3. Chronology: 1740 to 1760 4. Chronology: 1760 to 1780

Honours awarded to Leonhard Euler (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1747

Lunar features

Crater Euler and Rima Euler

Paris street names

Rue Euler (8th Arrondissement)

Other Web sites

1. Clark Kimberling 2. Rouse Ball 3. Euler-Mascheroni constant 4. Interactive Real Analysis 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 1998 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (10 of 11) [2/16/2002 11:09:12 PM]

Euler

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Euler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html (11 of 11) [2/16/2002 11:09:12 PM]

Eutocius

Eutocius of Ascalon Born: about 480 in Ascalon (now Ashqelon), Palestine Died: about 540 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eutocius of Ascalon was for a long time thought to have been born in 530. It is instructive to see how this came about for it shows how many pitfalls there are in the study of history. Eutocius wrote commentaries on three works of Archimedes. His commentary on Book II of On the Sphere and Cylinder ends with the statement:... the edition was revised by Isidorus of Miletus, the mechanical engineer, our teacher. From this it was thought that Eutocius was a pupil of Isidorus and his dates were deduced from this information. However, further investigation showed that this contradicted other information such the dedications that Eutocius makes in some of his other works. It was then realised that the comment at the end of Eutocius's commentary to Archimedes' On the Sphere and Cylinder was inserted by a later editor of the work who was indeed a pupil of Isidorus of Miletus. It is thought that the first of Eutocius's commentaries on Archimedes was written around 510. Ascalon, where Eutocius was born, had a long history and is mentioned in the Old Testament as Askelon. It received its Greek name after it was conquered by Alexander the Great in 332 BC and the city had fine public buildings built by Herod the Great. It seems likely that Eutocius left Ascalon and went to Alexandria to study. Paul Tannery has argued convincingly (see [5]) that Eutocius was almost certainly a pupil of Ammonius in Alexandria and it appears that he went on to become head of the Alexandrian School after Ammonius. Ammonius himself was a pupil of Proclus and Eutocius dedicates his commentary on Book I of Archimedes' On the sphere and cylinder to him. Eutocius addresses Ammonius in the preface asking him to [1]:... bear with him if he should have erred through youth. He explains that he found no satisfactory commentaries on Archimedes before his own time and promises further elucidation of the master if his work should meet with approval from Ammonius. Certainly this reads as if Eutocius is addressing his teacher and Paul Tannery's deduction seems secure. One has to assume that indeed Ammonius did approve, for Eutocius went on to write commentaries on other works by Archimedes, namely Measurement of the circle and On plane equilibria. However, Bulmer-Thomas in [1] is convinced that Eutocius did not know of certain other works by Archimedes such as Quadrature of a parabola and On spirals for he claims that he would have referred to them at

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eutocius.html (1 of 3) [2/16/2002 11:09:14 PM]

Eutocius

certain natural places in his commentaries rather than give much less suitable references at these points. Eutocius also edited and wrote commentaries on the first 4 books of the Conics of Apollonius. Heath writes [2]:Eutocius's commentary on Apollonius's "Conics" is extant for the first four Books, and it is probably owing to their having been commented on by Eutocius, as well as to their being more elementary than the rest, that these four Books alone survive in Greek. One sees immediately that commentators such as Eutocius are very important in the history of mathematics and many important works have only survived due to the work of the commentators. Eutocius does not appear to have done any original work. However, his commentaries contain much that is invaluable in the nature of historical information which might otherwise have been completely lost. Heath lists some of these important pieces of information [2]:(i) the account of the solutions of the problem of the duplication of the cube, or the finding of two mean proportionals, by Plato, Heron, Philon, Apollonius, Diocles, Pappus, Sporus, Menaechmus, Archytas, Eratosthenes, Nicomedes; (ii) the fragment discovered by Eutocius himself containing the missing solution, promised by Archimedes in On the Sphere and Cylinder Book II. 4, of the auxiliary problem amounting to the solution by means of conics of the cubic equation (a - x) x2 = b c2. (iii) the solutions (a) by Diocles of the original problem of II.4 without bringing in the cubic, (b) by Dionysodorus of the auxiliary cubic equation. As to contributions to astronomy, Eutocius did write an introduction to Book I of the Almagest but Neugebauer writes [3]:Eutocius did not write a "commentary" of the ordinary type that follows a given text chapter by chapter. ... the main part concerns methods of sexagesimal computation: multiplication, division, square roots etc. Another chapter concerns isoperimetric problems, followed by a short section about the shape and size of the earth, based on Ptolemy's norm of 500 stades for the equatorial degree. Obviously nothing of real astronomical interest has come down from Eutocius. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Doubling the cube 2. How do we know about Greek mathematicians?

Other references in MacTutor

Chronology: 500 to 900

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eutocius.html (2 of 3) [2/16/2002 11:09:14 PM]

Eutocius

Other Web sites

Vatican exhibition

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Eutocius.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Eutocius.html (3 of 3) [2/16/2002 11:09:14 PM]

Evans

Griffith Conrad Evans Born: 11 May 1887 in Boston, Massachusetts, USA Died: 8 Dec 1973 in Berkeley, California, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Griffith Evans' father was George William Evans and his mother was Mary Taylor. It was a family in which mathematics played a major role for George Evans was a mathematics teacher at the English and Latin High School in Boston. In fact George Evans had published a number of school level mathematics texts including Algebra for schools which had given him a good reputation as a teacher of mathematics. Griffith attended the school at which his father taught and completed his studies there in 1903 at the age of sixteen. He then entered Harvard University and showed outstanding ability. He was awarded his first degree, the A.B., in 1907 and he had the good fortune to have been taught by several outstanding mathematicians, including Osgood, Coolidge and Bôcher. It was under Bôcher's supervision that Evans began research at Harvard, being awarded his Master's degree in 1908 and his doctorate in 1910. His doctoral dissertation Volterra's integral equation of the second kind with discontinuous kernel was published in the Transactions of the American Mathematical Society in two parts in 1910 and 1911. Already before completing his Ph.D., Evans had been employed as an instructor at Harvard in session 1909-10. He won a Sheldon Travelling Fellowship from Harvard which enabled him to spend 1910 to 1912 studying in Europe. Most of this time Evans spent in Rome studying with Volterra but he also had an interest in applied mathematics and so he spent a summer in Berlin studying with Planck. Evans returned to the United States in 1912. He was now in the happy position of being sought by several of the top universities. In particular the Massachusetts Institute of Technology tried to encourage him to accept an appointment as did the University of California at Berkeley. However he accepted an offer form a third institution, namely the Rice Institute in Houston, Texas. The decision was not made

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Evans.html (1 of 4) [2/16/2002 11:09:16 PM]

Evans

lightly by Evans from which of these institutions to accept a post but he felt that Rice offered him the greatest opportunities. Appointed as assistant professor in 1912, Evans was promoted to full professor at Rice in 1916. In 1917 he married Isabel Mary John on 20 June and they had three children, all sons. This, of course, was the period of World War I and Evans was commissioned as a captain in the Air Branch of the U.S. Signal Corps during 1918-19. Rider writes in [1]:His scientific assignments concerning bomb trajectories and sights, and anti-aircraft defences took him to England, France, and Italy. With the help of Volterra, Evans facilitated the enrolment of U.S. military personnel in special wartime courses in Italian universities. Evans did a fine job of attracting visiting professors to Rice during his time there, including Menger and Rado. He also made many visits himself including Belgium, France and Italy. During a visit to the University of Minnesota in 1921 he met Fisher who convinced him of the need to develop statistics in the United States. The University of California at Berkeley continued to try to attract Evans, and he taught there during the summer terms of 1921 and 1928. He turned down an attractive offer from them in 1927 and he also turned down several offers from Harvard. However, in 1933 the University of California at Berkeley made him an offer which was so attractive that he accepted it. Basically he was given the explicit mandate to revitalise and improve the mathematics department and set up a programme for graduate studies. In the summer of 1934 Evans left Rice and began his task at Berkeley as chairman of the mathematics department. The article [3] shows why and how Evans was successful in building a world class centre there. Although at first he favoured appointing unemployed Americans, he soon changed to reap the benefits from the availability of mathematicians expelled from Europe and also from the changes to American science policy that resulted from World War II. In particular he appointed Lewy, Neyman and Tarski among fifteen appointments he made from 1934 to 1949. During World War II Evans again undertook work related to the war effort. As well as serving on the Applied Mathematics Panel of the National Defense Research Council, he also acted as a consultant on the design of guns for the Office of the Chief of Ordnance in Aberdeen, Maryland. For his war work he was awarded the Distinguished Assistance Award of the War Department and the Presidential Certificate of Merit. Evans was chairmen of the mathematics department at Berkeley for fifteen years, ending his term in 1949. He retired in 1955 but lived to see the new mathematics building at Berkeley named Evans Hall in 1971. His work dealt with potential theory, functional analysis, integral equations and the problem of minimal surfaces, the Plateau Problem. It was built on the foundations provided by Lebesgue, Volterra, Fréchet and Poincaré. Zund writes in [4] of his contributions to potential theory:He pioneered the use of general notions of integration and measure theory in this area, and his interests lay in application and development of new techniques rather than in deep structural theorems. These include the theories of harmonic and superharmonic functions, the Poisson-Stieltjes integral, the logarithmic potential, the discovery of the precursor of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Evans.html (2 of 4) [2/16/2002 11:09:16 PM]

Evans

Sobolev spaces in 1920 ..., generalisations of Heni Poincaré's "sweeping out" process, the theory of capacity, and finally an ingenious theory of multiple-valued harmonic functions (1947,1951). In fact the work referred to here on multiple-valued harmonic functions was a generalisation of his own work on the problem of minimal surfaces which he published on from the 1920s. He had proved, among many other results, that among the surfaces with a fixed boundary, there is a surface of minimal electric capacity. Evans also wrote on mathematical economics, in particular on monopolies, competition and cooperation, taxation, profit, prices, etc. Rider writes in [1]:Evans formulated a model of the economy as a whole and posed the problem of defining an aggregate variable in terms of microeconomics components. His 1924 paper on the dynamics of monopoly, which introduced time derivatives in demand relations, was recognised as the beginning of dynamic theories of economics. Among the important texts he wrote were Functional equations and their applications (1918), The logarithmic potential (1927), and Mathematical Introduction to economics (1930). Evans received many honours for his mathematical contribution. He was vice-president of the American Mathematical Society in 1924-26 and vice-president of the Mathematical Association of America in 1932. He served as vice-president for mathematics of the American Association for the Advancement of Science in 1931-32 and vice-president for economics of the American Association for the Advancement of Science in 1936-37. He was elected to the National Academy of Sciences in 1933 and was president of the American Mathematical Society in 1939-40. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Griffith Evans' (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1939 - 1940

AMS Colloquium Lecturer

1916

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Evans.html (3 of 4) [2/16/2002 11:09:16 PM]

Evans

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Evans.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Evans.html (4 of 4) [2/16/2002 11:09:16 PM]

Ezra

Abraham ben Meir ibn Ezra Born: 1092 in Tudela, Emirate of Saragossa (now Spain) Died: 1167 in Calahorra, Spain Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Rabbi Ben Ezra lived in Muslim Spain. Little is known of his life except that he was on friendly terms with the eminent poet and philosopher Judah ha-Levi, who some historians believe was ibn Ezra's father-in-law. Ibn Ezra made his reputation as a scholar and a poet. It is recorded that during this period of his life, up to 1140, he travelled to North Africa and possibly visited Egypt. From 1140 to 1160 ibn Ezra's life changed markedly. He was forced to wander throughout Europe during this period and he eventually settled down in Rome, then Lucca, for a few years before his death. It was during this latter period of his life that he composed his most famous works. In addition to his poetry, ibn Ezra wrote on [7]:... grammar, exegesis, philosophy, medicine, astronomy, and astrology. In addition to these topics, ibn Ezra wrote on permutations and combinations, the calendar, the astrolabe, and Biblical studies. He is of particular importance because he spread the learning of the Arabs through Europe at a time when scholarship in Christian Europe had been been neglected for five hundred or more years. Of the most interest to us in this archive devoted to the history of mathematics is ibn Ezra's work on numbers. He wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. The Book of the Unit is a work on the Indian symbols 1, 2, 3, 4, 5, 6, 7, 8, 9. A second work is the Book of the Number which describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Despite ibn Ezra's books, these ideas would not become accepted into mainstream European mathematics for several more centuries. Ibn Ezra translated al-Biruni's commentary on al-Khwarizmi's tables and made interesting comments on the introduction of Indian mathematics into Arabic science in the 8th century. Historians of science try today to quantify precisely how much Arabic mathematics was influenced by knowledge of Indian mathematics, so ibn Ezra's writing on this topic are carefully studied. Ibn Ezra's writings on grammar and poetry were often motivated by the "paytanim" [2]:Synagogues began ... to appoint official precentors, part of whose duty it was to compose poetical additions to the liturgy on special Sabbaths and festivals. The authors were called "paytanim" (from the Greek poietes, "poet"), their poems "piyyutim". The keynote was messianic fervour and religious exuberance. Besides employing the entire biblical, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ezra.html (1 of 2) [2/16/2002 11:09:17 PM]

Ezra

Mishnaic, and Aramaic vocabularies, the paytanim coined thousands of new words. ... Abraham ibn Ezra... attacked the language and style of the early paytanim; he [was] the first to use Arabic metres in religious poems. In fact to ibn Ezra there was no conflict between science and religion for he considered that science and astrology were at the basis of Jewish learning [7]:For ibn Ezra revelation and reason are ultimately perfectly congruent. His critical reading of the biblical text and his astrological interpretations of some biblical passages arise from his consistent application of a naturalist and rationalistic exegetical method and express his commitment to the view that rationality in inherent in revelation itself. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) Mathematicians born in the same country Cross-references to History Topics

A history of Zero

Honours awarded to Rabbi Ben Ezra (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater A ben Ezra

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ezra.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ezra.html (2 of 2) [2/16/2002 11:09:17 PM]

Faa_di_Bruno

Francesco Faà di Bruno Born: 29 March 1825 in Alessandria, Piemonte, Italy Died: 27 March 1888 in Turin, Italy

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Faà di Bruno studied at the Royal Military Academy of Turin with the aim of making a career in the army. He began his education in 1841, then he was commissioner in the army in 1847. However by 1853 he had decided to leave the army and take up the study of mathematics. Faà di Bruno travelled to Paris where he studied at the Sorbonne under Cauchy who [1]:... he admired, not only for his genius, but also for his religious fervour and his philanthropy. At the Sorbonne he was in the same classes as Hermite. After graduating he returned to Turin where he studied for his doctorate, which he obtained in 1861. In 1871 Faà di Bruno became a professor at the University of Turin, being appointed to the Chair of Higher Analysis there in 1876. His most famous mathematical work was one on binary forms which he published in 1876. This became better known in 1881 when Max Noether published a German edition of Faà di Bruno's work. However, Faà di Bruno had interests other than mathematics which took up much of his time and undoubtedly prevented him undertaking more mathematical research. After his time studying in the Sorbonne, Faà di Bruno did much charity work on his return to Turin. At this time Faà di Bruno came in contact with Giovanni Bosco. Bosco had been ordained a Roman Catholic priest in 1841 in Turin and began to work there to help boys who came to look for work in the city. Bosco provided boys with education, religious instruction, and recreation. He eventually he headed

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faa_di_Bruno.html (1 of 3) [2/16/2002 11:09:19 PM]

Faa_di_Bruno

a large establishment containing a grammar school, a technical school, and a church, all built through his efforts. In Turin Bosco and others founded the Society of St Francis de Sales in 1859. Influenced by Giovanni Bosco, Faà di Bruno was ordained a Roman Catholic priest in Rome on 22 October 1876. Faà di Bruno founded a religious order (Suore Minime di Nostra Signora del Suffragio) in order to direct and work for girls gathered in a house (similar to those founded by Giovanni Bosco) called Conservatorio del Suffragio. In order to provide work for the girls, Faà di Bruno had the idea that they could train as typesetters. He purchased a printing press and set up the Tipographia Suffragio. There a number of mathematics books were published including one by Faà di Bruno himself on elliptic functions. In 1898 the printing press was purchased by Peano for 407 lire and he printed the Rivista di Matematica on it for several years. Bosco was made a Saint on 1 April 1934. Already by this time there was a movement to canonise Faà di Bruno and in 1955 the Sacred Congregation of Rites officially accepted the claim for Faà di Bruno to be canonised. Faà di Bruno was declared a Saint by Pope John Paul II in St. Peter's Square in Rome on 25 September 1988. In [1] Faà di Bruno is described as follows:Faà di Bruno was tall and not always well dressed, but he was simple and good natured. He was of a solitary disposition and spoke seldom (and not always successfully in the classroom). He cultivated music and was said to be a good pianist. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (51 books/articles) Mathematicians born in the same country Other Web sites

1. The Catholic Encyclopedia 2. The Faa di Bruno web-site

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faa_di_Bruno.html (2 of 3) [2/16/2002 11:09:19 PM]

Faa_di_Bruno

JOC/EFR December 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Faa_di_Bruno.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faa_di_Bruno.html (3 of 3) [2/16/2002 11:09:19 PM]

Faber

Georg Faber Born: 5 April 1877 in Germany Died: 7 March 1966 in Germany Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Faber studied mathematics and physics at the universities of Munich and Göttingen between 1896 and 1901. In 1902 he received a doctorate from Munich for a thesis on the series expansion of analytical functions. He received his teaching qualification, the Habilitationsschrift, from the University of Würzburg in 1905 for a thesis on the series expansion of analytical functions. After working at a number of universities he was appointed to a chair at the Technische Hochschule in Munich in 1916. This was a post he held until he retired in 1946. Faber's most important work was on the polynomial expansion of functions. This is the problem of expanding an analytical function in an area bounded by a smooth curve as a sum of polynomials, where the polynomials are determined by the area. These polynomials are now known as Faber polynomials. Most of Faber's publications are in function theory. However, he also edited the collected works of Christoffel and volumes 14, 15, 16 of Euler's collected works. Faber was also interested in mathematical education and he worked with his colleague von Dyck at Munich on the mathematical education of engineers, physicists and mathematicians. In addition to his research areas, Faber lectured on complex analysis, probability theory, the theory of relativity and analytical mechanics. When World War II ended in 1945, Faber was appointed rector of the Technische Hochschule in Munich by the government. He organised restarting teaching in the university before retiring the following year. Faber had many interests outside mathematics. He was a great linguist, loved music and art, and enjoyed long walks. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country Other Web sites

Munich, Germany

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faber.html (1 of 2) [2/16/2002 11:09:20 PM]

Faber

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Faber.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faber.html (2 of 2) [2/16/2002 11:09:20 PM]

Fabri

Honoré Fabri Born: 5 April 1607 in Virieu-le-Grand, Dauphiné, France Died: 8 March 1688 in Rome, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Honoré Fabri entered the Jesuit Order in 1626 spending two years at Avignon. In 1628 he entered the Collège de la Trinité in Lyon where he studied philosophy, going on to study theology at Lyon from 1632 to 1636. He was ordained in 1635. His first position was in a Jesuit college, namely as professor of philosophy in Arles from 1636 to 1638. Further positions in Jesuit colleges followed. He was professor of logic in Aix- en- Provence for a year from 1638 then, for six years from 1640, professor of logic and mathematics at the Collège de la Trinité in Lyon. In 1646 Fabri went to Rome where he met Ricci. He joined the Penitentiary College being involved with the Inquisition. He was unable to avoid religious problems himself and he was accused of believing the philosophy of Descartes. After spending a year back in France in 1668/69 he returned to Rome and was put in prison. Through Ricci he had made the acquaintance of Grand Duke Leopold II and the Grand Duke saw that Fabri was released fairly soon from prison. Fabri worked on astronomy, physics and mathematics. He studied Saturn's rings in 1660, a topic on which he became involved in a dispute with Huygens which ran for five years. He also discovered the Andromeda nebula. Fabri developed a theory of tides which was based on the action of the moon. He also studied magnetism, optics and calculus. In calculus he was closer to Newton than to Cavalieri and his notation was cumbersome. His work on the calculus appeared in his major mathematical publication Opusculum geometricum. This book was written

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fabri.html (1 of 2) [2/16/2002 11:09:22 PM]

Fabri

because of the controversy about the cycloid which arose from Pascal's challenge. In this work Fabri computed xnsin x dx, sinnx dx and other integrals. He had a major influence on the development of the calculus through Leibniz. Fabri's students include Cassini and La Hire at the Collège de la Trinité and he also worked with Dechales. He was a friend and correspondent of Gassendi whom he first met while he was at Aix-en-Provence and he also corresponded with Huygens, Leibniz, Descartes, Mersenne and others. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country Other Web sites

1. Jesuit Scientists 2. The Galileo Project 3. The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fabri.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fabri.html (2 of 2) [2/16/2002 11:09:22 PM]

Fagnano_Giovanni

Giovanni Francesco Fagnano dei Toschi Born: 31 Jan 1715 in Sinigaglia (now Senigallia), Italy Died: 14 May 1797 in Sinigaglia (now Senigallia), Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giovanni Fagnano was the son of Giulio Fagnano dei Toschi. Giovanni was born into one of the leading families in Sinigaglia. The town of Sinigaglia, now known as Senigallia, is in central Italy and at the time of Giulio's birth was part of the Papal States. In fact the family went back very many generations in their association with Sinigaglia and one of the members of the family in the 12th century had been Lamberto Scannabecchi who became Pope Honorius II in 1124. Giovanni's father Giulio Fagnano held high office in Sinigaglia. He was appointed gonfaloniere in 1723 when Giovanni was eight years old. Gonfaloniere literally means "standard bearer" and it was a title of high civic magistrates in the medieval Italian city-states such as Sinigaglia. Giovanni was one of many children in his family but the only one to follow his father's interest in mathematics. He entered the Church being ordained priest, then appointed as canon of the cathedral in Sinigaglia in 1752. In 1755 Fagnano was appointed as archpriest, a very high rank to achieve. Fagnano continued his father's work on the triangle and wrote an unpublished treatise on the topic. One theorem on the triangle which he discovered is worth quoting. He proved that given any triangle T, then the triangle whose vertices are the bases of the altitudes of T has these altitudes as the bisectors of its angles. Fagnano also considered integration computing the integrals of xnsin(x) and xncos(x) by parts. In addition he calculated the integral of tan(x) as -log cos(x) and of cot(x) as log sin(x). Some of Fagnano's publications appear in the Nova acta eruditorum in 1774. However, he never achieved the international standing of his father although he did publish some work outside Italy. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fagnano_Giovanni.html (1 of 2) [2/16/2002 11:09:23 PM]

Fagnano_Giovanni

Cross-references to Famous Curves

Lemniscate of Bernoulli

Other Web sites

The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fagnano_Giovanni.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fagnano_Giovanni.html (2 of 2) [2/16/2002 11:09:23 PM]

Fagnano_Giulio

Giulio Carlo Fagnano dei Toschi Born: 6 Dec 1682 in Sinigaglia (now Senigallia), Italy Died: 26 Sept 1766 in Sinigaglia (now Senigallia), Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giulio Fagnano's father was Francesco Fagnano and his mother was Camilla Bartolini. Giulio was born into one of the leading families in Sinigaglia. The town of Sinigaglia, now known as Senigallia, is in central Italy and at the time of Giulio's birth was part of the Papal States. In fact the family went back very many generations in their association with Sinigaglia and one of the members of the family in the 12th century had been Lamberto Scannabecchi who became Pope Honorius II in 1124. Fagnano was brought up to follow the family tradition of high office in Sinigaglia. He was appointed gonfaloniere in 1723. Gonfaloniere literally means "standard bearer" and it was a title of high civic magistrates in the medieval Italian city-states such as Sinigaglia. Such offices were not easy in these times and Fagnano was subjected to many false charges made against him by envious citizens who were maliciously trying to damage his reputation. Fagnano had many children, one of whom was Giovanni Fagnano who followed in his father's footsteps becoming interested in mathematics. Giulio Fagnano was self educated in mathematics and treated the subject as a hobby. However, he achieved considerable international fame as a mathematician, and rightly so given the outstanding contributions which he made on a number of different topics. Fagnano suggested new methods of solving equations of degree 2, 3 and 4. He improved Bombelli's work on complex numbers giving a famous formula /4 =

log[(1 - i)/(1 + i)].

One of the topics for which Fagnano is best known is his work on triangles. Natucci in [1] writes:... he may well be considered the founder of the geometry of the triangle. Considering triangles he looked at interesting problems such as: Given ABC find P minimising PA2 + PB2 + PC2. For a quadrilateral ABCD find P minimising AP + BP + CP + DP. He also discovered that if X is the centre of gravity of the triangle ABC then XA2 + XB2 + XC2 = (AB2 + BC2 + CA2)/3. In his study of the rectification of the lemniscate, Carlo Fagnano introduced ingenious analytic

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fagnano_Giulio.html (1 of 3) [2/16/2002 11:09:24 PM]

Fagnano_Giulio

transformations that laid the foundation for the theory of elliptic integrals and his work was to lead to elliptic functions. Fagnano collected many of his published works, and a few unpublished ones, and produced the two volume treatise Produzioni matematiche in 1750. In 1751 Euler was asked to examine Produzioni matematiche and he found in this treatise relations between special types of elliptic integrals, that express the length of an arc of a lemniscate, which were quite unexpected to him. Generalising Fagnano's results, Euler went on to create a general theory of these integrals, in particular giving the famous addition formula for elliptic integrals. Fagnano had proved the duplication formula, a particular case of the addition formula, for the integrals. In fact Fagnano had proved remarkable properties of the lemniscate, including the fact that its arcs may be divided in n equal parts using a ruler and compass construction, where n = 2 2m, 3 2m, or 5 2m. He also found the area of the lemniscate. Fagnano made many other major contributions but his mathematical work was not without controversy. He was involved in priority disputes with Nicolaus(I) Bernoulli and, not surprisingly, the big dispute of the day which was between the supporters of Newton and those of Leibniz. Brook Taylor issued a challenge which Bernoulli and Fagnano both answered; the background and details of this controversy are studied in [3]. Fagnano received many honours. He had the title of count conferred on him by Louis XV in 1721, was elected to the Royal Society of London in 1723, and was made a marquis of Sant' Onofrio in 1745. In addition he was elected to the Berlin Academy of Sciences and was proposed for the Paris Académie des Sciences in 1766 but died before he could be elected. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

A visit to James Clerk Maxwell's house

Other references in MacTutor

Chronology: 1740 to 1760

Honours awarded to Giulio Fagnano (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1723

Other Web sites

The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fagnano_Giulio.html (2 of 3) [2/16/2002 11:09:24 PM]

Fagnano_Giulio

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fagnano_Giulio.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fagnano_Giulio.html (3 of 3) [2/16/2002 11:09:24 PM]

Faltings

Gerd Faltings Born: 28 July 1954 in Gelsenkirchen-Buer, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gerd Faltings studied for his doctorate at the University of Münster, being awarded his Ph.D. in 1978. Following the award of his doctorate, Faltings went to the United States where he spent a year doing postdoctoral work as a research fellow at Harvard University in 1978-79. In 1979 Faltings returned to Germany, taking up an appointment as professor of mathematics at the University of Wuppertal. In 1985 Faltings was appointed to the faculty at Princeton. Faltings proved conjectures by Mordell, Shafarevich and Tate during 1983. In the same year he received the Danny Heinemen Prize from the Akademie der Wissenschaften, Göttingen. In 1986 Faltings received the highest honour that a young mathematician can receive when he was awarded a Fields Medal at the International Congress of Mathematicians at Berkeley. At the Congress B Mazur gave an address describing the work by Faltings which had led to the award. He received the medal primarily for his proof of the Mordell Conjecture which he achieved using methods of arithmetic algebraic geometry. Faltings has been closely linked with the work leading to the final proof of Fermat's Last Theorem by Andrew Wiles. In 1983 Faltings proved that for every n > 2 there are at most a finite number of coprime integers x, y, z with xn + yn = zn. This was a major step but a proof that the finite number was 0 in all cases did not seem likely to follow by extending Falting's arguments. However, Faltings was the natural person that Wiles turned to when he wanted an opinion on the correctness of his repair of his proof of Fermat's Last Theorem in 1994. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faltings.html (1 of 2) [2/16/2002 11:09:26 PM]

Faltings

Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

Fermat's last theorem

Other references in MacTutor

Chronology: 1980 to 1990

Honours awarded to Gerd Faltings (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1986

Other Web sites

1. AMS (An article about the proof of Fermat's last theorem 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Faltings.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faltings.html (2 of 2) [2/16/2002 11:09:26 PM]

Fano

Gino Fano Born: 5 Jan 1871 in Mantua, Italy Died: 8 Nov 1952 in Verona, Italy

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gino Fano's father was Ugo Fano and his mother was Angelica Fano. Ugo Fano came from a wealthy family and he had no need for employment. Ugo Fano was a follower of Giuseppe Garibaldi and strongly in favour of Italian unification. In fact on 17 March 1861, ten years before Gino's birth, the Kingdom of Italy was formally created but it was only just before Gino was born that Italian troops captured Rome. Gino grew up in the newly created country which suffered many problems but also had a new confidence in education. Fano studied at the University of Turin which he entered in 1888. His studies there were directed by Corrado Segre and he was also influenced by Castelnuovo. In fact Castelnuovo had been appointed as D'Ovidio's assistant in Turin the year before Fano began his studies and Corrado Segre had been appointed to the chair of higher geometry in Turin the year that Fano entered the University of Turin. This was an exciting place for research in geometry and it is not surprising that Fano was led to specialise in this area. In 1892 Fano graduated from Turin and then, in 1893, he went to Göttingen to undertake research and of course to study under Felix Klein. Twenty years earlier, in 1872, Klein had produced his synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programme (1872). The Erlanger Programme gave the unified approach to geometry that is now the standard accepted view. Corrado Segre corresponded regularly with Klein and in this way Fano had been brought to Klein's attention. In fact this had led to Fano translating the Erlanger Programme into Italian while he was an undergraduate at Turin and it had been published in the Annali di matematica in 1890. Of course Fano did indeed study with Klein in Göttingen where he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fano.html (1 of 3) [2/16/2002 11:09:28 PM]

Fano

attended his lectures. Fano became Castelnuovo's assistant in Rome in 1894, a position he held for four years. Following this assistantship, Fano went to Messina in the extreme northeastern Sicily where he worked from 1899 to 1901. He had left that city well before an earthquake struck Messina on 28 December 1908, almost totally destroying the city and killing 78000. By this time Fano was far away in Turin where he had been appointed as professor at the University in 1901. In 1911 Fano married Rosetta Cassin and their two children, both of the sons, became professors in the United States. He taught at Turin from 1901 until 1938 when he was deprived of his chair by the Fascist Regime. After this Fano went to Switzerland where he taught Italian students at an international camp near Lausanne. After the end of World War II Fano, who was seventy-four years old by this time, continued to travel and lecture on mathematics. In particular he visited the United States where he lectured and he also lectured in his native Italy during the remaining six years of his life. Fano's work was mainly on projective and algebraic geometry. Fano was a pioneer in finite geometry and one of the first people to try to set geometry on an abstract footing. Before Hilbert was to make such abstract statements Fano said:As a basis for our study we assume an arbitrary collection of entities of an arbitrary nature, entities which, for brevity, we shall call points, but this is quite independent of their nature. Struik, in [1], describes Fano's contribution:Early studies deal with line geometry and linear differential equations with algebraic coefficients ... . Later work is on algebraic and especially cubic surfaces, as well as on manifolds with a continuous group of Cremona transformations. He showed the existence of irrational involutions in three-space S3, i.e., of "unirational" manifolds not birationally representable on S3. He also studied birational contact transformations and non-euclidean and non-archimedean geometries. Fano wrote many textbooks, examples of which are his famous geometry texts Lezioni di geometria descrittiva (1914) and Lezioni di geometria analitica e proiettiva (1930). This last text was written jointly with Alessandro Terracini who wrote the obituaries [5] and [6] of Fano. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fano.html (2 of 3) [2/16/2002 11:09:28 PM]

Fano

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fano.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fano.html (3 of 3) [2/16/2002 11:09:28 PM]

Faraday

Michael Faraday Born: 22 Sept 1791 in Newington Butts, Surrey (now London) England Died: 25 Aug 1867 in Hampton Court, Middlesex, England

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Michael Faraday did not directly contribute to mathematics so should not really qualify to have his biography in this archive. However he was such a major figure and his science had such a large impact on the work of those developing mathematical theories that it is proper that he is included. We say more about this below. Faraday's father, James Faraday, was a blacksmith who came from Yorkshire in the north of England while his mother Margaret Hastwell, also from the north of England, was the daughter of a farmer. Early in 1791 James and Margaret moved to Newington Butts, which was then a village outside London, where James hoped that work was more plentiful. They already had two children, a boy Robert and a girl, before they moved to Newington Butts and Michael was born only a few months after their move. Work was not easy to find and the family moved again, remaining in or around London. By 1795, when Michael was around five years, the family were living in Jacob's Wells Mews in London. They had rooms over a coachhouse and, by this time, a second daughter had been born. Times were hard particularly since Michael's father had poor health and was not able to provide much for his family. The family were held closely together by a strong religious faith, being members of the Sandemanians, a form of the Protestant Church which had split from the Church of Scotland. The Sandemanians believed in the literal truth of the Bible and tried to recreate the sense of love and community which had characterised the early Christian Church. The religious influence was important for Faraday since the theories he developed later in his life were strongly influenced by a belief in a unity of the world.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faraday.html (1 of 5) [2/16/2002 11:09:30 PM]

Faraday

Michael attended a day school where he learnt to read, write and count. When Faraday was thirteen years old he had to find work to help the family finances and he was employed running errands for George Riebau who had a bookselling business. In 1805, after a year as an errand-boy, Faraday was taken on by Riebau as an apprentice bookbinder. He spent seven years serving his apprenticeship with Riebau. Not only did he bind books but he also read them. Riebau wrote a letter in 1813 in which he described how Faraday spent his days as an apprentice (see for example [3]):After the regular hours of business, he was chiefly employed in drawing and copying from the Artist's Repository, a work published in numbers which he took in weekly. ... Dr Watts's Improvements of the mind was then read and frequently took in his pocket, when he went an early walk in the morning, visiting some other works of art or searching for some mineral or vegetable curiosity. ... His mind ever engaged, besides attending to bookbinding which he executed in a proper manner. His mode of living temperate, seldom drinking any other than pure water, and when done his day's work, would set himself down in the workshop ... If I had any curious book from my customers to bind, with plates, he would copy such as he thought singular or clever ... Faraday himself wrote of this time in his life:Whilst an apprentice, I loved to read the scientific books which were under my hands ... From 1810 Faraday attended lectures at John Tatum's house. He attended lectures on many different topics but he was particularly interested in those on electricity, galvanism and mechanics. At Tatum's house he made two special friends, J Huxtable who was a medical student, and Benjamin Abbott who was a clerk. In 1812 Faraday attended lectures by Humphry Davy at the Royal Institution and made careful copies of the notes he had taken. In fact these lectures would become Faraday's passport to a scientific career. In 1812, intent on improving his literary skills, he carried out a correspondence with Abbott. He had already tried to leave bookbinding and the route he tried was certainly an ambitious one. He had written to Sir Joseph Banks, the President of the Royal Society, asking how he could become involved in scientific work. Perhaps not surprisingly he had received no reply. When his apprenticeship ended in October 1812, Faraday got a job as a bookbinder but still he attempted to get into science and again he took a somewhat ambitious route for a young man with little formal education. He wrote to Humphry Davy, who had been his hero since he attended his chemistry lectures, sending him copies of the notes he had taken at Davy's lectures. Davy, unlike Banks, replied to Faraday and arranged a meeting. He advised Faraday to keep working as a bookbinder, saying:Science [is] a harsh mistress, and in a pecuniary point of view but poorly rewarding those who devote themselves to her service. Shortly after the interview Davy's assistant had to be sacked for fighting and Davy sent for Faraday and invited him to fill the empty post. In 1813 Faraday took up the position at the Royal Institution. In October 1813 Davy set out on a scientific tour of Europe and he took Faraday with him as his assistant and secretary. Faraday met Ampère and other scientists in Paris. They travelled on towards Italy where they spent time in Genoa, Florence, Rome and Naples. Heading north again they visited Milan where Faraday met Volta. The trip was an important one for Faraday [3]:These eighteen months abroad had taken the place, in Faraday's life, of the years spent at university by other men. He gained a working knowledge of French and Italian; he had http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faraday.html (2 of 5) [2/16/2002 11:09:30 PM]

Faraday

added considerably to his scientific attainments, and had met and talked with many of the leading foreign men of science; but, above all, the tour had been what was most valuable to him at that time, a broadening influence. On his return to London, Faraday was re-engaged at the Royal Institution as an assistant. His work there was mainly involved with chemical experiments in the laboratory. He also began lecturing on chemistry topics at the Philosophical Society. He published his first paper in 1816 on caustic lime from Tuscany. In 1821 Faraday married Sarah Barnard whom he had met when attending the Sandemanian church. Faraday was made Superintendent of the House and Laboratory at the Royal Institution and given additional rooms to make his marriage possible. The year 1821 marked another important time in Faraday's researches. He had worked almost entirely on chemistry topics yet one of his interests from his days as a bookbinder had been electricity. In 1820 several scientists in Paris including Arago and Ampère made significant advances in establishing a relation between electricity and magnetism. Davy became interested and this gave Faraday the opportunity to work on the topic. He published On some new electro-magnetical motions, and on the theory of magnetism in the Quarterly Journal of Science in October 1821. Pearce Williams writes [1]:It records the first conversion of electrical into mechanical energy. It also contained the first notion of the line of force. It is Faraday's work on electricity which has prompted us to add him to this archive. However we must note that Faraday was in no sense a mathematician and almost all his biographers describe him as "mathematically illiterate". He never learnt any mathematics and his contributions to electricity were purely that of an experimentalist. Why then include him in an archive of mathematicians? Well it was Faraday's work which led to deep mathematical theories of electricity and magnetism. In particular the remarkable mathematical theories on the topic developed by Maxwell would not have been possible without Faraday's discovery of various laws. This is a point which Maxwell himself stressed on a number of occasions. In the ten years from 1821 to 1831 Faraday again undertook research on chemistry. His two most important pieces of work on chemistry during that period was liquefying chlorine in 1823 and isolating benzene in 1825. Between these dates, in 1824, he was elected a fellow of the Royal Society. Now this was a difficult time for Faraday since Davy was at this time President of the Royal Society and could not see the man who he still thought of as his assistant as becoming a Fellow. Although Davy opposed his election, he was over-ruled by the other Fellows. Faraday never held the incident against Davy, always holding him in the highest regard. Faraday introduced a series of six Christmas lectures for children at the Royal Institution in 1826. In 1831 Faraday returned to his work on electricity and made what is arguably his most important discovery, namely that of electro-magnetic induction. This discovery was the opposite of that which he had made ten years earlier. Now he showed that a magnet could induce an electrical current in a wire. Thus he was able to convert mechanical energy into electrical energy and discover the first dynamo. Again he made lines of force central to his thinking. He published his first paper in what was to become a series on Experimental researches on electricity in 1831. He read the paper before the Royal Society on 24 November of that year. In 1832 Faraday began to receive honours for his major contributions to science. In that year he received

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faraday.html (3 of 5) [2/16/2002 11:09:30 PM]

Faraday

an honorary degree from the University of Oxford. In February 1833 he became Fullerian Professor of Chemistry at the Royal Institution. Further honours such as the Royal Medal and the Copley Medal, both from the Royal Society, were to follow. In 1836 he was made a Member of the Senate of the University of London, which was a Crown appointment. During this period, beginning in 1833, Faraday made important discoveries in electrochemistry. He went on to work on electrostatics and by 1838 he [1]:... was in a position to put all the pieces together into a coherent theory of electricity. The extremely high workload eventually told on Faraday's health and in 1839 he suffered a nervous breakdown. He did recover his health and by 1845 he began intense research activity again. The work which he undertook at this time was the result of mathematical developments in the subject. Faraday's ideas on lines of force had received a mathematical treatment from William Thomson. He wrote to Faraday on 6 August 1845 telling him of his mathematical predictions that a magnetic field should affect the plane of polarised light. Faraday had attempted to detect this experimentally many years earlier but without success. Now, with the idea reinforced by Thomson, he tried again and on 13 September 1845 he was successful in showing that a strong magnetic field could rotate the plane of polarisation, and moreover that the angle of rotation was proportional to the strength of the magnetic field. Faraday wrote (see for example [1]):That which is magnetic in the forces of matter has been affected, and in turn has affected that which is truly magnetic in the force of light. He followed his line of experiments which led him to discover diamagnetism. By the mid 1850s Faraday's mental abilities began to decline. At around the same time Maxwell was building on the foundations Faraday had created developing a mathematical theory which would always have been out of reach for Faraday. However Faraday continued to lecture at the Royal Institution but declined the offer of the Presidency of the Royal Society in 1857. He continued to give the children's Christmas lectures. In 1859-60 he gave the Christmas lectures on the various forces of matter. At the following Christmas he gave the children's lectures on the chemical history of the candle. These two final series of lectures by Faraday were published and have become classics. The Christmas lectures at the Royal Institution, begun by Faraday, continue today but now reach a much greater audience since they are televised. I [EFR] have watched these lectures with great interest over many years. They are a joy for anyone interested as I am in the "public understanding of science". I particularly remember lectures by Carl Sagan on "the planets" and mathematics lectures by Chris Zeeman and Ian Stewart. The Royal Institution literature states:[Faraday's] magnetic laboratory, where many of his most important discoveries were made, was restored in 1972 to the form it was known to have had in 1854. A museum, adjacent to the laboratory, houses a unique collection of original apparatus arranged to illustrate the most important aspects of Faraday's immense contribution to the advancement of science in his fifty years at the Royal Institution. Martin, in [3], gives this indication of Faraday's character:He was by any sense and by any standard a good man; and yet his goodness was not of the kind that make others uncomfortable in his presence. His strong personal sense of duty did http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faraday.html (4 of 5) [2/16/2002 11:09:30 PM]

Faraday

not take the gaiety out of his life. ... his virtues were those of action, not of mere abstention ... Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Michael Faraday (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1824

Royal Society Royal Medal Royal Society Copley Medal Other Web sites

1. E Katz 2. IEEE (Exhibition on Maxwell and Faraday) 3. I Hutchison 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Faraday.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faraday.html (5 of 5) [2/16/2002 11:09:30 PM]

Farey

John Farey Born: 1766 in Woburn, Bedfordshire, England Died: 6 Jan 1826 in London, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Farey has been included in this archive despite being a geologist and not a mathematician. The reason we have included him is that he made one mathematical observation and, from this, the Farey series of fractions has been named. We shall discuss below Farey's contribution to mathematics and also look at others who contributed to Farey series. Farey attended a local school in Woburn until he was sixteen years of age when he went to a school in Halifax, Yorkshire, where he studied mathematics, drawing and surveying. He married in 1790 and, the following year, his first son (also called John Farey) was born. John Farey Jnr (1791-1851) went on to become a civil engineer and also has an entry in the Dictionary of National Biography immediately following that of his father. Francis, the fifth Duke of Bedford had extensive estates in Bedfordshire and, in 1792, he appointed Farey as the land steward for his Woburn estates. Farey held this post for ten years and it was during this time that he was able to gain expertise in geology. In October 1801 William Smith, the engineer and geologist who is best known for his development of the science of stratigraphy (which is the study of rock successions and relation to the historical time scale), was employed by the Duke of Bedford. Farey had already become interested in soils and rocks through carrying out his duties as land steward and he now took the opportunity to learn all that he could from Smith about stratification. When the Duke died suddenly in 1802, the Duke's brother John dismissed Farey from his post. At this point Farey went to London where he [4]:... established an extensive practice as a consulting surveyor and geologist. There are two aspects to Farey's contributions to science. On the one hand he applied the skills in geology which he had learnt from Smith and made some important contributions. Perhaps the word "important" here is overstating the case which Eyles in [1] espresses as follows:As a geologist Farey is entitled to respect for the work which he carried out himself, although it has scarcely been noticed in the standard histories of geology. Let us note that this work included producing a map of the strata visible between London and Brighton (a journey he made frequently to visit his brother who lived in Brighton), and his Survey of the county of Derbyshire which studied the soils and the succession of strata in that county.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Farey.html (1 of 3) [2/16/2002 11:09:33 PM]

Farey

The second aspect of Farey's work was his scientific writings. These are important, in part because he argued strongly for William Smith to be recognised as a major figure in geology and without his efforts Smith's contributions may not have been so readily appreciated. Farey published around sixty scientific articles between 1804 and 1824, most in Rees's Encyclopaedia, the Monthly Magazine and the Philosophical Magazine. His first article, written in 1804 in the Philosophical Magazine was On the mensuration of timber while the last, in the same publication, was On the velocity of sound and on the Encke planet. Neither the biographies [1] nor [4] mention Farey's contribution to mathematics. Farey's article which is relevant to our history of mathematics archive was also published in the Philosophical Magazine and appeared in 1816. It was called On a curious property of vulgar fractions and it was sent to the editor from Howland Street in London, the residence of Farey's eldest son where he spent the last years of his life (in fact he died in that house). The article consists of only four paragraphs. In the first paragraph Farey says that he noted the "curious property" while examining the tables of Complete decimal quotients produced by Henry Goodwin. In the second paragraph he defines the Farey series and states the "curious property". The Farey series (really a sequence) is defined as follows. For a fixed number n, consider all rationals between 0 and 1 which, when expressed in their lowest terms, have denominator not exceeding n. Write the sequence in ascending order of magnitude beginning with the smallest. Then the "curious property" is that each member of the sequence is equal to the rational whose numerator is the sum of the numerators of the fractions on either side, and whose denominator is the sum of the denominators of the fractions on either side. In the third paragraph of his article Farey gives an example. He takes n = 5. Then the Farey sequence is: F5 = Now Farey illustrates the "curious property" with the example further examples

,

and

. To help the reader we note the

.

The final paragraph of [5] reads:I am not acquainted, whether this curious property of vulgar fractions has been before pointed out?; or whether it may admit of some easy or general demonstration?; which are points on which I should be glad to learn the sentiments of some of your mathematical readers ... One mathematical reader (at least of a French translation) was Cauchy, and he gave the necessary proof in his Exercises de mathématique which was published in the same year as Farey's article. This might have been the end of the story but the there is more to tell. Farey was not the first to notice the property. Haros, in 1802, wrote a paper on the approximation of decimal fractions by common fractions. He explains how to construct what is in fact the Farey sequence for n = 99 and Farey's "curious property" is built into his construction. However, this is certainly not a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Farey.html (2 of 3) [2/16/2002 11:09:33 PM]

Farey

proof, nor for that matter a general statement of the "curious property". Historical references to the Farey sequence have been examined by the authors of [3]. The standard reference for the Farey sequence is [2] in which Hardy writes:[Farey] gave no proof, and it is unlikely that he had found one, since he seems to have been at the best an indifferent mathematician. The quotation we gave above shows that Farey explicitly states he has no proof. Hardy continues:Farey has a notice of twenty lines in the Dictionary of National Biography, where he is described as a geologist. As a geologist he is forgotten, and his biographer does not mention the one thing in his life which survives. This is, in my opinion, unnecessarily cruel and not strictly accurate. Even worse, and certainly completely inaccurate, is Hardy's comment in A mathematician's apology where he writes:... Farey is immortal because he failed to understand a theorem which Haros had proved perfectly fourteen years before ... The article [3] contains other interesting information on Farey's sequence, its relation to Pick's area theorem, and the inaccurate historical comments made about the sequence over many years. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Farey.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Farey.html (3 of 3) [2/16/2002 11:09:33 PM]

Fatou

Pierre Joseph Louis Fatou Born: 28 Feb 1878 in Lorient, France Died: 10 Aug 1929 in Pornichet, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Fatou entered the Ecole Normanle Supérieure in Paris in 1898 to study mathematics. He graduated in 1901 and then decided that the chance of obtaining a mathematics post was so low that he would apply for a position in the Paris Observatory. Having been appointed to the astronomy post, Fatou continued to work on mathematics for his thesis. He submitted his thesis in 1906 which was on integration theory and complex function theory. Fatou proved that if a function is Lebesgue integrable, then radial limits for the corresponding Poisson integral exist almost everywhere. This result led to generalisations by Privalov, Plessner and Marcel Riesz. Although not giving a complete solution, Fatou's work also made a major contribution to finding a solution to the related question of whether conformal mapping of Jordan regions onto the open disc can be extended continuously to the boundary. In 1907 Fatou received his doctorate for this important work. The book [2] presents a beautiful historical account of the global theory of iteration of complex analytic functions. Fatou enters this history in a rather complicated way and the book does an excellent job in explaining an interesting episode in the history of mathematics. In 1915, the Académie des Sciences in Paris gave the topic for its 1918 Grand Prix. The prize would be awarded for a study of iteration from a global point of view. The author of [2] suggests that mathematicians such as Appell, Emile Picard, and Koenigs had put forward the idea to the Académie des Sciences because they were hoping for developments of Montel's concept of normal families. Fatou wrote a long memoirs which did indeed use Montel's idea of normal families to develop the fundamental theory of iteration in 1917. Although we do not know for certain that he was intending to enter for the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fatou.html (1 of 3) [2/16/2002 11:09:35 PM]

Fatou

Grand Prix, it seems almost certain that he undertook the work with that in mind. Given that the topic had been proposed for the prize, it is not surprising that another mathematician would also work on the topic, and indeed Julia also produced a long memoir developing the theory in a similar way to Fatou. The two, however, chose different ways to go forward. During the later half of 1917 Julia deposited his results in sealed envelopes with the Académie des Sciences. Fatou, on the other hand, published an announcement of his results in a note in the December 1917 part of Comptes Rendus. It later became evident that they had discovered very similar results. Julia wrote a letter to Comptes Rendus concerning priority which was published on 31 December 1917. Julia had asked the Académie des Sciences to inspect his sealed envelopes and Georges Humbert had been asked to carry out the task. In the same 31 December 1917 part of Comptes Rendus Georges Humbert has a letter reporting on Julia's papers. Almost certainly as a result of these letters Fatou did not enter for the Grand Prix and it was awarded to Julia. Fatou did not lose out completely, however, and even though he had not entered for the prize, the Académie des Sciences gave him an award for his outstanding paper on the topic. Fatou was given the title of "astronomer" in 1928 and, as an astronomer, he also made contributions to that topic. Using existance theorems for the solutions to differential equations, Fatou was able to prove rigorously certian results on planetary orbits which Gauss had suggested by only verified with an intuitive argument. He also studied the motion of a planet in a resistant medium with the intention of explaining how twin stars would form with the capture of one moving in the atmosphere of the other. We have mentioned some of his important mathematical work above. We should also mention his work on Taylor series where he examined the convergence and the analytic extension of the series. Perhaps Fatou's most famous result is that a harmonic function u > 0 in a ball has a nontangential limit almost everywhere on the boundary. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fatou.html (2 of 3) [2/16/2002 11:09:35 PM]

Fatou

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fatou.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fatou.html (3 of 3) [2/16/2002 11:09:35 PM]

Faulhaber

Johann Faulhaber Born: 5 May 1580 in Ulm, Germany Died: 1635 in Ulm, Germany

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Faulhaber was trained as a weaver. However he was taught mathematics in Ulm and showed such promise that the City of Ulm appointed him city mathematician and surveyor. He opened his own school in Ulm in 1600 but he was in great demand because of his skill in fortification work. His expertise saw him working on fortifications for Basel, Frankfurt and many other cities. He also designed waterwheels in Ulm and made mathematical and surveying instruments, particularly ones with military applications. Among the scientists with whom Faulhaber collaborated were Kepler and van Ceulen. He was a Rosicrucian, a brotherhood combining elements of mystical beliefs with an optimism about the ability of science to improve the human condition. He made a major impression on Descartes with both his scientific and Rosicrucian beliefs and influenced his thinking. Faulhaber was a 'Cossist', an early algebraist. He is important for his work explaining logarithms associated with Stifel, Bürgi and Napier. He made the first German publication of Briggs' logarithms. Faulhaber's most major contribution, however, was in studying sums of powers of integers. Let N = n(n+1)/2. Define nk to be the sum ik where the sum is from 1 to n. Then N = n1. In 1631 Faulhaber published Academia Algebra in Augsburg. It was a German text despite the Latin title. In Academia Algebra Faulhaber gives nk as a polynomial in N, for k = 1, 3, 5, ... ,17. He also gives the corresponding polynomials in n. Faulhaber states that such polynomials in N exist for all k, but gave no proof. This was first proved by Jacobi in 1834. It is not known how much Jacobi was influenced by http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faulhaber.html (1 of 3) [2/16/2002 11:09:37 PM]

Faulhaber

Faulhaber's work, but we do know that Jacobi owned Academia Algebra since his copy of it is now in the University of Cambridge. Faulhaber did not discover the Bernoulli numbers but Jacob Bernoulli refers to Faulhaber in Ars Conjectandi published in Basel in 1713, eight years after Jacob Bernoulli died, where the Bernoulli numbers (so named by De Moivre) appear. Academia Algebra contains a generalisation of sums of powers. Faulhaber gave formulas for m-fold sums of powers defined as follows. Define 0 nk = nk and m+1 nk = m 1k + m 2k + ... +

m nk.

Faulhaber gives formulas for many of these m-fold sums including giving a polynomial for 11 n6. Knuth, in [7] remarks:His polynomial ... turns out to be absolutely correct, according to calculations with a modern computer. ... One cannot help thinking that nobody has ever checked these numbers since Faulhaber himself wrote them down, until today. At the end of Academia Algebra Faulhaber states that he has calculated polynomials for nk as far as k = 25. He gives the formulas in the form of a secret code, which was common practice at the time. Knuth, in [7], suggests he is the first to crack the code (the task [of cracking the code] is relatively easy with modern computers) and shows that Faulhaber had the correct formulas up to k = 23, but his formulas for k = 24 and k = 25 appear to be wrong. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) A Poster of Johann Faulhaber

Mathematicians born in the same country

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faulhaber.html (2 of 3) [2/16/2002 11:09:37 PM]

Faulhaber

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Faulhaber.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Faulhaber.html (3 of 3) [2/16/2002 11:09:37 PM]

Fefferman

Charles Louis Fefferman Born: 18 April 1949 in Washington, D.C., USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Fefferman was a child prodigy. It is claimed that he had mastered the calculus before the age of twelve. Fefferman entered the University of Maryland when he was very young and in 1966, at the age of 17, he graduated with the highest distinction. After graduating Fefferman undertook postgraduate work at Princeton University supervised by Elias Stein. He was awarded his PhD in 1969 for a thesis entitled Inequalities for Strongly Regular Convolution Operators. He lectured at Princeton for the years 1969-70. He moved to the University of Chicago in 1970 and, one year later in 1971, he was promoted to full professor there, earning him the distinction of becoming the youngest full professor every appointed in the United States. In 1973 Fefferman returned to Princeton and, in 1984, he was appointed Herbert Jones Professor at Princeton. Fefferman contributed several innovations that revised the study of multidimensional complex analysis by finding correct generalisations of classical low-dimensional results. Fefferman's work on partial differential equations, Fourier analysis, in particular convergence, multipliers, divergence, singular integrals and Hardy spaces earned him a Fields Medal at the International Congress of Mathematicians at Helsinki in 1978. In 1979 Fefferman described the way that he like to work:I like to lie down on the sofa for hours at a stretch thinking intently about shapes, relationships and change - rarely about numbers as such. I explore idea after idea in my mind, discarding most. When a concept finally seems promising, I'm ready to try it out on paper. But first I get up and change the baby's diaper. ... New ideas are not easy to find. If http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fefferman.html (1 of 3) [2/16/2002 11:09:39 PM]

Fefferman

you are lucky enough to be working on an idea which is actually right, it can take a long time before you know that it's right. Conversely, if you are going up a blind alley, it can also take a long time before you find out. You can end up saying 'Oops, I've been working for years on something wrong.' A good mathematician must have the courage to take a lot of work and throw it away. In 1992 Fefferman was awarded the Bergman Prize. The citation from the awarding committee read (see [2]):Charles Fefferman has made enormously important contributions to the study of the Bergman kernel and has initiated much of the activity in the topic. ... Each of [three of Fefferman's papers] is a 'tour-de-force', each contains not only results about the Bergman kernel and its applications but each also develops highly original ideas and techniques which are of great importance... In his reply Fefferman said:I am grateful to the selection committee for awarding me the Bergman Prize. Bergman's ideas have been a major influence in my work. They continue to provide deep, important problems for analysis. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1970 to 1980

Honours awarded to Charles Fefferman (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1978

AMS Colloquium Lecturer

1983

Other Web sites

1. Bellevue College, USA 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fefferman.html (2 of 3) [2/16/2002 11:09:39 PM]

Fefferman

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fefferman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fefferman.html (3 of 3) [2/16/2002 11:09:39 PM]

Feigenbaum

Mitchell Jay Feigenbaum Born: 19 Dec 1944 in Philadelphia, USA

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mitchell Feigenbaum's father is Abraham Joseph Feigenbaum, an analytic chemist whose parents had emigrated from a town near Warsaw in Poland to the United States. Mitchell's (or Mitch's as he is known) mother is Mildred Sugar whose parents emigrated to the United States from Kiev. Mitchell was the middle child of his parents three children, having an older brother Edward and a younger sister Glenda. Mitchell entered a public school for gifted children when he was five years old. Unlike Edward who displayed all the characteristics of a child prodigy, reading from a very young age, Mitchell could not read when he entered school and he needed tutoring from his mother to bring him up to the level of the other children. Moved to a different school, he became somewhat bored and had no friends among the other children. In fact up until the time he went to university Mitchell would not enjoy the company of his fellow pupils. Feigenbaum's mother taught him algebra when he was in the fifth form but reading continued to be something that he did not like much. Perhaps the reason was that he tried reading articles in Encyclopaedia Britannica which, given that he was so young, proved too difficult for him to understand. When he was twelve years old he started his high school education in Brooklyn. About the same time he began to develop certain obsessive tendencies such as excessive cleanliness which meant that he was continually washing his hands. He suffered these difficulties for quite a few years but overcame them when a university student. The school system seemed unable to provide Feigenbaum with the right stimulus for he tried as hard as he could to avoid classes despite making remarkable academic progress and scoring full marks in mathematics and science in the examinations covering the State. Even when he went to Tilden High http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feigenbaum.html (1 of 5) [2/16/2002 11:09:41 PM]

Feigenbaum

School in Brooklyn, a school with a fine reputation, Feigenbaum found the education there no more enjoyable, despite once again excelling in examinations. In [1] Feigenbaum described how his love of calculating started at school:... starting in junior high school, I decided that I could calculate the logarithm table myself, and later the trigonometric tables. I loved Newton's method for solving transcendentals, and in high school I already knew that starting values can make a big difference and lead to non-convergent jumps up to the limit of patience of manual arithmetic. My father showed me his beautiful ivory-on-mahogany slide rule in junior high school, and I quickly realised its idea. I was allowed to use the new Friden calculating machine which, shortly before its transformation into a relic, could also extract square roots. I love numbers and always as an amusement, and more seriously than that, invented new algorithms to calculate them. In fact while at school Feigenbaum had usually learnt more in studying by himself than in the formal lessons. He had already taught himself to play the piano when he was about 12 years old, but at high school he taught himself calculus. Also at high school a friend of his father gave him a mechanical devise with switching circuits that could play nim and other games. The machine came with a paper by Shannon on Boolean logic which fascinated Feigenbaum with his self-learning attitude. In February 1960, at the age of sixteen, Feigenbaum entered the City College of New York. There he studied electrical engineering but attended all the mathematics courses and the physics courses in addition to those in electrical engineering. Completing the five year course in less than four years he graduated with a Bachelor's degree in 1964. In the summer of that year he began his graduate studies at Massachusetts Institute of Technology. He entered MIT with the intention of researching in electrical engineering for his doctorate but after only one term he changed to physics and began to study general relativity. Now again general relativity was a topic which he studied on his own, reading the book Course of Theoretical Physics by Lev Landau and Evgenii Lifshitz. His official courses were on quantum mechanics, classical mechanics, and complex function theory. It was while he was at MIT that Feigenbaum first used a computer but not as part of his studies there. It was when he was visiting Brooklyn Polytechnic that he found they had a programmable digital computer. He writes [1]:This was the first computer I ever used, and within an hour had programmed it to take square roots by Newton's method. At MIT Feigenbaum's doctoral studies were supervised by Francis Low and he was awarded a doctorate in 1970 for a dissertation on dispersion relations. Following this he went to Cornell as an instructor/research associate, a post which was half funded by an NSF postdoctoral grant, and half funded as a teaching post. During his two years at Cornell he taught courses on variational techniques and on quantum mechanics. He used a HP computer at Cornell which perhaps could be better described as a programmable calculator. The machine had only one other user, Ken Wilson, so he was able to spend time mastering its use. After the two years at Cornell, Feigenbaum went to Virginia Polytechnic Institute as a postdoctoral worker, again with a two year position. He again taught, giving courses on Banach spaces and C*-algebras. Certainly these short term posts were not ideal. As Feigenbaum said (see [7]):These two year positions made serious work almost impossible. After one year you had to

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feigenbaum.html (2 of 5) [2/16/2002 11:09:41 PM]

Feigenbaum

start worrying about where you could go next. After the two years at Virginia Polytechnic Institute, Feigenbaum was offered a long term position on the staff of the theory division at Los Alamos. He writes [1]:When I arrived at Los Alamos, the theory division head, P Carruthers, felt that the time was right, and I was the appropriate person, to see if Wilson's renormalisation group ideas could solve the century and a half old problem of turbulence. In a nutshell, it couldn't - or so far hasn't - but led me off in wonderful directions. The 'wonderful directions' that Feigenbaum refers to here involve the study of chaos where he was to make a remarkable discovery. It was made since data was available from computing and, as Feigenbaum himself has noted, only became obvious because the computers he used calculated so slowly that he could see the intermediate steps of the calculation. Feigenbaum's involvement with computers moved forward in December 1974 when he got his own programmable calculator for the first time, the HP65. With this machine [1]:In swift order, I invented new ODE solvers, minimisation routines, interpolation methods, etc. For someone who cares for numbers, much of the tedium was eliminated. In 1976 Sir Robert May, then a professor of biology at Princeton, pointed out that the logistic map led to chaotic dynamics. The logistic mapping g is defined by xn+1 = g(xn) =

xn(1 - xn).

It models the relative population xn which is the ratio of the actual population to the maximum population. Each iteration gives the new relative population in terms of the old one. The parameter the effective growth rate. We must have 0 < xn 1 and 0 For

< 1, xn tends to 0. For 1

is

4.

3, xn tends to 1 - 1/ . Beyond 3 a bifurcation occurs

(corresponding to high and low populations in alternate years). Further bifurcations occur until at 3.53... chaotic dynamics sets in.

=

In 1973 it had been conjectured that the behaviour of the logistic equation was the same in a qualitative sense for all g(x) which have a maximum value and decrease monotonically on either side of this maximum. The remarkable result obtained by Feigenbaum was to show that not only was the behaviour qualitatively similar but there was a very precise mathematical result which held for all such logistic equations. Feigenbaum did not actually work with the precise logistic equation which May studied and in fact his work was independent of that by May. What Feigenbaum pointed out, if we state it in terms of the notation set up above, was that if (

n

-

n-1)/(

n+1

-

n)

n

is the parameter value at which the nth bifurcation occurs then

4.669201660910... as n

.

When Feigenbaum first found 4.669 in August 1975, which he only found to three places due to the limit of the accuracy of his HP65, he spend some time trying to see if it was a simple combination of 'well-known' numbers. He did not find anything. Of course, now the number is 'well-known' and called the Feigenbaum number.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feigenbaum.html (3 of 5) [2/16/2002 11:09:41 PM]

Feigenbaum

Now this in itself was surprising but in October 1975 Feigenbaum found that this number is the same for a large class of period doubling mappings. This was indeed remarkable and Feigenbaum realised the significance of it immediately [1]:I called my parents that evening and told them that I had discovered something truly remarkable, that, when I had understood it, would make me a famous man. By April 1976 Feigenbaum had completed his first paper on the topic. He submitted it to a journal but after taking six months to referee the paper they rejected it. By 1977 he had been asked by over a 1000 scientists for a copy of it. He eventually managed to get it published in 1978. His second, more technical, paper finished in November 1976, suffered a similar fate and was rejected when first submitted. It eventually appeared in print in 1979. Feigenbaum presents an elementary review on period-doubling bifurcations in nonlinear dynamical systems in [4]. Feigenbaum has made other contributions to the theory of chaos and he has also written two papers on the mathematics of making maps. In one of these (the paper [2]) Feigenbaum writes:Constructing maps from a digital database requires the development of a number of special tools. These, amongst others, include methods for generalising linework and for the automated placement of type. Additionally, granted the numerical power of a computer with its attendant indifference to whether it plots lines and circles or analytically much more complicated curves, an opportunity exists to craft projections of much higher fidelity than have previously been possible. Thus, one should develop tools to capitalise on this power and modernise cartography. ... The modernisation of cartography done to archival standards poses many problems, the solutions for which are strongly illuminated by the ideas and methods of nonlinear systems. The maps constructed with these methods all appeared for the first time in The Hammond Atlas of the World, published exactly one year ago. The Introduction to the Hammond Atlas notes [6]:Using fractal geometry to describe natural forms such as coastlines, mathematical physicist Mitchell Feigenbaum developed software capable reconfiguring coastlines, borders, and mountain ranges to fit a multitide of map scales and projections. Dr Feigenbaum also created a new computerised type placement program which places thousands of map labels in minutes, a task which previously required days of tedious labour. It might at this point be reasonable to wonder whether Feigenbaum considers himself a mathematician or a physicist. His view is that there is no hard distinction between physics and mathematics. We agree with him and certainly in constructing this archive we have taken the view that mathematics includes theoretical physics. In 1982 Feigenbaum left Los Alamos when he was appointed to a professorship at Cornell. Four years later he became the first Toyota professor at Rockefeller University. In the same year that he was appointed to Rockefeller University he was awarded the Wolf Prize in physics. The citation for the prize said that it was awarded to Feigenbaum:... for his pioneering theoretical studies demonstrating the universal character of non-linear systems, which has made possible the systematic study of chaos. The press release made at the time that he was awarded the prize, sums up nicely his contribution:The impact of Feigenbaum's discoveries has been phenomenal. It has spanned new fields of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feigenbaum.html (4 of 5) [2/16/2002 11:09:41 PM]

Feigenbaum

theoretical and experimental mathematics ... It is hard to think of any other development in recent theoretical science that has had so broad an impact over so wide a range of fields, spanning both the very pure and the very applied. Article by: J J O'Connor and E F Robertson List of References (7 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1970 to 1980

Other Web sites

1. Feigenbaum's home page 2. H Berland (The Feigenbaum fractal)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Feigenbaum.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feigenbaum.html (5 of 5) [2/16/2002 11:09:41 PM]

Feigl

Georg Feigl Born: 13 Oct 1890 in Hamburg, Germany Died: 25 April 1945 in Wechselburg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Feigl attended school in Hamburg then, in 1909, he began his studies at the University of Jena. Due to ill health his studies took longer than expected. He obtained a degree from Jena, then in 1919 he obtained a doctorate there working under Koebe. His doctoral dissertation was on conformal mappings. In 1919 Feigl became Schmidt's assistant at Berlin and Feigl's later work was to be greatly influenced by him. Feigl later becoming professor at Berlin himself. From 1928 to 1935 Feigl was managing editor of Jahrbuch über Fortschritte der Mathematik, the only reviewing journal at that time. Feigl worked on geometry, in particular the foundations of geometry and topology. He was also interested in teaching and he introduced many teaching reforms. Through him the modern approach of Hilbert and Klein was introduced into universities and even secondary schools. In 1935 Feigl became professor of mathematics at Breslau. During World War II he led a computing team working on aeronautical research. In 1945 the advancing Russian army meant that his team had to move and Feigl could not obtain his medication. This caused his death some months later. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feigl.html (1 of 2) [2/16/2002 11:09:43 PM]

Feigl

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Feigl.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feigl.html (2 of 2) [2/16/2002 11:09:43 PM]

Fejer

Lipót Fejér Born: 9 Feb 1880 in Pécs, Hungary Died: 15 Oct 1959 in Budapest, Hungary

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lipót Fejér was born Leopold Weiss but changed his name around 1900 to make himself more Hungarian. This was a standard practice carried out at that time to show solidarity with Hungarian culture. In 1897 Fejér won a prize in one of the first mathematics competitions to be held in Hungary. From that year until 1902 Fejér studied mathematics and physics at the University of Budapest and at the University of Berlin where he was a student of Schwarz. After he changed his name from Weiss to Fejér, Schwarz refused to talk to him! In 1900 Fejér published a fundamental summation theorem for Fourier series. This work was the basis of his doctoral thesis which he presented to the University of Budapest in 1902. From 1902 to 1905 Fejér taught at the University of Budapest and from 1905 until 1911 he taught at Kolozsvár in Hungary (now Cluj in Romania). In 1911 Fejér was appointed to the chair of mathematics at the University of Budapest and he held that post until his death. However there were problems regarding his appointment to the chair as is recounted in [4]:Although already world famous and warmly endorsed by Poincaré on the occasion of the awarding of the Bolyai Prize, Fejér's appointment to a chair at the University had been opposed by anti-semites on the Faculty. One of them, knowing full well that Fejér's original name had been Weiss, asked during the occasion of Fejér's candidacy: 'Is this Leopold Fejér related to our distinguished colleague on the Faculty of Theology, Father Ignatius Fejér?' Without missing a beat Loránd Eötvös, Professor of Physics, answered "Illegitimate http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fejer.html (1 of 2) [2/16/2002 11:09:44 PM]

Fejer

son". After that the appointment sailed through smoothly. During his period in the chair at Budapest Fejér led a highly successful Hungarian school of analysis. Fejér's main work was in harmonic analysis. He worked on power series and on potential theory. Much of his work is on Fourier series and their singularities but he also contributed to approximation theory. Fejér collaborated to produce important papers, one with Carathéodory on entire functions in 1907 and another major work with Riesz in 1922 on conformal mappings. One of Fejér's students described his lecturing style in the following way (see [4]):Fejér gave very short, very beautiful lectures. They lasted less than an hour. You sat there for a long time before he came. When he came in, he would be in a sort of frenzy. He was very ugly-looking when you first examined him, but he had a very lively face with a lot of expression. The lecture was thought out in very great detail, with dramatic denouement. He seemed to relive the birth of the theorem; we were present at the creation. He made his famous contemporaries equally vivid; they rose from the pages of the textbooks. That made mathematics appear as a social as well as an intellectual activity. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

A Quotation

A Poster of Lipót Fejér

Mathematicians born in the same country

Other references in MacTutor

1. Chronology: 1900 to 1910 2. Chronology: 1920 to 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fejer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fejer.html (2 of 2) [2/16/2002 11:09:44 PM]

Feller

William Feller Born: 7 July 1906 in Zagreb, Croatia Died: 14 Jan 1970 in New York, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Feller was educated by private tutors and had no secondary schooling. He entered the University of Zagreb and was awarded his first degree in 1925. His Ph.D. was awarded by the University of Göttingen in 1926. Another two years were spent at Göttingen before he accepted an appointment from the University of Kiel where he worked until 1933. Because of his Jewish background, Hitler's policies forced Feller out of Germany in 1933. He went to Copenhagen until 1934, then he moved to the University of Stockholm where he joined the probability group. Feller went to the USA in 1939 and became professor of mathematics at Brown University. The Nazis had taken over the German mathematical reviewing journal and there was a need for another such journal to be set up out of their control. Feller became the first executive editor of Mathematical Reviews which was set up at this time. In 1945 Feller accepted a professorship at Cornell university. He was to work there for five years until he was appointed Eugene Professor of Mathematics at Princeton in 1950. Feller worked on mathematical probability using Kolmogorov's measure theoretic formulation. His approach was pure mathematical but he did study applications of probability, particularly to genetics. He transformed the relation between Markov processes and partial differential equations. Later he put his results in a functional analysis framework. Feller made notable contributions to the mathematical theory of Brownian motion and diffusion processes during the years 1930-1960.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feller.html (1 of 2) [2/16/2002 11:09:46 PM]

Feller

Feller's most important work was Introduction to Probability Theory and its Applications (1950-61), a two volume work which he frequently revised and improved with new approaches, new examples and new applications. J L Doob wrote the following tribute to Feller:Those who knew him personally remember Feller best for his gusto, the pleasure with which he met life, and the excitement with which he drew on his endless fund of anecdotes about life and its absurdities, particularly the absurdities involving mathematics and mathematicians. To listen to him lecture was a unique experience, for no one else could lecture with such intense excitement. Feller received many honours. He was president of the Institute of Mathematical statistics and he was a member of the Royal Statistical Society in the UK. He was awarded the 1969 National Medal for Science but died before the presentation. his wife received the medal on his behalf. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles)

Some Quotations (2)

Mathematicians born in the same country Other Web sites

Feller's Coin Tossing Constants

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Feller.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feller.html (2 of 2) [2/16/2002 11:09:46 PM]

Fermat

Pierre de Fermat Born: 17 Aug 1601 in Beaumont-de-Lomagne, France Died: 12 Jan 1665 in Castres, France

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Fermat's father was a wealthy leather merchant and second consul of Beaumont- de- Lomagne. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. Although there is little evidence concerning his school education it must have been at the local Franciscan monastery. He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius's Plane loci to one of the mathematicians there. Certainly in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Etienne d'Espagnet who clearly shared mathematical interests with Fermat. From Bordeaux Fermat went to Orléans where he studied law at the University. He received a degree in civil law and he purchased the offices of councillor at the parliament in Toulouse. So by 1631 Fermat was a lawyer and government official in Toulouse and because of the office he now held he became entitled to change his name from Pierre Fermat to Pierre de Fermat. For the remainder of his life he lived in Toulouse but as well as working there he also worked in his home town of Beaumont-de-Lomagne and a nearby town of Castres. From his appointment on 14 May 1631 Fermat worked in the lower chamber of the parliament but on 16 January 1638 he was appointed to a higher chamber, then in 1652 he was promoted to the highest level at the criminal court. Still further promotions seem to indicate a fairly meteoric rise through the profession but promotion was done mostly on seniority and the plague struck the region in the early 1650s meaning that many of the older men died.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html (1 of 7) [2/16/2002 11:09:48 PM]

Fermat

Fermat himself was struck down by the plague and in 1653 his death was wrongly reported, then corrected:I informed you earlier of the death of Fermat. He is alive, and we no longer fear for his health, even though we had counted him among the dead a short time ago. The following report, made to Colbert the leading figure in France at the time, has a ring of truth:Fermat, a man of great erudition, has contact with men of learning everywhere. But he is rather preoccupied, he does not report cases well and is confused. Of course Fermat was preoccupied with mathematics. He kept his mathematical friendship with Beaugrand after he moved to Toulouse but there he gained a new mathematical friend in Carcavi. Fermat met Carcavi in a professional capacity since both were councillors in Toulouse but they both shared a love of mathematics and Fermat told Carcavi about his mathematical discoveries. In 1636 Carcavi went to Paris as royal librarian and made contact with Mersenne and his group. Mersenne's interest was aroused by Carcavi's descriptions of Fermat's discoveries on falling bodies, and he wrote to Fermat. Fermat replied on 26 April 1636 and, in addition to telling Mersenne about errors which he believed that Galileo had made in his description of free fall, he also told Mersenne about his work on spirals and his restoration of Apollonius's Plane loci. His work on spirals had been motivated by considering the path of free falling bodies and he had used methods generalised from Archimedes' work On spirals to compute areas under the spirals. In addition Fermat wrote:I have also found many sorts of analyses for diverse problems, numerical as well as geometrical, for the solution of which Viète's analysis could not have sufficed. I will share all of this with you whenever you wish and do so without any ambition, from which I am more exempt and more distant than any man in the world. It is somewhat ironical that this initial contact with Fermat and the scientific community came through his study of free fall since Fermat had little interest in physical applications of mathematics. Even with his results on free fall he was much more interested in proving geometrical theorems than in their relation to the real world. This first letter did however contain two problems on maxima which Fermat asked Mersenne to pass on to the Paris mathematicians and this was to be the typical style of Fermat's letters, he would challenge others to find results which he had already obtained. Roberval and Mersenne found that Fermat's problems in this first, and subsequent, letters were extremely difficult and usually not soluble using current techniques. They asked him to divulge his methods and Fermat sent Method for determining Maxima and Minima and Tangents to Curved Lines, his restored text of Apollonius's Plane loci and his algebraic approach to geometry Introduction to Plane and Solid Loci to the Paris mathematicians. His reputation as one of the leading mathematicians in the world came quickly but attempts to get his work published failed mainly because Fermat never really wanted to put his work into a polished form. However some of his methods were published, for example Hérigone added a supplement containing Fermat's methods of maxima and minima to his major work Cursus mathematicus. The widening correspondence between Fermat and other mathematicians did not find universal praise. Frenicle de Bessy became annoyed at Fermat's problems which to him were impossible. He wrote angrily to Fermat but although Fermat gave more details in his reply, Frenicle de Bessy felt that Fermat was almost teasing http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html (2 of 7) [2/16/2002 11:09:48 PM]

Fermat

him. However Fermat soon became engaged in a controversy with a more major mathematician than Frenicle de Bessy. Having been sent a copy of Descartes' La Dioptrique by Beaugrand, Fermat paid it little attention since he was in the middle of a correspondence with Roberval and Etienne Pascal over methods of integration and using them to find centres of gravity. Mersenne asked him to give an opinion on La Dioptrique which Fermat did describing it as groping about in the shadows. He claimed that Descartes had not correctly deduced his law of refraction since it was inherent in his assumptions. To say that Descartes was not pleased is an understatement. Descartes soon found reason to feel even more angry since he viewed Fermat's work on maxima, minima and tangents as reducing the importance of his own work La Géométrie which Descartes was most proud of and which he sought to show that his Discours de la méthod alone could give. Descartes attacked Fermat's method of maxima, minima and tangents. Roberval and Etienne Pascal became involved in the argument and eventually so did Desargues who Descartes asked to act as a referee. Fermat proved correct and eventually Descartes admitted this writing:... seeing the last method that you use for finding tangents to curved lines, I can reply to it in no other way than to say that it is very good and that, if you had explained it in this manner at the outset, I would have not contradicted it at all. Did this end the matter and increase Fermat's standing? Not at all since Descartes tried to damage Fermat's reputation. For example, although he wrote to Fermat praising his work on determining the tangent to a cycloid (which is indeed correct), Descartes wrote to Mersenne claiming that it was incorrect and saying that Fermat was inadequate as a mathematician and a thinker. Descartes was important and respected and thus was able to severely damage Fermat's reputation. The period from 1643 to 1654 was one when Fermat was out of touch with his scientific colleagues in Paris. There are a number of reasons for this. Firstly pressure of work kept him from devoting so much time to mathematics. Secondly the Fronde, a civil war in France, took place and from 1648 Toulouse was greatly affected. Finally there was the plague of 1651 which must have had great consequences both on life in Toulouse and of course its near fatal consequences on Fermat himself. However it was during this time that Fermat worked on number theory. Fermat is best remembered for this work in number theory, in particular for Fermat's Last Theorem. This theorem states that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2. Fermat wrote, in the margin of Bachet's translation of Diophantus's Arithmetica I have discovered a truly remarkable proof which this margin is too small to contain. These marginal notes only became known after Fermat's son Samuel published an edition of Bachet's translation of Diophantus's Arithmetica with his father's notes in 1670. It is now believed that Fermat's 'proof' was wrong although it is impossible to be completely certain. The http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html (3 of 7) [2/16/2002 11:09:48 PM]

Fermat

truth of Fermat's assertion was proved in June 1993 by the British mathematician Andrew Wiles, but Wiles withdrew the claim to have a proof when problems emerged later in 1993. In November 1994 Wiles again claimed to have a correct proof which has now been accepted. Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries. Fermat's correspondence with the Paris mathematicians restarted in 1654 when Blaise Pascal, Etienne Pascal's son, wrote to him to ask for confirmation about his ideas on probability. Blaise Pascal knew of Fermat through his father, who had died three years before, and was well aware of Fermat's outstanding mathematical abilities. Their short correspondence set up the theory of probability and from this they are now regarded as joint founders of the subject. Fermat however, feeling his isolation and still wanting to adopt his old style of challenging mathematicians, tried to change the topic from probability to number theory. Pascal was not interested but Fermat, not realising this, wrote to Carcavi saying:I am delighted to have had opinions conforming to those of M Pascal, for I have infinite esteem for his genius... the two of you may undertake that publication, of which I consent to your being the masters, you may clarify or supplement whatever seems too concise and relieve me of a burden that my duties prevent me from taking on. However Pascal was certainly not going to edit Fermat's work and after this flash of desire to have his work published Fermat again gave up the idea. He went further than ever with his challenge problems however:Two mathematical problems posed as insoluble to French, English, Dutch and all mathematicians of Europe by Monsieur de Fermat, Councillor of the King in the Parliament of Toulouse. His problems did not prompt too much interest as most mathematicians seemed to think that number theory was not an important topic. The second of the two problems, namely to find all solutions of Nx2 + 1 = y2 for N not a square, was however solved by Wallis and Brouncker and they developed continued fractions in their solution. Brouncker produced rational solutions which led to arguments. Frenicle de Bessy was perhaps the only mathematician at that time who was really interested in number theory but he did not have sufficient mathematical talents to allow him to make a significant contribution. Fermat posed further problems, namely that the sum of two cubes cannot be a cube (a special case of Fermat's Last Theorem which may indicate that by this time Fermat realised that his proof of the general result was incorrect), that there are exactly two integer solutions of x2 + 4 = y3 and that the equation x2 + 2 = y3 has only one integer solution. He posed problems directly to the English. Everyone failed to see that Fermat had been hoping his specific problems would lead them to discover, as he had done, deeper theoretical results. Around this time one of Descartes' students was collecting his correspondence for publication and he turned to Fermat for help with the Fermat - Descartes correspondence. This led Fermat to look again at the arguments he had used 20 years before and he looked again at his objections to Descartes' optics. In particular he had been unhappy with Descartes' description of refraction of light and he now settled on a principle which did in fact yield the sine law of refraction that Snell and Descartes had proposed. However Fermat had now deduced it from a fundamental property that he proposed, namely that light http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html (4 of 7) [2/16/2002 11:09:48 PM]

Fermat

always follows the shortest possible path. Fermat's principle, now one of the most basic properties of optics, did not find favour with mathematicians at the time. In 1656 Fermat had started a correspondence with Huygens. This grew out of Huygens interest in probability and the correspondence was soon manipulated by Fermat onto topics of number theory. This topic did not interest Huygens but Fermat tried hard and in New Account of Discoveries in the Science of Numbers sent to Huygens via Carcavi in 1659, he revealed more of his methods than he had done to others. Fermat described his method of infinite descent and gave an example on how it could be used to prove that every prime of the form 4k + 1 could be written as the sum of two squares. For suppose some number of the form 4k + 1 could not be written as the sum of two squares. Then there is a smaller number of the form 4k + 1 which cannot be written as the sum of two squares. Continuing the argument will lead to a contradiction. What Fermat failed to explain in this letter is how the smaller number is constructed from the larger. One assumes that Fermat did know how to make this step but again his failure to disclose the method made mathematicians lose interest. It was not until Euler took up these problems that the missing steps were filled in. Fermat is described in [9] as Secretive and taciturn, he did not like to talk about himself and was loath to reveal too much about his thinking. ... His thought, however original or novel, operated within a range of possibilities limited by that [1600 - 1650] time and that [France] place. Carl B Boyer, writing in [2], says:Recognition of the significance of Fermat's work in analysis was tardy, in part because he adhered to the system of mathematical symbols devised by François Viète, notations that Descartes' Géométrie had rendered largely obsolete. The handicap imposed by the awkward notations operated less severely in Fermat's favourite field of study, the theory of numbers, but here, unfortunately, he found no correspondent to share his enthusiasm. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (46 books/articles)

Some Quotations (3)

A Poster of Pierre Fermat

Mathematicians born in the same country

Some pages from publications

The first page of Ad locos planos et solidos isagoge An extract from Maxima et minima (1679).

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html (5 of 7) [2/16/2002 11:09:48 PM]

Fermat

Cross-references to History Topics

1. Fermat's Last Theorem 2. The rise of Calculus 3. Mathematical games and recreations 4. An overview of the history of mathematics 5. Prime numbers 6. Abstract linear spaces

Cross-references to Famous Curves

1. Cissoid of Diocles 2. Cycloid 3. Fermat's spiral 4. Witch of Agnesi

Other references in MacTutor

1. A triangle theorem by Fermat 2. Work on Minimal Paths 3. Chronology: 1625 to 1650 4. Chronology: 1650 to 1675

Honours awarded to Pierre Fermat (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Fermat

Paris street names

Passage Fermat and Rue Fermat (14th Arrondissement) 1. The Galileo Project

Other Web sites

2. Rouse Ball 3. Kevin Brown (Fermat primes) 4. Kevin Brown (The method of descent) 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html (6 of 7) [2/16/2002 11:09:48 PM]

Fermat

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fermat.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fermat.html (7 of 7) [2/16/2002 11:09:48 PM]

Ferrar

William Leonard Ferrar Born: 21 Oct 1893 in Bristol, England Died: 22 Jan 1990 in Oxford, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bill Ferrar entered Queen's College Oxford in 1912 after a good education at Bristol Grammar School where his mathematics teacher had inspired him with a love of pure mathematics. In 1914 he won the Junior Mathematical Scholarship but his studies were interrupted by World War I. Bill served as a telephonist in an artillery unit and in an intelligence unit in France. He returned to Oxford in 1919, graduated in 1920 with the best First Class degree, and took up a post in Bangor which he had been offered before he took his finals. He held this post for 4 years, and although happy there, he was keen enough to move to a more research oriented place that he made several applications. Bill was therefore delighted to accept Whittaker's invitation of a senior lectureship in Edinburgh where, in addition to Whittaker, his colleagues were Copson and Aitken. The death of J E Campbell left a vacancy at Oxford which Ferrar filled in 1925. At Oxford, although his main aim was to be heavily involved in research, he had to spend much time teaching and examining. His salary was such that he really had to supplement it to provide enough to support his family and he did this by setting and marking school examination papers. Ferrar wrote many research papers which deal with the convergence of series, an interest which came from working with G N Watson at Cambridge for during a summer vacation while an undergraduate. He worked on interpolation theory, a topic which was suggested to him by Whittaker. From about 1930 his interests turned towards number theory and he examined the convergence of series and the evaluation of singular integrals. These come from a study of Bessel functions which arise from applying summation formulas. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferrar.html (1 of 3) [2/16/2002 11:09:50 PM]

Ferrar

Hardy had been influential in setting up the Quarterly Journal of Mathematics at Oxford and Ferrar served as its editor from 1930 to 1933. In addition Ferrar published many papers in the Quarterly Journal of Mathematics during this period. He moved towards administration with the post of Senior Tutor at his College in 1934 and then, in 1937, he became Bursar of Hertford College, Oxford. He held this post for 22 years. Ferrar is also famed for his 10 outstanding textbooks including A textbook of convergence (1938), Algebra: a textbook of determinants, matrices and quadratic forms (1941), and Finite matrices (1951). Perhaps because of the disruption of his undergraduate course by World War I, Ferrar never obtained a doctorate. Copson, who was a friend and collaborator, suggested that he obtain a doctorate. In 1947 Ferrar submitted 35 papers and 2 books for the degree of Doctor of Science at Oxford and made many of his colleagues who had begun to think of him as solely an administrator realise what an outstanding research record he had. In 1959 he gave up the position of Bursar, hoping for a quite couple of years before he retired, but he was invited to become Principal of Hertford College so the last years of his career were anything but quiet. He continued to write textbooks and help with school examinations in his retirement. He published Mathematics for science (1965), Calculus for beginners (1967) and Advanced mathematics for science (1969) all when over the age of 70. In a personal communication, Michael Ferrar, W L Ferrar's son, comments on his father's interest in lecturing:In a letter to his Uncle at University College Nottingham my father was very critical of the standard of lecturing at Oxford in 1912. It was perhaps in Edinburgh that the foundations were laid of his own high reputation as a lecturer. In those days many Edinburgh undergraduates had little money to buy textbooks and so were very dependent on lectures. If a lecturer overran his time they walked out because they had to get to their next lecture. If the lecture was good they applauded. My father was particularly proud of an occasion when he overran by several minutes and his audience stayed both to listen and to applaud. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferrar.html (2 of 3) [2/16/2002 11:09:50 PM]

Ferrar

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ferrar.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferrar.html (3 of 3) [2/16/2002 11:09:50 PM]

Ferrari

Lodovico Ferrari Born: 2 Feb 1522 in Bologna, Papal States (now Italy) Died: 5 Oct 1565 in Bologna, Papal States (now Italy) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lodovico Ferrari was a remarkable young man. Born in Bologna in 1522, he arrived at Cardan's house as a fourteen year old to become a servant. Cardan, upon the discovery that the lad could read and write, exempted him from menial tasks and appointed the youngster as his secretary. It was soon clear to Cardan that his secretary was an exceptionally gifted young man and he decided to teach him mathematics. Ferrari repaid his master by helping him with his manuscripts and, when Cardan generously resigned his post at the Piatti Foundation in Milan to make way for him in 1540, Ferrari easily defeated his only rival for the post in a debate and thus, at the age of eighteen, became a public lecturer in geometry. Cardan and Ferrari had made remarkable progress on the foundations that Tartaglia had unwillingly given them. Ferrari discovered the solution of the quartic equation in 1540 with a quite beautiful argument but it relied on the solution of cubic equations so could not be published before the solution of the cubic had been published and there was no way to make this public without the breaking the sacred oath made by Cardan. Despairing of ever publishing their ground breaking work, Cardan and Ferrari travelled to Bologna to call upon their mathematical colleague, della Nave. Cardan and Ferrari satisfied della Nave that they could solve the ubiquitous cosa and cube problem, and della Nave showed them in return the papers of the late del Ferro, proving that Tartaglia was not the first to discover the solution of the cubic. Cardan published both the solution to the cubic and Ferrari's solution to the quartic in Ars Magna (1545) convinced that he could break his oath since Tartaglia was not the first to solve the cubic. Tartaglia was furious and Ferrari wrote to Tartaglia, berating him mercilessly and challenging him to a public debate. Tartaglia was extremely reluctant to dispute with Ferrari, still a relatively unknown youngster, against whom even a victory would do little material good. Tartaglia wrote back to Ferrari, trying to bring Cardan into the debate. Ferrari and Tartaglia wrote fruitlessly to each other for about a year, trading the most offensive personal insults but achieving little in the way of resolving the dispute. Things seemed to fizzle out, when suddenly in 1548, Tartaglia received an impressive offer of lecturing in his home town, Brescia. To establish he was the man for the job, Tartaglia was asked to journey to Milan and conclude the contest with Ferrari. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferrari.html (1 of 3) [2/16/2002 11:09:51 PM]

Ferrari

On 10 August 1548, the contest which all Italy wanted to see, for the correspondence between the two antagonists had taken the form of open letters, took place in the Church in the Garden of the Frati Zoccolanti in Milan. A huge crowd had gathered, and the Milanese celebrities came out in force, with Don Ferrante di Gonzaga, governor of Milan, the supreme arbiter. Ferrari was confident of success, despite his inexperience in such matters, and brought a large crowd of friends and supporters. Alone but for his brother, Tartaglia was a vastly experienced disputant and also fancied his chances. By the end of the first day, it was clear that things were not going Tartaglia's way. He was unwilling to give Ferrari time to respond to his criticisms and when he did, it was Ferrari who got in the more telling blows. Ferrari clearly understood the cubic and quartic equations more thoroughly than his opponent who decided that he would leave Milan that very night and thus leave the contest unresolved, so victory went to Ferrari. On the strength of this challenge, Ferrari's fame soared and he was inundated with offers of employment, including a request from the emperor himself, who wanted a tutor for his son. Ferrari fancied a more financially rewarding position though, and took up an appointment as tax assessor to the governor of Milan. After transferring to the service of the church, he retired as a young and very rich man. He moved back to his home town of Bologna, and was called to a professorship of mathematics in 1565 but, sadly, Ferrari died in 1565. It is claimed that he died of white arsenic poisoning, administered by his own sister. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles)

Mathematicians born in the same country

Cross-references to History Topics

1. Quadratic, cubic and quartic equations 2. An overview of the history of mathematics

Other references in MacTutor

Chronology: 1500 to 1600

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferrari.html (2 of 3) [2/16/2002 11:09:51 PM]

Ferrari

JOC/EFR June 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ferrari.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferrari.html (3 of 3) [2/16/2002 11:09:51 PM]

Ferrel

William Ferrel Born: 29 Jan 1817 in Bedford County, Pennsylvania, USA Died: 18 Sept 1891 in Maywood, Kansas, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Ferrel did not write a scientific paper until he was 36. In 1857 Ferrel joined the staff of The American Ephemeris and Nautical Almanac. In 1867 Benjamin Peirce persuaded him to join the US Coastal Survey. Ferrel conjectured that tides should retard the Earth's rotation, an effect dismissed by Laplace. Ferrel found that Laplace had neglected 2nd order terms which give the retarding. As well as research on tides Ferrel studied currents, storms and invented a machine to predict tidal maxima and minima. He became the chief founder of the subject of geophysical fluid dynamics. Among his works were Popular Essays on the Movements of the Atmosphere (1882), Temperature of the Atmosphere and the Earth's Surface (1884), Recent Advances in Meteorology (1886), and A Popular Treatise on the Winds (1889). Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferrel.html (1 of 2) [2/16/2002 11:09:52 PM]

Ferrel

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ferrel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferrel.html (2 of 2) [2/16/2002 11:09:52 PM]

Ferro

Scipione del Ferro Born: 6 Feb 1465 in Bologna, Italy Died: 5 Nov 1526 in Bologna, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Scipione del Ferro is sometimes known as Ferreo, sometimes as Ferro, and sometimes as dal Ferro. His role in the history of mathematics is an important one and he deserves great credit for solving one of the outstanding ancient problems of mathematics. In one sense he is well known, for his role in solving cubic equations is explained in almost every general work on the history of mathematics ever written, and yet, surprisingly, his name remains relatively unknown. Scipione del Ferro's parents were Floriano and Filippa Ferro. Floriano Ferro was employed in paper making which, because of the invention of printing in the 1450s, became an important trade at this time due naturally to a vastly increased demand for paper. Of Scipione del Ferro's education little is known but it is probable that it was at the University of Bologna which was founded in the 11th century and so was a long established and famous university four hundred years before del Ferro was born. We know that del Ferro was appointed as a lecturer in arithmetic and geometry at the University of Bologna in 1496 and that he retained this post for the rest of his life. However he was not only involved in academic activities for records have survived which show that he was involved in business transactions in the latter part of his life. No writings of del Ferro have survived. This must be due, at least in part, to his reluctance to make his results widely known, preferring to communicate them only to a few close friends and students. We do know however that he kept a notebook in which he recorded his most important discoveries. This notebook passed to del Ferro's son-in-law Hannibal Nave when del Ferro died in 1526. Hannibal Nave was also a mathematician and he had married del Ferro's daughter Filippa, who of course was named after del Ferro's mother. Hannibal Nave took over del Ferro's lecturing duties at the University of Bologna in 1526 and also his name since he adopted the name of dalla Nave alias dal Ferro. Nave still had the notebook in 1543, for in that year Cardan and Ferrari travelled to Bologna to see him and his father-in-law's notebook for Ferrari records this in his writings. We quote the relevant passage from Ferrari below. The outstanding problem which del Ferro solved was to find a formula to solve a cubic equation similar to the formula which had been known since the time of the Babylonians for solving quadratic equations. Today we write the solutions to ax2 + bx + c = 0 as

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferro.html (1 of 4) [2/16/2002 11:09:53 PM]

Ferro

x = [-b + (b2 - 4ac)]/2a and x = [-b - (b2 - 4ac)]/2a. In del Ferro's time, although such solutions were known, they were not known in this form. Firstly, in the middle of the 16th century in Europe, zero was not in use; secondly negative numbers were not in use; and thirdly there was no understanding of a quadratic having two roots. Mathematicians in the time of del Ferro knew that the problem of solving the general cubic could be reduced to solving the two cases x3 + mx = n and x3 = mx + n, where m and n are positive numbers. (The term in x2 can always be removed by means of a suitable substitution.) Of course, if negative coefficients had been in use then there would have been only one case. There has been much conjecture as to whether del Ferro came to work on the solution to cubic equations as a result of a visit which Pacioli made to Bologna. Pacioli taught at the University of Bologna during 1501-02 and discussed mathematical problems with del Ferro at that time. It is not known whether the two discussed the algebraic solution of cubic equations, but certainly Pacioli had included this topic in his famous treatise the Summa which he had published seven years earlier. Some time after Pacioli's visit to Bologna, del Ferro solved one of the two cases of this classic problem (but as we mention below, he may have solved both cases). The subsequent developments in the story of the solution of the cubic, namely the contest in 1535 between Antonio Maria Fior (a student of del Ferro) and Tartaglia, then the involvement of Cardan, are told in detail in our biographies of Tartaglia and of Cardan. As far as this biography of del Ferro is concerned we should stress that it was Cardan's discovery that del Ferro had been the first to solve the cubic and not Tartaglia which made him feel that he could honour his oath to Tartaglia not to divulge his method and still publish the solution in Ars Magna for there Cardan considered he is giving del Ferro's method, not that of Tartaglia. Ferrari, a student of Cardan's wrote (on 1 April 1547) about their earlier trip to see Hannibal della Nave (see for example [3]):Four years ago when Cardano was going to Florence and I accompanied him, we saw at Bologna Hannibal della Nave, a clever and humane man who showed us a little book in the hand of Scipione del Ferro, his father-in-law, written a long time ago, in which that discovery [solution of cubic equations] was elegantly and learnedly presented. In Ars Magna Cardan writes with great respect for the achievements of del Ferro (see for example [1]):Scipione Ferro of Bologna, almost thirty years ago, discovered the solution of the cube and things equal to a number [which in today's notation is the case x3 + mx = n], a really beautiful and admirable accomplishment. In distinction this discovery surpasses all mortal ingenuity, and all human subtlety. It is truly a gift from heaven, although at the same time a proof of the power of reason, and so illustrious that whoever attains it may believe himself capable of solving any problem. The story that Fior was the only person to whom del Ferro divulged his solution is common in most histories of mathematics, yet it is false. As we have seen above the solution was written down by del Ferro and certainly was known to Nave. Pompeo Bolognetti, who lectured at the University of Bologna on mathematics from 1554 to 1568, also had access to the original solution by del Ferro as well as the solution as given by Cardan in Ars Magna which had been published by then. Bombelli, who published his Algebra in 1572, also had access to details of del Ferro's work which no longer exists today. Bombelli, like Cardan, expressed wonder at the genius of del Ferro and describes him as:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferro.html (2 of 4) [2/16/2002 11:09:53 PM]

Ferro

... a man uniquely gifted in this art [of algebra]... Around 1925, Bortolotti (see [2]) examined sixteenth century manuscripts reproducing work by Bolognetti, Cardan and Bombelli. One important manuscript is headed:Dal Ferro's rule for the solution of cubic equations. From the Cavaliere Bolognetti, who had it from the Bolognese master of former days, Scipione dal Ferro. On unknowns and cubes equal to numbers. The manuscript gives a method of solution which is applied to the equation 3x3 + 18x = 60. From research on this and the other manuscripts, Bortolotti concluded that, contrary to the widely held belief that del Ferro only solved one case of the cubic, that indeed he solved both cases. However Crossley in [3] believes that the evidence from the Bolognetti manuscript adds weight to the belief that del Ferro solved only one case. We know a little about other work by del Ferro. He made an important contribution to rationalising fractions, extending methods to rationalise fractions which had square roots in the denominator (which were know to Euclid) to fractions whose denominators were the sum of three cube roots. We also know that del Ferro worked on another problem which was popular in his time, namely examining which geometrical problems could be solved with a compass set in a fixed position. Ferrari, in a letter to Tartaglia, states the del Ferro worked on such problems but he did not give any details of del Ferro's results. It is sad that del Ferro's notebook has not survived. Indeed it is probable that he would have attained considerably more fame had we been able to give details of the problems which he solved and wrote down in his notebook. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

Quadratic, cubic and quartic equations

Other references in MacTutor

Chronology: 1500 to 1600

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferro.html (3 of 4) [2/16/2002 11:09:53 PM]

Ferro

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ferro.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ferro.html (4 of 4) [2/16/2002 11:09:53 PM]

Feuerbach

Karl Wilhelm Feuerbach Born: 30 May 1800 in Jena, Germany Died: 12 March 1834 in Erlangen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karl Feuerbach was a brilliant student. By the age of 22 he had been awarded his doctorate, been appointed to a professorship at the Gymnasium at Erlangen and had published an extremely important mathematics paper. His life, however, did not go well. His career as a teacher only lasted six years and even these were years of great difficulty due to ill health. In 1828 Feuerbach retired from teaching, unable to cope any longer with teaching given his state of health. He only lived for a further six years and these he spent in Erlangen living as a recluse. Feuerbach was a geometer who discovered the nine point circle of a triangle. This is sometimes called the Euler circle but this incorrectly attributes the result. Feuerbach also proved that the nine point circle touches the inscribed and three escribed circles of the triangle. These results appear in his 1822 paper, and it is on the strength of this one paper that Feuerbach's fame is based. He wrote in that paper:The circle which passes through the feet of the altitudes of a triangle touches all four of the circles which are tangent to the three sides of the triangle; it is internally tangent to the inscribed circle and externally tangent to each of the circles which touch the sides of the triangle externally. The nine point circle which is described here had also been described in work of Brianchon and Poncelet the year before Feuerbach's paper appeared. The point where the incircle and the nine point circle touch is now called the Feuerbach point. You can see a diagram showing the Feuerbach point. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feuerbach.html (1 of 2) [2/16/2002 11:09:55 PM]

Feuerbach

Feuerbach did publish a further work in 1827. This is a second major work and it has been studied carefully by Moritz Cantor. In this work, Moritz Cantor has discovered, Feuerbach introduces homogeneous coordinates. He must therefore be considered as the joint inventor of homogeneous coordinates since Möbius, in his work Der barycentrische Calkul also published in 1827, introduced homogeneous coordinates into analytic geometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Karl Feuerbach

Mathematicians born in the same country

Other references in MacTutor

1. Feuerbach's theorem 2. Chronology: 1820 to 1830

Other Web sites

Clark Kimberling

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Feuerbach.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feuerbach.html (2 of 2) [2/16/2002 11:09:55 PM]

Feynman

Richard Phillips Feynman Born: 11 May 1918 in New York, USA Died: 15 Feb 1988 in Los Angeles, California, USA

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Richard Feynman studied at MIT and received his doctorate from Princeton in 1942. His doctoral work developed a new approach to quantum mechanics using the principle of least action. He replaced the wave model of electromagnetics of Maxwell with a model based on particle interactions mapped into space time. Feynman worked on the atomic bomb project at Princeton University (1941-42) and then at Los Alamos (1943-45). After World War II he was appointed to the chair of theoretical physics at Cornell University, then, in 1950, to the chair of theoretical physics at Caltech. He remained at Caltech for the rest of his career. Feynman's main contribution was to quantum mechanics, following on from the work of his doctoral thesis. He introduced diagrams (now called Feynman diagrams) that are graphic analogues of the mathematical expressions needed to describe the behaviour of systems of interacting particles. For this work he was awarded the Nobel Prize in 1965, jointly with Schwinger and Tomonoga. Other work on particle spin and the theory of 'partons' which led to the current theory of quarks were fundamental in pushing forward an understanding of particle physics. Feynman's books include many outstanding ones which evolved out of lecture courses. For example Quantum Electrodynamics (1961) and The Theory of Fundamental Processes (1961), The Feynman Lectures on Physics (1963-65) (3 volumes), The Character of Physical Law (1965) and QED: The Strange Theory of Light and Matter (1985). In [3] Gleick described Feynman's approach to science:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feynman.html (1 of 3) [2/16/2002 11:09:57 PM]

Feynman

So many of his witnesses observed the utter freedom of his flights of thought, yet when Feynman talked about his own methods not freedom but constraint ... For Feynman the essence of scientific imagination was a powerful and almost painful rule. What scientists create must match reality. It must match what is already known. scientific imagination, he said, is imagination in a straitjacket ... The rules of harmonic progression made for Mozart a cage as unyielding as the sonnet did for Shakespeare. As unyielding and as liberating - for later critics found the creator's genius in the counterpoint of structure and freedom, rigour and inventiveness. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles)

Some Quotations (7)

A Poster of Richard Feynman

Mathematicians born in the same country

Cross-references to History Topics

A visit to James Clerk Maxwell's house

Honours awarded to Richard Feynman (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1965

Fellow of the Royal Society

Elected 1965

Other Web sites

1. Feynman online 2. Nobel prizes site (A biography of Feynman and his Nobel prize presentation speech) 3. Boston Globe (Obituary) 4. John Talbot (The role of doubt in science) 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feynman.html (2 of 3) [2/16/2002 11:09:57 PM]

Feynman

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Feynman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Feynman.html (3 of 3) [2/16/2002 11:09:57 PM]

Fibonacci

Leonardo Pisano Fibonacci Born: 1170 in (probably) Pisa (now in Italy) Died: 1250 in (possibly) Pisa (now in Italy)

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leonardo Pisano is better known by his nickname Fibonacci. He was the son of Guilielmo and a member of the Bonacci family. Fibonacci himself sometimes used the name Bigollo, which may mean good-for-nothing or a traveller. As stated in [1]:Did his countrymen wish to express by this epithet their disdain for a man who concerned himself with questions of no practical value, or does the word in the Tuscan dialect mean a much-travelled man, which he was? Fibonacci was born in Italy but was educated in North Africa where his father, Guilielmo, held a diplomatic post. His father's job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia. Bejaia is a Mediterranean port in northeastern Algeria. The town lies at the mouth of the Wadi Soummam near Mount Gouraya and Cape Carbon. Fibonacci was taught mathematics in Bugia and travelled widely with his father, recognising and the enormous advantages of the mathematical systems used in the countries they visited. Fibonacci writes in his famous book Liber abaci (1202):When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms. Fibonacci ended his travels around the year 1200 and at that time he returned to Pisa. There he wrote a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html (1 of 6) [2/16/2002 11:09:59 PM]

Fibonacci

number of important texts which played an important role in reviving ancient mathematical skills and he made significant contributions of his own. Fibonacci lived in the days before printing, so his books were hand written and the only way to have a copy of one of his books was to have another hand-written copy made. Of his books we still have copies of Liber abaci (1202), Practica geometriae (1220), Flos (1225), and Liber quadratorum. Given that relatively few hand-made copies would ever have been produced, we are fortunate to have access to his writing in these works. However, we know that he wrote some other texts which, unfortunately, are lost. His book on commercial arithmetic Di minor guisa is lost as is his commentary on Book X of Euclid's Elements which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view. One might have thought that at a time when Europe was little interested in scholarship, Fibonacci would have been largely ignored. This, however, is not so and widespread interest in his work undoubtedly contributed strongly to his importance. Fibonacci was a contemporary of Jordanus but he was a far more sophisticated mathematician and his achievements were clearly recognised, although it was the practical applications rather than the abstract theorems that made him famous to his contemporaries. The Holy Roman emperor was Frederick II. He had been crowned king of Germany in 1212 and then crowned Holy Roman emperor by the Pope in St Peter's Church in Rome in November 1220. Frederick II supported Pisa in its conflicts with Genoa at sea and with Lucca and Florence on land, and he spent the years up to 1227 consolidating his power in Italy. State control was introduced on trade and manufacture, and civil servants to oversee this monopoly were trained at the University of Naples which Frederick founded for this purpose in 1224. Frederick became aware of Fibonacci's work through the scholars at his court who had corresponded with Fibonacci since his return to Pisa around 1200. These scholars included Michael Scotus who was the court astrologer, Theororus the court philosopher and Dominicus Hispanus who suggested to Frederick that he meet Fibonacci when Frederick's court met in Pisa around 1225. Johannes of Palermo, another member of Frederick II's court, presented a number of problems as challenges to the great mathematician Fibonacci. Three of these problems were solved by Fibonacci and he gives solutions in Flos which he sent to Frederick II. We give some details of one of these problems below. After 1228 there is only one known document which refers to Fibonacci. This is a decree made by the Republic of Pisa in 1240 in which a salary is awarded to:... the serious and learned Master Leonardo Bigollo .... This salary was given to Fibonacci in recognition for the services that he had given to the city, advising on matters of accounting and teaching the citizens. Liber abaci, published in 1202 after Fibonacci's return to Italy, was dedicated to Scotus. The book was based on the arithmetic and algebra that Fibonacci had accumulated during his travels. The book, which went on to be widely copied and imitated, introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe. Indeed, although mainly a book about the use of Arab numerals, which became known as algorism, simultaneous linear equations are also studied in this work. Certainly many of the problems that Fibonacci considers in Liber abaci were similar to those appearing in Arab sources.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html (2 of 6) [2/16/2002 11:09:59 PM]

Fibonacci

The second section of Liber abaci contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China. A problem in the third section of Liber abaci led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered today:A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? The resulting sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... (Fibonacci omitted the first term in Liber abaci). This sequence, in which each number is the sum of the two preceding numbers, has proved extremely fruitful and appears in many different areas of mathematics and science. The Fibonacci Quarterly is a modern journal devoted to studying mathematics related to this sequence. Many other problems are given in this third section, including these types, and many many more: A spider climbs so many feet up a wall each day and slips back a fixed number each night, how many days does it take him to climb the wall. A hound whose speed increases arithmetically chases a hare whose speed also increases arithmetically, how far do they travel before the hound catches the hare. Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given. There are also problems involving perfect numbers, problems involving the Chinese remainder theorem and problems involving summing arithmetic and geometric series. Fibonacci treats numbers such as 10 in the fourth section, both with rational approximations and with geometric constructions. A second edition of Liber abaci was produced by Fibonacci in 1228 with a preface, typical of so many second editions of books, stating that:... new material has been added [to the book] from which superfluous had been removed... Another of Fibonacci's books is Practica geometriae written in 1220 which is dedicated to Dominicus Hispanus who we mentioned above. It contains a large collection of geometry problems arranged into eight chapters with theorems based on Euclid's Elements and Euclid's On Divisions. In addition to geometrical theorems with precise proofs, the book includes practical information for surveyors, including a chapter on how to calculate the height of tall objects using similar triangles. The final chapter presents what Fibonacci called geometrical subtleties [1]:Among those included is the calculation of the sides of the pentagon and the decagon from the diameter of circumscribed and inscribed circles; the inverse calculation is also given, as well as that of the sides from the surfaces. ... to complete the section on equilateral triangles, a rectangle and a square are inscribed in such a triangle and their sides are algebraically calculated ... In Flos Fibonacci give an accurate approximation to a root of 10x + 2x2 + x3 = 20, one of the problems that he was challenged to solve by Johannes of Palermo. This problem was not made up by Johannes of Palermo, rather he took it from Omar Khayyam's algebra book where it is solved by means of the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html (3 of 6) [2/16/2002 11:09:59 PM]

Fibonacci

intersection a circle and a hyperbola. Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction. He then continues:And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation. Without explaining his methods, Fibonacci then gives the approximate solution in sexagesimal notation as 1.22.7.42.33.4.40 (this is written to base 60, so it is 1 + 22/60 + 7/602 + 42/603 + ...). This converts to the decimal 1.3688081075 which is correct to nine decimal places, a remarkable achievement. Liber quadratorum, written in 1225, is Fibonacci's most impressive piece of work, although not the work for which he is most famous. The book's name means the book of squares and it is a number theory book which, among other things, examines methods to find Pythogorean triples. Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula n2 + (2n+1) = (n+1)2. Fibonacci writes:I thought about the origin of all square numbers and discovered that they arose from the regular ascent of odd numbers. For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers. To construct the Pythogorean triples, Fibonacci proceeds as follows:Thus when was wish to find two square numbers whose addition produces a square number, I take any odd square number as one of the two square numbers and I find the other square number by the addition of all the odd numbers from unity up to but excluding the odd square number. For example, I take 9 as one of the two squares mentioned; the remaining square will be obtained by the addition of all the odd numbers below 9, namely 1, 3, 5, 7, whose sum is 16, a square number, which when added to 9 gives 25, a square number. Fibonacci also proves many interesting number theory results such as: there is no x, y such that x2 + y2 and x2 - y2 are both squares. and x4 - y4 cannot be a square. He defined the concept of a congruum, a number of the form ab(a + b)(a - b), if a + b is even, and 4 times this if a + b is odd. Fibonacci proved that a congruum must be divisible by 24 and he also showed that for x, c such that x2 + c and x2 - c are both squares, then c is a congruum. He also proved that a square cannot be a congruum. As stated in [2]:... the Liber quadratorum alone ranks Fibonacci as the major contributor to number theory between Diophantus and the 17th-century French mathematician Pierre de Fermat. Fibonacci's influence was more limited than one might have hoped and apart from his role in spreading the use of the Hindu-Arabic numerals and his rabbit problem, Fibonacci's contribution to mathematics has been largely overlooked. As explained in [1]:Direct influence was exerted only by those portions of the Liber abaci and of the Practica that served to introduce Indian-Arabic numerals and methods and contributed to the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html (4 of 6) [2/16/2002 11:09:59 PM]

Fibonacci

mastering of the problems of daily life. Here Fibonacci became the teacher of the masters of computation and of the surveyors, as one learns from the "Summa" of Luca Pacioli ... Fibonacci was also the teacher of the "Cossists", who took their name from the word causa which was first used in the West by Fibonacci in place of res or radix. His alphabetic designation for the general number or coefficient was first improved by Viète ... Fibonacci's work in number theory was almost wholly ignored and virtually unknown during the Middle ages. Three hundred years later we find the same results appearing in the work of Maurolico. The portrait above is from a modern engraving and is believed to not be based on authentic sources.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (21 books/articles)

A Quotation

A Poster of Fibonacci

Mathematicians born in the same country

Cross-references to History Topics

1. Mathematical games and recreations 2. A chronology of pi 3. An overview of the history of mathematics 4. The trigonometric functions 5. Prime numbers 6. A history of Zero 7. Arabic numerals

Other references in MacTutor

1. Fibonacci numbers and the Euclidean algorithm 2. Continued fractions and Fibonacci numbers 3. Chronology: 1100 to 1300

Other Web sites

1. R Knott (Fibonacci numbers and other links) 2. R Knott (A biography) 3. Pass Magazine 4. Karen H Parshall 5. Clark Kimberling (A picture of a statue of Fibonacci) 6. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html (5 of 6) [2/16/2002 11:09:59 PM]

Fibonacci

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fibonacci.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html (6 of 6) [2/16/2002 11:09:59 PM]

Fields

John Charles Fields Born: 14 May 1863 in Hamilton, Ontario, Canada Died: 9 Aug 1932 in Toronto, Ontario, Canada

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Fields received his B.A. in mathematics from the University of Toronto in 1884. After a Ph.D. at Johns Hopkins University, Fields was appointed Professor of Mathematics at Allegheny College in 1889. However from 1892 Fields studied in Europe with Fuchs, Frobenius, Hensel, Schwarz and Planck. This period was clearly important for his future development as a research mathematician. In 1902 Fields was appointed to the position of lecturer at the University of Toronto where he remained until his death. In 1923 he was promoted to research professor at the University of Toronto. His main research topic was on algebraic functions. Fields received several important honours. He was elected a fellow of the Royal Society of Canada in 1907 and, in 1913, he was elected a fellow of the Royal Society of London. In 1924 the International Congress of Mathematicians was held at Toronto and Fields was honoured by being President of the Congress. However Fields is best remembered for conceiving the idea of, and for providing funds for, an international medal for mathematical distinction. Adopted at the International Congress of Mathematicians at Zurich in 1932, the first medals were awarded at the Oslo Congress of 1936. Field's Medals are awarded to no fewer than two and no more than four mathematicians under 40 years of age every four years at the International Congress of Mathematicians. These conditions were set down to recognise Fields' wish, set out in his Will, that the awards recognise both work completed and point to the potential for future achievement.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fields.html (1 of 2) [2/16/2002 11:10:01 PM]

Fields

The first Field's Medals were awarded to Lars Ahlfors and Jesse Douglas in 1936. No awards were made during World War II, then beginning in 1950 the Medals have been awarded every four years. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) A Poster of John C Fields

Mathematicians born in the same country

Honours awarded to John C Fields (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1913

Other Web sites

1. Toronto University 2. FAQ's in Mathematics (Details of the Fields medals, including a list of recipients) 3. Berlin, Germany (More information about the prizes) 4. FAQ's in Mathematics (Why there is no Nobel prize in mathematics) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fields.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fields.html (2 of 2) [2/16/2002 11:10:01 PM]

Finck

Pierre-Joseph-Etienne Finck Born: 15 Oct 1797 in Lauterbourg, France Died: 27 July 1870 in Strasbourg, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre-Joseph-Etienne Finck was left an orphan in 1810, just under the age of 13, when his father and mother both died within a few months of each other. Finck was then adopted and brought up in the town of Landau in der Pfalz. Finck entered the Ecole Polytechnique in 1815, in sixth place in the ranking of students admitted that year. He graduated in 1817 and was admitted to the Artillery School, ranked third from 25 students on entry. The work of the Artillery School did not interest Finck and, in March 1818, he requested that he be allowed to enter the cavalry regiment of the Royal Guard. His request was turned down. Finck tried again with his request in July of the same year, this time saying that if he was not allowed the move he would resign. His request was turned down and Finck resigned. However, by March 1819 Finck seems to have decided that his actions had been mistaken since he wrote asking to be readmitted to the Artillery School. This request was also turned down: Finck was clearly not very highly regarded at the Artillery School. Returning to university, Finck entered the University of Strasbourg in 1821 studying mathematics in the Faculty of Science. He studied for his doctorate, receiving this for a dissertation Sur les mouvements de l'equateur terrestre in 1829. Before completing his doctoral work, Finck began teaching mathematics at the Artillery School of Strasbourg in 1825, which does have a certain irony after his own experiences as a student at Artillery School. He was promoted to Professor of Mathematics there in 1827. He also taught at the Collège de Strasbourg from 1827, becoming a professor there in 1829. In 1842 Finck was appointed to the chair of mathematics at the University of Strasbourg. He began to suffer from ill health in 1862 and by 1866 he was forced to take sick leave. He did not return from sick leave but retired in 1868. Finck wrote over 20 papers and seven textbooks on mathematics. His texts include books on algebra, mechanics, geometry and analysis. In [1] Finck's work on algorithms is discussed, in particular his work on analysing the Euclidean algorithm.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Finck.html (1 of 2) [2/16/2002 11:10:02 PM]

Finck

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Finck.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Finck.html (2 of 2) [2/16/2002 11:10:02 PM]

Fincke

Thomas Fincke Born: 6 Jan 1561 in Flensburg, Denmark (now Germany) Died: 24 April 1656 in Copenhagen, Denmark Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Fincke attended a school in Flensburg until he was 16. After this he spent five years studying at Strasbourg. He then attended seven universities over the next five years; Jena, Wittenberg, Heidelberg, Leipzig, Basel, Padua and Pisa. Only in Padua did he spend a number of years. He studied medicine in Basel and he practised as a doctor from 1587 until 1591. The first three of these years were spent in medical practise in his home town of Flensburg. In 1591 Fincke became professor of mathematics at Copenhagen. He became professor of rhetoric at Copenhagen in 1602, then the following year professor of medicine. His most famous book Geometriae rotundi (1583), was intended as a textbook and the reader is referred to Regiomontanus for more details. Based on works of Ramus from whom he took the word 'radius', the book introduces the terms 'tangents' and 'secants' and Fincke devised new formulas such as the law of tangents. Fincke's book was recommended by Clavius, Napier and Pitiscus all of whom adopted much from it. His other books on astronomy and astrology are of much less interest despite the fact that he was in touch with Brahe and Kepler. Erasmus Bartholin was Fincke's grandson. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Cross-references to History Topics

The trigonometric functions

Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fincke.html (1 of 2) [2/16/2002 11:10:03 PM]

Fincke

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fincke.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fincke.html (2 of 2) [2/16/2002 11:10:03 PM]

Fine

Oronce Fine Born: 20 Dec 1494 in Briançon, France Died: 8 Aug 1555 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oronce Fine was educated in Paris obtaining a medical degree from the Collège de Navarre in 1522. He spent a while in prison in 1518 before completing his degree and again in 1524 he spent some time in prison. Whether he worked on a sundial while in prison is uncertain but he certainly constructed an ivory sundial in 1524 which still exists. Like many mathematicians of his time Fine was an expert on fortifications. He worked on fortifications of Milan. He was appointed to the chair of mathematics at Collège Royal in Paris in 1531 where he taught until his death. Fine wrote on astronomical instruments and astronomy suggesting that eclipses of the moon could be used to determine the longitude of places. He invented a map projection and, around 1519, he produced a map of the world using his projection. He also produced maps of France in 1525 and on another map of the world which he produced in 1531 the name "Terra Australia" appears for the first time. Fine also wrote on arithmetic and geometry. He gave the value of gave 47/15 and, in De rebus mathematicis (1556), he gave 3 11/78.

to be (22 2/9)/7 in 1544, later he

Article by: J J O'Connor and E F Robertson List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Longitude and the Académie Royale 2. Squaring the circle

Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fine.html (1 of 2) [2/16/2002 11:10:04 PM]

Fine

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fine.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fine.html (2 of 2) [2/16/2002 11:10:04 PM]

Fine_Henry

Henry Burchard Fine Born: 14 Sept 1858 in Chambersburg, Pennsylvania , USA Died: 22 Dec 1928 in Princeton, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henry Fine received private coaching before entering the College of New Jersey in 1876, almost exactly 100 years after it was founded. The College of New Jersey's name was changed to Princeton University in 1896. Fine entered the College with the intention of studying classics and he began to study these subjects as well as Sanskritt. However, at Princeton he came across Halsted who had studied under Sylvester at Johns Hopkins University and spent the years 1878 to 1881 as an instructor in postgraduate mathematics at Princeton. Halsted inspired Fine to turn towards mathematics. After receiving his A.B. in 1880, Fine was appointed a fellow in experimental science at Princeton but, never happy with experimental work, he happily changed to be a tutor in mathematics in 1881. Fine then, as was the custom of the day, decided to study in Germany. He travelled to Leipzig in 1884 and there attended lectures by Klein, Carl Neumann and others. He worked for his doctorate on a topic suggested by Study, and approved by Klein, and the degree was awarded for the dissertation On the singularities of curves of double curvature in May 1885. Fine spent the summer of 1885 in Berlin attending Kronecker's lectures on eliminants which made a strong impression on him. Returning to the United States from Berlin, he was appointed assistant professor at Princeton. Despite great promise as a research mathematician, Fine moved into other areas. As Archibald writes in [1]:... Fine published a few research papers (1887-1890), and another of some importance as late as 1916. But his time was mainly devoted to teaching, administration, and the logical exposition of elementary mathematics.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fine_Henry.html (1 of 2) [2/16/2002 11:10:05 PM]

Fine_Henry

Among the elementary texts he wrote are Number system of algebra treated theoretically and historically (1891), A college algebra (1905) and Calculus (1927). Fine's most important contributions were to the American Mathematical Society and to Princeton University. He served the American Mathematical Society as vice-president 1992-93 and as president 1911-12. He chaired the committee set up in 1925 to obtain funds for research in the sciences at Princeton. Largely due to his efforts three million dollars were raised by 1928. His other interests are described in [1]:In early days Fine played the flute in the college orchestra ... His knowledge of music was extensive... He took keen interest in games and in the daily life of the undergraduates. Fine died after a bicycle accident:In the uncertain evening light he was riding his bicycle on a road in the outskirts of Princeton and was struck from behind by an automobile, the driver of which failed to see that he was starting to make a left turn. He died the next morning without having recovered consciousness. Fine Hall, at the Institute for Advanced Study at Princeton, is a memorial which keeps his name before mathematicians at one of the most important centres of mathematics in the United States, and the world. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Henry Fine (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1911 - 1912

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fine_Henry.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fine_Henry.html (2 of 2) [2/16/2002 11:10:05 PM]

Finsler

Paul Finsler Born: 11 April 1894 in Heilbronn, Neckar, Germany Died: 29 April 1970 in Zurich, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Finsler attended a grammar school in Urach, the between 1908 and 1912 he attended a secondary school in Cannstatt. After leaving school he entered the Technische Hochschule in Stuttgart where Kutta was among his teachers. Then in 1913 he entered Göttingen to undertake graduate studies. Among his teachers at Göttingen were a host of top mathematicians including Hecke, Hilbert, Klein, Landau, Runge, Born and Carathéodory. Finsler' doctoral dissertation was supervised by Carathéodory on Curves and surfaces in general spaces. This secured Finsler a name for himself as a differential geometer. In fact in 1934 Cartan wrote a book Les espaces de Finsler which established Finsler's name in differential geometry. A Finsler space is a generalisation of a Riemannian space where the length function is defined differently and Minkowski's geometry holds locally. Differential geometry was not Finsler's research topic for long since he moved to take up set theory. Finsler's habilitation thesis was submitted to the University of Cologne in 1922 and the following year he have his inaugural lecture on Are there contradictions in mathematics. This attempted to remove contradictions, as stated in [2]:Concerning ... Russell's paradox, Finsler points out that one needs to distinguish between satisfiable and unsatisfiable circular definitions. Russell's definition of the set of all sets which do not contain themselves is a non-satisfiable circular definition. In 1927 Finsler was appointed to the University of Zurich, becoming an Ordinary Professor there in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Finsler.html (1 of 3) [2/16/2002 11:10:07 PM]

Finsler

1944. At Zurich, in addition to his work on set theory he also worked on differential geometry, number theory, probability theory and the foundations of mathematics. Finsler's set theory was in the spirit of Cantor. Both were Platonists and as described in [2]:He believed in the reality of pure concepts. Together they form the purely conceptual realm which encompasses all mathematical objects, structure and patterns. ... Mathematicians do not invent or construct their structures and propositions, they recognise or discover, how these objects in the conceptual realm are interrelated with each other. In 1926 Finsler produced the first part of a major work on set theory On the foundations of set theory. He intended to publish the second part as a continuation of his theories but his plan changed when the first part came under attack. In the end he wrote part two as a defence of part one in 1965 rather than what he originally intended. We quote from [2]:... Finsler develops his approach to the paradoxes, his attitude towards formalised theories and his defence of Platonism in mathematics. He insisted on the existence of a conceptual realm within mathematics which transcends formal systems. From the foundational point of view, Finsler's et theory contains a strengthened criterion for set identity and a coinductive specification of the universe of sets. ... Combinatorially, Finsler considers sets as generalised numbers to which one may apply arithmetical techniques. Of course, as mentioned above, the set paradoxes were of particular significance to Finsler. Again quoting from [2]:[Finsler] maintained that consistency is sufficient for the existence of mathematical objects. Furthermore, he thought that the antinomies which led to the foundational crisis, could be solved without the notion that existence is equivalent to formal constructability. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Finsler.html (2 of 3) [2/16/2002 11:10:07 PM]

Finsler

JOC/EFR June 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Finsler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Finsler.html (3 of 3) [2/16/2002 11:10:07 PM]

Fischer

Ernst Sigismund Fischer Born: 12 July 1875 in Vienna, Austria Died: 14 Nov 1954 in Cologne, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernst Fischer studied at Vienna under Mertens from 1894. He spent 1899 in Berlin, then studied at Zurich and Göttingen with Minkowski. From 1902 he was assistant to E Waelsch at the University of Brünn (now Brno), becoming professor there after a few years. From 1911 until 1920, Fischer was professor at Erlangen, then from 1920 he worked at Cologne. In 1907 Ernst Fischer studied orthonormal sequences of functions and gave necessary and sufficient conditions for a sequence of constants to be the Fourier coefficients of a square integrable function. This led to the concept of a Hilbert space. F Riesz published a similar result in the same year. The theorem, now called the Riesz-Fischer theorem, is one of the great achievements of the Lebesgue theory of integration. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Ernst Fischer

Mathematicians born in the same country

Other Web sites

Clark Kimberling

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fischer.html (1 of 2) [2/16/2002 11:10:09 PM]

Fischer

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fischer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fischer.html (2 of 2) [2/16/2002 11:10:09 PM]

Fisher

Sir Ronald Aylmer Fisher Born: 17 Feb 1890 in London, England Died: 29 July 1962 in Adelaide, Australia

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ronald Fisher received a B.A. in astronomy from Cambridge in 1912. There he studied the theory of errors under Stratton using Airy's manual on the Theory of Errors. It was Fisher's interest in the theory of errors in astronomical observations that eventually led him to investigate statistical problems. Fisher gave up being a mathematics teacher in 1919 to work at the Rothamsted Agricultural Experiment Station where he worked as a biologist and made many contributions to both statistics and genetics. He had a long dispute with Pearson and he turned down a post under him, choosing to go to Rothamsted instead. There he studied the design of experiments by introducing the concept of randomisation and the analysis of variance, procedures now used throughout the world. In 1921 he introduced the concept of likelihood. The likelihood of a parameter is proportional to the probability of the data and it gives a function which usually has a single maximum value, which he called the maximum likelihood. In 1922 he gave a new definition of statistics. Its purpose was the reduction of data and he identified three fundamental problems. These are (i) specification of the kind of population that the data came from (ii) estimation and (iii) distribution. The contributions Fisher made included the development of methods suitable for small samples, like those of Gosset, the discovery of the precise distributions of many sample statistics and the invention of analysis of variance. He introduced the term maximum likelihood and studied hypothesis testing. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fisher.html (1 of 2) [2/16/2002 11:10:10 PM]

Fisher

Fisher is considered one of the founders of modern statistics because of his many important contributions. He was elected a Fellow of the Royal Society in 1929, was awarded the Royal Medal of the Society in 1938 and he was awarded the Darwin Medal of the Society in 1948:... in recognition of his distinguished contributions to the theory of natural selection, the concept of its gene complex and the evolution of dominance. Then, in 1955, he was awarded the Copley Medal of the Royal Society:... in recognition of his numerous and distinguished contributions to developing the theory and application of statistics for making quantitative a vast field of biology. Article by: J J O'Connor and E F Robertson List of References (13 books/articles)

Some Quotations (2)

A Poster of Ronald Fisher

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1920 to 1930

Honours awarded to Ronald Fisher (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1929

Royal Society Copley Medal

Awarded 1955

Royal Society Royal Medal

Awarded 1938

Other Web sites

1. University of Minnesota 2. Lloyd Allison 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fisher.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fisher.html (2 of 2) [2/16/2002 11:10:10 PM]

Fiske

Thomas Scott Fiske Born: 12 May 1865 in New York, USA Died: 10 Jan 1944 in Poughkeepsie, New York, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Fiske attended the Old Trinity Church School in New York, then the Pingry School in Elizabeth, New Jersey. Fiske entered Columbia College in 1882, thirty years before it became Columbia University, receiving his A.B. in 1885 and his A.M. in 1886. From 1886 to 1888 he was both an assistant at Columbia College and undertaking research for his doctorate. In 1887, in Fiske's second year of graduate studies, Van Amringe suggested that he should spend at least six months in England at the University of Cambridge. He arrived with letters of introduction written by G L Rives, a trustee of Columbia College who had been a wrangler at Cambridge in 1872. With letters addressed to Cayley, Glaisher, Forsyth and Darwin, Fiske was well placed to take advantage of his time at Cambridge. He writes in [3]:Scientifically I benefited most from the instruction and advice of Forsyth and from my reading with Dr H W Richmond, who consented to give me private lessons. However, from Dr J W L Glaisher, who made me an intimate friend, who spent many an evening with me in heart to heard talks, who took me to meetings of the London Mathematical Society and the Royal Astronomical Society, and entertained me with gossip about scores of contemporary and earlier mathematicians, I gained more in a general way than from anyone else. I had attended only a few lectures by Cayley on 'The calculus of the extraordinaires' when, slipping on the ice, he suffered a fracture of the leg, which brought the lectures to an end. Back in the United States, Fiske completed his research being awarded his doctorate in 1888. He was appointed a tutor in mathematics at Columbia University in 1888 and was successively promoted to instructor in 1891, adjunct professor in 1894, and full professor in 1897. He held the post of professor at http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fiske.html (1 of 2) [2/16/2002 11:10:12 PM]

Fiske

Columbia from then until he retired in 1936. Fiske is of little importance as a research mathematician. He published a few papers on elliptic integrals and surface integrals during his career, as well as a number of papers on mathematical education. Later in has career he wrote some encyclopaedia articles on elliptic functions, functions of real and complex variables and a couple of others. His real importance, however, is that he was the founder of the American Mathematical Society in 1888. He served the new Society in a variety of ways, for example as secretary from 1888 to 1895, as treasurer from 1890 to 1891, as editor-in-chief of the Bulletin of the American Mathematical Society from 1891 to 1899, as vice-prisident of the Society from 1898 to 1901, as editor of the Transactions of the American Mathematical Society from 1899 to 1905 and as president of the Society from 1903 to 1904. Archibald [1] talks of:... the enormous debt which the Society owes to its able Founder, whose enthusiastic activities on her behalf during the first fifteen years of her existence, were so unremitting and so wise. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Honours awarded to Thomas Fiske (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1903 - 1904

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fiske.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fiske.html (2 of 2) [2/16/2002 11:10:12 PM]

FitzGerald

George Francis FitzGerald Born: 3 Aug 1851 in Dublin, Ireland Died: 22 Feb 1901 in Dublin, Ireland

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George FitzGerald was a brilliant mathematical physicist who today he is known by most scientists as one of the proposers of the FitzGerald-Lorentz contraction in the theory of relativity. However, this suggestion by FitzGerald, as we shall see below, was not in the area in which he undertook most of his research, and he would certainly not have rated this his greatest contribution. George FitzGerald's parents were William FitzGerald and Anne Frances Stoney. His father William was a minister in the Irish Protestant Church and rector of St Ann's Dublin at the time of George's birth. William, although having no scientific interests himself, was an intellectual who went on to become Bishop of Cork and later Bishop of Killaloe. It seems that George's later interest in metaphysics came from his father's side of the family. George's mother was the daughter of George Stoney from Birr in King's County and she was also from an intellectual family. George Johnstone Stoney, who was Anne's brother, was elected a Fellow of the Royal Society of London and George FitzGerald's liking for mathematics and physics seems to have come mainly from his mother's side of the family. William and Anne had three sons, George being the middle of the three. Maurice FitzGerald, one of George's two brothers, also went on to achieve academic success in the sciences, becoming Professor of Engineering at Queen's College Belfast. George's schooling was at home where, together with his brothers and sisters, he was tutored by M A Boole, who was George Boole's sister. It is doubtful whether Miss Boole realised what enormous potential her pupil George had, for although he showed himself to be an excellent student of arithmetic and algebra, he was no better than an average pupil at languages and had rather a poor verbal memory. However, when the tutoring progressed to a study of Euclid's Elements George showed himself very able indeed, and he also exhibited a great inventiveness for mechanical http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/FitzGerald.html (1 of 7) [2/16/2002 11:10:14 PM]

FitzGerald

constructions, having great dexterity. He was also an athletic boy but he had no great liking for games. Miss Boole prepared her pupils very well for their university studies. She noticed one remarkable talent in her pupil George: that was his skill as an observer. Many years later FitzGerald, clearly thinking of his own youth, wrote:The cultivation and training of the practical ability to do things and to learn from observation, experiment and measurement, is a part of education which the clergyman and the lawyer can maybe neglect, because they have to deal with emotions and words, but which the doctor and the engineer can only neglect at their own peril and that of those who employ them. These habits should be carefully cultivated from the earliest years while a child's character is being developed. As the twig is bent so the tree inclines. FitzGerald certainly showed that he had acquired the ability to learn from observation, experiment and measurement. He entered Trinity College Dublin at the young age of 16 to study his two best subjects which were mathematics and experimental science and he was soon putting the training he had received at home to good use. At Trinity College, FitzGerald [7]:... attained all the distinctions that lay in his path with an ease, and wore them with a grace, that endeared him to his rivals and contemporaries. It was not a undergraduate career devoted entirely to study, however, for FitzGerald played a full part in literary clubs and social clubs. He also continued his athletic interests, taking to gymnastics and to racquet sports. In 1871 he graduated as the best student in both mathematics and experimental science. He won a University Studentship and two First Senior Moderatorships in his chosen topics. The aim of FitzGerald was now to win a Trinity College Fellowship but at this time these were few and far between. He was to spend six years studying before he obtained the Fellowship he wanted, but during these years he laid the foundation of his research career. He studied the works of Lagrange, Laplace, Franz Neumann, and those of his own countrymen Hamilton and MacCullagh. In addition he absorbed the theories put forward by Cauchy and Green. Then, in 1873, a publication appeared which would play a major role in his future. This was Electricity and Magnetism by Maxwell which, for the first time, contained the four partial differential equations, now known as Maxwell's equations. FitzGerald immediately saw Maxwell's work as providing the framework for further development and he began to work on pushing forward the theory. It is worth noting that FitzGerald's reaction to Maxwell's fundamental paper was not that of most scientists. Very few seemed to see the theory as a starting point, rather most saw it only a means to produce Maxwell's own results. It is a tribute to FitzGerald's insight as a scientist that he saw clearly from the beginning the importance of Electricity and Magnetism. Maxwell's theory was for many years, in the words of Heaviside, "considerably underdeveloped and little understood" but a few others were to see it in the same light as FitzGerald including Heaviside, Hertz and Lorentz. FitzGerald would exchange ideas over the following years with all three of these scientists. During the six years he spent working for the Fellowship, FitzGerald also studied metaphysics, a topic which he had not formally studied as an undergraduate, and he was particularly attracted to Berkeley's philosophy. His liking for metaphysics and his deep understanding of the topic combined with his other great talents in his future career. He won his Fellowship and became a tutor at Trinity College Dublin in 1877. This was not his first attempt at winning a Fellowship, rather it was his second since he failed to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/FitzGerald.html (2 of 7) [2/16/2002 11:10:14 PM]

FitzGerald

win a Fellowship at his first attempt. At Trinity College he was attached to the Department of Experimental Physics and soon he was the major influence on the teaching of the physical sciences in the College. In 1881 John R Leslie, the professor of natural philosophy at Dublin, died and FitzGerald succeeded him to the Erasmus Smith Chair of Natural and Experimental Philosophy. At the time of his appointment he gave up his duties as College tutor, a role in which he had been extremely successful, to concentrate on his duties as a professor. One of FitzGerald's long running battles at Trinity College Dublin was to increase the amount of teaching of experimental physics. He soon set up classes in an old chemical laboratory that he was able to obtain for his use, and he gathered round him colleagues who would help in the practical aspects of the subject. As is so often the case in universities, however, he was restricted in the progress he could make from a lack of funds. In a lecture which he gave to the Irish Industrial League in 1896 FitzGerald emphasised his lifelong belief in practical studies:The fault of our present system is in supposing that learning to use words teaches us to use things. This is at its best. If really does not even teach children to use words, it only teaches them to learn words, to stuff their memories with phrases, to be a pack of parrots, to suffocate thought with indigestible verbiage. Take the case of experimenting. How can you teach children to make careful experiments with words? Yet it is great importance that they should be able to learn from experiments. However, practical applications are built on theoretical foundations and FitzGerald fully understood this. In his inaugural lecture on 22 February 1900 as President of the Dublin Section of the Institution of Electrical Engineers, he spoke of how electricity had been applied to the benefit of mankind during the nineteenth century. Behind a practical invention such as telegraphy there was a wealth of theoretical work:... telegraphy owes a great deal to Euclid and other pure geometers, to the Greek and Arabian mathematicians who invented our scale of numeration and algebra, to Galileo and Newton who founded dynamics, to Newton and Leibniz who invented the calculus, to Volta who discovered the galvanic coil, to Oersted who discovered the magnetic actions of currents, to Ampère who found out the laws of their action, to Ohm who discovered the law of resistance of wires, to Wheatstone, to Faraday, to Lord Kelvin, to Clerk Maxwell, to Hertz. Without the discoveries, inventions, and theories of these abstract scientific men telegraphy, as it now is, would be impossible. We should also look at FitzGerald's idea of the purpose of a university for it was, like his other educational beliefs, the driving force to how he carried out his professorial duties. He believed that the primary purpose of a university was not to teach the few students who attended but, through research, to teach everyone. He wrote in 1892:The function of the University is primarily to teach mankind. .. at all times the greatest men have always held that their primary duty was the discovery of new knowledge, the creation of new ideas for all mankind, and not the instruction of the few who found it convenient to reside in their immediate neighbourhood. ... Are the Universities to devote the energies of the most advanced intellects of the age to the instruction of the whole nation, or to the instruction of the few whose parents can afford them an - in some places fancy - education

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/FitzGerald.html (3 of 7) [2/16/2002 11:10:14 PM]

FitzGerald

that can in the nature of things be only attainable by the rich? As can be seen from the quotations we have given from FitzGerald's writing, his interest in education went well beyond the narrow confines of his own department. It was not merely a theoretical interest for, true to his own beliefs, he took a very practical role in education. He was an examiner in physics at the University of London beginning in 1888 and he served as a Commissioner of National Education in Ireland in 1898 being concerned with reforming primary education in Ireland. As part of this task he travelled to the United States on a fact finding tour in the autumn of 1898. As one might have expected, his aim was to bring far more practical topics into the syllabus of primary schools. At the time of his death he was involved in the reform of intermediate education in Ireland and he also served on the Board which was considering technical education. In 1883 FitzGerald married Harriette Mary Jellett. She was the daughter of the Rev J H Jellett, the Provost of Trinity College and an outstanding scientist who had been awarded the Royal Medal by the Royal Society of London. It was through his personal friendship with Jellett, and also their joint scientific studies, that FitzGerald got to know Harriette. Although the couple had been married just under eight years at the time of FitzGerald's death, they had eight children during this time; three sons and five daughters. FitzGerald was elected a Fellow of the Royal Society of London in 1883 and, like his father-in-law, he was to receive its Royal medal. This was in 1899 when the prestigious award was made to FitzGerald for his contributions to theoretical physics, especially to optics and electrodynamics. We should now examine the research for which FitzGerald received these honours. Beginning in 1876, before he obtained his Fellowship, FitzGerald began to publish the results of his research. His first work On the equations of equilibrium of an elastic surface filled in cases of a problem studied by Lagrange. His second paper in the same year was on magnetism and he then, still in 1876, published On the rotation of the plane of polarisation of light by reflection from the pole of a magnet in the Proceeding of the Royal Society. He had already begun to contribute to Maxwell's theory and, as well as theoretical contributions, he was conducting experiments in electromagnetic theory. His first major theoretical contribution was On the electromagnetic theory of the reflection and refraction of light which he sent to the Royal Society in October 1878. Maxwell, in reviewing the paper, noted that FitzGerald was developing his ideas in much the same general direction as was Lorentz. At a meeting of the British Association in Southport in 1883, FitzGerald gave a lecture discussing electromagnetic theory. He suggested a method of producing electromagnetic disturbances of comparatively short wavelengths:... by utilising the alternating currents produced when an accumulator is discharged through a small resistance. It would be possible to produce waves of as little as 10 metres wavelength or less. So FitzGerald, using his own studies of electrodynamic, suggested in 1883 that an oscillating electric current would produce electromagnetic waves. However, as he later wrote:... I did not see any feasible way of detecting the induced resonance. In 1888 FitzGerald addressed the Mathematical and Physical Section of the British Association in Bath as its President. He was able to report to British Association that Heinrich Hertz had, earlier that year, verified this experimentally. Hertz had verified that the vibration, reflection and refraction of electromagnetic waves were the same as those of light. In this brilliant lecture, given to a general audience, FitzGerald described how Hertz:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/FitzGerald.html (4 of 7) [2/16/2002 11:10:14 PM]

FitzGerald

... has observed the interference of electromagnetic waves quite analogous to those of light. After his appointment to the chair, FitzGerald had continued to produce many innovative ideas but no major theories. For example despite his ideas on electromagnetic waves he had not followed through the research and the final experimental verification had been achieved by Hertz. The reason for this is perhaps best understood with a quotation from a letter which FitzGerald sent to Heaviside on 4 February 1889 (see for example [1]):I admire from a distance those who contain themselves till they worked to the bottom of their results but as I am not in the very least sensitive to having made mistakes I rush out with all sorts of crude notions in hope that they may set others thinking and lead to some advance. Although FitzGerald is modestly talking down his contributions in this quotation, the comment he made about himself is essentially correct. O J Lodge [8] gives a similar, but fairer, analysis of FitzGerald's work:... the leisure of long patient analysis was not his, nor did his genius altogether lie in this direction: he was at his best when, under the stimulus of discussion, his mind teemed with brilliant suggestions, some of which he at once proceeded to test by rough quantitative calculation, for which he was an adept in discerning the necessary data. The power of grasping instantly all the bearings of a difficult problem was his to an extraordinary degree ... Again Heaviside wrote (see for example [7]):He had, undoubtedly, the quickest and most original brain of anybody. That was a great distinction; but it was, I think, a misfortune as regards his scientific fame. He saw too many openings. His brain was too fertile and inventive. I think it would have been better for him if he had been a little stupid - I mean not so quick and versatile, but more plodding. He would have been better appreciated, save by a few. Finally we should examine the contribution for which FitzGerald is universally known today. There had been many attempts to detect the motion of the Earth relative to the aether, a medium in space postulated to carry light waves. A A Michelson and E W Morley conducted an accurate experiment to compare the speed of light in the direction of the Earth's motion and the speed of light at right angles to the Earth's motion. Despite the difference in relative motion to the aether, the velocity of light was found to be the same. In 1889, two years after the Michelson-Morley experiment, FitzGerald suggested that the shrinking of a body due to motion at speeds close to that of light would account for the result of that experiment. Lodge [8] writes that the idea:... flashed on him in the writer's study at Liverpool as he was discussing the meaning of the Michelson-Morley experiment. Lorentz, independently in 1895, gave a much more detailed description of the same kind. It was typical of these two great men that both was more than ready to acknowledge the contribution of the other, but there is little doubt that each had the idea independently of the other. The FitzGerald-Lorentz contraction now plays an important role in relativity. Sadly FitzGerald died at the age of only 49 years. Maxwell, whose work had proved so fundamental for FitzGerald, had died at the age of 48 while Hertz died at the age of 36. In fact in 1896 FitzGerald had reviewed the publication of Hertz's Miscellaneous Papers for Nature after Hertz's death. Four years later,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/FitzGerald.html (5 of 7) [2/16/2002 11:10:14 PM]

FitzGerald

in September 1900, FitzGerald began to complain of indigestion and began to have to be careful what he ate. A few weeks later he complained that he was finding it difficult to concentrate on a problem. His health rapidly deteriorated and despite having an operation the end came quickly. W Ramsay, on hearing of FitzGerald's death wrote (see[7]):... to me, as to many others, FitzGerald was the truest of true friends; always interested, always sympathetic, always encouraging, whether the matter discussed was a personal one, or one connected with science or with education. And yet I doubt if it were these qualities alone which made his presence so attractive and so inspiring. I think it was the feeling that one was able to converse on equal terms with a man who was so much above the level of one's self, not merely in intellectual qualities of mind, but in every respect. ... he had no trace of intellectual pride; he never put himself forward, and had no desire for fame; he was content to do his duty. And he took this to be the task of helping others to do theirs. FitzGerald was described by Lord Kelvin (William Thomson) as (see [9]):... living in an atmosphere of the highest scientific and intellectual quality, but always a comrade with every fellow-worker of however humble quality.... My scientific sympathy and alliance with him have greatly ripened during the last six or seven years over the undulatory theory of light and the aether theory of electricity and magnetism. Article by: J J O'Connor and E F Robertson List of References (11 books/articles) Mathematicians born in the same country Cross-references to History Topics

Special relativity

Other references in MacTutor

Chronology: 1880 to 1890

Honours awarded to George FitzGerald (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1883

Royal Society Royal Medal

Awarded 1899

Fellow of the Royal Society of Edinburgh Lunar features

Crater FitzGerald

Other Web sites

1. M J D Coey 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/FitzGerald.html (6 of 7) [2/16/2002 11:10:14 PM]

FitzGerald

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/FitzGerald.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/FitzGerald.html (7 of 7) [2/16/2002 11:10:14 PM]

Flamsteed

John Flamsteed Born: 19 Aug 1646 in Denby (near Derby), Derbyshire, England Died: 31 Dec 1719 in Greenwich, London, England

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Flamsteed's father was a business man who was quite wealthy. Flamsteed's mother however died when he was still a child and this affected his upbringing. Flamsteed attended Derby free school which prepared children for a university education. However life did not go smoothly for Flamsteed who, at the age of 14, developed severe health problems. A chronic rheumatic condition led to his father deciding not to send him to university. Flamsteed was extremely disappointed but he did not let it prevent him from studying. Between 1662 and 1669 Flamsteed studied astronomy on his own without the help of teachers. In fact he does not seem to have missed the formal teaching but his father continued to oppose his studies and this made far more difficulties for Flamsteed than the fact that he could not attend lectures. Flamsteed's father always maintained that it was because of his son's ill health that he opposed his studying but Flamsteed, in his correspondence in later life, suggested that his father may have had other motives. Since Flamsteed's mother had died when he was young, Flamsteed was useful to his father as someone to look after the home. Whether or not this was his father's motive, certainly Flamsteed felt bitterness towards his father. Flamsteed began systematic observations in 1671. He also began corresponding with Henry Oldenburg and John Collins. These two arranged for Flamsteed to meet Jonas Moore during a visit Flamsteed made to the Royal Society in London in 1670. Jonas Moore became his patron and persuaded Charles II to grant a warrant so that Jesus College Cambridge could award an M.A. to Flamsteed in 1674. In February 1675 Flamsteed arrived in London to stay with Jonas Moore and Moore arranged that http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Flamsteed.html (1 of 3) [2/16/2002 11:10:16 PM]

Flamsteed

Flamsteed visit the King, Charles II, to ask for a Royal Observatory. In fact Flamsteed had to some extent paved the way to find favour with the King, having made a barometer and a thermometer for Charles II and the Duke of York in previous year. On 4 March 1675 the King appointed Flamsteed his astronomical observer by Royal Warrant. From his salary of 100 he had to pay 10 taxes and also provide all his own instruments. The Royal Observatory at Greenwich was built and equipped for his observations and he began observing there in 1676. Ordained in 1675, Flamsteed received the income of the living of Burstow, Surrey from 1684. In 1677 he was elected a fellow of the Royal Society. Flamsteed was a skilled observer and had a number of observing programmes at the Royal Observatory to answer major questions. Among his other achievements was the fact that Flamsteed invented the conical projection, an important projection of the sphere onto a plane which is used in cartography. Newton required data for his understanding of the orbit of the Moon, a difficult problem to which Newton applied his universal law of gravity. Flamsteed never quite seemed to understand what Newton required and the two were not on the best of terms, in fact Flamsteed was a perfectionist and was not an easy man to get on with. In [1] he is described as follows:Possessed of an attitude that can only be described as uncompromising, he was an intemperate man even by the standards of an intemperate age. The particular and enduring subject of his passion was Edmond Halley. The last thirty years of Flamsteed's extensive correspondence is infused with vituperative remarks about the man who should have been his most natural ally. It is hard to say exactly why Flamsteed was so bitter towards Halley but their personalities certainly clashed while there must have been a certain professional jealousy between them. The battle Flamsteed had with Halley over the publication of his carefully made observations is described in [2]:The latter part of Flamsteed's life passed in controversy over the publication of his excellent observations. He struggled to withhold them until completed, but they were urgently needed by Isaac Newton and Edmond Halley, among others. Newton, through the Royal Society, led the movement for their immediate publication. In 1704 Prince George of Denmark undertook the cost of publication, and, despite the prince's death in 1708 and Flamsteed's objections, the incomplete observations were edited by Halley, and 400 copies were printed in 1712. Flamsteed later managed to burn 300 of them. Flamsteed did publish his star catalogue Historia Coelestis Britannica in 1725 containing data on 3000 stars. It listed more stars and gave their positions considerably more accurately than any other previous publication had done. It was ironical that his greatest enemy, Halley, should succeed him as the second Astronomer Royal. Article by: J J O'Connor and E F Robertson List of References (8 books/articles) A Poster of John Flamsteed

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Flamsteed.html (2 of 3) [2/16/2002 11:10:16 PM]

Flamsteed

Cross-references to History Topics

English attack on the Longitude Problem

Honours awarded to John Flamsteed (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1676

Lunar features

Crater Flamsteed

Other Web sites

1. Museum of the History of Science, Oxford (An exhibition) 2. The Galileo Project 3. Linda Hall Library (Star Atlas) 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Flamsteed.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Flamsteed.html (3 of 3) [2/16/2002 11:10:16 PM]

Flugge-Lotz

Irmgard Flügge-Lotz Born: 16 July 1903 in Hameln, Germany Died: 22 May 1974 in Stanford, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Irmgard Flügge-Lotz's father was a journalist. Her mother's family were construction engineers and through visiting building sites as a child she became interested in construction. She attended a number of schools as her father moved between different towns; in Frankenthal, Mönchen- Gladbach and Hanover. After graduating from school in 1923 she entered the Technical University of Hanover to study mathematics and engineering. She obtained her first degree in 1927 and a doctorate in 1929 for a thesis an the mathematical theory of heat. She then went to the aerodynamics research institute in Göttingen. Here she applied her mathematical skills in solving differential equations to solve an important problem on the distribution of lift on wings. She published what is now known as the 'Lotz method' on 1931. In 1938 she married Wilhelm Flügge, a civil engineer from Göttingen. In 1944 they moved to a region of Germany which became France at the end of World War II and Flügge-Lotz and her husband both were offered posts at the National Office for Aeronautical Research in Paris. In 1948 both received offers of posts at Stanford in the United States and accepted. There Flügge-Lotz undertook research in numerical methods to solve boundary layer problems in fluid dynamics. Her pioneering work involved finite difference methods and the use of computers. Later she worked on automatic control theory and published important works Discontinuous Automatic Control (1953) and Discontinuous and Optimal Control (1958). Flügge-Lotz became Stanford's first woman Professor of Engineering in 1961. She received many http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Flugge-Lotz.html (1 of 2) [2/16/2002 11:10:17 PM]

Flugge-Lotz

honours including being chosen to give the von Kármán lecture to the American Institute of Aeronautics and Astronautics in 1971 and an honorary degree from the University of Maryland in 1973. The citation read at the ceremony to award the honorary degree states:Professor Flügge-Lotz has acted in a central role in the development of the aircraft industry in the Western world. Her contributions have spanned a lifetime during which she demonstrate, in a field dominated by men, the value and quality of a woman's intuitive approach in searching for and discovering solutions to complex engineering problems. Her work manifests unusual personal dedication and native intelligence. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of Irmgard Flügge-Lotz

Mathematicians born in the same country

Other Web sites

●

Agnes Scott College

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Flugge-Lotz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Flugge-Lotz.html (2 of 2) [2/16/2002 11:10:17 PM]

Fomin

Sergei Vasilovich Fomin Born: 9 Dec 1917 in Moscow, Russia Died: 17 Aug 1975 in Vladivostok, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sergei Vasilovich Fomin's father was a professor of medicine at the University of Moscow. He began his schooling in 1925 at the age of seven and he was placed directly into the second form. His early education was much influenced by Chaplygin who was a friend of the family. While Fomin was still at school Chaplygin, who had spotted his talents at a young age, advised Fomin to attend lectures at Moscow University. When he was still 15 years old Fomin decided that he did not wish to study at school any longer and that he would enter Moscow University at the age of 16. He had no school certificate but he sat the university entrance examination and passed with very high marks. Despite his age he became a student and his first interest was in abstract algebra. It was not long before Fomin had proved some new results in the theory of infinite abelian groups, examining conditions for such a group to be the direct product of a periodic subgroup and a torsion free subgroup. These results were published in his first paper while he was 19 years old. Fomin graduated in 1939 and began to undertake research at the University of Moscow under Kolmogorov's supervision. Kolmogorov suggested problems in the theory of dynamical systems for Fomin to investigate, but Fomin was also advised by Aleksandrov to look at some problems in point-set topology and he also began to work in this area. Aleksandrov and Urysohn had made a conjecture in 1923 concerning necessary and sufficient conditions for a Hausdorff space to be compact and this was not proved until 1935 when M H Stone gave an http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fomin.html (1 of 3) [2/16/2002 11:10:19 PM]

Fomin

exceedingly complicated proof using representation theory of Boolean algebras. Aleksandrov asked Fomin to try to find a simpler proof and he succeeded with this task, the result becoming his second publication which appeared in 1940. The authors of [1] and [2] write:Fomin's topological papers are not numerous, but they are undoubtedly classical pieces of general topology. The work for his papers in topology was still going on when World War II broke out and Fomin was conscripted into the Red Army. Quite how he managed to continue with his mathematical researches under difficult army conditions is almost impossible to understand, but indeed he did continue to work for his doctorate. In 1942 the Red Army gave him an assignment in Kazan which was rather fortunate. At the beginning of World War II the Steklov Mathematical Institute had been moved from Moscow to Kazan and it remained there until the spring of 1943 when it was moved back to Moscow. While Fomin was in Kazan in 1942 he managed to arrange with the Steklov Mathematical Institute for him to defend his thesis and he did so brilliantly. When the war ended Fomin returned to Moscow University and joined Tikhonov's Department. He was awarded his habilitation for a dissertation On dynamical systems with invariant measure in 1951. Two years later he was appointed professor. In 1964 Fomin became professor in the Department of the Theory of Functions and Functional Analysis and two years later he was appointed as a professor in the Department of General Control Problems. We have already mentioned Fomin's work on topology. There are several other areas with which he is associated and in which he made major contributions. One of the areas with which he is particularly associated is ergodic theory. He worked on this early in his career while still influenced by Kolmogorov, publishing a number of important papers such as On dynamical systems in a function space in 1950. Fomin wrote a couple of papers with Gelfand and in the first of these, also published in 1950, they apply the theory of infinite dimensional representations of Lie groups to the theory of dynamical systems. In 1959 Fomin began to look at applications to mathematical biology. Although he continued his work at Moscow University, and his research in other areas of mathematics, he was appointed Head of the Laboratory of Mathematical Methods in Biology in 1960. Within mathematical biology he worked in a number of areas: the excitation in nerve fibres; receptors in the visual system; control of the human motor system; and artificial intelligence and robots. In 1973 he published an important text with M B Berkinblit Mathematical problems in biology which included many of his own results in the subject. While he was carrying out his work in mathematical biology, Fomin was also studying global analysis which he did from 1966 until his death. The reason for his interest in this subject came from another of his many interests, namely quantum physics. In this area he examined the theory of differentiable measures in infinite dimensional spaces and the theory of distributions. He worked with a number of collaborators from 1973 on the writing of a monograph on measure theory and differential equations. Halmos, who first met Fomin in the spring of 1965, writes:Some of the mathematical interests of Sergei Vasilovich were always close to some of mine (measure and ergodic theory); he supervised the translation of a couple of my books into Russian. We had corresponded before we met, and it was a pleasure to shake hands with a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fomin.html (2 of 3) [2/16/2002 11:10:19 PM]

Fomin

man instead of reading a letter. Three or four years later he came to visit me in Hawaii, and it was a pleasure to see him enjoy, in contrast to Moscow, the warm sunshine. A number of important texts written by Fomin ran into several editions and have been translated into English. He wrote Elements of the theory of functions and functional analysis in two volumes. The first appeared in 1954 and covered metric and normed spaces. The second volume on measure, the Lebesgue integral and Hilbert space appeared in 1960. A second revised edition, written with Kolmogorov, was published in 1968, a third edition in 1972, and a fourth in 1976. Concerning this text the authors of [1] and [2] write:Fomin was a master of mathematical style; he knew how to set out complex mathematical questions simply and explicitly. ... This excellent work is used and will be used for the study of functional analysis by many generations of students. Fomin's interests outside mathematics are described in [1] and [2]:Fomin was a man of high inner culture. His spirit responded in a lively manner to scientific problems, to artistic events, and to human stories. He loved music, literature, and painting; he drew well, played a good game of tennis, was an excellent skier, and to the end of his life did not cease to take part in sport. He died in Vladivostok while taking part in a Mathematical Summer School there. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fomin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fomin.html (3 of 3) [2/16/2002 11:10:19 PM]

Fontaine_des_Bertins

Alexis Fontaine des Bertins Born: 13 Aug 1704 in Claveyson, Drôme, France Died: 21 Aug 1771 in Cuiseaux, Saône-et-Loire, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alexis Fontaine's father was Jacques Fontaine and his mother was Madeleine Seytres. Jacques Fontaine was a royal notary, so he served the king in a legal capacity. Alexis enjoyed an upbringing in a fairly well off family and he was educated at the Collège de Tournon. In 1732 Fontaine went to live near Paris, where he had acquired a residence, and he began to study mathematics under Castel. Around this time he became friends with Clairaut and Maupertuis, and he began to submit memoirs to the Académie des Sciences. As a result of these papers Fontaine was elected to the Academy in 1733 as an adjoint mécanicien and he was promoted geometer (a term used to mean mathematician at this time) in 1739. Although associated with the Academy for the rest of his life he did not participate in the work of the Academy, rather preferring to pursue his own agenda. He led a solitary life showing little interest in the work of others. His papers are rather confused, and ignorant of the work of others, but do contain some very original ideas in the calculus of variations, differential equations and the theory of equations. Taton writes in [1]:Fontaine's work is of limited scope, often obscure, and willfully ignorant of the contributions of other mathematicians. Nevertheless, its inspiration is often original and it presents, amid confused developments, a number of ideas that proved fertile ... In 1732 Fontaine gave a solution to the brachistochrone problem, in 1734 he gave a solution of the tautochrone problem which was more general than that given by Huygens, Newton, Euler or Jacob Bernoulli, and in 1737 he gave a solution to an orthogonal trajectories problem. The methods which he developed to solve these problems led to the calculus of variations. He used what he called the "fluxio-differential" method, so called because it used two independent first-order Leibniz type differential operators. This technique was praised by Johann Bernoulli, Euler and d'Alembert. Fontaine then used differential coefficients instead of differentials and Greenberg in [5] shows how Fontaine progressed from a calculus of variations to a calculus of several variables. However Fontaine rather spoilt this fine contribution by, in 1767 and 1768, unjustly criticising Lagrange's method of variation presented in 1762. Fontaine had retired in 1765 to a country estate in Burgundy the purchase of which had stretched his finances to the point of almost leaving him bankrupt. In [3] Greenberg considers Fontaine's work and that of his contemporaries who are usually given credit http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fontaine_des_Bertins.html (1 of 2) [2/16/2002 11:10:20 PM]

Fontaine_des_Bertins

for laying the foundations for the calculus of several variables. Greenberg discusses the question of priority in [3] and also in [4]. One of the reasons that Fontaine has come off badly was his apparent attempts to gain credit for ideas which had first been presented by others. For example [1]:In his work of 1764 Fontaine included a study of dynamics dated 1739 and based on a principle closely analogous to the one that d'Alembert had made the foundation of his treatise of 1743. Although Fontaine did not raise any claim of priority, he attracted the hostility of a powerful rival who subsequently took pains to destroy the reputation of his work, which - without being of the first rank - still merits mention for its original inspiration and for certain fecund ideas that it contains. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fontaine_des_Bertins.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fontaine_des_Bertins.html (2 of 2) [2/16/2002 11:10:20 PM]

Fontenelle

Bernard le Bouyer de Fontenelle Born: 11 Feb 1657 in Rouen, France Died: 9 Jan 1757 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bernard de Fontenelle was educated in a Jesuit College in Rouen and became friends with Varignon and de l'Hôpital. He wrote on the history of mathematics and the philosophy of mathematics and science. He evaluated the works of others extremely well and his works contain a wonderful source of information about the scientists of his era. Fontenelle's most famous work was Entretiens sur la pluralité des mondes (1686). He was elected to the Académie Française in 1691 and became permanent secretary of the Académie des Sciences from 1697. Fontenelle presented many obituary notices to the Académie, those of Newton and Leibniz being particularly notable. In 1699 Fontenelle wrote Of the Usefulness of Mathematical Learning. In it he wrote To what purpose should People become fond of the Mathematicks and Natural Philosophy? ... People very readily call Useless what they do not understand. It is a sort of Revenge ... . One would think at first that if the Mathematicks were to be confin'd to what is useful in them, they ought only to be improv'd in those things which have an immediate and sensible Affinity with Arts, and the rest ought to be neglected as a Vain Theory. But this would be a very wrong Notion. As for Instance, the Art of Navigation hath a necessary Connection with Astronomy, and Astronomy can never be too much improv'd for the Benefit of Navigation. Astronomy cannot be without Optics by reason of Perspective Glasses: and both, as all parts of the Mathematicks are grounded upon Geometry ... . Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fontenelle.html (1 of 2) [2/16/2002 11:10:22 PM]

Fontenelle

List of References (7 books/articles)

Some Quotations (5)

A Poster of Bernard de Fontenelle

Mathematicians born in the same country

Honours awarded to Bernard de Fontenelle (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1733

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fontenelle.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fontenelle.html (2 of 2) [2/16/2002 11:10:22 PM]

Forsyth

Andrew Russell Forsyth Born: 18 June 1858 in Glasgow, Scotland Died: 2 June 1942 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Andrew Forsyth's father was John Forsyth and his mother was Christina Glenn. John Forsyth was an engineer working in the Glasgow shipyards and the family came from Paisley. However John Forsyth moved to Liverpool and, taking his family with him, Andrew was soon to show his exceptional mathematical abilities at secondary school in that city. Forsyth entered Trinity College of the University of Cambridge in 1877 where he studied under Cayley, graduating in 1881. Taking the Mathematical Tripos in that year he was placed first in the ranked list of first class graduates (First Wrangler) and he was appointed to a fellowship at Trinity College. This type of fellowship was competitive and candidates had to submit a thesis; Forsyth's thesis proved deep results on double theta functions. His remarkable talent saw him leave Cambridge in the following year when he was appointed to the chair of mathematics at the University of Liverpool at the remarkably young age of 24. Although Forsyth was back in Liverpool, the city which had become his home, he did not remain there for very long, accepting a lectureship at Cambridge in 1884. Two years later, at the age of 28, he was elected a Fellow of the Royal Society of London. In 1893 he published Theory of Functions of a complex variable which had such an impact at Cambridge that function theory dominated there for many years. Whittaker writes in [5] that this text:... had a greater influence on British mathematics than any work since Newton's Principia. However the reputation of the book outside Britain was not high. In fact this is not surprising since the whole thrust of the book was to bring the great advances of Continental mathematics to Cambridge http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Forsyth.html (1 of 3) [2/16/2002 11:10:24 PM]

Forsyth

which Forsyth rightly saw as living in the past. He was well equipped to undertake this task for he travelled widely and, being a good linguist, was able to appreciate the advances made by authors writing in French and German. On Cayley's death Forsyth was appointed to his chair in 1895 becoming the Sadlerian professor of Pure Mathematics. However his preference for technical mastery rather than rigorous analysis meant that he failed to inspire future pure mathematicians. In fact one would have to say that Forsyth was unlucky, for although he saw the importance of Continental mathematics, at the same time his greatest strengths lay in his ability to handle complex formulas. He therefore excelled at precisely the style of mathematics which he himself campaigned successfully to replace at Cambridge. He had a love affair with Marion Amelia Boys, the wife of C V Boys, and the scandal of 1910 forced him to resign his chair at Cambridge. After marrying Marion Boys, he left the country for a while spending some time in Calcutta before he eventually found another post in England, being appointed to the chair in Imperial College London in 1913. He retired from his chair in London in 1923 when reaching the age of sixty-five but continued to publish mathematical texts until he was close to eighty years of age. Famous texts which Forsyth published before his 1893 work Theory of functions of a complex variable, are A treatise on differential equations (1885), and Theory of differential equations published in six volumes between 1890 and 1906. After his 1893 treatise he published many other texts, the most important of which are Lectures on the differential geometry of curves and surfaces (1912), Lectures introductory to the theory of functions of two complex variables (1914), Calculus of variations (1927), Geometry of four dimensions which was in two volumes and published in 1930, and Intrinsic geometry of ideal space also in two volumes, published in 1935. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles)

Some Quotations (2)

Mathematicians born in the same country Honours awarded to Andrew Forsyth (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1886

Royal Society Royal Medal

Awarded 1897

London Maths Society President

1904 - 1906

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Forsyth.html (2 of 3) [2/16/2002 11:10:24 PM]

Forsyth

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Forsyth.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Forsyth.html (3 of 3) [2/16/2002 11:10:24 PM]

Fourier

Jean Baptiste Joseph Fourier Born: 21 March 1768 in Auxerre, Bourgogne, France Died: 16 May 1830 in Paris, France

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Fourier's father was a tailor in Auxerre. After the death of his first wife, with whom he had three children, he remarried and Joseph was the ninth of the twelve children of this second marriage. Joseph's mother died went he was nine years old and his father died the following year. His first schooling was at Pallais's school, run by the music master from the cathedral. There Joseph studied Latin and French and showed great promise. He proceeded in 1780 to the Ecole Royale Militaire of Auxerre where at first he showed talents for literature but very soon, by the age of thirteen, mathematics became his real interest. By the age of 14 he had completed a study of the six volumes of Bézout's Cours de mathematique. In 1783 he received the first prize for his study of Bossut's Méchanique en général. In 1787 Fourier decided to train for the priesthood and entered the Benedictine abbey of St Benoit-sur-Loire. His interest in mathematics continued, however, and he corresponded with C L Bonard, the professor of mathematics at Auxerre. Fourier was unsure if he was making the right decision in training for the priesthood. He submitted a paper on algebra to Montucla in Paris and his letters to Bonard suggest that he really wanted to make a major impact in mathematics. In one letter Fourier wrote Yesterday was my 21st birthday, at that age Newton and Pascal had already acquired many claims to immortality. Fourier did not take his religious vows. Having left St Benoit in 1789, he visited Paris and read a paper on algebraic equations at the Académie Royale des Sciences. In 1790 he became a teacher at the Benedictine college, Ecole Royale Militaire of Auxerre, where he had studied. Up until this time there

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fourier.html (1 of 5) [2/16/2002 11:10:26 PM]

Fourier

had been a conflict inside Fourier about whether he should follow a religious life or one of mathematical research. However in 1793 a third element was added to this conflict when he became involved in politics and joined the local Revolutionary Committee. As he wrote:As the natural ideas of equality developed it was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests, and to free from this double yoke the long-usurped soil of Europe. I readily became enamoured of this cause, in my opinion the greatest and most beautiful which any nation has ever undertaken. Certainly Fourier was unhappy about the Terror which resulted from the French Revolution and he attempted to resign from the committee. However this proved impossible and Fourier was now firmly entangled with the Revolution and unable to withdraw. The revolution was a complicated affair with many factions, with broadly similar aims, violently opposed to each other. Fourier defended members of one faction while in Orléans. A letter describing events relates:Citizen Fourier, a young man full of intelligence, eloquence and zeal, was sent to Loiret. ... It seems that Fourier ... got up on certain popular platforms. He can talk very well and if he put forward the views of the Society of Auxerre he has done nothing blameworthy... This incident was to have serious consequences but after it Fourier returned to Auxerre and continued to work on the revolutionary committee and continued to teach at the College. In July 1794 he was arrested, the charges relating to the Orléans incident, and he was imprisoned. Fourier feared the he would go to the guillotine but, after Robespierre himself went to the guillotine, political changes resulted in Fourier being freed. Later in 1794 Fourier was nominated to study at the Ecole Normale in Paris. This institution had been set up for training teachers and it was intended to serve as a model for other teacher-training schools. The school opened in January 1795 and Fourier was certainly the most able of the pupils whose abilities ranged widely. He was taught by Lagrange, who Fourier described as the first among European men of science, and also by Laplace, who Fourier rated less highly, and by Monge who Fourier described as having a loud voice and is active, ingenious and very learned. Fourier began teaching at the Collège de France and, having excellent relations with Lagrange, Laplace and Monge, began further mathematical research. He was appointed to a position at the Ecole Centrale des Travaux Publiques, the school being under the direction of Lazare Carnot and Gaspard Monge, which was soon to be renamed Ecole Polytechnique. However, repercussions of his earlier arrest remained and he was arrested again imprisoned. His release has been put down to a variety of different causes, pleas by his pupils, pleas by Lagrange, Laplace or Monge or a change in the political climate. In fact all three may have played a part. By 1 September 1795 Fourier was back teaching at the Ecole Polytechnique. In 1797 he succeeded Lagrange in being appointed to the chair of analysis and mechanics. He was renowned as an outstanding lecturer but he does not appear to have undertaken original research during this time. In 1798 Fourier joined Napoleon's army in its invasion of Egypt as scientific adviser. Monge and Malus were also part of the expeditionary force. The expedition was at first a great success. Malta was occupied on 10 June 1798, Alexandria taken by storm on 1 July, and the delta of the Nile quickly taken. However, on 1 August 1798 the French fleet was completely destroyed by Nelson's fleet in the Battle of the Nile, so http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fourier.html (2 of 5) [2/16/2002 11:10:26 PM]

Fourier

that Napoleon found himself confined to the land that he was occupying. Fourier acted as an administrator as French type political institutions and administration was set up. In particular he helped establish educational facilities in Egypt and carried out archaeological explorations. While in Cairo Fourier helped found the Cairo Institute and was one of the twelve members of the mathematics division, the others included Monge, Malus and Napoleon Bonaparte. Fourier was elected secretary to the Institute, a position he continued to hold during the entire French occupation of Egypt. Fourier was also put in charge of collating the scientific and literary discoveries made during the time in Egypt. Napoleon abandoned his army and returned to Paris in 1799, he soon held absolute power in France. Fourier returned to France in 1801 with the remains of the expeditionary force and resumed his post as Professor of Analysis at the Ecole Polytechnique. However Napoleon had other ideas about how Fourier might serve him and wrote:... the Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place. Fourier was not happy at the prospect of leaving the academic world and Paris but could not refuse Napoleon's request. He went to Grenoble where his duties as Prefect were many and varied. His two greatest achievements in this administrative position was overseeing the operation to drain the swamps of Bourgoin and to oversee the construction of a new highway from Grenoble to Turin. He also spent much time working on the Description of Egypt which was not completed until 1810 when Napoleon made changes, rewriting history in places, to it before publication. By the time a second edition appeared every reference to Napoleon would have been removed. It was during his time in Grenoble that Fourier did his important mathematical work on the theory of heat. His work on the topic began around 1804 and by 1807 he had completed his important memoir On the Propagation of Heat in Solid Bodies. The memoir was read to the Paris Institute on 21 December 1807 and a committee consisting of Lagrange, Laplace, Monge and Lacroix was set up to report on the work. Now this memoir is very highly regarded but at the time it caused controversy. There were two reasons for the committee to feel unhappy with the work. The first objection, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what we now call Fourier series. Further clarification by Fourier still failed to convince them. As is pointed out in [4]:All these are written with such exemplary clarity - from a logical as opposed to calligraphic point of view - that their inability to persuade Laplace and Lagrange ... provides a good index of the originality of Fourier's views. The second objection was made by Biot against Fourier's derivation of the equations of transfer of heat. Fourier had not made reference to Biot's 1804 paper on this topic but Biot's paper is certainly incorrect. Laplace, and later Poisson, had similar objections. The Institute set as a prize competition subject the propagation of heat in solid bodies for the 1811 mathematics prize. Fourier submitted his 1807 memoir together with additional work on the cooling of infinite solids and terrestrial and radiant heat. Only one other entry was received and the committee set up to decide on the award of the prize, Lagrange, Laplace, Malus, Haüy and Legendre, awarded Fourier

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fourier.html (3 of 5) [2/16/2002 11:10:26 PM]

Fourier

the prize. The report was not however completely favourable and states:... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. With this rather mixed report there was no move in Paris to publish Fourier's work. When Napoleon was defeated and on his way to exile in Elba, his route should have been through Grenoble. Fourier managed to avoid this difficult confrontation by sending word that it would be dangerous for Napoleon. When he learnt of Napoleon's escape from Elba and that he was marching towards Grenoble with an army, Fourier was extremely worried. He tried to persuade the people of Grenoble to oppose Napoleon and give their allegiance to the King. However as Napoleon marched into the town Fourier left in haste. Napoleon was angry with Fourier who he had hoped would welcome his return. Fourier was able to talk his way into favour with both sides and Napoleon made him Prefect of the Rhône. However Fourier soon resigned on receiving orders, possibly from Carnot, that the was to remove all administrators with royalist sympathies. He could not have completely fallen out with Napoleon and Carnot, however, for on 10 June 1815, Napoleon awarded him a pension of 6000 francs, payable from 1 July. However Napoleon was defeated on 1 July and Fourier did not receive any money. He returned to Paris. Fourier was elected to the Académie des Sciences in 1817. In 1822 Delambre, who was the Secretary to the mathematical section of the Académie des Sciences, died and Fourier together with Biot and Arago applied for the post. After Arago withdrew the election gave Fourier an easy win. Shortly after Fourier became Secretary, the Academy published his prize winning essay Théorie analytique de la chaleur in 1822. This was not a piece of political manoeuvring by Fourier however since Delambre had arranged for the printing before he died. During Fourier's eight last years in Paris he resumed his mathematical researches and published a number of papers, some in pure mathematics while some were on applied mathematical topics. His life was not without problems however since his theory of heat still provoked controversy. Biot claimed priority over Fourier, a claim which Fourier had little difficulty showing to be false. Poisson, however, attacked both Fourier's mathematical techniques and also claimed to have an alternative theory. Fourier wrote Historical Précis as a reply to these claims but, although the work was shown to various mathematicians, it was never published. Fourier's views on the claims of Biot and Poisson are given in the following, see [4]:Having contested the various results [Biot and Poisson] now recognise that they are exact but they protest that they have invented another method of expounding them and that this method is excellent and the true one. If they had illuminated this branch of physics by important and general views and had greatly perfected the analysis of partial differential equations, if they had established a principal element of the theory of heat by fine experiments ... they would have the right to judge my work and to correct it. I would submit with much pleasure .. But one does not extend the bounds of science by presenting, in a form said to be different, results which one has not found oneself and, above all, by forestalling the true author in publication.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fourier.html (4 of 5) [2/16/2002 11:10:26 PM]

Fourier

Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles)

Some Quotations (5)

A Poster of Joseph Fourier

Mathematicians born in the same country

Cross-references to History Topics

1. Topology enters mathematics 2. orbits and gravitation 3. An overview of the history of mathematics

Other references in MacTutor

1. Chronology: 1800 to 1810 2. Chronology: 1820 to 1830

Honours awarded to Joseph Fourier (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1823

Lunar features

Crater Fourier

Commemorated on the Eiffel Tower Other Web sites

1. Rouse Ball 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fourier.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fourier.html (5 of 5) [2/16/2002 11:10:26 PM]

Fowler

Ralph Fowler Born: 17 Jan 1889 in Fedsden, Roydon, Essex, England Died: 28 July 1944 in Cambridge, England

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ralph Howard Fowler was the eldest son of Howard Fowler and Frances Eva (the daughter of George Dewhurst the Manchester cotton merchant). He was described as being quite athletic and possessing a loud, though affable, laugh. He apparently inherited his athleticism from his father who was a star at rugby and cricket. His father was an Oxford man who was called to the bar, but instead of becoming a barrister went into business. Ralph's early education was handled at home by a governess. When he was ten he matriculated at Evans' preparatory school at Horris Hill where he excelled at athletics, in particular cricket and football (soccer). At the age of 13, in 1902, Ralph won a scholarship to Winchester College placing second in the entrance examination (which apparently annoyed him despite the fact that he was sick at the time). He spent the next six years at Winchester where he became Prefect of Hall (Head of School) and won school prizes in mathematics and natural science. Once again, athletics was a major focus of Ralph's life, though despite his success at golf, he was outdone by his sister Dorothy who was a champion golfer. Ralph's academic ability was exceptional at Winchester. His classical sixth form master, Frank Carter, believed he could have been as good a classical scholar as he was a mathematical scholar. But it was mathematics that attracted him. And his ability at mathematics was matched by his affable character. At Winchester Ralph made numerous friends. Despite his penchant for speaking plainly, he always made up for it after and this seemed to have an affect on people who, both at Winchester and later, seemed to gather around him. His family life appeared to be equally well-balanced. He could often be found golfing with his father, sister, and brother Christopher. Family life at the Fowlers was apparently quite happy and Ralph's parents "spared nothing that their children's education should be as perfect as possible, with well-occupied

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fowler.html (1 of 5) [2/16/2002 11:10:29 PM]

Fowler

vacations" [3]. During Ralph's stay at Winchester his family moved from Essex to Norfolk, taking up residence in Weybourne, near Sheringham which was where the Fowlers golfed regularly. A bit later they moved on to Glebelands, Burnham-on-Sea, Somerset, but not before Ralph got in a bit of cricket, playing for Norfolk County for some time. In December of 1906, Ralph won a Major Scholarship to Trinity College, Cambridge, and he left for Trinity during Michaelmas term 1908. At Trinity he read mathematics completing Part I of the Mathematical Tripos in 1909, taking a first class. In 1911 he became a wrangler in Part II of the Tripos. He received his BA degree in 1911 upon completion of Part II and received his MA in 1915. In 1913 he was awarded a Rayleigh Prize in Mathematics. Needless to say, Cambridge was quite good to Ralph - or, rather, Ralph was quite good to Cambridge. After completion of his degree, Ralph took to researching pure mathematics. His adeptness and insight won him a Trinity Fellowship in October 1914. However, World War I had just broken out and he obtained a commission in the Royal Marine Artillery. He could often be seen at Trinity wearing his gown over his military khakis which was apparently quite an unusual site. The War was less unusual, being rather typical in handing out suffering. Ralph was not immune. In 1917 he lost his brother Christopher on the Somme which was quite a blow to him. Later he was to lose his two greatest friends at Winchester, A D Gillespie and R H Hutchison, in a similar manner. Ralph himself was severely wounded in the shoulder at Gallipoli. The wound turned out to be a bit of a mixed bag, however, as it caused him to be introduced to A V Hill, Captain in the Cambridgeshires, and a Fellow at King's College (and former Fellow at Trinity). For it was Hill who was ultimately the catalyst that brought Ralph's mathematical ability into the realm of physics. Hill, in collaboration with Horace Darwin (of the ever-prolific Darwin family that included the physicist C G Darwin, Frances Galton, and the most famous of all, Horace's father, Charles Darwin, the evolutionist), had invented a mirrored system originally designed to determine the flight paths of aircraft and later used to target German zeppelins for anti-aircraft gunneries. Fowler came in just as they began to test the instrument in the field. As talented young scientists joined the group, it came to be known as "Hill's brigands" and it was here that Fowler made some of his most endearing friendships, not the least of which was E A Milne, who wrote several articles and obituaries on Fowler both before and after his death. Fowler soon became Hill's second in command working with the Experimental Department of HMS Excellent on Whale Island. Fowler, being Assistant Director, was resident in Portsmouth while Hill traveled often to London often for commune with higher ranks. Hill eventually became a brevet Major while Fowler was made a Captain, RMA. Fowler recruited a long list of able mathematicians to join the group and, combined with Hill's inspirations, Fowler's mathematical ability led the group to a number of important works. Many were published in journals including two now classic papers that appeared in the Philosophical Transactions of the Royal Society that were to have a profound impact on the field of ballistics both in Britain and in North America, particularly in World War II. Far from being a "paper-pusher," Fowler apparently was active in both the experiments as well as the laborious paper-writing. His work in this field led him, in particular, to consider wind structure and temperature structure at high altitudes which could have been the catalyst for his later interest in thermodynamics and statistical mechanics. For his ballistics work, Ralph was awarded the OBE in 1918. In 1919 Fowler left the service and returned to Trinity, though he was to have a part in the newly formed http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fowler.html (2 of 5) [2/16/2002 11:10:29 PM]

Fowler

Ordnance Board during the Second World War later on. This was when Fowler came under the influence of Lord Rutherford who had just been appointed Cavendish Professor. The two became very good friends and Fowler was eventually appointed College Lecturer in Mathematics in 1920. Here he jumped into a variety of mathematical problems and eventually began moving to more recent problems in mathematical physics including work on various kinetic theories of gases, again leading him toward thermodynamics and statistical mechanics. In 1921, Ralph married Eileen, the only daughter of Lord Rutherford, Ralph's good friend and colleague at Cambridge. The two were to have four children in the following nine years with Eileen dying just after the birth of the last. In 1922, Ralph became a Proctor at Cambridge which, being a Marine, he was well-suited for, finding himself chasing after undergraduates frequently and, on one occasion, injuring himself doing so. It was also in 1922 that Ralph began what would be his most seminal work. It began as a collaboration with C G Darwin (another of the famous Darwin clan). The two began working on the problem of the partition of energy, inspired by works of Ehrenfest and Trkal. Having developed a new technique for approaching physical chemistry through statistical mechanics, the two, and later Fowler alone, justified a number of formulae and calculations performed by the likes of Saha, Lindemann, and Chapman. In 1922-23, Ralph established the validity of the dissociation formula for high temperature ionization. In early 1923, Ralph along with E A Milne, wrote a seminal work on stellar spectra, temperatures, and pressures. This work continued in a series of papers through the 1920s leading to the Adams Prize of the University of Cambridge in 1923-24 and was published in 1929 as the seminal volume, Statistical Mechanics, which had a second edition, minus the astrophysical applications, published in 1936. In 1939 a successor volume, entitled Statistical Thermodynamics, was co-authored and published with E A Guggenheim. 1926 marked the publication of his most seminal individual paper which linked the gaseous degenerate state (obeying quantum statistics, co-discovered by P A M Dirac, who was introduced to quantum theory by Fowler himself) to white dwarf stars. It is rumored that he was annoyed that he did not, at the same time, apply Fermi-Dirac statistics to conductors, something later done by Sommerfeld. Fowler's range of interests kept him going throughout the next two decades as he produced papers on spectroscopy, physical chemistry, what is now known as condensed matter physics (or solid state physics), and magnetism in materials. He eventually took up a post in the Cavendish Laboratory at Cambridge and, in 1932, he was elected to the newly created Plummer Chair of Theoretical Physics. Working closely with his father-in-law, Lord Rutherford, he examined a number of interesting problems and delivered the 1935 Bakerian Lecture on specific heats of crystals and the 1934 Liversidge Lecture on the heavy isotope of hydrogen. In 1925 he was elected as a Fellow of the Royal Society and became a Fellow at Winchester in 1933. In 1936 he was awarded one of the Royal Medals and was appointed as Director of the National Physical Laboratory in 1938, though he was unable to take up the post due to ill health. According to Milne, the transformation from pure mathematician prior to World War I to physicist, engineer, and administrator by Word War II was nothing short of astonishing and a great tribute to Fowler's ability in practical matters. In 1938, upon taking ill and not being able to take up the National Laboratory Directorship, he chose to remain at Cambridge in the Plummer Chair. In 1939 when war broke out, he immediately resumed his work with the Ordnance Board, despite his health, and was eventually chosen to become a scientific

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fowler.html (3 of 5) [2/16/2002 11:10:29 PM]

Fowler

liaison to Canada and later the United States, two countries which he had formerly been familiar through visits to Toronto and visiting professorships at Princeton and the University of Wisconsin. For his work as this liaison, he was knighted in 1942. He returned to Britain later in the war and was reportedly working for the Ordnance Board and the Admiralty up until just a few weeks prior to his very premature death in 1944. It must be said that Sir Ralph Fowler was a brilliant physicist. But it may be for his influence upon others that he is best known. In fact, no less than fifteen Fellows of the Royal Society and three Nobel Laureates were supervised by Fowler between 1922 and 1939. The total number supervised during this time was a staggering sixty-four giving him an average of eleven research students at any given time. One might be led to believe that this did not allow for any depth of relationship to form between him and his students. However, this was far from the truth of the matter. Those who studied under Fowler had a tremendous admiration for him. In particular, E A Milne [1] was especially taken by the man whom he fondly referred to as "the kind of man you can still remain friendly with, even when he has sold you a motor-bike; it is not possible to say more" and whom he called a "prince amongst men" [2]. Aside from Milne, on whom he had a profound impact, he also had the opportunity of influencing the likes of Sir Arthur Eddington, Subramanian Chandrasekhar, Paul Dirac, Sir William McCrea, Lady Jeffreys and others either directly through supervision or indirectly through collaboration. Even in his personal life he was intimately connected with brilliant people having married Eileen, the only daughter of Lord Rutherford whom he met through Rutherford's Cavendish Laboratory at Cambridge. Sometimes his influence was simply the fact that he was known to so many people. It was Fowler who ultimately introduced Paul Dirac to the burgeoning field of quantum theory in 1923 leading Dirac to the forefront of its ultimate discovery in 1925. Fowler also put Dirac and Werner Heisenberg in touch with each other through Niels Bohr. As Sir William McCrea simply put it [3]: "he was the right man in the right place at the right time." Fowler's influence was far-reaching, extending beyond the hallowed halls of Cambridge and into government, both British and foreign. This aspect of his life was once again partly a matter of timing. While serving in the Royal Marines during Gallipoli he was seriously wounded. During his recovery he met A V Hill whose work in anti-aircraft gunnery had spawned "Hill's Brigands," a group of talented scientists charged with developing better techniques for targeting German Zeppelins. This work led to two classic papers on the subject of the aerodynamics of spinning shells and, according to Milne, the work had a tremendous impact on ballistics both in Britain and North America. This experience also made Fowler influential enough to affect the actions of Britain and her allies twenty years later in the second World War. During this later War, Fowler acted as a liaison between Britain and Canada and, later, Britain and the United States. In 1942 he was knighted, no doubt partly due to his heroic service to his country during both wars. Early in his career, after receiving his degree, Fowler took to examining the behavior of the solutions to certain second-order differential equations. In particular, he studied Emden's differential equation: 1/ 2 d /d ( 2d /d ) = - n. Sir Arthur Eddington had originally shown that the equilibrium of gaseous stars could be found using the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fowler.html (4 of 5) [2/16/2002 11:10:29 PM]

Fowler

above equation with n = 3. From this he had found a formula for the luminosity of the star in terms of mass. Milne wondered what would happen if the star did not have this particular rate of energy generation. He rightly deduced Emden's equation must have other solutions. When Milne divulged his thoughts to Fowler, Ralph immediately developed a new solution for different values of n and all types of boundary solutions. The resulting general equation, which had considerable later influence on stellar astrophysics, was: ' n = 0. d /d ) d /d ( This was later to be related to his most original paper on the degenerate state of white dwarf stars. In this paper, Fowler showed that the material of white dwarf stars must consist of a gas obeying Fermi-Dirac statistics - that is, it must be in a degenerate state. The atoms are ionized so dramatically that they are virtually electron-free. These ions are closely packed leaving the free electrons to form a degenerate gas which Fowler described as "like a gigantic molecule in its lowest state." The equilibrium of the white dwarfs was later found to be described by a solution to Emden's equation as generalized by Fowler in the above equation with n = 3/2.

Fowler's genius was in his ability to apply intense mathematical rigour to a variety of physical problems. But perhaps his greatest trait was his ability to make friends and acquaintances exemplified both by his numerous "fans" and the web of influences he wove both scientifically and politically. It is indeed a shame that the world lost such a great man at such a young age. Article by: Ian Durham, University of St Andrews. Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fowler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fowler.html (5 of 5) [2/16/2002 11:10:29 PM]

Fox

Charles Fox Born: 17 March 1897 in London, England Died: 30 April 1977 in Montreal, Canada

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Fox entered Sidney Sussex College Cambridge in 1915. After two years there he interrupted his studies to be part of the British Expeditionary Forces in France and was wounded in 1918. After completing his Cambridge studies he was appointed to a lectureship in Imperial College, London in 1919. Fox emigrated to Canada in 1949 to take up an appointment at McGill University. In 1956 he was promoted to professor at McGill, a post he held until he retired in 1967. His next eight years were spent as visiting professor at Sir George Williams University (now Concordia University) Montreal where he continued to teach mathematics until he was close to 80 years of age. Fox's main contributions were on hypergeometric functions, integral transforms, integral equations and the mathematics of navigation. In the theory of special functions he introduced an H-function with a formal definition. It is a type of generalisation of a hypergeometric function and related ideas can be found in the work of Pincherle, Mellin, Ferrar, Bochner and others. He wrote only one book An introduction to the calculus of variations (1950). He wrote it because:During my many years of teaching at London University I felt that none of the existing texts covered the subject as I would like to teach it and so I undertook the task of writing one of my own. How many texts have been written for this reason! Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fox.html (1 of 2) [2/16/2002 11:10:31 PM]

Fox

Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fox.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fox.html (2 of 2) [2/16/2002 11:10:31 PM]

Fraenkel

Adolf Abraham Halevi Fraenkel Born: 17 Feb 1891 in Munich, Germany Died: 15 Oct 1965 in Jerusalem, Israel

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adolf Fraenkel, in common with most students in Germany in his time, studied for periods at different universities. He spent some time at the University of Munich, the University of Marburg, the University of Berlin and the University of Breslau. From 1916 he lectured at the University of Marburg, being promoted to professor there in 1922. In 1928 Fraenkel left Marburg and spent one year teaching at the University of Kiel. He was a fervent Zionist and, after leaving Kiel, he taught at the Hebrew University of Jerusalem from 1929. Fraenkel was to spend the rest of his career at the Hebrew University. Fraenkel's first work was on Hensel's p-adic numbers and on the theory of rings. However he is best known for his work on set theory, writing his first major work on the topic Einleitung in die Mengenlehre in 1919. He made two attempts, in 1922 and 1925, to put set theory into an axiomatic setting that avoided the paradoxes. He tried to improve the definitions of Zermelo and, within his axiom system, he proved the independence of the axiom of choice. His system of axioms was modified by Skolem in 1922 to give what is today known as the ZFS system. This is named after Zermelo, Fraenkel and Skolem. Within this system it is harder to prove the independence of the axiom of choice and this was not achieved until the work of Cohen in 1963. Fraenkel was also interested in the history of mathematics and wrote a number of important works on the topic. He wrote on Gauss's work in algebra in 1920, then in 1930, he published an important biography

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fraenkel.html (1 of 2) [2/16/2002 11:10:33 PM]

Fraenkel

of Cantor. In 1960 he published Jewish mathematics and astronomy. A number of Fraenkel's students have made important contributions to mathematics including Robinson who succeeded him when he retired from his chair at the Hebrew University. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1920 to 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fraenkel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fraenkel.html (2 of 2) [2/16/2002 11:10:33 PM]

Francais_Francois

François Joseph Français Born: 7 April 1768 in Saverne, Bas-Rhin, France Died: 30 Oct 1810 in Mainz, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

François Français was the brother of Jacques Français. The brothers were the children of Jacques Frédéric Français, who was a grocer in Saverne, and Maria Barbara Steib. In 1789 François became a seminarist, that is a student in an institution for training candidates for the priesthood. Two years later, in June 1791, he was appointed professor in the Collège at Colmar. Just over a year later, in September 1792, he was appointed to the chair of mathematics in the Collège at Strasbourg. Political events produced an interlude in Français' career. The French Revolution of 1789 had not gained universal support and there were attempted counter-revolutions in parts of France. In the Vendée region in the west of France there was an uprising which was sparked off by the introduction of conscription in February 1793. By the middle of March there was an alliance of Royalists and peasants with a fair size army. Français joined the government army in May 1793 which was moving to put down the rebellion. The rebel army at this stage numbered around 30 000 and they took the towns of Thouars, Parthenay, and Fontenay in May. Crossing the Loire the rebels moved east taking Angers in the middle of June but their progress was stopped when they failed to take the town of Nantes. During the summer months there was confused fighting with the government forces somewhat fragmented. Français continued to fight with the government forces and as the autumn approached they became a more cohesive force under a single command and heavily defeated the rebel army (by this time numbering around 65000) in October. The fighting did not stop there and it was not until the end of December that the government forces mopped up all resistance. Français, however, left the government army in October and returned to teaching. The period of army life must have been attractive to Français for he quickly rejoined the army as an officer and served for a further four years until October 1797 when he was appointed professor of mathematics at the Ecole Centrale du Haut-Rhin in Colmar. Français never seemed to be one to stay in any one post for long but this appointment in Colmar lasted longer than any of his others. He remained there for six years but, once he moved on in September 1803 to the Lycée in Mainz, he was to teach mathematics in a number of different institutions over the next few years. In 1804 he taught at the Ecole d'Artillierie in La Fère, then he returned to Mainz and taught at the Ecole d'Artillierie there. Much of François Français's work was published after his death by his brother who added to it in a way

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Francais_Francois.html (1 of 2) [2/16/2002 11:10:34 PM]

Francais_Francois

to make the contribution of each hard to distinguish. François worked on partial differential equations and his memoir of 1795 on this topic was developed further and presented to the Académie des Sciences in 1797. Lacroix praised Français' work and described it as making a major contribution to the study of partial differential equations; however, it was not published. Français was friendly with Arbogast and together they worked on the calculus of derivations. After Arbogast died in 1803, Français inherited his mathematical papers and continued to work on the calculus of derivations. He presented a memoir on this topic, in particular applying the methods to study projectiles in a resistant medium, to the Académie des Sciences in 1804. This memoir was very highly praised by Biot in a report of 22 April 1805, but again the work was not published. After this Français did work which was praised by Legendre, Lagrange, Lacroix and Biot but submitted no further memoirs during his lifetime. However, as described above, his brother published much of his work after his death, publishing four memoirs of François Français's work. Taton, writes in [1]:While not of the first rank, the mathematical activity of the Français brothers merits mention for its originality and diversity. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Francais_Francois.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Francais_Francois.html (2 of 2) [2/16/2002 11:10:34 PM]

Francais_Jacques

Jacques Frédéric Français Born: 20 June 1775 in Saverne, Bas-Rhin, France Died: 9 March 1833 in Metz, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jacques Français was the brother of François Français. The brothers were the children of Jacques Frédéric Français, who was a grocer in Saverne, and Maria Barbara Steib. Jacques entered the College at Strasbourg and excelled in his studies there. He volunteered for the army in 1793. The reason why volunteers were needed in that year was that the French Revolution of 1789 had not gained universal support and there were attempted counter-revolutions in parts of France. In the Vendée region in the west of France there was an uprising which was sparked off by the introduction of conscription in February 1793. By the middle of March there was an alliance of Royalists and peasants with a fair size army. Jacques' brother François Français also joined the government army in 1793 which was being assembled to put down the rebellion. In September 1794 Jacques Français became an assistant in the engineering corps. In the autumn of 1797, when François Français left the army, Jacques Français entered the Ecole Polytechnique and in the spring of the next year he moved to the Ecole du Génie. By 1801 he had reached the rank of first lieutenant and he was sent by Napoleon to Egypt in January of that year. Once back in France he was stationed at Toulon and promoted to captain of the sappers. Further promotions followed, and in November 1802 he became second in command at the staff headquarters of the engineering corps. He went on to participate, under the command of Admiral Villeneuvre, in the naval battles of Cape Finisterre and Trafalgar. In 1807 he was stationed at Strasbourg and his commander there was Malus. We discuss below the mathematics which Jacques Français produced at this time due mainly to the encouragement of Malus. By 1810 Français had reached the position of first in command at the staff headquarters of the Ecole d'Application in Metz and, in 1811, he was appointed professor of military art in Metz. The first mathematics memoir which Jacques Français seems to have written was submitted in 1800. It was a work on the integration of first order partial differential equations, but the memoir had been lost so there are few details as to its precise contents. He then appears to have lost interest in mathematics until, under the command of Malus, he was encouraged to prepare his work on analytic geometry for publication. During 1807-08 he wrote works on the straight line and the plane in oblique coordinates, also considering transformations between systems of oblique coordinates. He applied his methods to the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Francais_Jacques.html (1 of 3) [2/16/2002 11:10:35 PM]

Francais_Jacques

famous problem of finding a sphere tangent to four given spheres, publishing a number of notes on the topic between 1808 and 1812, and giving a complete solution in the 1812 paper which appeared in Gergonne's Journal. Although with encouragement from Malus Jacques Français had begun to publish mathematics, a second boost to his mathematics occurred after the death of his brother François Français. He worked on his brother's manuscripts after his death, and published a mixture of his own work and his brother's over the next few years. It is fair to say that he was inspired by studying his brother's manuscripts. In September 1813 Français published a work in which he gave a geometric representation of complex numbers with interesting applications. This was based on Argand's paper which had been sent, without disclosing the name of the author, by Legendre to François Français. Although Wessel had published an account of the geometric representation of complex numbers in 1799, and then Argand had done so again in 1806, the idea was still little known among mathematicians. This changed after Français' paper since a vigorous discussion between Français, Argand and Servois took place in Gergonne's Journal. In this argument Français and Argand believed in the validity of the geometric representation, while Servois argued that complex numbers must be handled using pure algebra. After this burst of mathematical activity, Français appears to have given up mathematics at the end of 1815. Taton, writes in [1]:While not of the first rank, the mathematical activity of the Français brothers merits mention for its originality and diversity. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Francais_Jacques.html (2 of 3) [2/16/2002 11:10:35 PM]

Francais_Jacques

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Francais_Jacques.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Francais_Jacques.html (3 of 3) [2/16/2002 11:10:35 PM]

Francesca

Piero della Francesca Born: 1412 in Borgo San Sepolero (now Sansepolcro), Italy Died: 12 Oct 1492 in Borgo San Sepolero, Italy

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Piero della Francesca, who came from a family of fairly prosperous merchants, is recognised as one of the most important painters of the Renaissance. In his own time he was also known as a highly competent mathematician. In his Lives of the most famous painters ..., Giorgio Vasari (1511 - 1572) says that Piero showed mathematical ability in his earliest youth and went on to write 'many' mathematical treatises. Of these, three are now known to survive. The titles by which they are known are: Abacus treatise (Trattato d'abaco), Short book on the five regular solids (Libellus de quinque corporibus regularibus) and On perspective for painting (De prospectiva pingendi). Piero almost certainly wrote all three works in the vernacular (his native dialect was Tuscan), and all three are in the style associated with the tradition of 'practical mathematics', that is, they consist largely of series of worked examples, with rather little discursive text. The Abacus treatise is similar to works used for instructional purposes in 'Abacus schools'. It deals with arithmetic, starting with the use of fractions, and works through series of standard problems, then it turns to algebra, and works through similarly standard problems, then it turns to geometry and works through rather more problems than is standard before (without warning) coming up with some entirely original three-dimensional problems involving two of the 'Archimedean polyhedra' (those now known as the truncated tetrahedron and the cuboctahedron). Four more Archimedeans appear in the Short book on the five regular solids : the truncated cube, the truncated octahedron, the truncated icosahedron and the truncated dodecahedron. (All these modern

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Francesca.html (1 of 3) [2/16/2002 11:10:37 PM]

Francesca

names are due to Johannes Kepler (1619).) Piero appears to have been the independent re-discoverer of these six solids. Moreover, the way he describes their properties makes it clear that he has in fact invented the notion of truncation in its modern mathematical sense. On perspective for painting is the first treatise to deal with the mathematics of perspective, a technique for giving an appearance of the third dimension in two-dimensional works such as paintings or sculptured reliefs. Piero is determined to show that this technique is firmly based on the science of vision (as it was understood in his time). He accordingly starts with a series of mathematical theorems, some taken from the optical work of Euclid (possibly through medieval sources) but some original to Piero himself. Some of these theorems are of independent mathematical interest, but on the whole the work is conceived as a manual for teaching painters to draw in perspective, and the detailed drawing instructions are mind-numbing in their repetitiousness. There are many diagrams and illustrations, but unfortunately none of the known manuscripts has illustrations actually drawn by Piero himself. None of Piero's mathematical work was published under his own name in the Renaissance, but it seems to have circulated quite widely in manuscript and became influential through its incorporation into the works of others. Much of Piero's algebra appears in Pacioli's Summa (1494), much of his work on the Archimedeans appears in Pacioli's De divina proportione (1509), and the simpler parts of Piero's perspective treatise were incorporated into almost all subsequent treatises on perspective addressed to painters. Article by: J. V. Field, London Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles) A Poster of Piero della Francesca Other Web sites

Mathematicians born in the same country 1. Vatican exhibition 2. The Piero della Francesca Web site 3. Mark Harden's Artchive (Works by Piero della Francesca) 4. George W Hart (Piero della Francesca's polyhedra) 5. Kevin Brown (Piero della Francesca's tetrahedron formula) 6. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Francesca.html (2 of 3) [2/16/2002 11:10:37 PM]

Francesca

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR August 1995

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Francesca.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Francesca.html (3 of 3) [2/16/2002 11:10:37 PM]

Francoeur

Louis Benjamin Francoeur Born: 16 Aug 1773 in Paris, France Died: 15 Dec 1849 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Louis Francoeur entered the College d'Harcourt at Navarre in 1790. From 1793 he served in the army. He entered the Ecole Polytechnique in 1794 and in 1798 he taught an analysis course for Lacroix. Francoeur carried on with both a military and an academic career. He was theoretically made a sub-lieutenant in the artillery in 1802. In 1804 he was appointed professor of mathematics at the Ecole Polytechnique, then in 1805 he became professor of mathematics at the Lycée Charlemagne. In 1808 he was made professor of mathematics at the Faculté des Sciences, a post he was to hold in addition to others until 1845. The French, under Napoleon, fought inconclusive battles and campaigns in various parts of Spain and Portugal during 1809-10. Francoeur was sent on this Peninsular Campaign but he broke a leg in an accident with a carriage. During 1810 he spent four months in bed as a result of this accident. In 1811 a comet appeared causing great interest with the public. Francoeur became interested in astronomy as a result of this comet. In 1816 he entered the Society which was to promote elementary instruction in schools. He later became secretary of the Society. Francoeur encouraged learning in schools and his he did his part writing books. Francoeur is famed as a writer of texts, publishing his mechanics book Traité de mécanique élémentaire in 1800, an elementary course of mathematics in 1809 and an astronomy text in 1812. His other books include Cours complet de mathematique pures (1819) in two volumes, La goniométrie (1820), L'enseignment du dessin linéaire (1827), Astronomie practique (1830), Elements de technologie (1833), Géodésie (1835) and Traité d'arithmétique appliquée à la banque (1845). From 1840 an illness curtailed his activities forcing him to retire from his chair in the Faculty of Sciences in 1845. Article by: J J O'Connor and E F Robertson List of References (3 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Francoeur.html (1 of 2) [2/16/2002 11:10:38 PM]

Francoeur

Mathematicians born in the same country Honours awarded to Louis Francoeur (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Francoeur (18th Arrondissement)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Francoeur.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Francoeur.html (2 of 2) [2/16/2002 11:10:38 PM]

Frank

Philipp Frank Born: 20 March 1884 in Vienna, Austria Died: 21 July 1966 in Cambridge, Massachusetts, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Philipp Frank's father was Ignaz Frank and his mother was Jenny Feilendorf. Philipp studied physics at the University of Vienna obtaining a doctorate in theoretical physics in 1907 after working under Boltzmann. He described his student days as follows (see for example [1]):... the domain of my most intensive interest was the philosophy of science. I used to associate with a group of students who assembled every Thursday night in one of the old Viennese coffee houses ... We returned again and again to the central problem: How can we avoid the traditional ambiguity and obscurity of philosophy? How can we bring about the closest possible rapprochement between philosophy and science? The group of students that Frank is describing in this quotation is the group who would eventually become known as the Vienna Circle. Other members of the group at this time were Hahn, von Mises, and an economist and sociologist, Otto Neurath. The group developed the philosophy of logical positivism, investigating scientific language and scientific methodology. During this time Frank became a friend of von Mises, who obtained his doctorate from Vienna in the 1907, the same year as Frank. It was a friendship which would last throughout their lives and involve joint work. In 1907 Frank wrote an important paper on causality. Einstein was impressed by Frank's ideas which he put forward in this paper and the resulting discussions led to another life long friendship, this time between Frank and Einstein. Both loved the philosophy of science and the ideas of each would influence the other. Frank received his habilitation and was appointed a lecturer in the University of Vienna in 1910. On Einstein's recommendation Frank succeeded him to the chair of theoretical physics in the German University of Prague in 1912. Frank, Hahn and von Mises became part of the somewhat larger group active during the 1920s in the Vienna Circle of Logical Positivists. Important influences on their thinking came from several other mathematicians and scientists interested in philosophy: Riemann, Helmholtz, Hertz, Boltzmann, Poincaré, Hilbert, and Einstein. More on the philosophy side, influences came from Frege, Russell and Whitehead. Frank, describing how he felt that science, mathematics and philosophy were linked, explained that [1]:... he sought always to achieve a balanced outlook on man and nature; and for him physics not only provided reliable answers to particular technical problems but also raised and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frank.html (1 of 3) [2/16/2002 11:10:39 PM]

Frank

illuminated important questions concerning the nature, scope, and validity of human knowledge. ... [he] believed that a stable perspective on life can best be achieved through the critical, intellectual method of modern natural science. The friendship between Frank and von Mises developed into a collaboration in the mid 1920s. They were joint authors of the lengthy two volume book Differentiagleichungen und Integralgleichungen der Mechanik und Physik which was published in 1925. Frank had married on 16 November 1920, his wife being Hania Gerson. Frank remained at the German University in Prague until 1938. The Munich Agreement in that year saw large parts of the Czechoslovak republic surrendered to Germany. German troops along with Hitler himself had entered Austria on 12 March 1938, and a Nazi government had been set up there. Political pressure was put on Frank and other members of the Vienna Circle, and the group disbanded with many of its members including Frank fleeing to the United States. In the United States Frank was first appointed as a visiting lecturer, then made a lecturer in physics and mathematics at Harvard. He was joined at Harvard by his friend von Mises. In 1947 Frank wrote an excellent biography Einstein: His Life and Times. Frank worked on a wide range of topics in mathematics, and when one takes into account his publications on physics and philosophy it was a truly remarkable breadth. In mathematics he worked on the calculus of variations, Fourier series, function spaces, Hamiltonian geometrical optics, Schrödinger wave mechanics, and relativity. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frank.html (2 of 3) [2/16/2002 11:10:39 PM]

Frank

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Frank.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frank.html (3 of 3) [2/16/2002 11:10:39 PM]

Franklin

Philip Franklin Born: 5 Oct 1898 in New York, USA Died: 27 Jan 1965 in Belmont, Massachusetts, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Philip Franklin attended the College of the City of New York receiving his B.S. in 1918, the year after Post. After graduating Franklin went to the Princeton University to undertake doctoral studies. He was awarded his Ph.D. in 1921 for a thesis The Four Color Problem written under Veblen's supervision. On completing his doctorate Franklin remained at Princeton where he was an instructor in mathematics 1921-22. Then in 1922 he went to Harvard where he was Benjamin Peirce Instructor until 1924. In 1924 Franklin was appointed Instructor in Mathematics at Massachusetts Institute of Technology. He was to remain at MIT, being promoted to assistant professor in 1925. Then, during 1927-28 he held a Guggenheim Fellowship, befor being promoted to associate professor in 1930 and full professor seven years later. He worked on the four colour problem and also published books on calculus, differential equations, complex variable and Fourier series. In particular he wrote Differential equations for electrical engineers (1933), Treatise on advanced calculus (1940), The four colour problem (1941), Methods of advanced calculus (1944), Fourier methods (1949), Differential and integral calculus (1953), Functions of a complex variable (1958) and Compact calculus (1963). In addition to this impressive collection of books, Franklin was editor of the Journal of Mathematics and Physics from 1929. In 1943 he was honoured by his old College where the College of the City of New York awarded him the Townsend Harris Medal. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country Cross-references to History Topics

The four colour theorem

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Franklin.html (1 of 2) [2/16/2002 11:10:40 PM]

Franklin

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Franklin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Franklin.html (2 of 2) [2/16/2002 11:10:40 PM]

Franklin_Benjamin

Benjamin Franklin Born: 30 Jun 1706 in Boston, Massachusetts, British colony Died: 16 Mar 1790 in Philadelphia, Pennsylvania, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Benjamin Franklin was born in Boston, but spent most of his life in Philadelphia (punctuated by residences in London and Passy, France). His formal education lasted less than two years, and so he was mainly self-educated, aided by the convenient access to books provided by an apprenticeship in the printing business. Franklin is best known in the popular imagination for his nonscientific pursuits: printer, American revolutionary, ambassador, to mention only a few roles he played. His scientific reputation rests mainly on his accomplishments as an inventor and as a pioneering theorist in the physics of electricity, but his interests were also mathematical. His version of magic square - a variant now termed the Franklin magic square - was inspired by the work of Stifel and Frénicle, both of whose magic squares were of a more traditional variety. He also drew magic circles. His first published magic square and his only published magic circle appeared in a 1767 book which also included unrelated excerpts from work by Thomas Simpson. Two years later, Franklin published this square: 52 61 14

3 62 51 46 35 30 19

53 60 11

4 13 20 29 36 45

5 12 21 28 37 44

6 59 54 43 38 27 22

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Franklin_Benjamin.html (1 of 3) [2/16/2002 11:10:42 PM]

Franklin_Benjamin

55 58 9

8 57 56 41 40 25 24

50 63 16

7 10 23 26 39 42

2 15 18 31 34 47

1 64 49 48 33 32 17

Aside from the row and column sums being constant, this square also has many other "magical" properties. Further examples of Franklin's magic squares would not be published until two centuries after his death, and doubtless many more were lost. Those which do survive are quite impressive. Franklin enjoyed close personal and professional relationships with quite a few of the important thinkers of his day, such as Hume, Priestley, Lavoisier and Condorcet. He was a member of the learned societies of many nations. Among these were the Royal Society, which awarded him its prestigious Copley medal for his work in electricity, and the American Philosophical Society, of which he was a founder. He received several honorary degrees, including a doctorate from St. Andrews. Article by: Paul C Pasles, Villanova University,USA. Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Benjamin Franklin (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1756

Royal Society Copley Medal

Awarded 1753

Other Web sites

1. Paul C Pasles 2. Franklin's autobiography 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Franklin_Benjamin.html (2 of 3) [2/16/2002 11:10:42 PM]

Franklin_Benjamin

JOC/EFR June 2001

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Franklin_Benjamin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Franklin_Benjamin.html (3 of 3) [2/16/2002 11:10:42 PM]

Frattini

Giovanni Frattini Born: 8 Jan 1852 in Rome, Italy Died: 21 July 1925 in Rome, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giovanni Frattini attended school in Rome and, completing his schooling in 1869, entered the University of Rome in November of that year to study mathematics. Frattini was taught by some outstanding mathematicians at the University of Rome, being tutored by the geometers Guiseppe Battaglini, Eugenio Beltrami (who had just published his masterpiece on non-euclidean geometry). In 1873 Luigi Cremona arrived in Rome and also taught Frattini who obtained his doctorate in 1875. After graduating Frattini went to Caltanissetta in central Sicily where he taught at the Liceo, taking up his appointment in 1876. At this school Frattini was the head of mathematics but it was a school he was only to teach in for two years for, in November 1878, he moved to Viterbo in central Italy. This was much nearer to home for Frattini for Viterbo is situated at the foot of the Cimini Mountains to the northwest of Rome. He taught at the Technical Institute there, becoming Head of Mathematics and Descriptive Geometry in the year following his appointment. Frattini's road back to Rome was completed in February 1881 when his request for a transfer to the Technical Institute there accepted. In 1884 a Military College was founded in Rome and Frattini lectured there from the time that the College opened. It was shortly after joining the College that Frattini published three papers on group theory which today make his name familiar to anyone who has studied the topic. The route that Frattini had taken to undertake research in group theory had been to study Camille Jordan's papers on the topic. As a result of this study, Frattini published two major papers on transitive groups, the first in 1883 and the second in the following year. These are not two of the three papers which have made him famous, the latter being three papers on the generators of finite groups one of which he published in 1885 and the remaining two in 1886. In the first of these papers Intorno alla generazione dei gruppi di operazioni Frattini defined the subgroup which today is known as the Frattini subgroup. His definition was as the subgroup generated by all the non-generators of the group (elements which if included in a generating set for the group can always be omitted to still leave a generating set). He showed that the Frattini subgroup is nilpotent and, in so doing, used the beautiful method of proof known today as the "Frattini argument". The quality of this work led to Frattini being offered a university chair in Naples, but he declined the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frattini.html (1 of 3) [2/16/2002 11:10:43 PM]

Frattini

offer not wishing to leave Rome for family reasons. Although he was offered a lectureship in algebra at the University of Rome in 1914 he never took up the appointment. By that time he was 62 years of age. Before commenting on the final years of Frattini's life it is worth noting that he contributed to other areas of mathematics in addition to group theory. His work on differential geometry is important as is his papers on the analysis of second degree indeterminates. On this latter topic he simplified the classical work by Euler, Lagrange and Gauss (anyone would be proud to improve on the work of these three mathematicians!). The First World War was a difficult time for Frattini who found the events very troubling. In 1915 Italy turned from the Triple Alliance to make a treaty with Britain, France, and Russia. Italy declared war on Austria-Hungary in May of 1915 and some rather indecisive battles were fought in the north. Many Italian soldiers were killed, 500,000 in 1916 alone. Frattini's son was wounded in the war and, in order to be able to support him in the difficult economic conditions, Frattini continued working at a time when he wished to retire. His wife died and poor Frattini suffered a number of unhappy years in his old age. A glimpse of Frattini's character can be gained by looking at one of his eccentricities. He was a great admirer of the poems of Giuseppe Gioacchino Belli, a Roman poet who wrote around 2,000 sonnets in the Roman dialect. Belli's sonnets:... ... express his revolt against literary tradition, the academic mentality, and the social injustices of the papal system. The ritualism of the church and the accepted principles of commonplace morality were also objects of his derision. Frattini recited the sonnets to his pupils in the Roman dialect in which Belli wrote them. This was reported to the authorities in Rome and a ministerial enquiry was set up. However [1]:[Frattini] declared himself willing to replace these sonnets by the poetry of a highest undersecretary of education. Without further ado, he was allowed to continue his recitation of Belli's poetry. One can feel from this episode that Frattini was probably an outstanding teacher and indeed this was the case. His belief was that to learn mathematics a student had to do more and read less. I [EFR] absolutely agree with Frattini that one can only learn mathematics by doing mathematics, not by reading about how to do mathematics. Despite Frattini's belief that one should do mathematics rather than read mathematics, he did write a number of excellent books. These do indeed present mathematics in a concrete way [1]:... bereft of superfluous abstraction and rich in elegant brilliance. Those who have seen the "Frattini argument" will agree that "elegant brilliance" is an apt phrase to describe that part of Frattini's work too. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frattini.html (2 of 3) [2/16/2002 11:10:43 PM]

Frattini

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Frattini.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frattini.html (3 of 3) [2/16/2002 11:10:43 PM]

Frechet

Maurice René Fréchet Born: 2 Sept 1878 in Maligny, Yonne, Bourgogne, France Died: 4 June 1973 in Paris, France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Maurice Fréchet was a student of Hadamard's and, under his supervision, Fréchet wrote an outstanding dissertation in 1906, introducing the concept of a metric space. He did not invent the name 'metric space' which is due to Hausdorff. A versatile mathematician, Fréchet served as professor of mechanics at the University of Poitiers (1910-19) and professor of higher calculus at the University of Strasbourg (1920-27). He held several different positions in the field of mathematics at the University of Paris (1928-48) including lecturer of the calculus of probabilities, professor of differential and integral calculus and professor of the calculus of probabilities. Fréchet made major contributions to the topology of point sets and defined and founded the theory of abstract spaces. Fréchet also made important contributions to statistics, probability and calculus. In his dissertation of 1906, mentioned above, he investigated functionals on a metric space and formulated the abstract notion of compactness. In 1907 he discovered an integral representation theorem for functionals on the space of quadratic Lebesgue integrable functions. A similar result was discovered independently by Riesz. Fréchet's most important work includes (i) Les Espaces abstrait (1928), (ii) Récherchés théoretiques modernes sur la théorie des probabilités (1937-38), (iii) Pages choisies d'analyse générale (1953), http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frechet.html (1 of 2) [2/16/2002 11:10:45 PM]

Frechet

(iv) Les Mathématiques et le concret (1955). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) A Poster of Maurice Fréchet

Mathematicians born in the same country

Cross-references to History Topics

Topology enters mathematics

Other references in MacTutor

Chronology: 1900 to 1910

Honours awarded to Maurice Fréchet (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh Honorary Fellow of the Edinburgh Maths Society

Elected 1938

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Frechet.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frechet.html (2 of 2) [2/16/2002 11:10:45 PM]

Fredholm

Erik Ivar Fredholm Born: 7 April 1866 in Stockholm, Sweden Died: 17 Aug 1927 in Stockholm, Sweden

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ivar Fredholm was a student at Uppsala, then later at Stockholm and undertook research into mathematical physics. His Ph.D. from Uppsala was awarded in 1896. He was appointed to a chair in mechanics and mathematical physics at Stockholm in 1906. He made important contributions to mathematical physics. A student of Mittag-Leffler, Fredholm is best remembered for his work on integral equations and spectral theory. He developed the theory of Fredholm integral equations in Sur une nouvelle méthode pour la résolution du problème de Dirichlet (1900). Volterra had studied some aspects of integral equations but before Fredholm little had been done. Fredholm wrote papers with great care and attention so he produced work of high quatity which quickly gained him a high reputation through Europe. However his papers required so much effort from him that he wrote only a few. Hilbert extended Fredholm's work to include a complete eigenvalue theory for the Fredholm integral equation. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Ivar Fredholm

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fredholm.html (1 of 2) [2/16/2002 11:10:47 PM]

Fredholm

Cross-references to History Topics

An overview of the history of mathematics

Other references in MacTutor

Chronology: 1900 to 1910

Honours awarded to Ivar Fredholm (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Fredholm

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fredholm.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fredholm.html (2 of 2) [2/16/2002 11:10:47 PM]

Freedman

Michael Hartley Freedman Born: 21 April 1951 in Los Angeles, California, USA

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Michael Freedman entered the University of California at Berkeley in 1968 and continued his studies at Princeton University in 1969. He was awarded a doctorate by Princeton in 1973 for his doctoral dissertation entitled Codimension-Two Surgery. His thesis supervisor was William Browder. After graduating Freedman was appointed a lecturer in the Department of Mathematics at the University of California at Berkeley. He held this post from 1973 until 1975 when he became a member of the Institute for Advanced Study at Princeton. In 1976 he was appointed as assistant professor in the Department of Mathematics at the University of California at San Diego. Freedman was promoted to associate professor at San Diego in 1979. He spent the year 1980/81 at the Institute for Advanced Study at Princeton returning to the University of California at San Diego where he was promoted to professor on 1982. He holds this post in addition to the Charles Lee Powell Chair of Mathematics which he was appointed to in 1985. Freedman was awarded a Fields Medal in 1986 for his work on the Poincaré conjecture. The Poincaré conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere. The higher dimensional Poincaré conjecture claims that any closed n-manifold which is homotopy equivalent to the n-sphere must be the n-sphere. When n = 3 this is equivalent to the Poincaré conjecture. Smale proved the higher dimensional Poincaré conjecture in 1961 for n at least 5. Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remains open. Milnor, describing Freedman's work which led to the award of a Fields Medal at the International Congress of Mathematicians in Berkeley in 1986, said:Michael Freedman has not only proved the Poincaré hypothesis for 4-dimensional

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freedman.html (1 of 4) [2/16/2002 11:10:49 PM]

Freedman

topological manifolds, thus characterising the sphere S4, but has also given us classification theorems, easy to state and to use but difficult to prove, for much more general 4-manifolds. The simple nature of his results in the topological case must be contrasted with the extreme complications which are now known to occur in the study of differentiable and piecewise linear 4-manifolds. ... Freedman's 1982 proof of the 4-dimensional Poincaré hypothesis was an extraordinary tour de force. His methods were so sharp as to actually provide a complete classification of all compact simply connected topological 4-manifolds, yielding many previously unknown examples of such manifolds, and many previously unknown homeomorphisms between known manifolds. Freedman has received many honours for his work. He was California Scientist of the Year in 1984 and, in the same year, he was made a MacArthur Foundation Fellow and also was elected to the National Academy of Sciences. In 1985 he was elected to the American Academy of Arts and Sciences. In addition to being awarded the Fields Medal in 1986, he also received the Veblen Prize from the American Mathematical Society in that year. The citation for the Veblen Prize reads (see [3]):After the discovery in the early 60s of a proof for the Poincaré conjecture and other properties of simply connected manifolds of dimension greater than four, one of the biggest open problems, besides the three dimensional Poincaré conjecture, was the classification of closed simply connected four manifolds. In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincaré conjecture. The major innovation was the solution of the simply connected surgery problem by proving a homotopy theoretic condition suggested by Casson for embedding a 2-handle, i.e. a thickened disc in a four manifold with boundary. Besides these results about closed simply connected four manifolds, Freedman also proved: (a) Any four manifold properly equivalent to R4 is homeomorphic to R4; a related result holds for S3 R. (b) There is a nonsmoothable closed four manifold. (c) The four-dimensional Hauptvermutung is false; i.e. there are four manifolds with inequivalent combinatorial triangulations. Finally, we note that the results of the above mentioned paper, together with Donaldson's work, produced the startling example of an exotic smoothing of R4. In his reply Freedman thanked his teachers (who he said included his students) and also gave some fascinating views on mathematics [3]:My primary interest in geometry is for the light it sheds on the topology of manifolds. Here it seems important to be open to the entire spectrum of geometry, from formal to concrete. By spectrum, I mean the variety of ways in which we can think about mathematical structures. At one extreme the intuition for problems arises almost entirely from mental pictures. At the other extreme the geometric burden is shifted to symbolic and algebraic thinking. Of course this extreme is only a middle ground from the viewpoint of algebra, which is prepared to go much further in the direction of formal operations and abandon http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freedman.html (2 of 4) [2/16/2002 11:10:49 PM]

Freedman

geometric intuition altogether. In the same reply Freedman also talks about the influence mathematics can have on the world and the way that mathematicians should express their ideas:In the nineteenth century there was a movement, of which Steiner was a principal exponent, to keep geometry pure and ward off the depredations of algebra. Today I think we feel that much of the power of mathematics comes from combining insights from seemingly distant branches of the discipline. Mathematics is not so much a collection of different subjects as a way of thinking. As such, it may be applied to any branch of knowledge. I want to applaud the efforts now being made by mathematicians to publish ideas on education, energy, economics, defence, and world peace. Experience inside mathematics shows that it isn't necessary to be an old hand in an area to make a contribution. Outside mathematics the situation is less clear, but I cannot help feeling that there, too, it is a mistake to leave important issues entirely to experts. In June 1987 Freedman was presented with the National Medal of Science at the White House by President Ronald Reagan. The following year he received the Humboldt Award and, in 1994, he received the Guggenheim Fellowship Award. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1980 to 1990

Honours awarded to Michael Freedman (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1986

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freedman.html (3 of 4) [2/16/2002 11:10:49 PM]

Freedman

JOC/EFR April 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Freedman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freedman.html (4 of 4) [2/16/2002 11:10:49 PM]

Frege

Friedrich Ludwig Gottlob Frege Born: 8 Nov 1848 in Wismar, Mecklenburg-Schwerin (now Germany) Died: 26 July 1925 in Bad Kleinen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gottlob Frege was one of the founders of modern symbolic logic putting forward the view that mathematics is reducible to logic. Frege received his education at the universities of Jena (1869-71) and Göttingen (1871-1873) where he studied mathematics, physics and chemistry. He then taught at Jena in the department of mathematics where he remained, first as a lecturer and then a professor, for the rest of his working life. Frege lectured on all branches of mathematics although his mathematical publications outside the field of logic are few. His writings on the philosophy of logic, philosophy of mathematics, and philosophy of language are of major importance. He once said Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician. He was the first to fully develop the main thesis of logicism, that mathematics is reducible to logic. His works The Foundations of Arithmetic (1884) andThe Basic Laws of Arithmetic,Volume1 (1893) are devoted to this project. His work was not particularly well received, mainly it was ignored. While volume 2 of The Basic Laws of Arithmetic was at the printers he received a letter (on June 16 1902) from Bertrand Russell. Russell pointed out, with great modesty, that the Russell paradox gave a contradiction in Frege's system of axioms. After many letters between the two Frege modified one of his axioms and explains in an appendix to the book that this was done to restore the consistency of the system. However with this modified axiom, many of the theorems of Volume 1 do not go through and Frege must have known this. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frege.html (1 of 2) [2/16/2002 11:10:50 PM]

Frege

He probably never realised that even with the modified axiom the system is inconsistent since this was not shown until after Frege's death by Leshniewski. Frege was a major influence on Peano and Bertrand Russell. Article by: J J O'Connor and E F Robertson List of References (10 books/articles)

Some Quotations (3)

A Poster of Gottlob Frege

Mathematicians born in the same country

Some pages from publications

Appendix from Frege's Grundgesetze der arithmetik (1903) The first paragraph can be translated as: Hardly anything more unwelcome can befall a scientific writer than that one of the foundations of his edifice be shaken after the work is finished. I have been placed in this position by a letter of Mr Bertrand Russell just as the printing of the second volume was nearing completion...

Cross-references to History Topics

The beginnings of set theory

Other references in MacTutor

Chronology: 1880 to 1890

Other Web sites

●

Stanford Encyclopedia of Philosophy

●

B W Carver (More links)

●

Metaphysics Research Lab, Stanford

●

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Frege.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frege.html (2 of 2) [2/16/2002 11:10:50 PM]

Freitag

Herta Taussig Freitag Born: 6 Dec 1908 in Vienna, Austria Died: 25 Jan 2000 in Roanoke, Virginia, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Herta Freitag received the degree Magister Rerum Naturalium, in Mathematics and Physics, from the University of Vienna in 1934, and graduated M.A. (1948), Ph.D. (1953) from Columbia University, USA. After the invasion of Austria on 11 March 1938, life became very difficult for the Taussig family. Her father, who as editor of "Die Neue Freie Presse" had written several editorials warning of the dangers of Nazism, was dismissed from his post. Months later, Herta Taussig was granted immigration to England where she had jobs as a housemaid, governess, waitress and finally as a teacher, before obtaining the visa which took her to the United States in 1944. Her mathematics career was thus rescued, although effectively suspended for six years. She taught at a school in upstate New York from 1944 to 1948 and there met Arthur Freitag, whom she married in 1950. In 1948 she moved to Hollins College (now University), where her career progressed from instructor to full Professor and departmental chairman, until her formal retirement in 1971. During her years at Hollins College, and throughout her even longer period of retirement, she received many awards. She was the first woman to become President of the Virginia, Maryland, and District of Columbia Section of the Mathematical Association of America. Her lectures, always meticulously crafted and beautifully illustrated in her inimitably artistic calligraphy, are delivered so enthusiastically and yet so modestly, as if she fears that her personality might take any of the glory or attention away from Mathematics. One of her most inspired remarks concerns mathematicians' fondness for generalizing results:A mathematician is like a lover - give him a little finger and he wants the whole hand! http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freitag.html (1 of 2) [2/16/2002 11:10:52 PM]

Freitag

A colleague, with echoes of Gauss's description of Mathematics as the Queen of the Sciences, and Number Theory as the Queen of Mathematics, named Herta Freitag as the Queen of the Fibonacci Association. For she has attended and given a paper at every International Conference of the Association since the first one in 1984. She has also contributed prodigiously to the Elementary Problems and Solutions Section of the Fibonacci Quarterly and published many papers in that journal. Most appropriately, the Fibonacci Quarterly chose to honour her not on her 90th birthday, but on the threshold of her 89th year, since 89 is a Fibonacci sequence. Article by: G M Phillips, St Andrews Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Freitag.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freitag.html (2 of 2) [2/16/2002 11:10:52 PM]

Frenet

Jean Frédéric Frenet Born: 7 Feb 1816 in Périgueux, France Died: 12 June 1900 in Périgueux, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean Frenet entered Ecole Normale Supérieure in 1840, then studied at Toulouse where he wrote a doctoral thesis in 1847. Part of the thesis contains the theory of space curves and contains the formulas now known as the Frenet-Serret formulas. Frenet gave only six formulas while Serret gave all nine. Frenet published this part of his thesis in the Journal de mathematique pures et appliques in 1852. Frenet became professor at Toulouse, then in 1848 he was appointed professor of mathematics at Lyon. He was also director of the astronomical observatory there and, in this capacity, he conducted meteorological observations. Frenet's exercise book on the calculus, first published in 1856 ran to seven editions, the seventh being published in 1917. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frenet.html (1 of 2) [2/16/2002 11:10:53 PM]

Frenet

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Frenet.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frenet.html (2 of 2) [2/16/2002 11:10:53 PM]

Frenicle_de_Bessy

Bernard Frenicle de Bessy Born: 1605 in Paris, France Died: 17 Jan 1675 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Frenicle de Bessy was an excellent amateur mathematician who held an official position as counsellor at the Court of Monnais in Paris. He corresponded with Descartes, Fermat, Huygens and Mersenne. Most of the correspondence between these men and Frenicle de Bessy was on number theory but not exclusively so. He does comment on applied mathematical problems such as the trajectory of a body which falls from a starting position with an initial horizontal component. In a letter which he wrote at Dover in England to Mersenne on 7 June 1634, Frenicle describes an experiment to study the trajectory of a body released from the top of the mast of a moving ship. The data which he presents in the letter is quite accurate. Again on a more applied mathematical topic, Frenicle wrote an article which makes comments on Galileo's Dialogue. Frenicle de Bessy is best known, however, for his contributions to number theory. He solved many of the problems posed by Fermat introducing new ideas and posing further questions. We shall look at some of the problems which were typical of those he worked on. On 3 January 1657 Fermat made a challenge to the mathematicians of Europe and England. He posed two problems (in words rather than using notation as we shall do) involving S(n), the sum of the proper divisors of n: 1. Find a cube n such that n + S(n) is a square. 2. Find a square n such that n + S(n) is a cube. We know that Frenicle found four solutions to the first of these problems on the day that he was given the problem, and found another six solutions the next day. He gave solutions to both problems in Solutio duorm problematum ... (1657). In this work he posed some problems of his own, including the following: Find an integer n such that S(n) = 5n, and S(5n) = 25n. Find an integer n such that S(n) = 7n, and S(7n) = 49n. Find n such that n3 - (n-1)3 is a cube. Frenicle solved other problems posed by Fermat. For example he showed that if a right angled triangle has sides integers a, b, c then its area bc/2 can never be a square. He also showed that the area of a right

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frenicle_de_Bessy.html (1 of 2) [2/16/2002 11:10:54 PM]

Frenicle_de_Bessy

angled triangle is never twice a square. Frenicle de Bessy also worked on magic squares and published Des quassez ou tables magiques. He was elected to the Académie Royale des Sciences in 1666. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1650 to 1675

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Frenicle_de_Bessy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frenicle_de_Bessy.html (2 of 2) [2/16/2002 11:10:54 PM]

Frenkel

Jacov Il'ich Frenkel Born: 10 Feb 1894 in Rostov-on-Don, Russia Died: 23 Jan 1952 in Leningrad, Russia

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jacov Il'ich Frenkel's father, Il'ya Abramovich Frenkel spent six years deported to Siberia because of his revolutionary views. Soon after his release in 1890 he married Rosaliya Abramovna Batkina. Jacov Il'ich Frenkel was the eldest of their children. The family moved to Kazan in the early 1900s and by this time there were four children. After one year in a Gymnasium, Jacov Il'ich was sent to Switzerland in 1905 because of the situation in Russia. Of his childhood he wrote [1]:As a child I was very talented at music and painting. This impelled my parents to arrange for me to have lessons in playing the violin (at age 8) and drawing (at age 12). Both these activities have remained my favourite hobbies ... In 1906 the family moved to Minsk, then in 1909 to St Petersburg. There Frenkel entered the K May Gymnasium. By this stage he was already interested in mathematics:I first became interested in mathematics and physics at age 14. By the end of the fifth grade I had learnt the whole mathematics course, and by the time I graduated from the Gymnasium, most of the university course in mathematics, mechanics and physics. Unfortunately, all along there was nobody to guide me, so I had to study on my own. By 1910 Frenkel was discovering mathematics for himself and working extremely hard. he became ill with heart problems which his parents put down to overwork and tried to stop him working during the summer of 1911, encouraging him to rest. His work was shown to J V Uspenskii, a student of A A Markov, who saw that Frenkel had discovered many results on finite differences for himself. By 1912 he had completed a 250 page paper on mathematical physics in which he studied the Earth's magnetism. Frenkel graduated from the K May http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frenkel.html (1 of 3) [2/16/2002 11:10:56 PM]

Frenkel

Gymnasium in 1913 with distinction and was awarded the Gold Medal. After this he visited the United States, returning to St Petersburg to enter University in August 1913. He graduated in 1916 from the Physics and Mathematics department, remaining at St Petersburg University to study for his doctorate. The political situation became very difficult, however, and he went to the Crimea where a new university had recently been set up. After graduating he began to teach at the University of Tavrichisky at Simferpol in November 1918. Frenkel married Sarra Isaakovna Gordina in the Crimea in December 1920. The couple went to St Petersburg (by this time called Petrograd) in early 1921. There he continued his research on mathematical physics at the Polytechnic Institute. In November 1925 Frenkel set out on a European tour, spending a year visiting Germany, France, and England. Returning to St Petersburg in 1926 (by the called Leningrad) he taught both at the Polytechnic Institute and at the University. He had already published a number of major books: The structure of matter I (1922), The theory of relativity (1923), The structure of matter II (1924), Vector and tensor analysis (1925), Electricity and matter (1925), and Electrodynamics (1926). After his return to Leningrad he worked on a second volume of Electrodynamics. He attended the Como Conference in Italy in September 1927 presenting his work on the electron theory of metals. He also participated in the Congress of Physicists in Moscow in the following year. In 1930 he went to the United States for a year, returning to Leningrad to maintain his remarkable publication record. He published the first volume of Wave mechanics in 1932 with the second volume appearing in 1934. If you think this means that a whole year went by without Frenkel publishing a book then you would be wrong for between the two volumes of Wave mechanics he published Statistical physics in 1933. Other major books included Analytical mechanics (1935) and Theoretical mechanics based on vector and tensor analysis (1940). Of course this incredible productivity had its drawbacks. Kapitza once told him:You would be a genius if you published ten times less than you do. His son writes in [1]:... precision was not among his merits. he was always late - for a train, for a seminar, for a lecture. Frenkel knew his own weakness here and once joked:In reply to your enquiry I inform you that I was not late for my lecture since it did not start before I arrived. During World War II the Institute was evacuated in Leningrad and Frenkel left for Kazan in August 1941 to join the resited Institute. He only just made it for two days later the advancing German armies prevented further trains leaving Leningrad. In Kazan he worked on his book Kinetic theory of liquids which was, in many ways, his best work. It was published in 1945. He had become Head of Theoretical Physics at Kazan University in 1942 and from around this time his interests turned towards geophysics. He returned to Leningrad in the spring of 1944 but the remaining eight years of his life were very difficult ones. His work was criticised for not contributing to the construction of socialism. His book were seen to oppose the deterministic Socialist philosophy, particularly his work on quantum mechanics. He was forced into making various statements declaring that his views had been in error just to allow him to survive at all. there is no doubt that the worry of these years contributed to his early death. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frenkel.html (2 of 3) [2/16/2002 11:10:56 PM]

Frenkel

List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Frenkel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frenkel.html (3 of 3) [2/16/2002 11:10:56 PM]

Fresnel

Augustin Jean Fresnel Born: 10 May 1788 in Broglie, France Died: 14 July 1827 in Ville-d'Avray, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Augustin Fresnel was educated at the Ecole Polytechnique and was active in the Corps des Ponts et Chaussées. He lost his engineering post temporarily during Napoleon's return from Elba in 1814 and around this time he did important work on optics where he was one of the founders of the wave theory of light. By applying mathematical analysis to his work Fresnel removed many of the objections to the wave theory of light. With Arago he rediscovered interference of polarised light and gave a theory of diffraction of light. He obtained circularly polarised light and developed the use of compound lenses instead of mirrors for lighthouses. Article by: J J O'Connor and E F Robertson List of References (10 books/articles)

A Quotation

A Poster of Augustin Fresnel

Mathematicians born in the same country

Honours awarded to Augustin Fresnel (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1825

Lunar features

Promontorium Fresnel and Rimae Fresnel

Paris street names

Rue Fresnel (16th Arrondissement)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fresnel.html (1 of 2) [2/16/2002 11:10:58 PM]

Fresnel

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fresnel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fresnel.html (2 of 2) [2/16/2002 11:10:58 PM]

Freudenthal

Hans Freudenthal Born: 17 Sept 1905 in Luckenwalde, Germany Died: 13 Oct 1990 in Utrecht, The Netherlands

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hans Freudenthal was born into a Jewish family, a fact which would have unfortunate consequences for him during World War II. He was educated in Luckenwalde, the town of his birth, where he studied at the Gymnasium. Although he developed an interest in mathematics and science at this stage, he was also interested in literature and read widely from the classics of literature and poetry. In 1923 he entered the University of Berlin to study mathematics and physics. Two important events took place in 1927 which would influence the direction of his career. Brouwer lectured in Berlin in that year and the meeting between Freudenthal and Brouwer would lead to Freudenthal's career being entirely in The Netherlands. Also in 1927, Freudenthal spent the summer semester at the University of Paris broadening his already broad interests. At the University of Berlin his doctoral supervisor was Hopf and in 1931 Freudenthal was awarded his doctorate for a thesis on the theory of ends of groups Uber die Enden topologischer Räume und Gruppen. By the time he was awarded his doctorate, Freudenthal was already in Amsterdam having been invited to go there in 1930 as Brouwer's assistant. He soon progressed to become a lecturer at the Mathematical Institute of the University of Amsterdam. Bos writes [3]:After settling in The Netherlands, he soon commanded the language and developed a versatile, rich, and direct style of writing in Dutch. Being out of Germany had the advantage to Freudenthal that, when the Nazis came to power in 1933 and passed legislation to deprive Jews of their jobs, he could continue with his teaching and research in Amsterdam. He wrote important papers on a spectral theorem for Riesz spaces in 1936 and on the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freudenthal.html (1 of 3) [2/16/2002 11:11:00 PM]

Freudenthal

suspension theorems in 1937. He was working on the algebraic characterisation of the topology of the real semisimple Lie groups in 1940 when Germany invaded The Netherlands. Now being of Jewish background became highly significant. The Nazi invaders now did not allow Freudenthal to continue to undertake his duties at the University. Bos writes [3]:He spent the war years with his young family in Amsterdam in often difficult circumstances. Indeed the circumstances were difficult and during this period of German occupation of that city, 70,000 Jewish inhabitants were deported, many going to their deaths in the concentration camps. Freudenthal and his family had to remain in hiding. Bos recounts a story in [3] which illustrates both Freudenthal's literary ability and the difficult circumstances of the war years:He competed in several literary contests. In one such competition in 1944, Freudenthal's work, a novel, was awarded first prize. But because of the occupation he could not reveal himself as the author. Therefore, the novel was sent in by a friend who had to play, with considerable risk, the role of a prizewinner; at interviews dinners and speeches. But the ruse succeeded and the prize money reached Freudenthal, a most welcome support during the last war year. In May 1945 Amsterdam was liberated by Canadian troops and soon after this Freudenthal was able to resume his duties at the university. He was offered the chair of pure and applied mathematics and foundations of mathematics at Utrecht University and he took up his duties there in 1946. He would hold this chair until he retired in 1975. In 1971 Freudenthal was appointed as the first director of the Institute for the Development of Mathematical Education in Utrecht. It was an Institute which he founded, and he was to remain its enthusiastic leader for many years. The Institute became part of the Faculty of Mathematics and Computer Science at Utrecht University in 1981 and, in September 1991, it was renamed the Freudenthal Institute. As we have indicated, Freudenthal's early work was on topology and algebra. In addition to the topics we mentioned above, we should single out his work on the characters of the semisimple Lie groups between 1954 and 1956. However, he later moved into broad areas including the history of mathematics and mathematical education. Adda writes in [2]:His culture was unbounded in scope and he always struggled (in many languages) against obscurantism. His thoughts and his works went in many complementary directions: mathematics, history of mathematics, mathematics education, philosophy ... . He worked to open mathematics education to everyone and never lost the intellectual requirements of a great scientific thinker. But he was also a man of action and had a great influence on the development of mathematics education research, not only in the Netherlands but also all around the world. Freudenthal's work on the history of mathematics included contributing articles to the Dictionary of Scientific Biography. In this archive we have included in our references articles in that publication on Arbuthnot, Cauchy, Haar, Heine, Hermite, Hilbert, Hopf, Hurwitz, Kerékjártó, Knopp, Lie, Loewner, Pringsheim, Quetelet, Riemann, Schönflies, Schottky, Sylow, and Christian Wiener. In addition we have referred to articles he has written on Huygens, Leibniz, von Staudt, Einstein, Brouwer, Weyl, and Dantzig. A particular interest that Freudenthal had in the history of mathematics was geometry. Bos, in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freudenthal.html (2 of 3) [2/16/2002 11:11:00 PM]

Freudenthal

[4], discusses his contributions to the history of geometry around 1900:In the late 1950s Freudenthal published several articles on the history of geometry around 1900, in particular on Hilbert's innovative approach to the foundations of geometry. In particular, his essay-review of the eighth edition of Hilbert's Grundlagen der Geometrie has become a standard reference in historical studies of geometry. He made many contributions to mathematical education writing books and papers, and giving lectures on "the learning of mathematics" and "the development of mathematical instruction". He was strongly opposed to the ideas behind the introduction of the "new maths". Freudenthal studied the relation between axiomatic mathematics and reality, and this study led him to contribute to intuitionism, as well as to the application of mathematics to linguistics. On this latter topic we should note his work on Lincos, a language designed for cosmic intercourse which he developed between 1957 and 1960. Veldkamp in [17] pays a tribute to Freudenthal, writing:Many people who have known Freudenthal will recall his vivid and inspiring personality. He died peacefully, sitting on a park bench while out for his morning walk near his home in Utrecht. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (17 books/articles) Mathematicians born in the same country Cross-references to History Topics

Indian numerals

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Freudenthal.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freudenthal.html (3 of 3) [2/16/2002 11:11:00 PM]

Freundlich

Erwin Finlay Freundlich Born: 29 May 1885 in Biebrich, Germany Died: 24 July 1964 in Wiesbaden, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Finlay Freundlich's father was E Philip Freundlich and his mother was Ellen Elizabeth Finlayson. Philip Freundlich was a German businessman and his wife Ellen was British. Perhaps it is worth explaining that Finlay Freundlich only called himself "Finlay" after he came to live in Scotland (this being a Scottish name); he was known as Erwin Freundlich for the first fifty years of his life. Finlay Freundlich was one of a family of seven children, five boys and two girls. He attended primary school in his home town of Biebrich, as did his brothers, and then he went to secondary school in the nearby large town of Wiesbaden where he attended Dilthey school. At eighteen years of age, in 1903, Freundlich completed his school education and went to work in the dockyards in Stettin. His aim at this stage was not related to mathematics or astronomy for he aimed to make a career in naval studies. He entered the Technische Hochschule of Charlottenburg and began a course of study in naval architecture but he had a health problem which forced him to take a break in his studies. The health problem was a heart condition and, when he had recovered, Freundlich decided not to continue his course on naval architecture but rather to enter the University of Göttingen to study mathematics, physics and astronomy. At Göttingen, Freundlich was a student of Klein. He spent the winter semester of 1905-06 at the Leipzig University but spent the rest of his university studies at Göttingen. This meant that he moved rather less around different universities than was the custom in Germany at this time, but the fact that he had changed course from naval architecture to mathematics, physics and astronomy almost certainly was a factor in this. Freundlich was awarded a doctorate by the University of Göttingen for a thesis on analytic function theory in 1910.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freundlich.html (1 of 4) [2/16/2002 11:11:02 PM]

Freundlich

Klein suggested to Freundlich that he might wish to apply for a post as an assistant at the Royal Observatory in Berlin and his appointment was confirmed on 1 July 1910. At this time Einstein was working on the general theory of relativity and, although he did not have the details of the theory worked out, he was beginning to understand some of its consequences. Already there was evidence that the orbit of Mercury did not fit that predicted by Newton's theory of gravitation and in 1911 Einstein asked Freundlich to make accurate observations of Mercury's orbit. Freundlich worked with Einstein in 1911 attempting to make the measurements of Mercury's orbit required to confirm the general theory of relativity. He confirmed it in a paper of 1913 but Freundlich had to go against the wishes of the Director of the Berlin Observatory who strongly advised him against publishing such a revolutionary idea. It is important to realise how daring this publication by Freundlich was for it claimed that Newton's theory of gravitation, so long held as one of the greatest achievements of the human mind, was wrong. Freundlich married Käte Hirschberg in 1913. She was Jewish while Freundlich was not, so the wedding was a civil one taking place in Herder House in Weimar. The observatory in Berlin moved to a new site at Neubabelsberg and a house was built for Freundlich and his wife close to the new observatory. Freundlich was interested in measuring the deflection in a light ray passing close to the sun since again Einstein's incomplete theory of relativity suggested that this test could be used to check the validity of the theory and show that Newton's theory was incorrect. The only way to make such measurements at this time was during an eclipse and Freundlich wanted to journey to somewhere within the path of totality of the eclipse which would happen in 1914. Such expeditions cost considerable amounts of money but Freundlich had the good fortune of being introduced to Gustav Krupp von Bohlen und Halbach by a friend. Gustav von Bohlen und Halbach was a German diplomat who had married the heiress Bertha Krupp of the Krupp family of industrialists and had taken over the family firm. He was impressed by Freundlich and, having considerable funds at his disposal, offered to finance an expedition to Feodosiya in the Crimea. Freundlich's good fortune came to an end, however, for World War I broke out before the time for the eclipse and the expedition was abandoned. Freundlich was interned for a while before being able to return to Berlin. He made other tests of general relativity based on gravitational redshift but these were inconclusive. He wrote his first book in 1916 following Einstein's publication of the general theory of relativity. Freundlich's book Grundlagen der Einsteinschen Gravitationstheorie discussed the ways that the general theory of relativity could be tested by astronomical observations. In 1918 Freundlich resigned his post in Berlin to work full time with Einstein. Forbes writes in [1]:He always modestly regarded himself as less of a collaborator with Einstein than as a butt for the latter's highly original ideas. His occasional inability to comprehend these ideas had the salutary effect of making Einstein seek to simplify their mathematical formulation, for if one of Felix Klein's pupils could not make sense of his equations who could? Through his intimate contact with Einstein, Freundlich was the first to become thoroughly acquainted with the fundamental principles of the new gravitational theory and, as Einstein himself remarks in the foreword of Freundlich's book, he was particularly well qualified as its exponent because he had been the first to attempt to put it to the test. In 1920 the Einstein Institute was created as the Astrophysical Observatory in Potsdam and Freundlich was appointed as observer there in 1921. He was later promoted to chief observer and professor of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freundlich.html (2 of 4) [2/16/2002 11:11:02 PM]

Freundlich

astrophysics. During this period Freundlich planned three further expeditions to observe an eclipse and measure the deflection of light passing close to the sun. Those of 1922 and 1923 failed because the weather did not permit observations to be made. However, one to Sumatra in 1929 was completely successful but the value which Freundlich found for the deflection of light was more than that predicted by Einstein's theory. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Freundlich himself was not affected by these laws but very reluctantly he resigned from his position in Potsdam and emigrated to Turkey. The reasons for this were two-fold. On the one hand his wife was Jewish and he could see that this might present problems for their family under the Nazis. The other, and perhaps even more important fact in Freundlich's mind, was that his wife's sister had died early in 1933 and Freundlich and his wife had become the guardians of his sister's two young children Hans and Renate (who of course were also Jewish). In Istanbul Freundlich helped create a modern observatory. He returned to Europe in 1937 when he was appointed professor of astronomy at the Charles University of Prague. However, he was forced out again by Hitler's policies in 1939 and escaped to Holland. While there he received an offer from the University of St Andrews to set up a department of astronomy at the university. Eddington had advised the Principal of the University of St Andrews that Freundlich was an outstanding person to both create the department of astronomy and to organise the construction of an observatory. In St Andrews Freundlich fitted in easily. An outstanding scientist, he commanded great respect for both his abilities and also for his exceptional personal qualities. His wife, however, found fitting in to the Scottish way of life somewhat harder than her husband. Walter Ledermann, who was a young lecturer in mathematics at St Andrews when Freundlich arrived, writes [4]:... Freundlich was very close to me. He was a fatherly friend of whom I have many fond memories ... During the war Freundlich and I taught navigation at the Initial Training Wing of the RAF which was stationed in St Andrews. We also published a joint paper in the Monthly Notices of the Royal Astronomical Society in 1944 ... . We had other interests in common apart from mathematics: Freundlich was a keen cellist, and we frequently played chamber music where I played the violin or viola. Once we went on holiday together to the West coast of Scotland, when Mrs Freundlich was unable to come. ... he was a tall impressive man, and when we walked side by side through the streets of St Andrews people would say: "Here come the Sun and Moon". Freundlich became the Napier Professor of Astronomy in St Andrews on 1 January 1951, a post he held until 1959. He delivered his inaugural lecture in the following year and the text of that lecture is given in [2]. In 1953 he suffered a heart attack but made a good recovery. During his years in St Andrews, as well as supervising the work of constructing a thirty-seven-inch Schmidt-Cassegrain telescope, he wrote another important text Celestial mechanics (1958). Freundlich, who had already suffered much as a result of the Nazis, was unfortunate in having an unhappy few final years. Forbes writes [1]:The closing years of Freundlich's life were marred by incidents arising out of the reluctance of his successor, D W N Stibbs, to grant him open access to the St Andrews observatory in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freundlich.html (3 of 4) [2/16/2002 11:11:02 PM]

Freundlich

order to witness the final stages of the work on the thirty-seven-inch Schmidt-Cassegrain telescope. The tensions that thus arose occasioned, inter alia, the resignation of his highly skilled technician Robert L Waland, before the optical components were satisfactorily completed and adjusted, and partly explain why that instrument has never yielded the results of which it might otherwise have been capable. Eventually Freundlich left for Wiesbaden where he was appointed honorary professor at the University of Mainz.

Article by: J J O'Connor and E F Robertson List of References (7 books/articles) A Poster of Finlay Freundlich

Mathematicians born in the same country

Cross-references to History Topics

General relativity

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Freundlich.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Freundlich.html (4 of 4) [2/16/2002 11:11:02 PM]

Friedmann

Aleksandr Aleksandrovich Friedmann Born: 16 June 1888 in St Petersburg, Russia Died: 16 Sept 1925 in Leningrad (now St Petersburg again), Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alexander Friedmann's date of birth is often given as 29 June. However this is an error which came about in converting the "Old Style" Russian date to the "New Style" date, which requires an addition of 12 days. Rather strangely Friedmann wrongly converted his own date of birth to 17 June (it should have been 4 + 12 = 16). Then, not realising that the date he gave had already been converted, it was converted again (17 + 12 = 29). Friedmann's father was a ballet dancer and his mother was a pianist. However the parents divorced when Alexander was nine years old. Records show that the church sided with the father and Alexander stayed with his father who soon remarried. Alexander entered the 2nd St Petersburg Gymnasium school in August 1897 and his record shows a quite ordinary school performance at first. Soon however Friedmann became one of the top two pupils in his class. The other outstanding pupil was Yakov Tamarkin, also an extraordinary mathematician, and the two boys were close friends, almost always together during their years at school and university. In 1905 Friedmann and Tamarkin wrote a paper on Bernoulli numbers and submitted the paper to Hilbert for publication in Mathematische Annalen. The paper was accepted and appeared in print in 1906. The year 1905 was not only one of great scientific importance for Friedmann, it was also one where he was extremely active politically. Friedmann and Tamarkin were student leaders of strikes at the school in protest at the government's repressive measures against schools. Friedmann graduated from school in 1906 and entered the University of St Petersburg in August of that year. There he was strongly influenced by Steklov who had taken up an appointment at St Petersburg in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Friedmann.html (1 of 6) [2/16/2002 11:11:04 PM]

Friedmann

the year Friedmann entered and shared Friedmann's political views. Friedmann was also influenced by Ehrenfest who moved to St Petersburg in 1906. By 1907 Ehrenfest had set up a modern physics seminar which was attended by a number of young physicists and by the two young mathematicians Friedmann and Tamarkin. This group discussed quantum theory, relativity and statistical mechanics. While Friedmann was an undergraduate at St Petersburg his father died. After he completed his studies in 1910 his scientific advisor, Steklov, wrote a reference for Friedmann to continue his studies. The death of Friedmann's father clearly had financial implications as the reference indicated, see [3]:Mr Friedmann is without means and when at the University made his living by private tutoring, proof-reading and earning a small salary. Steklov also wrote of Friedmann's mathematical work in his final year as an undergraduate:In his last year at the University he was working on an essay on the subject I assigned: 'Find all orthogonal substitutions such that the Laplace equation, transformed for the new variables, admits particular solutions in the form of a product of two functions, one of which depends only on one, and the other on the other two variables'. I touched on this problem in my doctoral thesis, but did not treat it in detail.. I suggested that Mr Friedmann should try to solve this problem, in view of his outstanding working capacity and knowledge compared with other persons of his age. In January of this year, Mr Friedmann submitted to me an extensive study of about 130 pages, in which he gave a quite satisfactory solution of the problem. ... Friedmann began to study for his Master's Degree and, in 1911, became involved with a circle formed to study mathematical analysis and mechanics. In addition to Friedmann, other members of the circle included Tamarkin, Smirnov, Petelin, Shokhat and, a little later, Besicovitch joined the circle. Friedmann lectured on Clebsch's work on elasticity and other topics including Goursat's books. While studying for his Master's Degree Friedmann lectured at the Mining Institute, cooperating there with Nikolai Krylov, and he also taught at the Railway Engineering. Through this work Friedmann became interested in aeronautics and in 1911 he published an article surveying the area describing, in particular, the contributions of Zhukovsky and Chaplygin. By 1913 Friedmann had completed the necessary examinations for the Master's Degree having been examined by Markov, Steklov and others. In February 1913 he was appointed to a position in the Aerological Observatory in Pavlovsk, a suburb of St Petersburg, where he was to study meteorology. In 1914 Friedmann went to Leipzig to study with Vilhelm Bjerknes, the leading theoretical meteorologist of the time. Friedmann left Leipzig in the summer of 1914 and took part in several flights in airships to made observations. When Austria gave Serbia an ultimatum after the June 1914 assassination of Archduke Francis Ferdinand, Russia supported Serbia, so Germany came to the support of Austria. World War I broke out on 1 August 1914 and Friedmann soon sought permission from the Head of the Observatory to join the volunteer aviation detachment. He began flying aircraft and was soon involved in bombing raids. He continued to study mathematics, writing and exchanging mathematical ideas with Steklov by letter. In a letter to Steklov written on 5 February 1915 Friedmann writes, see [3]:My life is fairly even, except such accidents as a shrapnel explosion twenty feet away, the explosion of an Austrian bomb within half a foot, which turned out almost happily, and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Friedmann.html (2 of 6) [2/16/2002 11:11:04 PM]

Friedmann

falling down on my face and head, which resulted in a ruptured upper lip and headaches. But one gets used to all this, of course, particularly seeing things all around which are a thousand times more awful. Also in this letter he asked Steklov's advice on integrating equations he had obtained from theoretically modelling bombs dropping. At this time the Russians were blockading the town of Przemysl, which was defended by Austrian troops, and Friedmann flew bombing missions over the town. He had used his mathematical skills, together with a suggestion from Steklov, to compute the trajectory that the bomb would take. In a letter of 28 February 1915 he wrote:I have recently had a chance to verify my ideas during a flight over Przemysl; the bombs turned out to be falling almost the way the theory predicts. To have conclusive proof of the theory I'm going to fly again in a few days. Friedmann was awarded the George Cross for bravery with his flights over Przemysl. In the summer of 1915 the Russian army retreated on its south west front. Friedmann was sent to Kiev and there he gave lectures on aeronautics for pilots. In March 1916 he was appointed Head of the Central Aeronautical Station in Kiev. In Kiev, Friedmann joined the Mathematical Society which had among its members Ch T Bialobzeski, P V Voronets, N B Delone, B N Delone, D A Grave, A P Kotelnikov, V P Linnik (I V Linnik's father) and O Yu Schmidt. In April 1917 the Central Aeronautical Station moved to Moscow, and Friedmann moved there. The Revolution of October 1917 became inevitable when Alexander Kerensky, the prime minister, sent troops to close down two Bolshevik newspapers. Lenin, who had been in hiding, made a public appearance telling the Bolsheviks to overthrow the Government. On the morning of October 26, after hardly any bloodshed, Lenin proclaimed that the Soviets were in power. After this, the work of the Central Aeronautical Station was stopped and Friedmann began to look for another post, but he was unsure of the direction he should take, particularly since his health had suffered as a result of the war. He wrote to Steklov saying:I'm very depressed; I often bitterly regret taking part in the war; it seems I achieved what I set out to do, but what's the use of it all now?. On 13 April 1918 Friedmann was elected an extraordinary professor in the Department of Mathematics and Physics at the University of Perm. Among the young colleagues he had there were A S Besicovitch, I M Vinogradov, N M Gunter and R O Kuzmin. At Perm Friedmann set up an Institute of Mechanics and became a member of the editorial board of the Journal of the newly founded Physico-Mathematical Society of Perm University. The Russian nation was plunged into civil war. The Red Army had been formed in February 1918 with Trotsky as its leader. The Reds opposed the White Army formed of anticommunists led by former imperial officers. In fact Friedmann had commented on the Red Army in Perm on 27 April 1918 when he wrote:Perm is surprisingly calm, and everything is done in the city in family fashion, in a good way, even the training of the Red army, which is 30-40 strong. However, on 20 December 1918, he wrote:Perm has come under an unlucky star. There is a rapid, overall and fairly chaotic evacuation. The University is in the second line of evacuation, but no transport or packing

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Friedmann.html (3 of 6) [2/16/2002 11:11:04 PM]

Friedmann

materials have been supplied so far, and the evacuation is at a standstill. ... I personally am not inclined to leave the city ... A week later the White Army occupied Perm. They controlled the town until August 1919 when the Red Army took control again. As the Red Army had approached Friedmann and all the staff, except Besicovitch, had left the University. Friedmann wrote:The only person who kept his head and saved the remaining property was Besicovitch, who is apparently A A Markov's disciple not only in mathematics but also with regard to resolute, precise definite actions. In the spring of 1920, with the Civil war still raging, Friedmann returned to St Petersburg (now named Petrograd) to take up a post at the Main Geophysical Observatory. Friedmann was never one to take life easy and he took up an impressive number of appointments in 1920 in Petrograd. He began teaching mathematics and mechanics at Petrograd University, became a professor in the Physics and Mathematics Faculty of the Petrograd Polytechnic Institute, worked in the Department of Applied Aeronautics at Petrograd Institute of Railway Engineering, worked at the Naval Academy and undertook research at the Atomic Commission at the Optical Institute. In 1922, nine years after completing the examinations for this Master's Degree, Friedmann submitted his Master's dissertation. The dissertation was entitled The Hydromechanics of a Compressible Fluid and was in two parts, the first on the kinematics of vortices and the second on the dynamics of a compressible fluid. Friedmann had taken up a new interest soon after returning to Petrograd. Einstein's general theory of relativity, although published in 1915, was not known in Russia due to World War I and the Civil War. By late 1920, Friedmann wrote in a letter to Ehrenfest:I have been working on the axiomatics of the relativity principle, proceeding from two underlying propositions: (1) uniform motion remains uniform for the uniformly moving world and (2) the velocity of light is constant (identical in the moving and stationary world), and have obtained formulae for a world with one space dimension which are more general than Lorentz's transformations ... Friedmann sent the article On the curvature of Space to Zeitschrift für Physik and it was received by the journal on 29 June 1922. In the paper Friedmann showed that the radius of curvature of the universe can be either an increasing or a periodic function of time. Friedmann, writing about the results of the paper in a book a little later, describes the results as follows:The stationary type of Universe comprises only two cases which were considered by Einstein and de Sitter. The variable type of Universe represents a great variety of cases; there can be cases of this type when the world's radius of curvature ... is constantly increasing in time; cases are also possible when the radius of curvature changes periodically ... Einstein quickly responded to Friedmann's article. His reply was received by Zeitschrift für Physik on 18 September 1922:The results concerning the non-stationary world, contained in [Friedmann's] work, appear to me suspicious. In reality it turns out that the solution given in it does not satisfy the field

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Friedmann.html (4 of 6) [2/16/2002 11:11:04 PM]

Friedmann

equations. On 6 December Friedmann wrote to Einstein:Considering that the possible existence of a non-stationary world has a certain interest, I will allow myself to present to you here the calculations I have made ... for verification and critical assessment. [The calculations are given] ... Should you find the calculations presented in my letter correct, please be so kind as to inform the editors of the Zeitschrift für Physik about it; perhaps in this case you will publish a correction to your statement or provide an opportunity for a portion of this letter to be published. However by the time the letter reached Berlin, Einstein had already left on a trip to Japan. He did not return to Berlin until March but he still did not seem to have read Friedmann's letter. Only Krutkov, when a colleague of Friedmann's from Petrograd, met Einstein at Ehrenfest's house in Leiden in May 1923 and told him of the details contained in Friedmann's letter did Einstein admit his error. He wrote immediately to Zeitschrift für Physik :In my previous note I criticised [Friedmann's work On the curvature of Space]. However, my criticism, as I became convinced by Friedmann's letter communicated to me by Mr Krutkov, was based on an error in my calculations. I consider that Mr Friedmann's results are correct and shed new light. In July 1923 Friedmann left Petrograd to visit Germany and Norway. In Germany he visited Berlin, Hamburg, Potsdam and Göttingen. In Norway he visited Christiania (Oslo). He discussed meteorology, aeronautics and mechanics. In Göttingen he talked to Prandtl and Hilbert, talking to Hilbert about his work in relativity. The following year, 1924, Friedmann was travelling again, this time to the First International Congress for Applied Mathematics held at Delft. He wrote about the congress:Everything went well at the congress, the attitude towards the Russians was wonderful; in particular, I was included among the members of the committee for convening the next conference. ... Courant from Göttingen got interested in Tamarkin's work. Blumenthal, Kármán and Levi-Civita got interested in my and my colleagues work. In July 1925 Friedmann made a record-breaking ascent in a balloon to 7400 metres to make meteorological and medical observations. He returned to Leningrad (Petrograd had been renamed Leningrad in 1924). Near the end of August 1925 Friedmann began to feel unwell. He was diagnosed as having typhoid and taken to hospital where he died two weeks later. In [3] Friedmann's contributions are summed up as follows:... Friedmann is seen as a profound, independent-minded, and daring thinker who destroys scientific prejudices, myths and dogmas; his intellect sees what others do not see, and will not see what others believe to be obvious but for which there are no grounds in reality. He rejects the centuries-old tradition which chose, prior to any experience, to consider the Universe eternal and eternally immutable. He accomplishes a genuine revolution in science. As Copernicus made the Earth go round the Sun, so Friedmann made the Universe expand. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Friedmann.html (5 of 6) [2/16/2002 11:11:04 PM]

Friedmann

List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

A brief history of cosmology

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Friedmann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Friedmann.html (6 of 6) [2/16/2002 11:11:04 PM]

Friedrichs

Kurt Otto Friedrichs Born: 28 Sept 1901 in Kiel, Germany Died: 31 Dec 1982 in New Rochelle, New York, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kurt Friedrichs solved a problem in relativity while still at school. His dissertation on the theory of elastic plates was written under Courant's supervision at Göttingen. He became Courant's assistant and helped with the Courant-Hilbert book. Friedrichs went to Aachen in 1929 and became professor in Braunschweig in 1932. His main work was on partial differential equations in mathematical physics. He used finite differences to prove existence of solutions. Friedrichs also worked on fluid dynamics. His girlfriend was Jewish so from 1933 he experienced problems. He went to the USA in 1937 to join Courant who had emigrated earlier, and was joined by his girlfriend whom he married. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to Kurt Friedrichs (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Friedrichs.html (1 of 2) [2/16/2002 11:11:05 PM]

Friedrichs

AMS Gibbs Lecturer

1954

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Friedrichs.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Friedrichs.html (2 of 2) [2/16/2002 11:11:05 PM]

Frisi

Paolo Frisi Born: 13 April 1728 in Milan, Austrian Habsburg (now Italy) Died: 22 Nov 1784 in Milan, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paolo Frisi entered the Barnabite Monastry around 1746, a religious Order devoted to the study of the Pauline Letters, where he was educated. The headquarters of the Order was in Milan, in the church of St Barnabas (hence the name of the Order). After being educated there Frisi joined the Barnabite Order. In addition Frisi was professor of philosophy at Casale Novara and Collegio Alessandro in Milan from 1753 to 1756. He left Milan in 1756 to take up a post as professor of philosophy at the University of Pisa. After holding the post in Pisa for eight years, Frisi returned to Milan becoming professor of mathematics at the Scuola Palatina in 1764. He was a leading authority on mathematics and science in his day. Frisi made a number of contributions to mathematics, physics and astronomy. In physics he worked on light and electricity but, although his work was very up to date for its time, his explanations were based on vibrations in the ether and did little to advance the topic. He was, however, the first to introduce the lightning conductor into Italy. His work on astronomy was based on Newton's theory of gravitation and is therefore of considerably more importance than his work in physics. He studied the motion of the earth and he was awarded a prize by the Berlin Academy for his outstanding memoir De moto diurno terrae. He also studied the physical causes for the shape and the size of the earth using the theory of gravity. Other astronomical phenomena which he studied included the difficult problem of the motion of the moon. He studied kinematics and hydraulics and he was responsible for drawing up plans for a canal between Milan and Pavia. In fact the work on this canal was not undertaken in Frisi's lifetime, but in 1819, thirty-five years after Frisi's death, the canal was built to his plan. His major work on hydraulics is Del modo di regolare i fiumi, e i torrenti written in 1762. In a paper of 1781 Frisi discussed isoperimetric problems. These were popular problems at this time with the Jacob and Johann Bernoulli having made important contributions and Euler, in 1744, having given a rule to determine a minimising arc between two points on a curve having continuous second derivatives. Frisi looked at problems involving both maximising and minimising. Frisi also wrote on the contributions of Galileo, Cavalieri, Newton and d'Alembert and brought their ideas to a wide audience. Such writings were widely read for Frisi was considered, in Italy, to be [1]:... a scientific authority and [he] was also well known abroad, so much so that his major http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frisi.html (1 of 2) [2/16/2002 11:11:07 PM]

Frisi

works (which he wrote in Latin) were translated into French and English. These major works are Algebra e geometrica analitica (1782), Meccanica (1783), and Cosmografia (1785) which contain much of Frisi's earlier work, but written up in a polished form. Frisi was also editor of the newspaper Il caffè. The paper was influenced by the ideas of the Illuminati (Enlightened Ones) which promoted free thought and democratic political theories. Through this paper Frisi had a major influence on the [1]:... cultural, social, and political life of Milan ... Letters written by Frisi and to Frisi are discussed in [4] and [6]. Letters between Teodorc Bonati and Frisi discussiong questions of mechanics and hydraulic mechanics are given in [4]. A letter by Frisi written in 1753 on mechanics and geometry is given and discussed in [6]. Frisi received many honours and his talents were recognised on an international scale. He was elected to the Académie des Sciences in Paris in 1753, then elected a Fellow of the Royal Society in London in 1757. In addition he was elected to the academies in St Petersburg, and Berlin. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) Mathematicians born in the same country Honours awarded to Paolo Frisi (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1757

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Frisi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frisi.html (2 of 2) [2/16/2002 11:11:07 PM]

Frobenius

Ferdinand Georg Frobenius Born: 26 Oct 1849 in Berlin-Charlottenburg, Prussia (now Germany) Died: 3 Aug 1917 in Berlin, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Frobenius's father was Christian Ferdinand Frobenius, a Protestant parson, and his mother was Christine Elizabeth Friedrich. Georg was born in Charlottenburg which was a district of Berlin which was not incorporated into the city until 1920. He entered the Joachimsthal Gymnasium in 1860 when he was nearly eleven years old and graduated from the school in 1867. In this same year he went to the University of Göttingen where he began his university studies but he only studied there for one semester before returning to Berlin. Back at the University of Berlin he attended lectures by Kronecker, Kummer and Weierstrass. He continued to study there for his doctorate, attending the seminars of Kummer and Weierstrass, and he received his doctorate (awarded with distinction) in 1870 supervised by Weierstrass. In 1874, after having taught at secondary school level first at the Joachimsthal Gymnasium then at the Sophienrealschule, he was appointed to the University of Berlin as an extraordinary professor of mathematics. For the description of Frobenius's career so far, the attentive reader may have noticed that no mention has been made of him receiving an habilitation before being appointed to a teaching position. This is not an omission, rather it is surprising given the strictness of the German system that this was allowed. Details of this appointment are given in [3] but we should say that it must ultimately have been made possible due to strong support from Weierstrass who was extremely influential and considered Frobenius one of his most gifted students. Frobenius was only in Berlin for a year before he went to Zürich to take up an appointment as an http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html (1 of 6) [2/16/2002 11:11:09 PM]

Frobenius

ordinary professor at the Eidgenössische Polytechnikum. For seventeen years, between 1875 and 1892, Frobenius worked in Zürich. He married there and brought up a family and did much important work in widely differing areas of mathematics. We shall discuss some of the topics which he worked on below, but for the moment we shall continue to describe how Frobenius's career developed. In the last days of December 1891 Kronecker died and, therefore, his chair in Berlin became vacant. Weierstrass, strongly believing that Frobenius was the right person to keep Berlin in the forefront of mathematics, used his considerable influence to have Frobenius appointed. However, for reasons which we shall discuss in a moment, Frobenius turned out to be something of a mixed blessing for mathematics at the University of Berlin. The positive side of his appointment was undoubtedly his remarkable contributions to the representation theory of groups, in particular his development of character theory, and his position as one of the leading mathematicians of his day. The negative side came about largely through his personality which is described in [5] as:... occasionally choleric, quarrelsome, and given to invectives. Biermann, in [3], looks more closely at his character (no pun intended!), and how it affected the success of mathematical education at the university. He describes the strained relationships which developed between Frobenius and his colleagues at Berlin. He had such high standards that in the end these did not serve Berlin well. He [3]:... suspected at every opportunity a tendency of the Ministry to lower the standards at the University of Berlin, in the words of Frobenius, to the rank of a technical school ... Even so, Fuchs and Schwarz yielded to him, and later Schottky, who was indebted to him alone for his call to Berlin. Frobenius was the leading figure, on whom the fortunes of mathematics at Berlin university rested for 25 years. Of course, it did not escape him, that the number of doctorates, habilitations, and docents slowly but surely fell off, although the number of students increased considerably. That he could not prevent this, that he could not reach his goal of maintaining unchanged the times of Weierstrass, Kummer and Kronecker also in their external appearances, but to witness helplessly these developments, was doubly intolerable for him, with his choleric disposition. We should not be too hard on Frobenius for, as Haubrich explains in [5], Frobenius's attitude was one which was typical of all professors of mathematics at Berlin at this time:They all felt deeply obliged to carry on the Prussian neo-humanistic tradition of university research and teaching as they themselves had experienced it as students. This is especially true of Frobenius. He considered himself to be a scholar whose duty it was to contribute to the knowledge of pure mathematics. Applied mathematics, in his opinion, belonged to the technical colleges. The view of mathematics at the University of Göttingen was, however, very different. This was a time when there was competition between mathematians in the University of Berlin and in the University of Göttingen, but it was a competition that Göttingen won, for there mathematics flourished under Klein, much to Frobenius's annoyance. In [3] Biermann writes that:The aversion of Frobenius to Klein and S Lie knew no limits ... Frobenius hated the style of mathematics which Göttingen represented. It was a new approach which

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html (2 of 6) [2/16/2002 11:11:09 PM]

Frobenius

represented a marked change from the traditional style of German universities. Frobenius, as we said above, had extremely traditional views. In a letter to Hurwitz, who was a product of the Göttingen system, he wrote on 3 February 1896 (see [4]):If you were emerging from a school, in which one amuses oneself more with rosy images than hard ideas, and if, to my joy, you are also gradually becoming emancipated from that, then old loves don't rust. Please take this joke facetiously. One should put the other side of the picture, however, for in [9] Siegel, who knew Frobenius for two years from 1915 when he became a student until Frobenius's death, relates his impression of Frobenius as having a warm personality and expresses his appreciation of his fast-paced varied and deep lectures. Others would describe his lectures as solid but not stimulating. To gain an impression of the quality of Frobenius's work before the time of his appointment to Berlin in 1892 we can do no better than to examine the recommendations of Weierstrass and Fuchs when Frobenius was elected to the Prussian Academy of Science in 1892. Fairly extensive quotes from this document, and another similar document from Fuchs and Helmholtz, are given in [4] but we quote a short extract to show the power, variety and high quality of Frobenius's work in his Zürich years. Weierstrass and Fuchs list 15 topics on which Frobenius had made major contributions:1. On the development of analytic functions in series. 2. On the algebraic solution of equations, whose coefficients are rational functions of one variable. 3. The theory of linear differential equations. 4. On Pfaff's problem. 5. 6. 7. 8.

Linear forms with integer coefficients. On linear substitutions and bilinear forms... On adjoint linear differential operators... The theory of elliptic and Jacobi functions...

9. On the relations among the 28 double tangents to a plane of degree 4. 10. On Sylow's theorem. 11. On double cosets arising from two finite groups. 12. On Jacobi's covariants... 13. On Jacobi functions in three variables. 14. The theory of biquadratic forms. 15. On the theory of surfaces with a differential parameter. In his work in group theory, Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups. He published über Gruppen von vertauschbaren Elementen in 1879 (jointly with Stickelberger, a colleague at Zürich) which looks at permutable elements in groups. This paper also gives a proof of the structure theorem for finitely generated abelian groups. In 1884 he published his next paper on finite groups in which he proved Sylow's theorems for abstract groups (Sylow had proved theorem as a result about permutation groups in his original paper). The proof which Frobenius gives is the one, based on conjugacy classes, still used http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html (3 of 6) [2/16/2002 11:11:09 PM]

Frobenius

today in most undergraduate courses. In his next paper in 1887 Frobenius continued his investigation of conjugacy classes in groups which would prove important in his later work on characters. In the introduction to this paper he explains how he became interested in abstract groups, and this was through a study of one of Kronecker's papers. It was in the year 1896, however, when Frobenius was professor at Berlin that his really important work on groups began to appear. In that year he published five papers on group theory and one of them über die Gruppencharactere on group characters is of fundamental importance. He wrote in this paper:I shall develop the concept [of character for arbitrary finite groups] here in the belief that through its introduction, group theory will be substantially enriched. This paper on group characters was presented to the Berlin Academy on July 16 1896 and it contains work which Frobenius had undertaken in the preceding few months. In a series of letters to Dedekind, the first on 12 April 1896, his ideas on group characters quickly developed. Ideas from a paper by Dedekind in 1885 made an important contribution and Frobenius was able to construct a complete set of representations by complex numbers. It is worth noting, however, that although we think today of Frobenius's paper on group characters as a fundamental work on representations of groups, Frobenius in fact introduced group characters in this work without any reference to representations. In was not until the following year that representations of groups began enter the picture, and again it was a concept due to Frobenius. Hence 1897 is the year in which the representation theory of groups was born. Over the years 1897-1899 Frobenius published two papers on group representations, one on induced characters, and one on tensor product of characters. In 1898 he introduced the notion of induced representations and the Frobenius Reciprocity Theorem. It was a burst of activity which set up the foundations of the whole of the machinery of representation theory. In a letter to Dedekind on 26 April 1896 Frobenius gave the irreducible characters for the alternating groups A4, A5, the symmetric groups S4 , S5 and the group PSL(2,7) of order 168. He completely determined the characters of symmetric groups in 1900 and of characters of alternating groups in 1901, publishing definitive papers on each. He continued his applications of character theory in papers of 1900 and 1901 which studied the structure of Frobenius groups. Only in 1897 did Frobenius learn of Molien's work which he described in a letter to Dedekind as "very beautiful but difficult". He reformulated Molien's work in terms of matrices and then showed that his characters are the traces of the irreducible representations. This work was published in 1897. Frobenius's character theory was used with great effect by Burnside and was beautifully written up in Burnside's 1911 edition of his Theory of Groups of Finite Order. Frobenius had a number of doctoral students who made important contributions to mathematics. These included Edmund Landau who was awarded his doctorate in 1899, Issai Schur who was awarded his doctorate in 1901, and Robert Remak who was awarded his doctorate in 1910. Frobenius collaborated with Schur in representation theory of groups and character theory of groups. It is certainly to Frobenius's credit that he so quickly spotted the genius of his student Schur. Frobenius's representation theory for finite groups was later to find important applications in quantum mechanics and theoretical physics which may not have entirely pleased the man who had such "pure" views about mathematics.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html (4 of 6) [2/16/2002 11:11:09 PM]

Frobenius

Among the topics which Frobenius studied towards the end of his career were positive and non-negative matrices. He introduced the concept of irreducibility for matrices and the papers which he wrote containing this theory around 1910 remain today the fundamental results in the discipline. The fact so many of Frobenius's papers read like present day text-books on the topics which he studied is a clear indication of the importance that his work, in many different areas, has had in shaping the mathematics which is studied today. Having said that, it is also true that he made fundamental contributions to fields which had already come into existence and he did not introduce any totally new mathematical areas as some of the greatest mathematicians have done. In [5] Haubrich gives the following overview of Frobenius's work:The most striking aspect of his mathematical practice is his extraordinary skill at calculations. In fact, Frobenius tried to solve mathematical problems to a large extent by means of a calculative, algebraic approach. Even his analytical work was guided by algebraic and linear algebraic methods. For Frobenius, conceptual argumentation played a somewhat secondary role. Although he argued in a comparatively abstract setting, abstraction was not an end in itself. Its advantages to him seemed to lie primarily in the fact that it can lead to much greater clearness and precision. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles)

A Quotation

A Poster of Georg Frobenius

Mathematicians born in the same country

Cross-references to History Topics

1. The development of group theory 2. The fundamental theorem of algebra 3. Matrices and determinants

Other references in MacTutor

1. Chronology: 1880 to 1890 2. Chronology: 1890 to 1900

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html (5 of 6) [2/16/2002 11:11:09 PM]

Frobenius

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Frobenius.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Frobenius.html (6 of 6) [2/16/2002 11:11:09 PM]

Fubini

Guido Fubini Born: 19 Jan 1879 in Venice, Italy Died: 6 June 1943 in New York, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Guido Fubini's father Lazzaro Fubini was a mathematics teacher at the Scuola Macchinisti in Venice so he came from a mathematical background and was influenced by his father towards mathematics when he was young. Guido attended secondary school in Venice where he showed that he was brilliant at mathematics and it was clear from this stage that he would follow career in the subject. In 1896 Fubini entered the Scuola Normale Superiore di Pisa. There he was taught by Dini and Bianchi who quickly influenced Fubini to undertake research in geometry. He presented his doctoral thesis Clifford's parallelism in elliptic spaces in 1900. Most young doctoral students take a few years to make themselves well known in their area. However, Fubini was lucky for his teacher Bianchi was about to publish an important work on differential geometry and he discussed the results of Fubini's thesis in his treatise which appeared in 1902. Fubini remained at Pisa to qualify as a university teacher. Most mathematicians at this stage in their careers extend the work they have begun in their doctoral thesis. Not so Fubini. He attacked a completely new topic to the one he had studied for his doctoral thesis studying the theory of harmonic functions in spaces of constant curvature. In October 1901 Fubini began teaching at the University of Catania in Sicily. There he won a competition to become a professor but soon he was heading north again when he was appointed to a chair at the University of Genoa. In 1908 Fubini moved to Turin where he taught both at the Politecnico and at the University of Turin. Fubini's interests were exceptionally wide moving from his early work on differential geometry towards http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fubini.html (1 of 3) [2/16/2002 11:11:11 PM]

Fubini

analysis. In this area he worked on differential equations, analytic functions and functions of several complex variables. He taught courses on these analysis topics at both the Politecnico and the University in Turin. During World War I Fubini studied the accuracy of artillery fire and these investigations led him on to work on acoustics and electricity. He was nearing the end of his career when the political situation in Italy suddenly put him in an exceptionally difficult position. All had seemed well and, despite the problems suffered by Jewish people in Germany from 1933, it seemed as though Italy would not follow that route. Up to 1934 Mussolini regarded Fascism as a development within Western civilisation and distrusted Germany and especially Hitler's National Socialism, which he declared to be:... one hundred percent racism: Against everything and everyone: Yesterday against Christian civilisation, today against Latin civilisation, tomorrow, who knows, against the civilisation of the whole world. However Mussolini soon found that he had to take the same line as the Nazis. In July 1938 he published the Manifesto of Fascist Racism and shortly after this anti-semitism became part of the official Fascist policy of Italy. A series of decrees removed Jews from positions of influence in government, banking and education. Fubini was forced to retire from his chair in Turin. Certainly Fubini had no wish to leave Italy but he had two sons who were engineers and, always a man who was devoted to his family, Fubini decided that his sons had no future in a country whose official policy was anti-semitism. When he received an invitation from the Institute for Advanced Study in Princeton in 1939 Fubini made the decision which he believed was best for his family. They emigrated to the United States immediately, although Fubini himself was in rather poor health by this time. Still, despite his health problem, he was able to teach for a few years in New York but, 5 years after emigrating he died of heart problems. As remarked above, Fubini's interests in mathematics were wide. In addition to the areas of analysis detailed above, he worked on the calculus of variations where he studied reducing Weierstrass's integral to a Lebesgue integral and also he worked on the expression of surface integrals in terms of two simple integrations. Another analysis topic he studied was non-linear integral equations. Fubini also worked on the theory of groups. In particular he studied linear groups and groups of automorphic functions. His interests included continuous groups where he looked at the question of putting a metric on the group. In non-euclidean spaces he extended results due to Appell and Mittag-Leffler. His most important work was on differential projective geometry where he used the absolute differential calculus. P Speziali, writing in [1], sums up Fubini's achievements as follows:Fubini was one of Italy's most fecund and eclectic mathematicians. His contributions opened new paths for research in several areas of analysis, geometry, and mathematical physics. Guided by an ever-alert geometric intuition and possessed of an absolute mastery of all the techniques of calculation, he was able to follow leads that had barely been glimpsed. His technical mastery often permitted him to discover simpler demonstrations of such theorems as those of Berstein and Pringsheim on the development of Taylor series. Since Fubini's sons trained as engineers he took an interest in the problems they were studying. This led

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fubini.html (2 of 3) [2/16/2002 11:11:11 PM]

Fubini

him to solve a whole series of engineering problems which he was writing up as a textbook towards the end of his life. The textbook, jointly authored with G Albenge, only appeared in 1954, eleven years after Fubini's death. This last textbook was one of an impressive collection of important textbooks on analysis which included books which described analysis courses which he had given and also books which were collections of problems. Speziali [1] describes Fubini in these terms:A man of great cultivation, fundamentally honourable and kind, Fubini possessed unequalled pedagogic talents. His witty banter and social charm made him delightful company; he was small in stature and his voice was vigorous and pleasant. He is rated highly for his wide ranging work and merits the high praise given in [1] where he is said to be:... one of the most luminous and original minds in mathematics during the first half of the twentieth century. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fubini.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fubini.html (3 of 3) [2/16/2002 11:11:11 PM]

Fuchs

Lazarus Immanuel Fuchs Born: 5 May 1833 in Moschin (near Posen), Germany (now Poznan, Poland) Died: 26 April 1902 in Berlin, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lazarus Fuchs attended the Friedrich Wilhelm Gymnasium in Berlin where his remarkable abilities at mathematics became very clear to his teachers while he was still young. Mathematics became the subject which, even at this early stage, Fuchs knew was going to dominate the rest of his life. After leaving the Gymnasium, he studied at the University of Berlin where he attended lectures by a number of famous mathematicians including Kummer and Weierstrass. Most significantly it was Weierstrass who introduced Fuchs to function theory and who went on to supervise his doctorate. The examiners for his doctoral dissertation were Kummer and Martin Ohm (the brother of Georg Simon) and Fuchs was awarded the degree by the University of Berlin 1858. After obtaining his doctorate, Fuchs was appointed to a teaching post at a Gymnasium. From there he moved on to a mathematics teaching position at Friedrich Werderschen Trade School. During this time he was undertaking research with the aim of becoming a university professor. He began his university teaching career when he was appointed as a Privatdozen at the University of Berlin 1865. He was promoted to extraordinary professor there in 1866 and taught at the University until winter semester 1868-69 when he accepted an appointment at Greifswald. Fuchs also held a second post in Berlin from 1867 when he was appointed as professor of mathematics at the Artillery and Engineering School. After spending five years in Greifswald he moved again, this time to Göttingen in 1874. Then in the following year he went to Heidelberg and taught there for nine years. In 1884 he returned to Berlin to fill Kummer's chair when his old teacher retired. Fuchs held this post for the rest of his life. He also http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fuchs.html (1 of 3) [2/16/2002 11:11:13 PM]

Fuchs

undertook important editorial duties in the final ten years of his life when he was the editor of Crelle's journal, the Journal für die reine und angewandte Mathematik. Fuchs worked on differential equations and the theory of functions. In [1] Manheim writes:Fuchs was a gifted analyst whose works form a bridge between the fundamental researches od Cauchy, Riemann, Abel, and Gauss and the modern theory of differential equations discovered by Poincaré, Painlevé, and Emile Picard. In 1865 Fuchs studied nth order linear ordinary differential equations with complex functions as coefficients. This is described by Bölling in [2]:Fuchs enriched the theory of linear differential equations with fundamental results. He discussed problems of the following kind: What conditions must be placed on the coefficients of a differential equation so that all solutions have prescribed proberties (e.g. to be regular or algebraic). This led him (1865, 1866) to introduce an important class of linear differential equations (and systems) in the complex domain with analytic coeffivcients, a class which today bears hios name (Fuchaian equations, equations of the Fuchsian class). ... He succeeded in characterising those differential equations the solutions of which have no essential singularity in the extended complex plane. Fuchs later also studied non-linear fifferential equations and moveable singularities. Fuchs' study (1876 with Hermite) of elliptic integrals as a function of a parameter marks an important step towards the theory of modular functions (Klein, Dedekind). In a series of papers (1880-81) Fuchs studied functions obtained by inverting the integrals of solutions to a second-order linear differential equation in a manner generalising Jacobi's inversion problem. It was Fuchs' work on this inverse function which led Poincaré to introduce what he called a Fuchsian group, and use this as a fundamental concept in the development of the theory of automorphic functions. Fuchs also investigated how to find the matrix connecting two systems of solutions of differential equations near two different points. A survey of Fuchs work appears in [3] where Gray also describes how this work influenced Klein, Jordan, Poincaré and others. In this interesting paper Gray also discusses the relationships between Fuchs' ideas and his mathematical tools, and illustrates how solutions of some problems led Fuchs to the study of further problems. In [2] Bölling describes Fuchs' character as follows:... Fuchs is a representative of both Berlin's classical and its post-classical era. His personality has been described as indecisive, timid, but at the same time humorous and full of kindness. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fuchs.html (2 of 3) [2/16/2002 11:11:13 PM]

Fuchs

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Fuchs.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fuchs.html (3 of 3) [2/16/2002 11:11:13 PM]

Fueter

Karl Rudolf Fueter Born: 30 June 1880 in Basel, Switzerland Died: 9 Aug 1950 in Brunnen, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Rudolf Fueter's father was Eduard Rudolf Fueter, who was an architect, and his mother was Adèle Gelzer. Rudolf studied in his home town of Basel before going to Göttingen in 1899 to study under Hilbert. He worked on quadratic number fields and his doctorate, supervised by Hilbert, was awarded in 1903 for a thesis entitled Der Klassenkörper der quadratischen Körper und die komplexe Multiplikation. After obtaining his doctorate Fueter travelled to various European centres of mathematical activity. He spent some time in Paris, then in Vienna and finally went to London while pursuing his studies. He was then appointed as a lecturer at the University of Marburg in 1907, and at the Mining Academy in Clausthal before being appointed as professor of mathematics at Basel in 1908. On 1 March 1908 he had married Amélie von Heusinger. They would have one daughter. Five years later, in 1913, Fueter left Basel to take the chair of mathematics at the Technical University of Karlsruhe, a post he only held for three years before moving to the chair at the University of Zurich. In [2] Burckhardt discusses the impact that Fueter had on the University of Zurich and demonstrates how he began the process of making it a cultural centre, even during a very difficult period of history. Fueter's first major publication was Synthetische Zahlentheorie which he published in 1917 soon after taking up the chair at Zurich. The work was a very successful one, a third edition being published in 1950, the year of Fueter's death. He did important work on the class formula for number fields summarising his work on the class formula for abelian number fields over an imaginary quadratic base field in the two volume work Vorlesungen über die singulären Moduln und die komplexe Multiplikation der elliptischen Funktionen. The first of the two volumes was published in 1924, with the second

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fueter.html (1 of 3) [2/16/2002 11:11:15 PM]

Fueter

following three years later. Between the two volumes ofVorlesungen über die singulären Moduln und die komplexe Multiplikation der elliptischen Funktionen Fueter published another major work, Das mathematische Werkzeug des Chemikers, biologen und Statistikers which was first published in 1926 and also ran to three editions, the third in 1947. Fueter was a cofounder of the Swiss Mathematical Society, which came into existence in 1910, becoming its first President. He also became Rector of the University of Zurich serving in this role during 1920-22. As a member of the Swiss Natural Science Society he served as an editor on the major project to publish the complete works of Euler. In addition he undertook other editorial work such at the editorship of Comentarii Mathematici Helvetici. Perhaps surprisingly Fueter also held the rank of colonel in the artillery of the Swiss army. The fact that Switzerland was neutral did not mean that the Swiss did nothing as Hitler came to power and the prospects of war became imminent. On the contrary they prepared themselves psychologically, economically, and militarily for involvement in a possible war. From the outbreak of World War II Fueter served in the Department of Press and Radio and was noted for his strong opposition to the Nazis. Despite being surrounded by Nazi and fascist controlled countries, Switzerland remained the only democratic state in central Europe and Fueter played an important role in that process. Kimche writes in [3]:Fueter restated the democratic rights of the press in almost classical form ... In his report of April 10, 1940 ... Fueter developed his argument more fully. "If is the duty of our press to reject the domestic and foreign policies of the National Socialists both clearly and forcefully." Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fueter.html (2 of 3) [2/16/2002 11:11:15 PM]

Fueter

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fueter.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fueter.html (3 of 3) [2/16/2002 11:11:15 PM]

Fuller

Richard Buckminster Fuller Born: 12 July 1895 in Milton, Massachusetts, USA Died: 1 July 1983 in Los Angeles, California, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Known as R Buckminster Fuller, he was an engineer, mathematician and architect. Twice expelled from Harvard University, business disasters and the death of his four year old daughter brought him close to suicide. However he decided to devote himself to proving that technology could save the World from itself, providing it is properly used. He examined a vectorial system of geometry, Energetic-Synergetic geometry, based on the tetrahedron which provides maximum strength with minimum structure. This led to his patent of a geodesic dome in 1947, a building the strength of which need only increase as the log of its size. Over 200 000 of such domes have been built. Fuller was research professor at Carbondale, Southern Illinois University, from 1959 to 1968. In 1968 he became a university professor and retired in 1975. Article by: J J O'Connor and E F Robertson List of References (6 books/articles)

Some Quotations (3)

A Poster of R Buckminster Fuller

Mathematicians born in the same country

Other Web sites

1. Portland (An account of some of Buckminster Fuller's innovations) 2. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fuller.html (1 of 2) [2/16/2002 11:11:17 PM]

Fuller

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fuller.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fuller.html (2 of 2) [2/16/2002 11:11:17 PM]

Fuss

Nicolaus Fuss Born: 30 Jan 1755 in Basel, Switzerland Died: 4 Jan 1826 in St Petersburg, Russia Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nikolai Fuss's mathematical abilities brought him to the attention of Daniel Bernoulli who recommended him for the post of Euler's secretary. Fuss went to St Petersburg in Russia in 1772 to take up this post and spent the rest of his life in Russia. His first papers, written under Euler's direction, were on problems in insurance. In 1776 he became and assistant at the St Petersburg Academy and, in 1783, academician in higher mathematics. From 1800 to 1826 he was permanent secretary to the Academy. Most of his papers are solutions to problems posed by Euler on spherical geometry, trigonometry, series, differential geometry and differential equations. His best papers are in spherical trigonometry, a topic he worked on with A J Lexell and F T Schubert. Fuss also worked on geometrical problems of Apollonius and Pappus. He made contributions to differential geometry and won a prize from the French Academy in 1778 for a paper Recherche sur le dérangement d'une comète qui passe près d'une planète. Fuss won other prizes from Sweden and Denmark. He contributed much in the field of education, writing many fine textbooks. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fuss.html (1 of 2) [2/16/2002 11:11:18 PM]

Fuss

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fuss.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Fuss.html (2 of 2) [2/16/2002 11:11:18 PM]

Galerkin

Boris Grigorievich Galerkin Born: 4 March 1871 in Polotsk, Belarus Died: 12 June 1945 in Moscow, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Boris Grigorievich Galerkin came from a poor family and this was to mean that he had a harder time through his years of education than would otherwise have been the case. He attended secondary school in Minsk, then in 1893 he entered the Petersburg Technological Institute. Here he studied mathematics and engineering but he needed to make money to survive so at first he took on private tutoring, then from 1896 he worked as a designer. After graduating from the Technological Institute in 1899 he got a job at the Kharkov Locomotive Plant. In 1903 Galerkin went to St Petersburg and there he became engineering manager at the Northern Mechanical and Boiler Plant. From 1909 Galerkin began to study building sites and construction works throughout Europe. In the same year he began teaching at the Petersburg Technological Institute. His first publication on longitudinal curvature also appeared in 1909, work which carried on from beginnings which had been laid by Euler. This paper was highly relevant to his study of construction sites since the results were applied to the construction of bridges and frames for buildings. His visits around European construction sites ended around 1914 but his academic work then turned to the area for which he is today best known, namely the method of approximate integration of differential equations known as the Galerkin method. He published his finite element method in 1915. In 1920 Galerkin was promoted to Head of Structural Mechanics at the Petersburg Technological Institute. By this time he also held two chairs, one in elasticity at the Leningrad Institute of Communications Engineers and one in structural mechanics at Leningrad University. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galerkin.html (1 of 2) [2/16/2002 11:11:20 PM]

Galerkin

In 1921 the St Petersburg Mathematical Society was reopened (it had closed in 1917 due to the Russian Revolution) as the Petrograd Physical and Mathematical Society. Galerkin played a major role in the Society along with Steklov, Sergi Bernstein, Friedmann and others. Other work for which Galerkin is famous is his work on thin elastic plates. His major monograph on this topic Thin Elastic Plates was published in 1937. From 1940 until his death, Galerkin was head of the Institute of Mechanics of the Soviet Academy of Sciences. A T Grigorian, writing in [1], describes other work:Galerkin's scientific research in the theory of casing (1934-45) revealed its broad application in industrial construction. His works in the field constitute a new direction in this important area. Galerkin was a consultant in the planning and building of many of the Soviet Union's largest hydrostations. In 1929, in connection with the building of the Dnepr dam and hydroelectric station, Galerkin investigated stress in dams and breast walls with trapezoidal profile. His results were used in planning the dam. Galerkin's important work on the finite element method is described in [2] and [3]. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Galerkin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galerkin.html (2 of 2) [2/16/2002 11:11:20 PM]

Galileo

Galileo Galilei Born: 15 Feb 1564 in Pisa (now in Italy) Died: 8 Jan 1642 in Arcetri (near Florence) (now in Italy)

Click the picture above to see eight larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Galileo Galilei's father, Vincenzo Galilei (c.1520 - 1591), who described himself as a nobleman of Florence, was a professional musician. He carried out experiments on strings to support his musical theories. Galileo studied medicine at the university of Pisa, but his real interests were always in mathematics and natural philosophy. He is chiefly remembered for his work on free fall, his use of the telescope and his employment of experimentation. After a spell teaching mathematics, first privately in Florence and then at the university of Pisa, in 1592 Galileo was appointed professor of mathematics at the university of Padua (the university of the Republic of Venice). There his duties were mainly to teach Euclid's geometry and standard (geocentric) astronomy to medical students, who would need to know some astronomy in order to make use of astrology in their medical practice. However, Galileo apparently discussed more unconventional forms of astronomy and natural philosophy in a public lecture he gave in connection with the appearance of a New Star (now known as 'Kepler's supernova') in 1604. In a personal letter written to Kepler (1571 - 1630) in 1598, Galileo had stated that he was a Copernican (believer in the theories of Copernicus). No public sign of this belief was to appear until many years later. In the summer of 1609, Galileo heard about a spyglass that a Dutchman had shown in Venice. From these reports, and using his own technical skills as a mathematician and as a workman, Galileo made a series of telescopes whose optical performance was much better than that of the Dutch instrument. The astronomical discoveries he made with his telescopes were described in a short book called Message from the stars (Sidereus Nuncius) published in Venice in May 1610. It caused a sensation. Galileo claimed to have seen mountains on the Moon, to have proved the Milky Way was made up of tiny stars, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galileo.html (1 of 4) [2/16/2002 11:11:21 PM]

Galileo

and to have seen four small bodies orbiting Jupiter. These last, with an eye on getting a job in Florence, he promptly named 'the Medicean stars'. It worked. Soon afterwards, Galileo became 'Mathematician and [Natural] Philosopher' to the Grand Duke of Tuscany. In Florence he continued his work on motion and on mechanics, and began to get involved in disputes about Copernicanism. In 1613 he discovered that, when seen in the telescope, the planet Venus showed phases like those of the Moon, and therefore must orbit the Sun not the Earth. This did not enable one to decide between the Copernican system, in which everything goes round the Sun, and the Tychonic (Tycho Brahe) one in which everything but the Earth (and Moon) goes round the Sun which in turn goes round the Earth. Most astronomers of the time in fact favoured the Tychonic system. However, Galileo showed a marked tendency to use all his discoveries as evidence for Copernicanism, and to do so with great verbal as well as mathematical skill. He seems to have made a lot of enemies by making his opponents look fools. Moreover, not all of them actually were fools. There eventually followed some expression of interest by the Inquisition. Prima facie, Copernicanism was in contradiction with Scripture, and in 1616 Galileo was given some kind of secret, but official, warning that he was not to defend Copernicanism. Just what was said on this occasion was to become a subject for dispute when Galileo was accused of departing from this undertaking in his Dialogue concerning the two greatest world systems, published in Florence in 1632. Galileo, who was not in the best of health, was summoned to Rome, found to be vehemently suspected of heresy, and eventually condemned to house arrest, for life, at his villa at Arcetri (above Florence). He was also forbidden to publish. By the standards of the time he had got off rather lightly. Galileo's sight was failing, but he had devoted pupils and amanuenses, and he found it possible to write up his studies on motion and the strength of materials. The book, Discourses on two new sciences, was smuggled out of Italy and published in Leiden (in the Netherlands) in 1638. Galileo wrote most of his later works in the vernacular, probably to distance himself from the conventional learning of university teachers. However, his books were translated into Latin for the international market, and they proved to be immensely influential. Article by: J. V. Field, London, Click on this link to see a list of the Glossary entries for this page List of References (174 books/articles)

Some Quotations (16)

A Poster of Galileo Galilei

Mathematicians born in the same country

Some pages from publications

The title page from Discorsi (1638) and another page and yet another page.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galileo.html (2 of 4) [2/16/2002 11:11:21 PM]

Galileo

Cross-references to History Topics

1. An Overview of the History of Mathematics 2. The mathematical discovery of planets Cosmology 3. Longitude and the Académie Royale 4. Thomas Harriot's manuscripts 5. General relativity 6. An overview of the history of mathematics

Cross-references to Famous Curves

1. cycloid 2. parabola 3. catenary

Other references in MacTutor

Chronology: 1600 to 1625

Honours awarded to Galileo Galilei (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Galilaei and Rima Galilaei

Paris street names

Rue Galilée (= Galileo) (16th Arrondissement)

Other Web sites

1. Science Museum, Florence 2. The Galileo Project 3. The Galileo Project. 4. CUNY 5. Karen H Parshall (Examples of Galileo's reasoning) 6. The Catholic Encyclopedia 7. Internet Encyclopedia of Philosophy 8. West Chester University 9. High Altitude Observatory 10. Linda Hall Library (Drawings of the Moon) 11. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galileo.html (3 of 4) [2/16/2002 11:11:21 PM]

Search Form

Galileo

Glossary index

Quotations index

Mathematicians of the day JOC/EFR August 1995

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Galileo.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galileo.html (4 of 4) [2/16/2002 11:11:21 PM]

Gallarati

Dionisio Gallarati Born: 8 May 1923 in Savona, Italy

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dionisio Gallarati entered the University of Pisa in 1941. However, this was the time of World War II and he studies were interrupted for two years because of the war. He recommenced his studies after the forced break and received his first degree from the University of Geneva. Gallarati undertook research at the l'Instituto Nazionale di Alta Matematica (The National Institute of Higher Mathematics) in Rome. There he was taught by most of the stars of Italian mathematics of this period: Giacomo Albanese, Leonard Roth (teaching in Rome as a visiting professor from Imperial College, London), L Tonelli, E G Togliatti, Beniamino Segre, and Francesco Severi. Gallarati was appointed to the University of Geneva in 1947 and he remained there until he retired in 1987. In [1], 33 of the 64 papers which Gallarati published between 1951 and 1996 are reproduced. These reflect the major area of his research which was mostly in algebraic geometry. Eight of the papers reproduced in [1] study surfaces in P3 with many isolated singularities. This was one of Gallarati's most important mathematical contributions. It is worth noting that some of lower bounds obtained by Gallarati for the maximal number of nodes of surfaces of degree n were not improved until comparatively recently and for large values of n the exact maximal number of nodes of surfaces of degree n is still unknown despite much recent work on this topic. Three papers given in [1] relate to Gallarati's contributions to Grassmannian geometry. His work on this topic extended in several ways the bound obtained by Beniamino Segre for the number of linearly independent linear complexes containing the curve in the Grassmannian corresponding to the tangent lines of a nondegenerate projective curve. Gallarati extended the results to the case of tangent spaces of varieties of arbitrary dimensions, to arbitrary curves in Grassmannians corresponding to nondegenerate scrolls, and to complexes of higher degree. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gallarati.html (1 of 2) [2/16/2002 11:11:23 PM]

Gallarati

Gallarati studied properties of varieties whose tangent spaces meet certain linear subspaces along spaces of dimension higher than expected. He gave an elegant characterization of the Veronese and Segre varieties in terms of their tangential properties. Another particularly notable research contribution was made by Gallarati in his work on multiple contacts of surfaces along a curve. He constructed counterexamples to conjectures of Babbage and Hochster and gave negative answers to questions posed by Mumford and Hartshorne. Other work by Gallarati included a classification of Fano varieties of the second kind, a study of irregularity of double spaces, and the finding of bounds for the class of a surface with ordinary singularities in terms of its degree. Zak, reviewing [1], remarks that republishing Gallarati's papers:... introduces the modern reader to the almost forgotten world of projective algebraic geometry, Italian style. Gallarati taught algebraic geometry for many years and notes from these courses still exist [1]:These notes reflect the unique style of their author and are rich in original observations. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gallarati.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gallarati.html (2 of 2) [2/16/2002 11:11:23 PM]

Galois

Evariste Galois Born: 25 Oct 1811 in Bourg La Reine (near Paris), France Died: 31 May 1832 in Paris, France

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Evariste Galois' father Nicholas Gabriel Galois and his mother Adelaide Marie Demante were both intelligent and well educated in philosophy, classical literature and religion. However there is no sign of any mathematical ability in any of Galois' family. His mother served as Galois' sole teacher until he was 12 years old. She taught him Greek, Latin and religion where she imparted her own scepticism to her son. Galois' father was an important man in the community and in 1815 he was elected mayor of Bourg-la-Reine. You can see a map of Paris in the 19th Century, showing Bourg-la-Reine. The starting point of the historical events which were to play a major role in Galois' life is surely the storming of the Bastille on 14 July 1789. From this point the monarchy of Louis 16th was in major difficulties as the majority of Frenchmen composed their differences and united behind an attempt to destroy the privileged establishment of the church and the state. Despite attempts at compromise Louis 16th was tried after attempting to flee the country. Following the execution of the King on 21 January 1793 there followed a reign of terror with many political trials. By the end of 1793 there were 4595 political prisoners held in Paris. However France began to have better times as their armies, under the command of Napoleon Bonaparte, won victory after victory. Napoleon became 1st Consul in 1800 and then Emperor in 1804. The French armies continued a conquest of Europe while Napoleon's power became more and more secure. In 1811 Napoleon was at the height of his power. By 1815 Napoleon's rule was over. The failed Russian campaign of 1812 was followed by defeats, the Allies entering Paris on 31 March 1814. Napoleon abdicated on 6 April and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galois.html (1 of 6) [2/16/2002 11:11:25 PM]

Galois

Louis XVIII was installed as King by the Allies. The year 1815 saw the famous one hundred days. Napoleon entered Paris on March 20, was defeated at Waterloo on 18 June and abdicated for the second time on 22 June. Louis XVIII was reinstated as King but died in September 1824, Charles X becoming the new King. Galois was by this time at school. He had enrolled at the Lycée of Louis-le-Grand as a boarder in the 4 th class on 6 October 1823. Even during his first term there was a minor rebellion and 40 pupils were expelled from the school. Galois was not involved and during 1824-25 his school record is good and he received several prizes. However in 1826 Galois was asked to repeat the year because his work in rhetoric was not up to the required standard. February 1827 was a turning point in Galois' life. He enrolled in his first mathematics class, the class of M. Vernier. He quickly became absorbed in mathematics and his director of studies wrote It is the passion for mathematics which dominates him, I think it would be best for him if his parents would allow him to study nothing but this, he is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments. Galois' school reports began to describe him as singular, bizarre, original and closed. It is interesting that perhaps the most original mathematician who ever lived should be criticised for being original. M. Vernier reported however Intelligence, marked progress but not enough method. In 1828 Galois took the examination of the Ecole Polytechnique but failed. It was the leading University of Paris and Galois must have wished to enter it for academic reasons. However, he also wished to enter this school because of the strong political movements that existed among its students, since Galois followed his parents example in being an ardent republican. Back at Louis-le-Grand, Galois enrolled in the mathematics class of Louis Richard. However he worked more and more on his own researches and less and less on his schoolwork. He studied Legendre's Géométrie and the treatises of Lagrange. As Richard was to report This student works only in the highest realms of mathematics. In April 1829 Galois had his first mathematics paper published on continued fractions in the Annales de mathématiques. On 25 May and 1 June he submitted articles on the algebraic solution of equations to the Académie des Sciences. Cauchy was appointed as referee of Galois' paper. Tragedy was to strike Galois for on 2 July 1829 his father committed suicide. The priest of Bourg-la-Reine forged Mayor Galois' name on malicious forged epigrams directed at Galois' own relatives. Galois' father was a good natured man and the scandal that ensued was more than he could stand. He hanged himself in his Paris apartment only a few steps from Louis-le-Grand where his son was studying. Galois was deeply affected by his father's death and it greatly influenced the direction his life was to take. A few weeks after his father's death, Galois presented himself for examination for entry to the Ecole Polytechnique for the second time. For the second time he failed, perhaps partly because he took it under the worst possible circumstances so soon after his father's death, partly because he was never good at communicating his deep mathematical ideas. Galois therefore resigned himself to enter the Ecole Normale, which was an annex to Louis-le-Grand, and to do so he had to take his Baccalaureate

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galois.html (2 of 6) [2/16/2002 11:11:25 PM]

Galois

examinations, something he could have avoided by entering the Ecole Polytechnique. He passed, receiving his degree on 29 December 1829. His examiner in mathematics reported: This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research. His literature examiner reported: This is the only student who has answered me poorly, he knows absolutely nothing. I was told that this student has an extraordinary capacity for mathematics. This astonishes me greatly, for, after his examination, I believed him to have but little intelligence. Galois sent Cauchy further work on the theory of equations, but then learned from Bulletin de Férussac of a posthumous article by Abel which overlapped with a part of his work. Galois then took Cauchy's advice and submitted a new article On the condition that an equation be soluble by radicals in February 1830. The paper was sent to Fourier, the secretary of the Academy, to be considered for the Grand Prize in mathematics. Fourier died in April 1830 and Galois' paper was never subsequently found and so never considered for the prize. Galois, after reading Abel and Jacobi's work, worked on the theory of elliptic functions and abelian integrals. With support from Jacques Sturm, he published three papers in Bulletin de Férussac in April 1830. However, he learnt in June that the prize of the Academy would be awarded the Prize jointly to Abel (posthumously) and to Jacobi, his own work never having been considered. July 1830 saw a revolution. Charles 10th fled France. There was rioting in the streets of Paris and the director of Ecole Normale, M. Guigniault, locked the students in to avoid them taking part. Galois tried to scale the wall to join the rioting but failed. In December 1830 M. Guigniault wrote newspaper articles attacking the students and Galois wrote a reply in the Gazette des Ecoles, attacking M. Guigniault for his actions in locking the students into the school. For this letter Galois was expelled and he joined the Artillery of the National Guard, a Republican branch of the militia. On 31 December 1830 the Artillery of the National Guard was abolished by Royal Decree since the new King Louis-Phillipe felt it was a threat to the throne. Two minor publications, an abstract in Annales de Gergonne (December 1830) and a letter on the teaching of science in the Gazette des Ecoles ( 2 January 1831) were the last publications during his life. In January 1831 Galois attempted to return to mathematics. He organised some mathematics classes in higher algebra which attracted 40 students to the first meeting but after that the numbers quickly fell off. Galois was invited by Poisson to submit a third version of his memoir on equation to the Academy and he did so on 17 January. On 18 April Sophie Germain wrote a letter to her friend the mathematician Libri which describes Galois' situation. .. the death of M. Fourier, have been too much for this student Galois who, in spite of his impertinence, showed signs of a clever disposition. All this has done so much that he has been expelled form Ecole Normale. He is without money... . They say he will go completely mad. I fear this is true. Late in 1830 19 officers from the Artillery of the National Guard were arrested and charged with

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galois.html (3 of 6) [2/16/2002 11:11:25 PM]

Galois

conspiracy to overthrow the government. They were acquitted and on 9 May 1831 200 republicans gathered for a dinner to celebrate the acquittal. During the dinner Galois raised his glass and with an open dagger in his hand appeared to make threats against the King, Louis-Phillipe. After the dinner Galois was arrested and held in Sainte-Pélagie prison. At his trial on 15 June his defence lawyer claimed that Galois had said To Louis-Phillipe, if he betrays but the last words had been drowned by the noise. Galois, rather surprisingly since he essentially repeated the threat from the dock, was acquitted. The 14th July was Bastille Day and Galois was arrested again. He was wearing the uniform of the Artillery of the National Guard, which was illegal. He was also carrying a loaded rifle, several pistols and a dagger. Galois was sent back to Sainte-Pélagie prison. While in prison he received a rejection of his memoir. Poisson had reported that:His argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigour. He did, however, encourage Galois to publish a more complete account of his work. While in Sainte-Pélagie prison Galois attempted to commit suicide by stabbing himself with a dagger but the other prisoners prevented him. While drunk in prison he poured out his soul Do you know what I lack my friend? I confide it only to you: it is someone whom I can love and love only in spirit. I have lost my father and no one has ever replaced him, do you hear me...? In March 1832 a cholera epidemic swept Paris and prisoners, including Galois, were transferred to the pension Sieur Faultrier. There he apparently fell in love with Stephanie-Felice du Motel, the daughter of the resident physician. After he was released on 29 April Galois exchanged letters with Stephanie, and it is clear that she tried to distance herself from the affair. The name Stephanie appears several times as a marginal note in one of Galois' manuscripts. Galois fought a duel with Perscheux d'Herbinville on 30 May, the reason for the duel not being clear but certainly linked with Stephanie. You can see a note in the margin of the manuscript that Galois wrote the night before the duel. The note reads There is something to complete in this demonstration. I do not have the time. (Author's note). It is this which has led to the legend that he spent his last night writing out all he knew about group theory. This story appears to have been exaggerated. Galois was wounded in the duel and was abandoned by d'Herbinville and his own seconds and found by a peasant. He died in Cochin hospital on 31 May and his funeral was held on 2 June. It was the focus for a Republican rally and riots followed which lasted for several days. Galois' brother and his friend Chevalier copied his mathematical papers and sent them to Gauss, Jacobi and others. It had been Galois' wish that Jacobi and Gauss should give their opinions on his work. No

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galois.html (4 of 6) [2/16/2002 11:11:25 PM]

Galois

record exists of any comment these men made. However the papers reached Liouville who, in September 1843, announced to the Academy that he had found in Galois' papers a concise solution ...as correct as it is deep of this lovely problem: Given an irreducible equation of prime degree, decide whether or not it is soluble by radicals. Liouville published these papers of Galois in his Journal in 1846. The theory that Galois outlined in these papers is now called Galois theory. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (19 books/articles)

Some Quotations (2)

A Poster of Evariste Galois

Mathematicians born in the same country

Some pages from publications

A page from Galois' Mémoire sur les conditions de resolubilité des equationspar radicaus (published in his collected works in 1897)

Cross-references to History Topics

1. The development of group theory 2. An overview of the history of mathematics 3. Abstract linear spaces

Other references in MacTutor

Chronology: 1820 to 1830

Honours awarded to Evariste Galois (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Galois

Paris street names

Rue Evariste Galois (20th Arrondissement)

Other Web sites

1. Kevin Brown (Galois' essay on his theory) 2. Geometry.net (Links to other Galois references) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galois.html (5 of 6) [2/16/2002 11:11:25 PM]

Galois

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Galois.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galois.html (6 of 6) [2/16/2002 11:11:25 PM]

Galton

Francis Galton Born: 16 Feb 1822 in Sparkbrook (near Birmingham), England Died: 17 Jan 1911 in Grayshott House, Haslemere, Surrey, England

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

An explorer and anthropologist, Francis Galton is known for his pioneering studies of human intelligence. He devoted the latter part of his life to eugenics, i.e. improving the physical and mental makeup of the human species by selected parenthood. Although weak in mathematics his ideas strongly influenced the development of statistics particularly his proof that a normal mixture of normal distributions is itself normal. Another of his major findings was reversion. This was his formulation of regression and its link to the bivariate normal distribution. He also made important contributions to the fields of meteorology, anthropometry, and physical anthropology. Galton was an indefatigable explorer and an investigator of human intelligence. Galton, the cousin of Charles Darwin, was convinced that pre-eminence in various fields was due almost entirely to hereditary factors. He opposed those who claimed intelligence or character were determined by environmental factors. He inquired into racial differences, something almost unacceptable today, and was one of the first to employ questionnaire and survey methods, which he used to investigate mental imagery in different groups of people. His work led him to advocate breeding restrictions. Galton was knighted in 1909. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galton.html (1 of 2) [2/16/2002 11:11:27 PM]

Galton

List of References (12 books/articles)

Some Quotations (3)

A Poster of Francis Galton

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1880 to 1890

Honours awarded to Francis Galton (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1860

Royal Society Copley Medal

Awarded 1910

Other Web sites

1. University of Minnesota 2. James Cook University, Australia 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Galton.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Galton.html (2 of 2) [2/16/2002 11:11:27 PM]

Gassendi

Pierre Gassendi Born: 22 Jan 1592 in Champtercier, Provence, France Died: 24 Oct 1655 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Gassendi attended school at Digne from 1599 to 1606 then continued his education at home supervised by his uncle. Then, in 1608, he entered the University of Aix where he studied philosophy for two years then theology for a further two years. Gassendi was Principal at the College of Digne from 1612 to 1614, then he received a doctorate in theology from Avignon and was ordained in 1615, one year later. He had already been appointed canon at a church in Digne in 1614. He held this post until 1634 when he was elevated to dean. In addition to these positions in the church, Gassendi was appointed professor of philosophy at the University of Aix in 1617. However this position only lasted until 1623 when the Jesuit order took control of the university of Aix and he was forced to leave. He did not hold any further academic posts until 1645 when he was appointed professor of mathematics at the Collège Royale in Paris. Gassendi first met Mersenne in 1624 when he visited Paris. Mersenne tried to persuade him to give up mathematics and theology in favour of philosophy. Gassendi rejected Descartes' philosophy, emphasising the inductive method. He believed in atomism and defended a mechanistic explanation of nature. Kepler had predicted a transit of Mercury would occur in 1631 and Gassendi was the first to observe such a transit. He wrote on astronomy, his own astronomical observations and on falling bodies. Of course Gassendi published on philosophy. His first work was Exercitationes paradoxicae (1624),

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gassendi.html (1 of 2) [2/16/2002 11:11:29 PM]

Gassendi

basically his lecture course at Aix written up for publication. In 1649 he published Animadversiones containing work on Epicurus. His Philosophical Treatise was published 3 years after his death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) A Poster of Pierre Gassendi

Mathematicians born in the same country

Honours awarded to Pierre Gassendi (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Gassendi and Rimae Gassendi

Paris street names

Rue Gassendi (14th Arrondissement)

Other Web sites

1. The Galileo Project 2. The Catholic Encyclopedia 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gassendi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gassendi.html (2 of 2) [2/16/2002 11:11:29 PM]

Gauss

Johann Carl Friedrich Gauss Born: 30 April 1777 in Brunswick, Duchy of Brunswick (now Germany) Died: 23 Feb 1855 in Göttingen, Hanover (now Germany)

Click the picture above to see eleven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

At the age of seven, Carl Friedrich Gauss started elementary school, and his potential was noticed almost immediately. His teacher, Büttner, and his assistant, Martin Bartels, were amazed when Gauss summed the integers from 1 to 100 instantly by spotting that the sum was 50 pairs of numbers each pair summing to 101. In 1788 Gauss began his education at the Gymnasium with the help of Büttner and Bartels, where he learnt High German and Latin. After receiving a stipend from the Duke of Brunswick- Wolfenbüttel, Gauss entered Brunswick Collegium Carolinum in 1792. At the academy Gauss independently discovered Bode's law, the binomial theorem and the arithmetic- geometric mean, as well as the law of quadratic reciprocity and the prime number theorem. In 1795 Gauss left Brunswick to study at Göttingen University. Gauss's teacher there was Kaestner, whom Gauss often ridiculed. His only known friend amongst the students was Farkas Bolyai. They met in 1799 and corresponded with each other for many years. Gauss left Göttingen in 1798 without a diploma, but by this time he had made one of his most important discoveries - the construction of a regular 17-gon by ruler and compasses This was the most major advance in this field since the time of Greek mathematics and was published as Section VII of Gauss's famous work, Disquisitiones Arithmeticae. Gauss returned to Brunswick where he received a degree in 1799. After the Duke of Brunswick had http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html (1 of 7) [2/16/2002 11:11:31 PM]

Gauss

agreed to continue Gauss's stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt. He already knew Pfaff, who was chosen to be his advisor. Gauss's dissertation was a discussion of the fundamental theorem of algebra. With his stipend to support him, Gauss did not need to find a job so devoted himself to research. He published the book Disquisitiones Arithmeticae in the summer of 1801. There were seven sections, all but the last section, referred to above, being devoted to number theory. In June 1801, Zach, an astronomer whom Gauss had come to know two or three years previously, published the orbital positions of Ceres, a new "small planet" which was discovered by G Piazzi, an Italian astronomer on 1 January, 1801. Unfortunately, Piazzi had only been able to observe 9 degrees of its orbit before it disappeared behind the Sun. Zach published several predictions of its position, including one by Gauss which differed greatly from the others. When Ceres was rediscovered by Zach on 7 December 1801 it was almost exactly where Gauss had predicted. Although he did not disclose his methods at the time, Gauss had used his least squares approximation method. In June 1802 Gauss visited Olbers who had discovered Pallas in March of that year and Gauss investigated its orbit. Olbers requested that Gauss be made director of the proposed new observatory in Göttingen, but no action was taken. Gauss began corresponding with Bessel, whom he did not meet until 1825, and with Sophie Germain. Gauss married Johanna Ostoff on 9 October, 1805. Despite having a happy personal life for the first time, his benefactor, the Duke of Brunswick, was killed fighting for the Prussian army. In 1807 Gauss left Brunswick to take up the position of director of the Göttingen observatory. Gauss arrived in Göttingen in late 1807. In 1808 his father died, and a year later Gauss's wife Johanna died after giving birth to their second son, who was to die soon after her. Gauss was shattered and wrote to Olbers asking him give him a home for a few weeks, to gather new strength in the arms of your friendship - strength for a life which is only valuable because it belongs to my three small children. Gauss was married for a second time the next year, to Minna the best friend of Johanna, and although they had three children, this marriage seemed to be one of convenience for Gauss. Gauss's work never seemed to suffer from his personal tragedy. He published his second book, Theoria motus corporum coelestium in sectionibus conicis Solem ambientium, in 1809, a major two volume treatise on the motion of celestial bodies. In the first volume he discussed differential equations, conic sections and elliptic orbits, while in the second volume, the main part of the work, he showed how to estimate and then to refine the estimation of a planet's orbit. Gauss's contributions to theoretical astronomy stopped after 1817, although he went on making observations until the age of 70. Much of Gauss's time was spent on a new observatory, completed in 1816, but he still found the time to work on other subjects. His publications during this time include Disquisitiones generales circa seriem infinitam, a rigorous treatment of series and an introduction of the hypergeometric function, Methodus nova integralium valores per approximationem inveniendi, a practical essay on approximate integration, Bestimmung der Genauigkeit der Beobachtungen, a discussion of statistical estimators, and Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractata. The latter work was inspired by geodesic problems and was principally concerned with potential theory. In fact, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html (2 of 7) [2/16/2002 11:11:31 PM]

Gauss

Gauss found himself more and more interested in geodesy in the 1820's. Gauss had been asked in 1818 to carry out a geodesic survey of the state of Hanover to link up with the existing Danish grid. Gauss was pleased to accept and took personal charge of the survey, making measurements during the day and reducing them at night, using his extraordinary mental capacity for calculations. He regularly wrote to Schumacher, Olbers and Bessel, reporting on his progress and discussing problems. Because of the survey, Gauss invented the heliotrope which worked by reflecting the Sun's rays using a design of mirrors and a small telescope. However, inaccurate base lines were used for the survey and an unsatisfactory network of triangles. Gauss often wondered if he would have been better advised to have pursued some other occupation but he published over 70 papers between 1820 and 1830. In 1822 Gauss won the Copenhagen University Prize with Theoria attractionis... together with the idea of mapping one surface onto another so that the two are similar in their smallest parts. This paper was published in 1825 and led to the much later publication of Untersuchungen über Gegenstände der Höheren Geodäsie (1843 and 1846). The paper Theoria combinationis observationum erroribus minimis obnoxiae (1823), with its supplement (1828), was devoted to mathematical statistics, in particular to the least squares method. From the early 1800's Gauss had an interest in the question of the possible existence of a non-Euclidean geometry. He discussed this topic at length with Farkas Bolyai and in his correspondence with Gerling and Schumacher. In a book review in 1816 he discussed proofs which deduced the axiom of parallels from the other Euclidean axioms, suggesting that he believed in the existence of non-Euclidean geometry, although he was rather vague. Gauss confided in Schumacher, telling him that he believed his reputation would suffer if he admitted in public that he believed in the existence of such a geometry. In 1831 Farkas Bolyai sent to Gauss his son János Bolyai's work on the subject. Gauss replied to praise it would mean to praise myself . Again, a decade later, when he was informed of Lobachevsky's work on the subject, he praised its "genuinely geometric" character, while in a letter to Schumacher in 1846, states that he had the same convictions for 54 years indicating that he had known of the existence of a non-Euclidean geometry since he was 15 years of age (this seems unlikely). Gauss had a major interest in differential geometry, and published many papers on the subject. Disquisitiones generales circa superficies curva (1828) was his most renowned work in this field. In fact, this paper rose from his geodesic interests, but it contained such geometrical ideas as Gaussian curvature. The paper also includes Gauss's famous theorema egregrium: If an area in E3 can be developed (i.e. mapped isometrically) into another area of E3, the values of the Gaussian curvatures are identical in corresponding points. The period 1817-1832 was a particularly distressing time for Gauss. He took in his sick mother in 1817, who stayed until her death in 1839, while he was arguing with his wife and her family about whether they should go to Berlin. He had been offered a position at Berlin University and Minna and her family were keen to move there. Gauss, however, never liked change and decided to stay in Göttingen. In 1831

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html (3 of 7) [2/16/2002 11:11:31 PM]

Gauss

Gauss's second wife died after a long illness. In 1831, Wilhelm Weber arrived in Göttingen as physics professor filling Tobias Mayer's chair. Gauss had known Weber since 1828 and supported his appointment. Gauss had worked on physics before 1831, publishing Über ein neues allgemeines Grundgesetz der Mechanik, which contained the principle of least constraint, and Principia generalia theoriae figurae fluidorum in statu aequilibrii which discussed forces of attraction. These papers were based on Gauss's potential theory, which proved of great importance in his work on physics. He later came to believe his potential theory and his method of least squares provided vital links between science and nature. In 1832, Gauss and Weber began investigating the theory of terrestrial magnetism after Alexander von Humboldt attempted to obtain Gauss's assistance in making a grid of magnetic observation points around the Earth. Gauss was excited by this prospect and by 1840 he had written three important papers on the subject: Intensitas vis magneticae terrestris ad mensuram absolutam revocata (1832), Allgemeine Theorie des Erdmagnetismus (1839) and Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und Abstossungskräfte (1840). These papers all dealt with the current theories on terrestrial magnetism, including Poisson's ideas, absolute measure for magnetic force and an empirical definition of terrestrial magnetism. Dirichlet's principle was mentioned without proof. Allgemeine Theorie... showed that there can only be two poles in the globe and went on to prove an important theorem, which concerned the determination of the intensity of the horizontal component of the magnetic force along with the angle of inclination. Gauss used the Laplace equation to aid him with his calculations, and ended up specifying a location for the magnetic South pole. Humboldt had devised a calendar for observations of magnetic declination. However, once Gauss's new magnetic observatory (completed in 1833 - free of all magnetic metals) had been built, he proceeded to alter many of Humboldt's procedures, not pleasing Humboldt greatly. However, Gauss's changes obtained more accurate results with less effort. Gauss and Weber achieved much in their six years together. They discovered Kirchhoff's laws, as well as building a primitive telegraph device which could send messages over a distance of 5000 ft. However, this was just an enjoyable pastime for Gauss. He was more interested in the task of establishing a world-wide net of magnetic observation points. This occupation produced many concrete results. The Magnetischer Verein and its journal were founded, and the atlas of geomagnetism was published, while Gauss and Weber's own journal in which their results were published ran from 1836 to 1841. In 1837, Weber was forced to leave Göttingen when he became involved in a political dispute and, from this time, Gauss's activity gradually decreased. He still produced letters in response to fellow scientists' discoveries usually remarking that he had known the methods for years but had never felt the need to publish. Sometimes he seemed extremely pleased with advances made by other mathematicians, particularly that of Eisenstein and of Lobachevsky. Gauss spent the years from 1845 to 1851 updating the Göttingen University widow's fund. This work gave him practical experience in financial matters, and he went on to make his fortune through shrewd investments in bonds issued by private companies.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html (4 of 7) [2/16/2002 11:11:31 PM]

Gauss

Two of Gauss's last doctoral students were Moritz Cantor and Dedekind. Dedekind wrote a fine description of his supervisor ... usually he sat in a comfortable attitude, looking down, slightly stooped, with hands folded above his lap. He spoke quite freely, very clearly, simply and plainly: but when he wanted to emphasise a new viewpoint ... then he lifted his head, turned to one of those sitting next to him, and gazed at him with his beautiful, penetrating blue eyes during the emphatic speech. ... If he proceeded from an explanation of principles to the development of mathematical formulas, then he got up, and in a stately very upright posture he wrote on a blackboard beside him in his peculiarly beautiful handwriting: he always succeeded through economy and deliberate arrangement in making do with a rather small space. For numerical examples, on whose careful completion he placed special value, he brought along the requisite data on little slips of paper. Gauss presented his golden jubilee lecture in 1849, fifty years after his diploma had been granted by Hemstedt University. It was appropriately a variation on his dissertation of 1799. From the mathematical community only Jacobi and Dirichlet were present, but Gauss received many messages and honours. From 1850 onwards Gauss's work was again of nearly all of a practical nature although he did approve Riemann's doctoral thesis and heard his probationary lecture. His last known scientific exchange was with Gerling. He discussed a modified Foucalt pendulum in 1854. He was also able to attend the opening of the new railway link between Hanover and Göttingen, but this proved to be his last outing. His health deteriorated slowly, and Gauss died in his sleep early in the morning of 23 February, 1855. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (67 books/articles)

Some Quotations (27)

A Poster of Carl Friedrich Gauss

Mathematicians born in the same country

Some pages from publications

A letter from Gauss to Taurinus discussing the possibility of non-Euclidean geometry. An extract from Theoria residuorum biquadraticorum (1828-32)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html (5 of 7) [2/16/2002 11:11:31 PM]

Gauss

Cross-references to History Topics

1. A comment from Thomas Hirst's diary 2. The development of group theory 3. Non-Euclidean geometry 4. Topology enters mathematics 5. The fundamental theorem of algebra 6. Orbits and gravitation 7. Memory, mental arithmetic and mathematics 8. Doubling the cube 9. Trisecting an angle 10. General relativity 11. An overview of the history of mathematics 12. Prime numbers 13. Matrices and determinants 14. An overview of Indian mathematics

Other references in MacTutor

1. 2. 3. 4.

Prime Number Theorem Gauss's estimate for the density of primes Comparison with Legendre's estimate Chronology: 1780 to 1800

5. Chronology: 1800 to 1810 6. Chronology: 1820 to 1830 7. Chronology: 1840 to 1850 Honours awarded to Carl Friedrich Gauss (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1804

Royal Society Copley Medal

Awarded 1838

Lunar features

Crater Gauss

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html (6 of 7) [2/16/2002 11:11:31 PM]

Gauss

Other Web sites

1. Nelly Cung 2. The Prime Pages (The Prime Number Theorem) 3. West Chester University 4. Kevin Brown (Constructing the 17-gon) 5. Kevin Brown (Geodesy) 6. Sonoma 7. S D Chambless (An obituary of Gauss's son) and an account of his life in the USA 8. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gauss.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gauss.html (7 of 7) [2/16/2002 11:11:31 PM]

Gegenbauer

Leopold Bernhard Gegenbauer Born: 2 Feb 1849 in Asperhofen (E of Herzogenburg), Austria Died: 3 June 1903 in Vienna, Austria

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leopold Gegenbauer studied at the University of Vienna from 1869 until 1873. He then went to Berlin where he studied from 1873 to 1875 working under Weierstrass and Kronecker. After graduating from Berlin, Gegenbauer was appointed to a position at the University of Czernowitz (then in the Austrian Empire but now Chernovtsy, Ukraine) in 1875. He remained in Czernowitz for three years before moving to the University of Innsbruck where he worked with Stolz. After three years teaching in Innsbruck Gegenbauer was appointed full professor in 1881, then he was appointed full professor at the University of Vienna in 1893. He remained there until his death. Gegenbauer had many mathematical interests but was chiefly an algebraist. He is remembered for the Gegenbauer polynomials. Article by: J J O'Connor and E F Robertson

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gegenbauer.html (1 of 2) [2/16/2002 11:11:33 PM]

Gegenbauer

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gegenbauer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gegenbauer.html (2 of 2) [2/16/2002 11:11:33 PM]

Geiringer

Hilda Geiringer von Mises Born: 28 Sept 1893 in Vienna, Austria Died: 22 March 1973 in Santa Barbara, California, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hilda Geiringer's father Ludwig Geiringer was born in Hungary. Her mother was Martha Wertheimer was from Vienna and Ludwig and Martha had married while he was working in Vienna as a textile manufacturer. It was a Jewish family which would later have a significant effect on Hilda's life. At Gymnasium Hilda showed great mathematical ability and her parents supported her financially so that she could study mathematics at the University of Vienna. After receiving her first degree, Geiringer continued her study of mathematics at Vienna, working under Wirtinger for her doctorate. This was awarded in 1917 for a thesis on Fourier series in two variables. She spent the following two years as Leon Lichtenstein's assistant editing the mathematics review journal Jahrbuch über die Fortschritte der Mathematik. In 1921 Geiringer moved to Berlin where she was employed as an assistant to von Mises in the Institute of Applied Mathematics. In this same year she married Felix Pollaczek who, like Geiringer, had been born in Vienna into a Jewish family but had studied in Berlin. He obtained his doctorate in 1922 working under Schur and went on to work for the Reichspost in Berlin applying mathematical methods to telephone connections. Hilda and Felix had a child, Magda Pollaczek, in 1922 but their marriage broke up. After the divorce Geiringer continued working for von Mises and at the same time brought up her child on her own. Although trained as a pure mathematician, Geiringer moved towards applied mathematics to fit in with the work being undertaken at the Institute of Applied Mathematics. Her work at this time was on statistics, in particular probability theory, and also on the mathematical theory of plasticity. She

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Geiringer.html (1 of 4) [2/16/2002 11:11:35 PM]

Geiringer

submitted a thesis for her habilitation to the University of Berlin but it was not immediately accepted. Siegmund- Schultze writes in [3]:The controversy surrounding Hilda Geiringer's application for Habilitation at the University of Berlin (1925 - 1927) sheds some light on the struggle of 'applied mathematics' for cognitive and institutional independence. The controversy and Geiringer's unpublished reminiscences reveal the decisive influence of Richard von Mises ... on both her career and the course of applied mathematics at the University of Berlin. ... The debate over Geiringer's theses for Habilitation opens up a chapter of the history of mathematical statistics, namely, expansions of a discrete distribution with an infinite number of values in a series in successive derivatives of the Poisson distribution with respect to the parameter. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Under this law, Geiringer lost the right to teach at the university in December 1933. In fact she had been proposed for appointment to the position of extraordinary professor in 1933 but the proposal had been put on hold after the Civil Service Law came into effect in April of that year. Geiringer left Germany after she was dismissed from the University of Berlin and, together with her child, she went to Brussels. There she was appointed to the Institute of Mechanics and worked on the theory of vibrations. Now von Mises, though a convert to Catholicism from Judaism, left Germany at the very end of 1933 to take up a newly founded chair of mathematics in Istanbul and, in 1934, Geiringer followed him to Istanbul. There she was appointed as professor of mathematics and Richards writes in [2]:In Turkey, Geiringer was part of a larger German community that was seeking refuge from Hitler's regime. Despite the obvious difficulties associated with this exile - for example, she had to learn Turkish in order to give her lectures - Geiringer continued to pursue her mathematical interests, particularly in plasticity. In 1938 Kemal Atatürk died and those in Turkey who had fled from the Nazis feared that their safe haven would become unsafe. In 1939 von Mises left Turkey for the United States. Geiringer feared that she might not find it so easy to obtain an entrance permit to the United States and she wrote to von Mises from Istanbul:Is there no way to marry pro cura? Here an emigrant who has a resident's permit has married his 'bride' and she was then allowed to come to him straight from Vienna. Her fears of not getting an entrance permit were unfounded, however, and together with her daughter she went to Bryn Mawr College where she was appointed as a lecturer. Again of course, Geiringer had to learn another language in order to teach. She also had to adjust to what she referred to as "the American form of teaching" and in this she was greatly assisted by Anna Wheeler. In addition to her lecturing duties at Bryn Mawr College, Geiringer undertook classified work for the National Research Council as part of the war effort. During 1942 she gave an advanced course in mechanics at Brown University, with the aim of raising the American standards of education to the level that had been attained in Germany. She wrote up her outstanding series of lectures on the geometrical foundations of mechanics and, although they were never properly published, these were widely used in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Geiringer.html (2 of 4) [2/16/2002 11:11:35 PM]

Geiringer

the United States for many years. One has to understand the problems that there were in the United States at this time as they tried, in general very successfully, to integrate many leading scientists fleeing from the Nazis into their system. Neyman, who himself had emigrated to the United States, wrote a report on Geiringer in April 1940, shortly after she arrived from Turkey. This is very fair in explaining where Geiringer fitted into the spectrum of professors of mathematics:Whether she is to be considered outstanding in ability or not depends on the standards of comparison. Among the present day mathematicians there are few whose names will remain in the history of mathematics ... As for the newcomers to this country, I have not the slightest doubt that von Mises is one of the men of such calibre. ... There will perhaps be a dozen or perhaps a score of such persons all over the world. ... and Mrs Geiringer does not belong in this category. But it may be reasonable to take another standard, that of a university professor of probability and statistics, perhaps an author of the now numerous books on statistical methods. In comparison with many of these people Mrs Geiringer is an outstanding person and I think it would be in the interests of American science and instruction to keep her in some university. Geiringer married von Mises in 1943 and the following year she left her lecturing post at Bryn Mawr College to be nearer to him. She accepted a post as professor and chairman of Wheaton College in Norton, Massachusetts. During the week she taught at the College, travelling to Cambridge every weekend to be with von Mises who worked at Harvard at this time. For many reasons this was not a good arrangement. There were only two members of the mathematics faculty at Wheaton College and Geiringer longed for a situation where she was among mathematicians who were carrying out research. She made many applications for other posts but these failed due to fairly open discrimination against women. As Richards writes in [2] one response she received was quite typical:I am sure that our President would not approve of a woman. We have some women on our staff, so it is not merely prejudice against women, yet it is partly that, for we do not want to bring in more if we can get men. For Geiringer who had been so discriminated against in Germany because of her Jewish background, to now be discriminated against because she was a woman must have been a difficult blow. However, she took it all remarkably calmly, believing that if she could do something for future generations of women then she would have achieved something positive. She also never gave up her research while at Wheaton College; in 1953 she wrote:I have to work scientifically, besides my college work. This is a necessity for me; I never stopped it since my student days, it is the deepest need of my life. In 1953 von Mises died and the following year Geiringer, although retaining her job at Wheaton College, began to work at Harvard on editing von Mises works. The year 1956 saw the University of Berlin elect her professor emeritus on full salary. In 1959 she formally retired from Wheaton College and the following year the College honoured her with the award of an honorary Doctorate of Science. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Geiringer.html (3 of 4) [2/16/2002 11:11:35 PM]

Geiringer

List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Geiringer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Geiringer.html (4 of 4) [2/16/2002 11:11:35 PM]

Geiser

Karl Friedrich Geiser Born: 26 Feb 1843 in Langenthal, Bern, Switzerland Died: 7 May 1934 in Küsnacht, Zurich, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karl Geiser's father was a butcher but his great uncle was Steiner which certainly helped him in his career, particularly when he went to the University of Berlin. He went to study at Berlin after first studying at the Polytechnikum in Zurich and, from 1863 to 1873, he taught at Berlin. In 1873 Geiser was appointed professor at Zurich Polytechnikum. This institution was later called the Eidgenössische Technische Hochschule and Geiser was to have as colleagues there, first Frobenius and then Hurwitz who filled Frobenius's chair. Geiser taught algebraic geometry (his own research topic), differential geometry and invariant theory at Zurich. He published on algebraic geometry and minimal surfaces. One of his most important results explains how the 28 double tangents of the plane quadric are related to the 27 straight lines of the cubic surface. He is also remembered by those working in algebraic geometry for his discovery of an involution, now named after him. However Geiser's most important contribution was not in his original research but rather in his political skills in organising the educational system in Switzerland. In this he was helped by having many contacts with political figures and also important mathematicians in other European countries. Although Geiser was helped in his career by his relationship with Steiner, he repaid the debt by editing Steiner's unpublished lecture notes and treatises. Another important contribution which Geiser made, that was not in the area of research, was to organise the first International Congress of Mathematicians held in Zurich in 1897. He was also the president of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Geiser.html (1 of 2) [2/16/2002 11:11:36 PM]

Geiser

Congress. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Geiser.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Geiser.html (2 of 2) [2/16/2002 11:11:36 PM]

Gelfand

Israil Moiseevic Gelfand Born: 2 Sept 1913 in Krasnye Okny, Odessa, Ukraine

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Israil Gelfand went to Moscow at the age of 16, in 1930, before completing his secondary education. There he took on a variety of different jobs such as door keeper at the Lenin library, but he also began to teach mathematics. There were many different institutes in Moscow where mathematics was taught in evening classes and Gelfand taught elementary mathematics in various of these institutes, then a little later progressing to teach more advanced mathematics. While he did this evening teaching he also attended lectures at Moscow University, the first course he attended being the theory of functions of a complex variable by Lavrentev. In 1932 Gelfand was admitted as a research student under Kolmogorov's supervision. His work was in functional analysis and he was fortunate to be in a strong school of functional analysis so he received much support from other mathematicians such as A E Plessner and L A Lyusternik. Gelfand presented his thesis Abstract functions and linear operators in 1935 which contains important results, but is perhaps even more important for the methods that he used, studying functions on normed spaces by applying linear functionals to them and using classical analysis to study the resulting functions. Gelfand's next major achievement was the theory of commutative normed rings which he created and studied in his D.Sc. thesis submitted in 1938. The importance of this work is brought out in [20] and [21]:... it was [Gelfand] who brought to light the fundamental concept of a maximal ideal which made it possible to unite previously uncoordinated facts and to create an interesting new theory. Gelfand's theory of normed rings revealed close connections between Banach's

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gelfand.html (1 of 4) [2/16/2002 11:11:38 PM]

Gelfand

general functional analysis and classical analysis. During the time that he was carrying out this work for his D.Sc., Gelfand taught at the Academy of Science of the USSR. He held a post at the Academy from 1935 until 1941 when he was appointed as professor at Moscow State University. In joint work with Naimark in the early 1940s, Gelfand worked on noncommutative normed rings with an involution. They showed that these rings could always be represented as a ring of linear operators on a Hilbert space. It is impossible to do any justice to the range of work covered by Gelfand in this short article. However, we should mention some of the main strands of his work. One important area which he started work on in the early 1940s was the theory of representations of non-compact groups. This work followed on from the representation theory of finite groups by Frobenius and Schur and the representation theory of compact groups by Weyl. Another important area of his work is that on differential equations where he worked on the inverse Sturm-Liouville problem. He saw the importance of the work of Sobolev and Schwartz on the theory of generalised functions and distributions, and he developed this theory in a series of monographs. He worked on computational mathematics, developing general methods for solving the equations of mathematical physics by numerical means. In this area he also worked on difference operators. Gelfand's work on group representations led him to study integral geometry (a term due to Blaschke) which in turn he was led to by a study of the Radon transform. Between 1968 and 1972 Gelfand produced a series of important papers on the cohomology of infinite dimensional Lie algebras. From 1958 onwards Gelfand became interested in problems in biology and medicine. In 1960, together with Fomin and other scientists, he set up the Institute of Biological Physics of the Academy of Sciences of the USSR. In particular he became interested in cell biology and also became interested in experimental work as well as the theoretical work which was his first interest. His work in biology is described in [9] and [10] as follows:On the basis of actual biological results, he developed important general principles of the organisation of control in complex multi-cell systems. These ideas, apart from their biological significance, served as a starting point for the creation of new methods of finding an extremum, which were succesfully applied to problems of X-ray structural analysis, problems of recognition, ..... Gelfand's interests were certainly not confined to research despite his incredible record of having published over 500 papers in mathematics, applied mathematics and biology. He established a correspondence school in mathematics which [4]:... helped to bring rich mathematical experiences to students all over the Soviet Union. His style of teaching is described in [20] and [21]:One of the characteristic features of Israil Moiseevic's activities has been the extremely close bond between his research work and his teaching. The formulation of new problems and unexpected questions, a tendency to look at even well known things from a new point of view characterises Gelfand as a teacher, regardless of whether at a given moment he is holding a conversation with schoolchildren or with his own colleagues. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gelfand.html (2 of 4) [2/16/2002 11:11:38 PM]

Gelfand

In 1973 Gelfand was awarded the Order of Lenin for the third time ([9] and [10]):For services in the development of mathematics, the training of scientific specialists and in celebration of his sixtieth birthday ... This is only one of a very large number of honours which have been given to Gelfand over many years of outstanding contributions. He was president of the Moscow Mathematical Society during 1968-70. He was elected an honorary member of the American National Academy of Science, the American Academy of Science and Arts, the Royal Irish Academy, the American Mathematical Society, the London Mathematical Society. He has been awarded many honorary doctorates including one fron the University of Oxford. In 1989-90 Gelfand taught at Harvard University and in 1990 he also taught at Massachusetts Institute of Technology. In that same year, 1990, he emigrated to the United States where he became Distinguished Visiting Professor at Rutgers. He also holds a chair in the departments of mathematics and biology at the Center for Mathematics, Science, and Computer Education in the Institute for Discrete Mathematics and Computer Science at Rutgers University. In 1992 Gelfand set up a programme in the United States similar to the correspondence school in mathematics which he had run in Russia. The Gelfand Outreach Program [4]:... fosters mathematical excellence in high school students. In 1994 Gelfand was awarded a MacArthur Fellowship from the John T and Catherine D MacArthur Foundation. The MacArthur Fellowships are [4]:... no-strings-attached awards that are intended to foster creativity in a wide range of human endeavours. The award to Gelfand is $375,000, to be paid over five years. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (21 books/articles) Mathematicians born in the same country Honours awarded to Israil Gelfand (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

1977

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gelfand.html (3 of 4) [2/16/2002 11:11:38 PM]

Gelfand

Glossary index

Quotations index

Mathematicians of the day JOC/EFR January 1999

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gelfand.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gelfand.html (4 of 4) [2/16/2002 11:11:38 PM]

Gelfond

Aleksandr Osipovich Gelfond Born: 24 Oct 1906 in St Petersburg, Russia Died: 7 Nov 1968 in Moscow, Russia

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aleksandr Gelfond taught mathematics at the Moscow Technological College (1929-30) and then from 1931 at Moscow State University where he held chairs of analysis, theory of numbers and the history of mathematics. In addition to his important work in the number theory of transcendental numbers (that is, numbers that are not the solution of an algebraic equation with rational coefficients) Gelfond made significant contributions to the theory of interpolation and the approximation of functions of a complex variable. In 1929 he conjectured that:If an and bn for 1 m n are algebraic numbers such that {ln an, 1 independent over Q, then b1 ln a1 + b2 ln a2 + ... + bn ln an 0.

m n} are linearly

In 1934 Gelfond proved a special case of his conjecture namely that ab is transcendental if a is algebraic (a 0, 1) and b is an irrational algebraic number. This result is now known as Gelfond's theorem and solved Problem 7 of the list of Hilbert 23 problems. It was solved independently by Schneider. In 1966 A Baker proved Gelfond's Conjecture in general. Gelfond's major contributions to transcendental numbers are in Transtendentnye i algebraicheskie chisla (1952) and to approximation and interpolation theories are in Ischislenie konechnykh raznostey (1952). Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gelfond.html (1 of 2) [2/16/2002 11:11:39 PM]

Gelfond

Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Aleksandr O Gelfond

Mathematicians born in the same country

Other references in MacTutor

1. Chronology: 1920 to 1930 2. Chronology: 1930 to 1940

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gelfond.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gelfond.html (2 of 2) [2/16/2002 11:11:39 PM]

Gellibrand

Henry Gellibrand Born: 17 Nov 1597 in London, England Died: 16 Feb 1636 in London, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henry Gellibrand entered Trinity College, Oxford in 1615 where he was introduced to mathematics by Savile. He received a B.A. from Trinity College in 1619 and an M.A. in 1623. He became a friend of Briggs while in Oxford. He entered the church becoming a curate in Chiddingstone, Kent a couple of years after receiving his M.A. from Oxford. Gellibrand succeeded Gunter to the chair of astronomy in Gresham College, London in 1627. It was largely through the influence of Briggs that he received this chair. Gellibrand's most famous scientific discovery was the change over the years in magnetic declination. He also made mathematical contributions to navigation, in particular working on methods to determine longitude. His methods were based on observing various celestial events and were published in Appendix concerning Longitude (1633). Gellibrand also published logarithm and trigonometrical tables. After his friend Briggs died in 1630, he worked to complete Briggs' Trigonometria Britannica which he did, publishing the work in 1633. Several of Gellibrand's publications appeared after he died. Institution Trigonometrical (1638) with an expanded version in 1658, applied trigonometry to navigation and astronomy. Epitome of Navigation first appeared 62 years after his death; a rather remarkable length of time. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gellibrand.html (1 of 2) [2/16/2002 11:11:41 PM]

Gellibrand

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gellibrand.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gellibrand.html (2 of 2) [2/16/2002 11:11:41 PM]

Geminus

Geminus Born: about 10 BC in (possibly) Rhodes, Greece Died: about 60 AD Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

It may be surprising that Geminus's name seems to be Latin rather than Greek but as Heath writes [3]:The occurrence of a Latin name in a centre of Greek culture need not surprise us, since Romans settled in such centres in large numbers during the last century BC. Geminus, however, in spite of his name, was thoroughly Greek. Geminus is believed by many historians to have worked in Rhodes. Certainly his astronomy text uses mountains on Rhodes to make specific points but, as Dicks points out in [1], this is not proof that he worked there. For example, Geminus refers to Mt Atabyrius (today called Mt Attaviros) without giving any indication of where it is but when he refers to Mt Cyllene he is careful to indicate that it is the Peloponnesus. However, since Rhodes was at this time the centre for astronomical research, and was taken as the reference point for latitude in astronomical observations, it is quite possible that Geminus would assume his reader were familiar with the reference points of Rhodes such as Mt Atabyrius without further comment. Geminus was a Stoic philosopher and either a pupil, or perhaps a later follower, of Posidonius. He wrote to defend the Stoic view of the universe, and in particular to defend mathematics from attacks which had been made on it by Sceptic philosophers and by Epicurean philosophers. Simplicius talks of a work by Geminus in which he merely reproduces the views of Posidonius but this is unfair on Geminus who, although holding similar views, shows his own independent point of view in many respects. Not all historians of science agree on the dates of Geminus that we have given. Some favour dates of 130 BC - 60 BC which are based largely on a calendar which appears in his Introduction to Astronomy called the Isagoge and seems to suggest a date of around 70 BC for the date when the text was written. Dicks in [1] seems convinced by this argument which becomes certain only if the date of the Egyptian Isis festival is known with certainty. Several Isis festivals took place in Egypt and to date Geminus correctly by this argument the proper festival must be selected. Neugebauer in [4] believes that the selection which give the date of 70 BC is incorrect and he favours a date for the Isis festival which leads to a date of 50 AD for Geminus's text:... it is clear that Geminus had in mind the Isis festivals which were celebrated in the Egyptian month Khoiak. This places him in the first century AD. The duration of the festival and the possible insecurity of the date of the Winter Solstice prevent us from establishing a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Geminus.html (1 of 4) [2/16/2002 11:11:42 PM]

Geminus

more accurate date, but ... the first half of the century [is] more likely that the latter part. ... we thus consider a date around 50 AD as fairly secure for the Isagoge ... Of course a date of 50 AD for the Isagoge means that Geminus could not have been a pupil of Posidonius who died in 50 BC as their lives would not have overlapped. Neugebauer comments in [4]:A much discussed question is the dependence of Geminus on the famous stoic philosopher Posidonius of Rhodes (who died around 50 BC). The assumption of such a dependence mainly rests on close parallels between passages in the Isagoge and the writings of Cleomedes who repeatedly refers to Posidonius as his source. In contrast the Isagoge itself contains not a single reference to this philosopher nor does the frequent mention of Rhodes (or its latitude) imply that Geminus was a pupil of Posidonius. Geminus wrote a number of astronomy texts, including the elementary text Isagoge or Introduction to Astronomy based on the work of Hipparchus which we referred to above. Geminus gave an historical account of earlier astronomical theories including those of Callippus and the Chaldeans. He made a significant comment on the stars, stating that:... we must not suppose that all the stars lie on one surface, but rather that some of them are higher and some are lower. The main part of the work contains little mathematical astronomy. It describes the main constellations, the variation of the length of night and day at different latitudes, the rising of the signs of the zodiac, and the length of the lunar month. The phases of the moon are explained and solar and lunar eclipses are also explained. The motion of the planets is discussed and certain geographical features are discussed as well as implications for the weather. The last chapter of Introduction to Astronomy (Chapter 18) seems rather different from the rest of the text being of a much more advanced nature. Dicks writes in [1] that this chapter:.... far more technical than the others and out of keeping with the rest of the book, may well be an unrelated fragment. The recent article [5] discusses Chapter 18 in detail. The authors claim this to be an important contribution to Greek astronomy introducing the use of mean motion. Geminus represents observational data for the motion of the moon in longitude by means of an arithmetical function. Geminus's mathematics text Theory of Mathematics is now lost but information about it is available from a number of sources. Proclus quotes extensively from it and Eutocius and Heron also give some information. In fact Proclus relies very heavily on the work of Geminus when he writes his own history of mathematics and it is fair to say that Geminus's books are the most valuable sources available to him. The Theory of Mathematics deals with the logical subdivisions of mathematics. Geminus considers the concepts of 'hypothesis', 'theorem', 'postulate', 'axiom', 'line', 'surface', 'figure', 'angle' etc. Not only did he examine the principles behind these ideas in depth but he also gave historical accounts of the development of the ideas. The work contains an explanation of where the name 'mathematics' came from. Geminus tells us that Pythagoras applied it to [3]:... geometry and arithmetic, sciences which deal with pure, the eternal and the unchangeable, but was extended by later writers to cover what we call 'mixed' or applied http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Geminus.html (2 of 4) [2/16/2002 11:11:42 PM]

Geminus

mathematics, which, though theoretical, has to do with sensible objects e.g. astronomy and optics. We should note that 'sensible' here has its older meaning of 'relating to the senses' rather than the modern meaning which is the opposite of 'stupid'. In fact the work does seem to have been a comprehensive work covering the whole of mathematics, perhaps even a mathematical encyclopaedia. Geminus is critical of Euclid's axioms in this work and he offers a 'proof' of the fifth postulate. Proclus quotes from Geminus (see for example [3]), saying that in the case of the parallel postulate:... when the right angles are lessened, [that] the straight lines converge is true and necessary; but the statement that, since they converge more and more as they are produced, they will sometime meet is plausible but not necessary... Geminus tried the following approach giving a definition of parallel lines:Parallel straight lines are straight lines situated in the same plane and such that the distance between them, if they are produced without limit in both directions at the same time is everywhere the same. The 'proof' which Geminus then gave of the parallel postulate is ingenious but it is false. He made an error right at the start of his argument for he assumed that the locus of points at a fixed distance from a straight line is itself a straight line and this cannot be proved without a further postulate. It is interesting, however, that Geminus attempts to prove the parallel postulate and, although it is unlikely to be the first such attempt, at least it is the earliest one for which details have survived. The helix, namely the curve cutting the generators of a right circular cylinder at a constant angle, appears in this work by Geminus. Proclus suggests, however, that the curve goes back to Apollonius 150 years before Geminus. But Geminus proves an interesting classification theorem, namely that the helix, the circle and the straight line are the only curves with the property that any part of the curve will coincide with any other part of the same length. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Cross-references to History Topics

How do we know about Greek mathematicians?

Other references in MacTutor

Chronology: 1AD to 500

Honours awarded to Geminus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Geminus

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Geminus.html (3 of 4) [2/16/2002 11:11:42 PM]

Geminus

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Geminus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Geminus.html (4 of 4) [2/16/2002 11:11:42 PM]

Gemma_Frisius

Regnier Gemma Frisius Born: 8 Dec 1508 in Dokkum, Friesland, Netherlands Died: 25 May 1555 in Louvain, Brabant (now Belgium)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Regnier Gemma Frisius was a native of Friesland, a coastal province in northern Netherlands, which explains his nickname of Frisius. He was educated at the University of Louvain, receiving a medical degree. He became the leading theoretical mathematician in the Low Countries and became professor of medicine and mathematics at the University of Louvain. He was also a practicing physician in Louvain. Gemma Frisius applied his mathematical expertise to geography, astronomy and map making. In Louvain he cooperated with his student Gerardus Mercator and with the engraver and goldsmith Gaspar à Myrica in the construction of maps, globes and astronomical instruments. In 1530 Gemma Frisius wrote De Principiis Astronomiae Cosmographicae which he published in Louvain. Chapter 19 of this work describes, for the first time, how the longitude of a place may be found using a clock to determine the difference in local and absolute times. He says (see [5]):... it is with the help of these clocks and the following methods that longitude is found. ... observe exactly the time at the place from which we are making our journey. ... When we have completed a journey ... wait until the hand of our clock exactly touches the point of an hour and, at the same moment by means of an astrolabe... find out the time of the place we now find ourselves. ... In this way I would be able to find the longitude of places, even if I was dragged off unawares across a thousand miles. Aware of the difficulties of keeping exact time he writes:... it must be a very finely made clock which does not vary with change of air. In a second edition of the work three years later he added some notes about finding the longitude at sea, the first time anyone had attacked the problem. It is worth noting that although there were many methods of finding longitude proposed in the 250 years following Gemma Frisius's work, ultimately the methods he proposed were to become the solution to finding the longitude at sea. He described the theory of trigonometric surveying in the 1533 edition of De Principiis Astronomiae Cosmographicae and in particular he was the first to triangulation as a method of accurately locating places. In 1534 he wrote Tractatus de Annulo Astronomicae. His work on applying trigonometric methods to astronomical problems led him to note correctly that comets displayed a proper motion against the background stars.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gemma_Frisius.html (1 of 2) [2/16/2002 11:11:44 PM]

Gemma_Frisius

In 1535-36 Gemma Frisius cooperated with Myrica and Gerardus Mercator in constructing a terrestrial globe and, in 1537, they constructed a celestial globe. Gemma Frisius's work on astronomical instruments was described in several of his books. For example in De Radio Astronomico (1545) he describes his work constructing a cross-staff about 1.5 metres long with one cross piece about 3/4 of a metre in length. It had brass sighting vanes and a sliding vane. He also invented a new astrolabe which he described in De Astrolabio which was published in 1556, after his death. Gemma Frisius made many astronomical observations. In particular he recorded comets in July 1533, January 1538 and 30 April 1539. Some of these comet observations are described in works by his son, Cornelius Gemma Frisius, who was born in 1533 and went on to become professor of medicine and astronomy at Louvain. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Regnier Gemma Frisius

Mathematicians born in the same country

Cross-references to History Topics

Longitude and the Académie Royale

Other references in MacTutor

Chronology: 1500 to 1600

Honours awarded to Regnier Gemma Frisius (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Gemma Frisius

Other Web sites

http://www-math.sci.kun.nl/math/werkgroepen/gmfw/bronnen/index.html Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gemma_Frisius.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gemma_Frisius.html (2 of 2) [2/16/2002 11:11:44 PM]

Genocchi

Angelo Genocchi Born: 5 March 1817 in Piacenza, Italy Died: 7 March 1889 in Turin, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Angelo Genocchi attended school in Piacenza and then studied law at the University of Piacenza. Despite his course being law he always had mathematics as his favourite subject and he became very knowledgeable in mathematics despite taking a law course. After graduating he practiced law for a number of years. However, as Kennedy explains in [1], he was not ideally suited to the practice of law:He was not a good debater. He quickly lost patience, would shake and become bitingly bitter - but he quickly grew calm again. When he was offered the chair of law at Piacenza University, a post he had not applied for, he accepted. However, he was not entirely successful as a teacher either [1]:...some of his students ... found him too cold and severe. Genocchi was, however, very active politically. Before the revolution of 1848, he was already part of a group of liberals in Piacenza who wanted to remove its Austrian rulers. At signs of unrest in the region the Austrian government reinforced its garrisons in Lombardy, arrested opposition leaders in Milan and suppressed student demonstrations. By 23 March 1845 the people had deposed their Austrian rulers from Piacenza. Within a few days the Austrian army lost nearly all of Lombardy. On 23 March Charles Albert of Sardinia-Piedmont declared war on Austria. Annexing Parma and Modena, whose Austrian rulers had been driven out, the Piedmontese won a number of victories before suffering reverses. Genocchi and his liberal friends had formed a provisional government in Piacenza but before they could make political progress the Austrians attacked the region.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Genocchi.html (1 of 3) [2/16/2002 11:11:46 PM]

Genocchi

The Piedmontese army was unable to withstand the Austrian counteroffensive. After a series of defeats, Charles Albert's army withdrew from Milan. On 6 August 1845 they crossed the Ticino River, leaving the city and its local controllers to the mercy of the returning Austrians. Before the Austrians arrived in Piacenza Genocchi, greatly disappointed at the turn of events, and other liberals, left the city. Genocchi went to live in Turin, refusing requests from his friends to return to Piacenza saying he would not return until freedom returned. In Turin he began to take mathematics seriously attending lectures by Plana and others. After a while he began teaching but he had to be tricked into entering the competition for the Chair of Algebra and Complementary Geometry at Turin. From 1859 Genocchi held the Chair of Algebra and Complementary Geometry at Turin, then the following year he moved to the Chair of Higher Analysis. In 1862 he moved chairs again, but remaining in Turin, to the Introduction to the Calculus and the following year to Infinitesimal Calculus. During the year 1881-82 Peano served as his assistant. The main research topics which Genocchi worked on were number theory, series and the integral calculus. He published 176 articles between 1851 and 1886. Kennedy writes in [1]:He did not adopt the methods of Riemann and Weierstrass, but rather worked in the tradition of Euler, Lagrange, Gauss and Cauchy. His major work was in number theory, of which he was the principal investigator in Italy. Genocchi's style as a teacher is also described in [1]:The qualities that stood out most in Genocchi as a teacher were learning and precision. ... He was scrupulously punctual and justly demanding of his students. ... His explanations were calm, with no repetitions, and he aimed at rigorously presenting the fundamental concepts and studying them so as to arrive at simple procedures and clear exposition. However, given his failing when he taught and practiced law, one would not expect him to be the perfect teacher, indeed:His austere character, along with his thin and monotonous voice, did not warm the classroom or allow the students to feel at ease with him. In 1882 Genocchi broke his kneecap and Peano took over his teaching. The kneecap was broken while Genocchi was on his holidays in September 1882. By this time his sight was poor and he fell over a post marking the edge of the road. In 1884 Differential Calculus and Fundamentals of Integral Calculus was published under Genocchi's name. This book was based on Genocchi's lectures but was largely the work of Peano. In [2] Peano explained how the publication arose:He was many times urged to publish his calculus course, but he never did. ... It was only in '83 that he agreed to let the firm of Fratelli Bocca publish his Leçons with my help, and this appeared in part the following year. He was ill at the time and wished to remain completely apart from the work. I used notes made by his students at his lessons, comparing them point by point with all the principal calculus texts, as well as with original memoirs... The result is that my publication does not exactly represent the professor's lessons. The book was important and widely aclaimed despite some unease over whether Peano had Genocchi's full agreement to publish the text with the additions he made. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Genocchi.html (2 of 3) [2/16/2002 11:11:46 PM]

Genocchi

Despite making a partial recovery, during which time he resumed his teaching duties, soon Genocchi's health failed again and he slowly gave up all his activities. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Genocchi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Genocchi.html (3 of 3) [2/16/2002 11:11:46 PM]

Gentzen

Gerhard Gentzen Born: 24 Nov 1909 in Greifswald, Germany Died: 4 Aug 1945 in Prague, Czechoslovakia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gerhard Gentzen's father was a lawyer who practiced law in Bergen on the Isle of Rügen. It was there that Gerhard spent his childhood years, attending first the elementary school there, and later the Realgymnasium. His father, however, was killed in World War I and in 1920 Gentzen's mother moved to Stralsund. Gentzen had already begun his secondary schooling at this stage but he continued his education at the Humanistische Gymnasium in Stralsund. Certainly moving schools did not affect Gentzen's academic achievements for when he received his Arbitur in 1928 it was with distinction and he was ranked top in his school. In [6] Robbel describes the intellectual world of the young Gentzen in particular examining the influences on him of his grandparents (especially A Bilharz) and his parents. The results of his 1928 Reifeprufung examination are given in an appendix to [6]. The headmaster of the Humanistische Gymnasium was certainly impressed with the results and, recognising his exceptional mathematical abilities, awarded him a university scholarship. Gentzen, as was usual at this time, moved between different German universities. He began his mathematical studies at the University of Greifswald in 1928 then, after studying there for two semesters, he entered the University of Göttingen on 22 April 1929. Again he spent only two semesters before moving on, this time to the University of Munich where he spent only one semester followed by one further semester at the University of Berlin. After this he returned to Göttingen where he worked under Weyl for his doctorate on the foundations of mathematics. He was taught by Bernays, Carathéodory, Courant, Hilbert, Kneser, Landau and, of course, his supervisor Weyl.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gentzen.html (1 of 4) [2/16/2002 11:11:48 PM]

Gentzen

In 1933 Gentzen was awarded his doctorate by Göttingen but the intense study in different environments had taken its toll so he was forced at this stage to return home to rest and recover his health. He returned to Göttingen, becoming Hilbert's assistant in 1934. M E Szabo writes in [2]:... he continued to work [at Göttingen] even after Hilbert's retirement. During these years Gentzen published some of his most important papers and was also given the responsible task of reviewing numerous works of eminent researchers from many countries for the Zentralblatt für Mathematik. These reviews attest his extraordinary range of interest and the great extent of his involvement in the international community of scholars. As we have mentioned, Gentzen's work was on logic and the foundations of mathematics. He submitted his first paper to Mathematische Annalen early in 1932. The paper studies the theory of 'sentence systems' and answers a major open problem in the subject by constructing a counterexample to show that not all sentence systems have independent axiom systems. However he also showed that linear sentence systems do have independent axiom systems. He introduced the notion of 'logical consequence' which provided a logic closer to mathematical reasoning than the systems proposed by Frege, Russell and Hilbert. This idea was later attributed to Tarski who introduced it in 1936, three years after Gentzen. In 1934 Gentzen gave the method of succinct Sequenzen, rules of consequents, which were particularly useful for deriving metalogical decidability results. Hilbert had him work on axiomatic methods and the classification of mathematics into levels. The idea of levels, probably first introduced by Weyl, considers number theory as the first level since it deals with the natural numbers, analysis as the second level since it deals with the real numbers, and set theory as the third level where the full extent of Cantor's cardinal and ordinal numbers would be studied. Gentzen wrote several papers on these concepts, particularly examining the occurrence of set theory paradoxes. Of course Gödel published his incompleteness theorem just at the time Gentzen was beginning his work. At first Gentzen worried that it affected what he wanted to achieve on the foundations of mathematics and he withdrew what would have been his second paper after he had corrected the final proofs because of worries about the significance of Gödel's theorems. Later, however, he wrote of Gödel's result saying:... this is undoubtedly a very interesting, but certainly not an alarming, result. We can paraphrase it by saying that for number theory no once-and-for-all sufficient system of forms of inference can be specified, but that on the contrary, new theorems can always be found whose proof requires a new form of inference. In a paper published in Mathematische Zeitschrift in 1935 Gentzen introduced two new versions of predicate logic now called the N-system and the L-system. In the following year he gave a consistency proof in terms of an N-type logic for the system S of arithmetic with induction. Gentzen wrote in the introduction to this paper:The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles. He then looks at why such consistency proofs are necessary:Mathematics is regarded as the most certain of all sciences. That it could lead to results which contradict one another seems impossible. This faith in the indubitable certainty of mathematical proofs was sadly shaken around 1900 by the discovery of the antinomies or paradoxes of set theory. It turned out that in this specialised branch of mathematics,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gentzen.html (2 of 4) [2/16/2002 11:11:48 PM]

Gentzen

contradictions arise without our being able to recognise any specific error in our reasoning. After discussing the paradoxes, in particular Russell's paradox, Gentzen writes:... I shall carry out such a consistency proof for elementary number theory. Yet even here we shall meet form of inference whose closer inspection will give us cause for concern. ... One point should, however, be made clear from the outset: these forms of inference which might possibly be considered disputable hardly ever occur in actual number theoretical proofs; we must not be misled and, because of the great self-evidence of these proofs, consider a consistency proof as superfluous. By Gödel's unprovability theorem, such a proof as Gentzen gave had to make use of tools stronger than those of S; extending ordinary mathematical induction, Gentzen employed transfinite induction up to Cantor's first epsilon number, and he also showed that this was the minimum required for such proof. Kleene wrote:... to what extent the Gentzen proof can be accepted as securing classical number theory in the sense of that problem formulation is in the present state of affairs a matter of individual judgement. Tarski wrote:Gentzen's proof of the consistency of arithmetic is undoubtedly a very interesting metamathematical result, which may prove very stimulating and fruitful. I cannot say, however, that the consistency of arithmetic is now much more evident to me ... than it was before the proof was given. Gentzen's was the most outstanding contribution to Hilbert's programme of axiomatising mathematics. In 1937 he addressed the Congress in Paris giving a talk with title Concept of infinity and the consistency of mathematics. His outstanding work, however, was cut short by the start of World War II. Gentzen remained on the staff at Göttingen until 1943, although he had to undertake military service in the years 1939 until 1941. He was conscripted into the army where he worked in telecommunications. He became ill, however, and spent three months recovering in a military hospital. His health was now too poor to allow him to continue with his military service and he returned to Göttingen. In the summer of 1942 he submitted his Habilitation thesis Provability and nonprovability of restricted transfinite induction in elementary number theory to Göttingen and, on the award of the degree, he became entitled to teach in universities. As part of the German war effort, he took up a teaching post as a Dozent in the Mathematical Institute of the German University of Prague and he taught there until arrested and taken into custody. The citizens of Prague rose in revolt against the occupying German forces on 5 May 1945, the day all the staff of the German University were arrested, and held the city until the Russian Army arrived four days later. One would have to mention the facts concerning Gentzen's political and military life that Vihan relates in [8], namely his association with the SA, NSDAP and NSD Dozentenbund. Gentzen was interned by the Russian forces and held in poor conditions. He died of malnutrition after 3 months in internment. A friend who was in prison with him described his last few days:I can see him lying on his wooden bunk thinking all day about the mathematical problems that preoccupied him. He once confided in me that he was really quite content since now he had at last time to think about a consistency proof for analysis... He also concerned himself http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gentzen.html (3 of 4) [2/16/2002 11:11:48 PM]

Gentzen

with other questions such as that of an artificial language, etc. Now and then he would give a short talk ... we were continually reassured that the formalities of our release would only take a few days longer.... he was hoping to be able to return to Göttingen and devote himself fully to the study of mathematical logic and the foundations of mathematics. He was dreaming of an Institute for this purpose ... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) A Poster of Gerhard Gentzen

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gentzen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gentzen.html (4 of 4) [2/16/2002 11:11:48 PM]

Gergonne

Joseph Diaz Gergonne Born: 19 June 1771 in Nancy, France Died: 4 May 1859 in Montpellier, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Gergonne's father was an architect and also a painter. However, he died when Joseph was twelve years old. Joseph was educated at the Collège de Nancy which was a religious establishment. After leaving the Collège he did some private tutoring but, like everyone else in France at this time, he was caught up in the events surrounding the French Revolution. In 1791 the French Assembly was at a difficult stage trying to stabilise the country following the French Revolution. The Assembly was not helped by the King, Louis XVI, attempting to flee the country in June of that year. After the King was returned to Paris, the Assembly reinforced the frontiers of France by calling for 100,000 volunteers from the National Guard. Gergonne gave his support becoming a captain in the National Guard. In April 1792 France went to war against Austria and Prussia. The French attack was quickly halted and then Prussian forces invaded France. The Assembly called for 100,000 military volunteers and Gergonne joined the French army being assembled to defend Paris against the Prussians. On 20 September 1792 Kellermann led the French forces at Valmy with Gergonne in his army. The French defeated the Prussians in an artillery duel and, following this, the Austrian and Prussian armies retreated from France. Following this great French victory, Gergonne went to Paris where he became a secretary to his uncle. It was a time of much military action, however, and Austria and Prussia were not going to simply accept the defeat at Valmy as the end of the war. By 1793 they were joined by Spain, Piedmont, and Britain. The French forces were soon in trouble, fighting on many different fronts and being defeated everywhere. Gergonne returned to the French army, this time as secretary to the general staff of the Moselle army. Gergonne spent a month at the Châlons artillery school in 1794 and after this he was commissioned as a lieutenant. With Spain having declared war on France in 1793, France sent forces to attack Spain in 1794. Gergonne was attached to these forces which entered Spain and laid siege to the town of Figueras capturing it. In 1795 Frederick William II of Prussia made a separate treaty, the Treaty of Basel, with the French and withdrew from the coalition against France. At this stage Gergonne was sent with his regiment to Nimes in southern France. In that city the Ecole Centrale had been set up shortly before and Gergonne was appointed to the chair of transcendental mathematics. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gergonne.html (1 of 3) [2/16/2002 11:11:49 PM]

Gergonne

This was not the end of problems in French politics, with the great military campaigns of Napoleon Bonaparte over the following years. Gergonne, however, settled down at this point. He married and began to concentrate on mathematics. His career was much influenced by Monge who, by this time, was Director of the Ecole Polytechnique in Paris. Finding problems getting his mathematics papers published, Gergonne established his own mathematics journal, the first part appearing in 1810. The Journal was officially called the Annales de mathématique pures et appliquées but became known as Annales de Gergonne . Gergonne's mathematical interests were in geometry so it is not surprising that this was this topic which figured most prominently in his journal. In fact many famous mathematicians published in the twenty-one volumes of the Annales de Gergonne which appeared during a period of twenty-two years. In addition to Gergonne himself (who published around 200 articles), Poncelet, Servois, Bobillier, Steiner, Plücker, Chasles, Brianchon, Dupin, Lamé, Galois and many others had papers appear in the Journal. In 1813 Gergonne wrote a prize winning essay for the Bordeaux Academy Methods of synthesis and analysis in mathematics. The essay has never been published but a summary of it appears in [5]. The essay tells us a lot about Gergonne's mathematical ideas. Despite the title of the essay, he suggests that the terms "analysis" and "synthesis" should not be used since everyone uses them with a different meaning. Perhaps surprisingly, since Gergonne was himself a geometer, he suggests that algebra is a more important topic than geometry. In a remarkable piece of foresight, he saw a time in the future when quasi-mechanical methods would be used to discover new results. We will examine Gergonne's contributions to geometry later in this article, but for the moment it is worth noting that he did publish on other topics. For example in 1815 he published Application de la methode des moindre quarres a l'interpolation des suites in his own journal. This paper examines the problem of observing values of a response function which depends upon a single independent variable. Gergonne looked at how to estimate the values of the response function, and of its derivatives, at a point when there are random errors in the observed values. Gergonne was appointed to the chair of astronomy at the University of Montpellier in 1816. There is a nice story told by Struik in [1] where he writes:... during the July Revolution of [1830], when rebellious students began to whistle in his class, he regained their sympathy by beginning to lecture on the acoustics of the whistle. It was in 1830 the Gergonne became rector of the University of Montpellier. He decided to end publication of his journal at this time although the monthly parts continued to appear until 1832. Gergonne retired in 1844 at the age of 73. Gergonne provided an elegant solution to the Problem of Apollonius in 1816. This problem is to find a circle which touches three given circles. Gergonne introduced the word polar and the principal of duality in projective geometry was one of his main contributions. Gergonne's first contributions to duality appear in a series of papers beginning in 1810. He noticed the fact that certain forms of geometry yielded theorems which appeared in related pairs, and the led him to a more detailed analysis of why this was so. Then [1]:... in three articles in the Annales [1824-27], Gergonne [gave] the general principle that every theorem in the plane, connecting points and lines, corresponds to another theorem in which points and lines are interchanged, provided no metrical relations are involved. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gergonne.html (2 of 3) [2/16/2002 11:11:49 PM]

Gergonne

We have already illustrated Gergonne's interest in the philosophy of mathematics with a description of his unpublished essay on the topic in 1813. He sprinkled charming philosophical comments around his papers. Chasles, in Aperçu historique ..., notes the comment by Gergonne:It is not possible to feel satisfied at having said the last word about some theory as long as it cannot be explained in a few words to any passer-by encountered in the street. Given this criterion, I doubt whether any theory has had the last word said about it! Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Other references in MacTutor

1. Gergonne's theorem 2. Chronology: 1810 to 1820

Other Web sites

Georgia (The Gergonne point)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gergonne.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gergonne.html (3 of 3) [2/16/2002 11:11:49 PM]

Germain

Marie-Sophie Germain Born: 1 April 1776 in Paris, France Died: 27 June 1831 in Paris, France

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Marie-Sophie Germain was the middle daughter of Ambroise-François, a prosperous silk-merchant, and Marie-Madelaine Gruguelin. Sophie's home was a meeting place for those interested in liberal reforms and she was exposed to political and philosophical discussions during her early years. At the age of thirteen, Sophie read an account of the death of Archimedes at the hands of a Roman soldier. She was moved by this story and decided that she too must become a mathematician. Sophie pursued her studies, teaching herself Latin and Greek. She read Newton and Euler at night while wrapped in blankets as her parents slept - they had taken away her fire, her light and her clothes in an attempt to force her away from her books. Eventually her parents lessened their opposition to her studies, and although Germain neither married nor obtained a professional position, her father supported her financially throughout her life. Sophie obtained lecture notes for many courses from Ecole Polytechnique. At the end of Lagrange's lecture course on analysis, using the pseudonym M. LeBlanc, Sophie submitted a paper whose originality and insight made Lagrange look for its author. When he discovered "M. LeBlanc" was a woman, his respect for her work remained and he became her sponsor and mathematical counsellor. Sophie's education was, however, disorganised and haphazard and she never received the professional training which she wanted. Germain wrote to Legendre about problems suggested by his 1798 Essai sur le Théorie des Nombres, and the subsequent Legendre - Germain correspondence became virtually a collaboration. Legendre included some of her discoveries in a supplement to the second edition of the Théorie. Several of her http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Germain.html (1 of 4) [2/16/2002 11:11:51 PM]

Germain

letters were later published in her Oeuvres Philosophique de Sophie Germain. However, Germain's most famous correspondence was with Gauss. She had developed a thorough understanding of the methods presented in his 1801 Disquisitiones Arithmeticae. Between 1804 and 1809 she wrote a dozen letters to him, initially adopting again the pseudonym "M. LeBlanc" because she feared being ignored because she was a women. During their correspondence, Gauss gave her number theory proofs high praise, an evaluation he repeated in letters to his colleagues. Germain's true identity was revealed to Gauss only after the 1806 French occupation of his hometown of Braunschweig. Recalling Archimedes' fate and fearing for Gauss's safety, she contacted a French commander who was a friend of her family. When Gauss learnt that the intervention was due to Germain, who was also "M. LeBlanc", he gave her even more praise. Among her work done during this period is work on Fermat's Last Theorem and a theorem which has become known as Germain's Theorem. This was to remain the most important result related to Fermat's Last Theorem from 1738 until the contributions of Kummer in 1840. In 1808, the German physicist Ernst F F Chladni had visited Paris where he had conducted experiments on vibrating plates, exhibiting the so-called Chladni figures. The Institut de France set a prize competition with the following challenge: formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence. A deadline of two years for all entries was set. Most mathematicians did not attempt to solve the problem, because Lagrange had said that the mathematical methods available were inadequate to solve it. Germain, however, spent the next decade attempting to derive a theory of elasticity, competing and collaborating with some of the most eminent mathematicians and physicists. In fact, Germain was the only entrant in the contest in 1811, but her work did not win the award. She had not derived her hypothesis from principles of physics, nor could she have done so at the time because she had not had training in analysis and the calculus of variations. Her work did spark new insights, however. Lagrange, who was one of the judges in the contest, corrected the errors in Germain's calculations and came up with an equation that he believed might describe Chladni's patterns. The contest deadline was extended by two years, and again Germain submitted the only entry. She demonstrated that Lagrange's equation did yield Chladni's patterns in several cases, but could not give a satisfactory derivation of Lagrange's equation from physical principles. For this work she received an honourable mention. Germain's third attempt in the re-opened contest of 1815 was deemed worthy of the prize of a medal of one kilogram of gold, although deficiencies in its mathematical rigour remained. To public disappointment, she did not appear as anticipated at the award ceremony. Though this was the high point in her scientific career, it has been suggested that she thought the judges did not fully appreciate her work and that

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Germain.html (2 of 4) [2/16/2002 11:11:51 PM]

Germain

the scientific community did not show the respect that seemed due to her . Certainly Poisson, her chief rival on the subject of elasticity and also a judge of the contest, sent a laconic and formal acknowledgement of her work, avoided any serious discussion with her and ignored her in public. As one biographer phrases it: Although it was Germain who first attempted to solve a difficult problem, when others of more training, ability and contact built upon her work, and elasticity became an important scientific topic, she was closed out. Women were simply not taken seriously. Germain attempted to extend her research, in a paper submitted in 1825 to a commission of the Institut de France, whose members included Poisson, Gaspard de Prony and Laplace. The work suffered from a number of deficiencies, but rather than reporting them to the author, the commission simply ignored the paper. It was recovered from de Prony's papers and published in 1880. Germain continued to work in mathematics and philosophy until her death. Before her death, she outlined a philosophical essay which was published posthumously as Considérations générale sur l'état des sciences et des lettres in the Oeuvres philosophiques. Her paper was highly praised by August Comte. She was stricken with breast cancer in 1829 but, undeterred by that and the fighting of the 1830 revolution, she completed papers on number theory and on the curvature of surfaces (1831). Germain died in June 1831, and her death certificate listed her not as mathematician or scientist, but rentier (property holder). The portrait above is taken from a commemorative medal. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles)

A Quotation

A Poster of Sophie Germain

Mathematicians born in the same country

Cross-references to History Topics

Fermat's last theorem

Other references in MacTutor

Chronology: 1800 to 1810

Honours awarded to Sophie Germain (Click a link below for the full list of mathematicians honoured in this way) Planetary features

Crater Germain on Venus

Paris street names

Rue Sophie Germain (14th Arrondissement)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Germain.html (3 of 4) [2/16/2002 11:11:51 PM]

Germain

Other Web sites

1. Agnes Scott College 2. PBS 3. San Diego 4. The Prime Pages (Germain primes) 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Germain.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Germain.html (4 of 4) [2/16/2002 11:11:51 PM]

Gherard

Gherard of Cremona Born: 1114 in Cremona, Italy Died: 1187 in Toledo, Spain Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gherard of Cremona's name is often written as Gerard or sometimes Gerhard. After being educated in Italy, he realised that European education was narrow and that he decided that he would try to make the riches of Arabic science available to European scholars through Latin translations of the major works in Arabic. For this reason Gherard went to Toledo in Spain where his intention was to learn Arabic so he could read Ptolemy's Almagest since no Latin translations existed at that time. Although we do not have detailed information of the date when Gherard went to Spain, he was certainly there by 1144. He remained there for most of the rest of his life and although he does not appear to have gathered a school around him, he certainly appears to have had quite a lot of assistance. He may have employed helpers who assisted him in the copying and checking of manuscripts and other chores associated with the great translation industry that he started. In all over a period of forty years, Gherard translated around eighty works from Arabic to Latin. The complete list of works which he translated is given in [1]. Some of these translations were of Arabic works while others were of Greek works which had been translated into Arabic. Often however, the works were a mixture in the sense that they were Arabic commentaries on Greek works. It was not the case that these works were all mathematical. Some were on science in general, others were on medicine. The most important, however, were on astronomy, geometry and other branches of mathematics. Gherard is mentioned in the archive as the translator of (i) works by the Banu Musa brothers, (ii) the Tabulae Jahen (to give them the Latin name as translated by Gherard) of al-Jayyani, (iii) al-Nayrizi's commentary on Euclid's Elements which themselves were based on al-Hajjaj's Arabic translation of the Elements from the Greek, (iv) work by Thabit ibn Qurra, (v) work by Abu Kamil, and (vi) Ahmed ibn Yusuf's work on ratio and proportion. In addition, of course, Gherard translated the Almagst but there is a slight puzzle over this. The translation which we have today appeared in 1175 (or at least this date is stated on the manuscript). We know that Gherard went to Toledo with the intention of translating the Almagest, and it seems beyond belief that such a prolific translator would have waited until he was sixty-one years old before completing what he considered his most important task. It must have been completed 25 years earlier, and this may be a revised edition or simply a new copy which has been given the date of copying instead http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gherard.html (1 of 3) [2/16/2002 11:11:52 PM]

Gherard

of the date of first completion. It seems hard to believe that, given the size of the task that he undertook, Gherard would have had much time for anything other than translating. He did however give public lectures and, in so doing, gained a high reputation as a man of great learning. One of the decisions made by Gherard in his translating was to render the Arabic word for sine into the Latin sinus, from where our sine function comes. It is interesting to realise that had Gherard made a different decision in his translation, this function, which is well-known to all who have made even a brief study of mathematics, would be known by a different name today. In [5] Gherard's contribution is summed up as follows:Because of the abundance and systematic nature of his production, his thoroughly critical approach to textual tradition, and his faithful adherence to literalness, together with a steady flow of the twelfth century, Gerard's translations soon came to obtain the preference of Latin scholars through the succeeding centuries. The tremendous upsurge of interest in Arabic and Greek science and philosophy in medieval universities from the start of the thirteenth century owes its stimulation in greater part to the work of Gerard of Cremona. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. The trigonometric functions 2. How do we know about Greek mathematics?

Other references in MacTutor

Chronology: 1100 to 1300

Other Web sites

The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gherard.html (2 of 3) [2/16/2002 11:11:52 PM]

Gherard

JOC/EFR November 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gherard.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gherard.html (3 of 3) [2/16/2002 11:11:52 PM]

Ghetaldi

Marino Ghetaldi Born: 1566 in Ragusa, Dalmatia (now Dubrovnik, Croatia) Died: 11 April 1626 in Ragusa, Dalmatia (now Dubrovnik, Croatia) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Marino Ghetaldi was educated in Ragusa, then he moved to Rome before travelling extensively in Europe. In Rome he was influenced by Clavius. He then studied at Antwerp and at Paris where he was greatly influenced by Viète. He then spent two years in England. Ghetaldi was offered the chair of mathematics at University of Louvain but he turned the offer down. Ghetaldi's first paper appeared in 1603 and it was on Archimedes. In this he gave an accurate table of specific weights of solids and liquids. In a second work he studied parabolas obtained as sections of a right circular cone. Viète had been working on constructing Apollonius's lost works. In fact Viète was often known as Apollonius Gallus because of this. Ghetaldi took over this work of Viète. He followed Pappus's description of the contents of certain lost books and to do this he had to solve the problems which the books were supposed to contain. In 1607 Ghetaldi produced a pamphlet with 42 problems with solutions Variorum problematum colletio. These contain early application of algebra to geometry. Ghetaldi's work is described in Herigone's 1634 work Cursus mathematicus. It is interesting to look at the kind of person Ghetaldi was. He turned down a chair at Louvain when he was a young man. Descriptions of him say he had the morals of an angel and to be a Ragusan gentleman of discernment. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ghetaldi.html (1 of 2) [2/16/2002 11:11:53 PM]

Ghetaldi

Other Web sites

1. The Galileo Project 2. Dubrovnik, Croatia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ghetaldi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ghetaldi.html (2 of 2) [2/16/2002 11:11:53 PM]

Gibbs

Josiah Willard Gibbs Born: 11 Feb 1839 in New Haven, Connecticut, USA Died: 28 April 1903 in New Haven, Connecticut, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

J Willard Gibbs' father, also called Josiah Willard Gibbs, was professor of sacred literature at Yale University. In fact the Gibbs family originated in Warwickshire, England and moved from there to Boston in 1658. However Gibbs is said to have taken after his mother in physical appearance. Gibbs was educated at the local Hopkins Grammar School where he was described as friendly but withdrawn. His total commitment to academic work together with rather delicate health meant that he was little involved with the social life of the school. In 1854 he entered Yale College where he won prizes for excellence in Latin and Mathematics. Remaining at Yale, Gibbs began to undertake research in engineering, writing a thesis in which he used geometrical methods to study the design of gears. When he was awarded a doctorate from Yale in 1863 it was the first doctorate of engineering to be conferred in the United States. After this he served as a tutor at Yale for three years, teaching Latin for the first two years and then Natural Philosophy in the third year. He was not short of money however since his father had died in 1861 and, since his mother had also died, Gibbs and his two sisters inherited a fair amount of money. From 1866 to 1869 Gibbs studied in Europe. He went with his sisters and spent the winter of 1866-67 in Paris, followed by a year in Berlin and, finally spending 1868-69 in Heidelberg. In Heidelberg he was influenced by Kirchhoff and Helmholtz. Gibbs returned to Yale in June 1869 and, two years later in 1871, he was appointed professor of mathematical physics at Yale. However, as Crowther points out in [2]:He returned more a European than an American scientist in spirit-one of the reasons why http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gibbs.html (1 of 4) [2/16/2002 11:11:55 PM]

Gibbs

general recognition in his native country came so slowly. Rather surprisingly his appointment to the chair at Yale came before he had published any work. Perhaps it is also surprising that Gibbs did not publish his first work until 1873 when he was 34 years old. Few scientists who produce such innovative work as Gibbs did are 34 years of age before producing signs of their genius. Gibbs' important 1873 papers were Graphical Methods in the Thermodynamics of Fluids and A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces. In 1876 Gibbs published the first part of the work for which he is most famous On the Equilibrium of Heterogeneous Substances, publishing the second part of this work in 1878. The first of these papers describes diagrams in thermodynamics. Bumstead (see [11] or [3]) writes:Of the new diagrams which he first described in this paper, the simplest, in some respects, is that in which entropy and temperature are taken as coordinates... the work or heat of any cycle is proportional to its area in any part of the diagram ... it has found most important applications in the study of the steam engine. The second paper extended the diagrams into three dimensions and this work impressed Maxwell so much that he constructed a three dimensional model of Gibbs's thermodynamic surface and, shortly before his death, sent the model to Gibbs. However the third paper is the most remarkable. Bumstead in [11] or [3] writes:It is universally recognised that its publication was an event of the first importance in the history of chemistry. ... Nevertheless it was a number of years before its value was generally known, this delay was due largely to the fact that its mathematical form and rigorous deductive processes make it difficult reading for any one, and especially so for students of experimental chemistry whom it most concerns... Gibbs' work on vector analysis was also of major importance in pure mathematics. He first produced printed notes for the use of his own students in 1881 and 1884 and it was not until 1901 that a properly published version appeared prepared for publication by one of his students. Using ideas of Grassmann, Gibbs produced a system much more easily applied to physics than that of Hamilton. He applied his vector methods to give a method of finding the orbit of a comet from three observations. The method was applied to find the orbit of Swift's comet of 1880 and involved less computation than Gauss's method. A series of five papers by Gibbs on the electromagnetic theory of light were published between 1882 and 1889. His work on statistical mechanics was also important, providing a mathematical framework for quantum theory and for Maxwell's theories. In fact his last publication was Elementary Principles in Statistical Mechanics and this work is a beautiful account putting the foundations of statistical mechanics on a firm foundation. Except for his early years and the three years in Europe, Gibbs spent his whole life living in the same house which his father had built only a short distance from the school Gibbs had attended, the College at which he had studied and the University where he worked the whole of his life. Crowther, in [2], sums up his life as follows:[Gibbs] remained a bachelor, living in his surviving sister's household. In his later years he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gibbs.html (2 of 4) [2/16/2002 11:11:55 PM]

Gibbs

was a tall, dignified gentleman, with a healthy stride and ruddy complexion, performing his share of household chores, approachable and kind (if unintelligible) to students. Gibbs was highly esteemed by his friends, but U.S. science was too preoccupied with practical questions to make much use of his profound theoretical work during his lifetime. He lived out his quiet life at Yale, deeply admired by a few able students but making no immediate impress on U.S. science commensurate with his genius. Bumstead describes Gibbs' personal character in the following glowing terms:Unassuming in manner, genial and kindly in his intercourse with his fellow-men, never showing impatience or irritation, devoid of personal ambition of the baser sort or of the slightest desire to exalt himself, he went far toward realising the ideal of the unselfish, Christian gentleman. In the minds of those who knew him, the greatness of his intellectual achievements will never overshadow the beauty and dignity of his life. The American Mathematical Society named a lecture series in honour of Gibbs. An annual lecture has been given by a distinguished mathematician most years since 1923. You can see a list of the lecturers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (25 books/articles)

Some Quotations (3)

A Poster of J Willard Gibbs

Mathematicians born in the same country

Cross-references to History Topics

An overview of the history of mathematics

Other references in MacTutor

1. Chronology: 1870 to 1880 2. Chronology: 1880 to 1890 3. Chronology: 1900 to 1910

Honours awarded to J Willard Gibbs (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1897

Royal Society Copley Medal

Awarded 1901

Lunar features

Crater Gibbs

Other Web sites

1. Gibbs' constant 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gibbs.html (3 of 4) [2/16/2002 11:11:55 PM]

Gibbs

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gibbs.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gibbs.html (4 of 4) [2/16/2002 11:11:55 PM]

Girard_Albert

Albert Girard Born: 1595 in St Mihiel, France Died: 8 Dec 1632 in Leiden, Netherlands Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Albert Girard was French but went as a religious refugee to the Netherlands. He attended the University of Leiden, where he studied mathematics, entering the University at the age of 22. In fact his first interest was music and he played the lute professionally. Albert Girard worked on algebra, trigonometry and arithmetic. In 1626 he published a treatise on trigonometry containing the first use of the abbreviations sin, cos, tan. He also gave formulas for the area of a spherical triangle. In algebra he had some early thoughts on the fundamental theorem of algebra and translated the works of Stevin in 1625. He is also famed for being the first to formulate the (now well known) inductive definition fn+2 = fn+1 + fn for the Fibonacci sequence. Like many mathematicians of his day Albert Girard was interested in military applications of mathematics and in particular studied fortifications. He translated several works on fortifications some from French to Flemish, others from Flemish to French. It appears that Girard spent some time as an engineer in the Dutch army although this was probably after he published his work on trigonometry. Gassendi, writing to a friend, talks about Girard and refers to his position in the Dutch army. He was described as an engineer rather than as a mathematician on his death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. The fundamental theorem of algebra 2. The trigonometric functions 3. Prime numbers

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Girard_Albert.html (1 of 2) [2/16/2002 11:11:56 PM]

Girard_Albert

Other references in MacTutor

Chronology: 1625 to 1650

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Girard_Albert.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Girard_Albert.html (2 of 2) [2/16/2002 11:11:56 PM]

Girard_Pierre

Pierre Simon Girard Born: 4 Nov 1765 in Caen, France Died: 30 Nov 1836 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Girard was an engineer at the Ecole des Ponts et Chaussés. There he was a close friend and collaborator of de Prony. In 1793 Girard was working on geometry problems along with de Prony and he wrote to him saying:I would have sent you sooner ... the numerical application ... but I have been in Amiens since the beginning of the year ... I thank you for the solution you sent me to the geometry problem... Girard had been in charge of planning and construction of the Amiens canal, which explains his reference to time spent in Amiens. After this de Prony and Girard decided to write a paper in which:... one may learn to find the equation for some solid as one finds the equation for a curved plane. Another project that Girard and de Prony collaborated on was the Dictionnaire des Ponts et Chaussés. In 1798 Girard wrote an important work on the strength of materials. In 1802 he was put in charge of the Ourcq canal project, one of many projects ordered by Napoleon to modernise Paris. While in charge of this project he was assigned a young assistant, namely Cauchy. Pierre Girard wrote extensively on fluids; in particular on flow in capillary tubes. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Girard_Pierre.html (1 of 2) [2/16/2002 11:11:57 PM]

Girard_Pierre

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Girard_Pierre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Girard_Pierre.html (2 of 2) [2/16/2002 11:11:57 PM]

Glaisher

James Whitbread Lee Glaisher Born: 5 Nov 1848 in Lewisham, Kent, England Died: 7 Dec 1928 in Cambridge, Cambridgeshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James Glaisher attended St Paul's School in London, winning a scholarship in 1867 to study at Trinity College, Cambridge. His mathematical researches began while he was still an undergraduate and he wrote a paper on the sine integral, cosine integral and exponential integral giving tables of these integrals which he had computed himself. The paper was communicated to the Royal Society by Cayley. In the final examination of 1871 Glaisher was placed second. Elected to a fellowship at Trinity College, he became a tutor and lecturer and taught at Cambridge all his life. In the same year in which he graduated Glaisher joined the Royal Astronomical Society and so began a long association with that Society. In 1872 he joined the London Mathematical Society. He went on to hold high office in both these Societies, being President of the Royal Astronomical Society from 1886 to 1888 and again from 1901 to 1903, and President of the London Mathematical Society 1884-1886. Glaisher wrote over 400 articles on his main interests of astronomy, special functions, calculation of numerical tables and the history of mathematics. His historical interests were on the early development of numerical computation, Stevin and the beginnings of the decimal system, Napier, Briggs and the beginnings of logarithms as well as the mathematical notation + and -. He applied special functions to problems in number theory, in particular representations of integers as sums of squares. The importance of Glaisher is less in the original research he did, much more in that he brought these mathematical topics into the Cambridge syllabus so setting it up to produce the outstanding English mathematicians who were educated there shortly afterwards. Forsyth, writes in [2]:The earliest years of his teaching at Cambridge were a time of transition in the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Glaisher.html (1 of 3) [2/16/2002 11:11:59 PM]

Glaisher

mathematical ideals of the University. Cayley was almost a voice in the wilderness; Glaisher himself described Cambridge pure mathematics as generals without armies. When he had ceased teaching, Cambridge pure mathematics had marched beyond his active vision mainly under men whom, as students, he had guided at the beginning. His voice was that of a teacher, yet not in the least similar to the great Cambridge coaches, for he contributed to his science and ranged far beyond conventional examination learning. He was a personality in his day; and he left a name, high among the noted names of his generation, in two widely different fields of constructive thought and human activity. Glaisher received many honours. He was elected a Fellow of the Royal Society in 1875. He received the De Morgan Medal of the London Mathematical Society in 1908. He was elected a Fellow of the Royal Society of Edinburgh, the National Academy of Sciences in the United States and was awarded an honorary degree by the University of Dublin on the occasion of their tercentenary. One honour, which it is less than certain he was offered, was the office of Astronomer Royal. It is believed that Glaisher was offered this post when Airy retired in 1871 but that he declined. One of Glaisher's hobbies was collecting works of art. Forsyth [2] describes the affect on his rooms at Cambridge:... his collections never ceased to grow, always under his earnest care. ... his collections outgrew available space, downstairs, upstairs, even in his remote bedroom. He was granted an additional set of rooms at the top of his staircase and next to the upper floor of his own set; they, too, soon were filled. He then hired a sort of warehouse, that also became filled in due course. ... the Fitzwilliam Museum ... granted him a room (also soon filled) in the new wing... As to Glaisher's character, Forsyth tells us [2] that:His smile of appreciation was delightful and infectious; when appreciation waxed into admiration, his attractive eyes would glow in his enthusiasm. Singularly fluent, he never aimed at eloquence in speech, yet dignified passages abound in his formal addresses. ... There was no shred of pomposity in his bearing, which was frank and simple. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles)

Some Quotations (2)

Mathematicians born in the same country Cross-references to History Topics

Mathematical games and recreations

Honours awarded to James Glaisher (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1875

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Glaisher.html (2 of 3) [2/16/2002 11:11:59 PM]

Glaisher

Royal Society Sylvester Medal

Awarded 1913

Fellow of the Royal Society of Edinburgh London Maths Society President

1884 - 1886

LMS De Morgan Medal

Awarded 1908

Other Web sites

Glaisher-Kinkelin Constant

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Glaisher.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Glaisher.html (3 of 3) [2/16/2002 11:11:59 PM]

Glenie

James Glenie Born: 1750 in Fife, Scotland Died: 23 Nov 1817 in Chelsea, London, England Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James Glenie was educated at St Andrews, where he was briefly an assistant in mathematics. Glenie then qualified as an engineer at the Royal Military Academy, Woolwich. He was an artillery officer in the American War of Independence and taught (1805-10) at the East India Company Royal Military College at Addiscombe. Glenie served as a member of the New Brunswick House of Assembly and failed in a business venture, dying in poverty. He wrote several journal articles, also books on Gunnery (1776) and Antecedental Calculus (1793, 1794), an attempt to base fluxional calculus on the binomial theorem rather than on the concept of motion. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Glenie.html (1 of 2) [2/16/2002 11:12:00 PM]

Glenie

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Glenie.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Glenie.html (2 of 2) [2/16/2002 11:12:00 PM]

Gnedenko

Boris Vladimirovich Gnedenko Born: 1 Jan 1912 in Simbirsk (now Ulyanovskaya), Russia Died: 27 Dec 1995 in Moscow, Russia

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Boris Vladimirovich Gnedenko was born in Simbirsk but his family, who were of Ukrainian origin, moved to Kazan when he was only three years old and it was in Kazan that he received his school education. He was a brilliant school pupil. In 1927, when he was only fifteen years old, Gnedenko tried to enter the University of Saratov. However, he was too young to meet the official entry requirements and only after he kept arguing his case did Lunacharskii, the Soviet Minister of Education, personally recommend that he be allowed to enter university. Gnedenko himself wrote of his reasons for studying mathematics at university. Most mathematicians study the subject because they develop such a deep love of the topic. Gnedenko, however, studied mathematics because [11]:... as an abstract science [it] was beyond the understanding of the Communist Party functionaries, thus remaining out of their control. He therefore selected mathematics as his life study. It is worth noting at this stage that Gnedenko's choice of mathematics did not save him from having the most severe problems with the Soviet authorities later in his life, as we describe below. It is also worth commenting that, even if he did not choose mathematics through a love of the subject when he was young, he certainly showed throughout his career that he deeply loved the topic. He graduated in 1930 after only three years of university study. This was because the normal five year course was shortened to three years on account of a decision by the government. They also introduced a strange method of examining students, namely to divide them into groups and have only the group leader sit the examinations for the whole group. Gnedenko, as might be expected, was one of the group leaders who

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gnedenko.html (1 of 5) [2/16/2002 11:12:02 PM]

Gnedenko

took the examinations and his whole group were awarded the high grades which he scored. He did not feel satisfaction, however, rather a feeling of humiliation at the whole procedure. From 1930 Gnedenko taught at the Textile Institute in Ivanovo. This town, east of Moscow, was a centre for the textile industry and it had figures highly in the history of mathematics with people such at Luzin and Khinchin teaching at the polytechnic there. It was during this period that Gnedenko published his first papers on probability and statistics. These resulted from problems he studied concerning the reliability of the machines used in textile manufacture and his first papers were published in 1933. He became deeply interested in probability theory after attending seminars by Kolmogorov and Khinchin. In 1934 Gnedenko decided to resume his university studies at postgraduate level. He was awarded a scholarship which allowed him to undertake research at the Institute of Mathematics at Moscow State University. There he studied under Kolmogorov and Khinchin. Kalashnikov writes in [11] that Gnedenko :... cared for them dearly and held them in the highest affection all his life. ... Kolmogorov was a connoisseur of art, and [Gnedenko and Kolmogorov] talked at length about ancient Russian icons and architecture, poetry and history. Gnedenko made other friendships at this time, becoming a close friend of Slutsky and other mathematicians, and his enthusiasm for his studies was not spoilt by the extremely difficult conditions [19]:... during his first year he and eleven friends shared a room that was so bitterly cold that water left in a glass overnight froze solid. Although Khinchin supervised Gnedenko's studies initially, he left in 1935 to spend two years at Saratov University, and Kolmogorov then took over as Gnedenko's supervisor. In June 1937 Gnedenko was examined on his doctoral dissertation on the theory of infinitely divisible distributions. After the award of his doctorate he was appointed as an assistant researcher in the Mathematics Institute of Moscow State University. During the summer of 1937 Gnedenko went on a hiking expedition to the Caucuses along with some fellow researchers. Although Kolmogorov did not spend as long as the others on the trip, he did take part for a while. It was a period when the researchers were able to talk about their mathematical ideas and profit greatly. However, they talked about many subjects and in the discussions on politics Gnedenko showed that he had little liking for the Soviet policies. This was to have the most severe consequences for him. After taking up his post as a research assistant in November 1937, Gnedenko was conscripted into the Red Army on 1 December. He was sent to Bryansk but on 5 December he was arrested. He had been denounced by one of the members of the Mathematics Department who had been with him on the summer trip. Gnedenko was imprisoned with 120 other prisoners in a cell built for six people and was constantly interrogated about statements he had made on the summer trip. His interrogators demanded that he [24]:... confirm that Kolmogorov was the ringleader of a group of "enemies of the people" centred in the mathematics department. Though interrogated daily over a six-month period, held in grim condition, and promised his release if he cooperated, he refused to admit even

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gnedenko.html (2 of 5) [2/16/2002 11:12:02 PM]

Gnedenko

the possibility of such an interpretation, knowing that there could be no hard evidence, and that the fate of all, himself included, depended on his resolution. Without warning he was released after six months and managed to return to his parent's home. After a great deal of difficulty, with strong support from Kolmogorov and Khinchin (which of course was a brave move by these two who could have paid dearly for giving such support), he was reinstated to his post of Research Assistant in 1938. He retained a "black mark" on his record indicating that he was not to be trusted and, because of this, he was not allowed to join the Soviet army in 1941 when the German forces attacked. In other respects, however, he was able to live a normal life despite this terrifying start to his career. From 1938 Gnedenko lectured at Moscow State University. He married Natalia Konstantinova in 1939 and they had two sons. He submitted his doctoral dissertation in 1941 and was awarded the doctorate in 1942. Of course this was during World War II and for most of the war years he undertook military related research. There was a period when he was evacuated from Moscow and worked further east in the Soviet Union. In 1945, on the recommendation of Kolmogorov, Gnedenko was elected to the Ukrainian Academy of Sciences. He left Moscow and, after a brief spell in Kiev, he became professor at Lvov University. There he met Banach and [11]:... he retained strong impressions of this meeting all his life. In 1949 Gnedenko was appointed as Head of the Physics, Mathematics and Chemistry Section of the Ukrainian Academy of Sciences in Kiev and he became Director of the Kiev Institute of Mathematics. He held these posts until 1960 when he returned to Moscow University, becoming Head of the Department of Probability Theory in 1966. He held this post for thirty years until his death. Gnedenko produced a remarkable number of papers and books during his lifetime. Certainly there are over 200 items in his list of publications, but this number increases substantially if further editions of books, translations into different languages etc. are taken into account. In fact he wrote several important books which we shall say a little about. In 1949 he published a work, jointly with Kolmogorov, Limit Distributions for Sums of Independent Random Variables which contains a description of much of his early research. The book is based on courses given by Gnedenko and Kolmogorov at Moscow and Lvov universities. It has three parts, the first part consisting of three introductory chapters while the second part is on general limit theorems. Included in this second part are sections on: general limit theorems for sums with independent summands; the concept of infinitely small summands; conditions necessary and sufficient that their sums have a given limiting distribution; convergence to the normal, Poisson, and unit distributions; and limit theorems for cumulative sums. The third part of the work is on summands with a common distribution function and includes discussion of principal limit theorems and convergence to the normal law. One of Gnedenko's most famous books is Course in the Theory of Probability which first appeared in 1950. Written in a clear and concise manner, the book was very successful in providing a first introduction to probability and statistics. It has gone through six Russian editions and has been translated into English, German, Polish and Arabic. In 1966, along with I V Kovalenko, he published Introduction to queuing theory. A reviewer described the work as follows:This is an attractively written systematic exposition of the basic probabilistic methods of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gnedenko.html (3 of 5) [2/16/2002 11:12:02 PM]

Gnedenko

congestion theory. At the same time the book has the unusual merit of going far enough into explicit computations to be of practical interest to the applied operations analyst. The book is written on a mature level, but little probability theory beyond very elementary concepts or very intuitive ones is presupposed. In his early work Gnedenko had been interested in probability as an abstract topic. However, in his later work his interests turned more towards applications to areas such as reliability and quality control. He wrote an important text Mathematical methods of reliability in 1965 with Belyaev and Solov'yev. Gnedenko was not only interested in research into mathematical topics, but he was also interested in the teaching of these topics. For example, with Solov'yev, he wrote the elementary work Mathematics and reliability theory in 1982 which aimed to give a popular account of the mathematical theory of reliability. In an elementary way the book describes the basic notions of reliability, lifetime distributions, redundant systems, renewal and maintenance theory, inclusion-exclusion principle, and the estimation of reliability. Another work, this time aimed at secondary school pupils, was An introduction to the speciality of mathematics published in 1991. The book was written for pupils who love mathematics and want to become mathematicians. The topics covered in this book are interesting in themselves and also, in the present context, because they tell us a good deal about Gnedenko's approach to mathematics. The topics include: the role of mathematics in science, technology, and life; a definition of "mathematics"; the abstract nature of the subject; mathematics as the language of science; interesting problems from various mathematical sciences; mathematical models; mathematical education; the history of mathematics including an appendix on Moscow University. Gnedenko's interest in the history of mathematics extended well beyond his text aimed at secondary school pupils. He published much on this topic (we list at least twelve articles in the References sections of the archive authored by Gnedenko) including the important Outline of the History of Mathematics in Russia which was not published until 1946 although he wrote it before the start of World War II. It is a fascinating book which looks at the history of mathematics in Russia in its cultural background. For example in discussing the pre-eighteenth century, Gnedenko considers the cultural influence of the medieval church and the role of the calendar. He discusses the founding of the St Petersburg Academy of Sciences, paying particular attention to the life and work of Euler. The work of many famous mathematicians is discussed in detail such as that of Lobachevsky, Bunyakovsky, Ostrogradski, Chebyshev, Markov, Lyapunov, and Kovalevskaya. Perhaps given Gnedenko's treatment in prison just before he wrote the book, it is not surprising that he makes every effort to be "politically correct" and plays down contributions by western mathematicians. For example he even manages to discuss non-euclidean geometry and Lobachevsky's contributions without even mentioning Bolyai. To complete our account of Gnedenko we should give a feeling for his personality. For this we quote from [11] (which is itself quoted in [24]). Gnedenko's favourite topics of conversation were:... the inter-relationship of mathematics and its fields of application, problems of education, history, books, poetry, art, and many others. Speaking on any topic, Gnedenko did not insist on his insights although he did not hide them. He knew many interesting stories and told them with a deep sense of humour. Gnedenko enjoyed classical music, and had a large collection of records. Sometimes, when he was especially proud of some new purchase, he would propose that you listen to it. His taste in music was traditional, and he was not afraid to confess that he could not understand some modern composers. On other occasions, he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gnedenko.html (4 of 5) [2/16/2002 11:12:02 PM]

Gnedenko

would ask you to look through a new album of painting of old masters. This was, in fact, the method of communication typical of the older generation of Russian intellectuals. Article by: J J O'Connor and E F Robertson List of References (24 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gnedenko.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gnedenko.html (5 of 5) [2/16/2002 11:12:02 PM]

Godel

Kurt Gödel Born: 28 April 1906 in Brünn, Austria-Hungary (now Brno, Czech Republic) Died: 14 Jan 1978 in Princeton, New Jersey, USA

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kurt Gödel attended school in Brünn, completing his school studies in 1923. His brother Rudolf Gödel said:Even in High School my brother was somewhat more one-sided than me and to the astonishment of his teachers and fellow pupils had mastered university mathematics by his final Gymnasium years. ... Mathematics and languages ranked well above literature and history. At the time it was rumoured that in the whole of his time at High School not only was his work in Latin always given the top marks but that he had made not a single grammatical error. Kurt entered the University of Vienna in 1923. He was taught by Furtwängler, Hahn, Wirtinger, Menger, Helly and others. As an undergraduate he took part in a seminar run by Schlick which studied Russell's book Introduction to mathematical philosophy. Olga Tausky-Todd, a fellow student of Gödel's, wrote:It became slowly obvious that he would stick with logic, that he was to be Hahn's student and not Schlick's, that he was incredibly talented. His help was much in demand. He completed his doctoral dissertation under Hahn's supervision in 1929 and became a member of the faculty of the University of Vienna in 1930, where he belonged to the school of logical positivism until 1938. He is best known for his proof of Gödel's Incompleteness Theorems. In 1931 he published these results in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme . He proved http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html (1 of 3) [2/16/2002 11:12:04 PM]

Godel

fundamental results about axiomatic systems showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved. This ended a hundred years of attempts to establish axioms to put the whole of mathematics on an axiomatic basis. One major attempt had been by Bertrand Russell with Principia Mathematica (1910-13). Another was Hilbert's formalism which was dealt a severe blow by Gödel's results. The theorem did not destroy the fundamental idea of formalism, but it did demonstrate that any system would have to be more comprehensive than that envisaged by Hilbert's. Gödel's results were a landmark in 20th-century mathematics, showing that mathematics is not a finished object, as had been believed. It also implies that a computer can never be programmed to answer all mathematical questions. Gödel met Zermelo in Bad Elster in 1931. Olga Taussky-Todd, who was at the same meeting, wrote:The trouble with Zermelo was that he felt he had already achieved Gödel's most admired result himself. Scholz seemed to think that this was in fact the case, but he had not announced it and perhaps would never have done so. ... The peaceful meeting between Zermelo and Gödel at Bad Elster was not the start of a scientific friendship between two logicians. In 1933 Hitler came to power. At first this had no effect on Gödel's life in Vienna. He had little interest in politics. However after Schlick, whose seminar had aroused Gödel's interest in logic, was murdered by a National Socialist student, Gödel was much affected and had his first breakdown. His brother Rudolf wrote This event was surely the reason why my brother went through a severe nervous crisis for some time, which was of course of great concern, above all for my mother. Soon after his recovery he received the first call to a Guest Professorship in the USA. In 1934 Gödel gave a series of lectures at Princeton entitled On undecidable propositions of formal mathematical systems. At Veblen's suggestion Kleene, who had just completed his Ph.D. this at Princeton, took notes of these lectures which have been subsequently published. He returned to Vienna, married Adele Porkert in 1938, but when the war started he was fortunate to be able to return to the USA although he had to travel via Russia and Japan to do so. In 1940 Gödel emigrated to the United States and held a chair at the Institute for Advanced Study in Princeton, from 1953 to his death. He received the National Medal of Science in 1974. His work Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory (1940) is a classic of modern mathematics. His brother Rudolf, himself a medical doctor, wrote:My brother had a very individual and fixed opinion about everything and could hardly be convinced otherwise. Unfortunately he believed all his life that he was always right not only in mathematics but also in medicine, so he was a very difficult patient for doctors. After severe bleeding from a duodenal ulcer ... for the rest of his life he kept to an extremely strict (over strict?) diet which caused him slowly to lose weight. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html (2 of 3) [2/16/2002 11:12:04 PM]

Godel

Towards the end of his life Gödel became convinced that he was being poisoned and, refusing to eat to avoid being poisoned, starved himself to death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles)

Some Quotations (4)

A Poster of Kurt Gödel

Mathematicians born in the same country

Cross-references to History Topics

The beginnings of set theory

Other references in MacTutor

Chronology: 1930 to 1940

Honours awarded to Kurt Gödel (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1968

AMS Gibbs Lecturer

1951

Other Web sites

1. Vienna, Austria (The home page of the Kurt Gödel Society) 2. Karlis Podnieks (A hypertext introduction to Gödel's theorem) 3. Lulea, Sweden (The incompletenes theorem) 4. Princeton University 5. The Guardian, UK 6. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR December 1996 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Godel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html (3 of 3) [2/16/2002 11:12:04 PM]

Gohberg

Israel Gohberg Born: 23 Aug 1928 in Russia

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Israel Gohberg came from a Jewish family. He attended school in the newly created Republic of Kirgiz. In 1926 the status of the autonomous Kirgiz oblast had been changed into that of an autonomous republic, and in 1936 a full union republic had been created, the Kirgiz Soviet Socialist Republic. Gohberg then entered the College of Education in Frunze. Frunze was named Bishkek up to 1926 when it became the capital of the Kirgiz Autonomous Soviet Socialist Republic and was named after the revolutionary leader Mikhail Frunze. Today it has been renamed Bishkek. Gohberg's teachers at the College of Education were so impressed with his abilities that persuaded him to move to a university. Thus, after two years at the College in Frunze, he went to the University of Kishinyov, in Kishinyov the capital of Moldova, which had opened in 1945. Today Kishinyov had reverted to its original name of Chisinau. After graduating from University of Kishinyov, Gohberg went to the University of Leningrad to study for his doctorate which he received in 1954. Then he moved to Moscow University where he received his habilitation. He taught at the Teachers' College in Soroki and later at the teacher-training institute in Balti in northern Moldova. Gohberg was appointed to the Academy of Sciences in Kishinyov where, in 1964, he became the Head of Functional Analysis. He also held a chair of mathematics at the University of Kishinyov. In [2] Gohberg describes some of the difficulties encountered by Jews in the Soviet academic world, particularly those he suffered from 1968. In [2] he points out that there were no Jews in the Mathematics section of the USSR Academy of Sciences between 1968 and 1984. By 1969 Gohberg had made the decision to try to emigrate to Israel but only after great difficulty, and a long wait, did Gohberg and his family obtain the visa necessary for them to leave. After obtaining the visa in 1974, he emigrated to Israel. Once in Israel, he was appointed to a professorship at Tel Aviv University and he also was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gohberg.html (1 of 3) [2/16/2002 11:12:06 PM]

Gohberg

appointed to the Weizman Institute in Rehovot. In 1983 he was appointed to the Free University (Vrije Universiteit) of Amsterdam. He has been a visiting professor at many universities such as the University of Calgary in Canada, and College Park Maryland in the United States. In addition to Gohberg's outstanding work in analysis and in particular in operator theory and matrix methods, he founded the major international journal Integral equations and operator theory in the late 1980s. The journal:... is devoted to the publication of current research in integral equations, operator theory and related topics with emphasis on the linear aspects of the theory. Gohberg has an amazing publication record with over 460 papers listed in Mathematical Reviews. Notice that this is around 100 more than the list given in [4] which is consistent with his almost unequalled record of averaging 10 papers per year. He has supervised 40 doctoral students and exercised an amazing influence on his area. Israel Gohberg had been described as one of the exceptional mathematicians of our time, comparable in his manner perhaps only to Paul Erdös. Three qualities would appear to account for his unique influence: he is a charismatic and inspirational man who quickly pulls others into his orbit, keeping acolytes motivated for long periods of time; he possesses an uncanny instinct for what is mathematically possible and productive; and he has the talent and stamina to see that his ideas and those of his co-authors are realised, and then communicated through publications to the larger interested audience. Gohberg has received many awards for his outstanding work including the Alexander von Humboldt Prize in 1992 and an honorary degree from the Institute of Technology in Darmstadt in June 1997. His awards prior to 1988 are listed in [3]. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gohberg.html (2 of 3) [2/16/2002 11:12:06 PM]

Gohberg

JOC/EFR November 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gohberg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gohberg.html (3 of 3) [2/16/2002 11:12:06 PM]

Goldbach

Christian Goldbach Born: 18 March 1690 in Königsberg, Prussia (now Kaliningrad, Russia) Died: 20 Nov 1764 in Moscow, Russia Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

In 1725 Christian Goldbach became professor of mathematics and historian at St Petersburg. Then, in 1728, he went to Moscow as tutor to Tsar Peter II. He travelled round Europe meeting mathematicians. He met Leibniz, Nicolaus(I) Bernoulli, Nicolaus(II) Bernoulli, de Moivre, Daniel Bernoulli and Hermann. Goldbach did important work in number theory, much of it in correspondence with Euler. He is best remembered for his conjecture, made in 1742 in a letter to Euler and still an open question, that every even integer greater than 2 can be represented as the sum of two primes. Goldbach also conjectured that every odd number is the sum of three primes. Vinogradov made progress on this second conjecture in 1937. Goldbach also studied infinite sums, the theory of curves and the theory of equations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Fermat's last theorem 2. Topology enters mathematics 3. The fundamental theorem of algebra 4. Prime numbers

Other references in MacTutor

Chronology: 1740 to 1760

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Goldbach.html (1 of 2) [2/16/2002 11:12:07 PM]

Goldbach

Other Web sites

1. The Prime Pages (Goldbach's conjecture) 2. The Times (London) (A $1 000 000 prize for proving it) 3. J Richstein including Goldbach's letter to Euler. 4. Linda Hall Library (Star Atlas) 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Goldbach.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Goldbach.html (2 of 2) [2/16/2002 11:12:07 PM]

Goldstein

Sydney Goldstein Born: 3 Dec 1903 in Hull, England Died: 22 Jan 1989 in Belmont, Massachusetts, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sydney Goldstein entered the University of Leeds in 1921 to study mathematics. He moved to St John's College, Cambridge graduating in 1925. Based on his very successful undergraduate career, Goldstein was awarded the Isaac Newton Studentship to continue undertaking research in applied mathematics under Harold Jeffreys. His doctorate was awarded for a thesis on the Mathieu (Emile Mathieu) functions. He was appointed Rockefeller Research Fellow and spent a year working in Göttingen. In 1929 he was made a fellow of St John's College, Cambridge and, in the same year, he was appointed to a Lectureship in Mathematics at the University of Manchester. Manchester had a profound influence on Goldstein. The influence of Reynolds and Lamb in fluid dynamics was still felt there and had a strong effect on Goldstein. He moved to Cambridge in 1931 and took over, on Lamb's death, the editorship of Modern Developments in Fluid Dynamics, this important work appearing in 1938. During the war years he worked on boundary layer theory at the National Physical Laboratory. Then in 1945, the University of Manchester made two inspiring appointments to the Department of Mathematics, Max Newman to the chair of Pure Mathematics and Goldstein to the chair of Applied Mathematics. However in 1950, Goldstein who was of strong Jewish beliefs, accepted the chairmanship of the department of mathematics at Technion in Israel. His stay in Israel was not very long however, and in 1955 he accepted a chair of Applied Mathematics at Harvard. He had made a major role in setting up he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Goldstein.html (1 of 2) [2/16/2002 11:12:08 PM]

Goldstein

academic framework of Technion but the administrative work was very heavy and led to his happily accepting the chair in the USA. Goldstein's work in fluid dynamics is of major importance. He is described in [1] as:... one of those who most influenced progress in fluid dynamics during the 20th century. He studied numerical solutions to steady-flow laminar boundary-layer equations in 1930. In 1935 he published work on the turbulent resistance to rotation of a disk in a fluid. His work was important in aerodynamics, a subject in which Goldstein was extremely knowledgeable. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Sydney Goldstein (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1937

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Goldstein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Goldstein.html (2 of 2) [2/16/2002 11:12:08 PM]

Gompertz

Benjamin Gompertz Born: 5 March 1779 in London, England Died: 14 July 1865 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Benjamin Gompertz came from a family of merchants who left Holland and settled in England. He was one of three sons born in England to the Dutch family which, although from Holland, was Jewish. Gompertz was self educated, learning mathematics by reading Newton and Maclaurin. He had to take this route since he was denied admission to universities since he was Jewish. In fact he was greatly helped in his mathematical education by the Society of Mathematicians of Spitalfields which was later to become the London Mathematical Society. Gompertz, writing to De Morgan, explained how he came to be a member of the Society [5]:As to the Mathematical Society, of which I was a member when only 18 years of age, having been, contrary to the rules, elected under the age of 21. How I came to be a member of that Society - and continued so until it joined the Astronomical Society, and then was the President - was: I happened to pass a bookseller's small shop of second hand books, kept by a poor taylor, but a good mathematician, John Griffiths. I was very pleased to meet a mathematician, and I asked him if he would give me some lessons; and his reply was that I was more capable to teach him, but he belonged to a society of mathematicians, and he would introduce me. I accepted the offer, and I was elected, and had many scholars then to teach, as one of the rules was, if a member asked for information, and applied to one who could give it, he was obliged to give it, or fine one penny. In 1810 Gompertz married Abigail Montefiore who came from a wealthy Jewish family with strong links with the stock exchange. Gompertz himself joined the stock exchange in 1810 and he became a Fellow of the Royal Society in 1819. The following year he read a paper to the Society which applied the differential calculus to the calculation of life expectancy. In 1824 he was appointed as actuary and head http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gompertz.html (1 of 3) [2/16/2002 11:12:10 PM]

Gompertz

clerk of the Alliance Assurance Company. Gompertz applied the calculus to actuarial questions and he is best remembered for Gompertz's Law of Mortality. Gompertz, in 1825, showed that the mortality rate increases in a geometric progression. Hence, when death rates are plotted on a logarithmic scale, a straight line known as the Gompertz function is obtained. It is the most informative actuarial function for investigating the ageing process. The slope of the Gompertz function line indicates the rate of actuarial ageing. The differences in longevity between species are the result primarily of differences in the rate of ageing and are therefore expressed in differences in slope of the Gompertz function. Tropp writes in [1]:[Gompertz's] rigid adherence to Newton's fluxional notation prevented wide recognition of his accomplishment, but he must be rated as a pioneer in actuarial science and one of the great amateur scholars of his day. Gompertz also wrote about scientific instruments, writing Theory of astronomical instruments (1822), A new instrument called the differential sextant (1825) and On the converted pendulum (1829). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1820 to 1830

Honours awarded to Benjamin Gompertz (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1819

Other Web sites

1. Statlib (Gompertz's Law of Mortality) 2. Argonne National Laboratory, USA

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gompertz.html (2 of 3) [2/16/2002 11:12:10 PM]

Gompertz

JOC/EFR September 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gompertz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gompertz.html (3 of 3) [2/16/2002 11:12:10 PM]

Goodstein

Reuben Louis Goodstein Born: 15 Dec 1912 in London, England Died: 8 March 1985 in Leicester, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Educated at St Paul's School London Reuben Goodstein won Scholarships and a prize for an essay on divergent series. He entered Magdalene College Cambridge in 1931 and his special subject in his undergraduate course was analysis. Goodstein then did research at Cambridge on transfinite numbers under Littlewood's supervision. After receiving his Master's Degree he accepted a post in pure and applied mathematics at Reading in 1935. During the war years he had to cover teaching in a wide range of pure and applied mathematics topics, among them engineering, applied mathematics, analysis and group theory. While undertaking this strenuous teaching load Goodstein carried out research which he submitted for a doctorate to the University of London in 1946. Goodstein was appointed professor at Leicester in 1948 and he remained there for the rest of his life. Goodstein worked on mathematical logic, in particular ordinal numbers, recursive arithmetic and analysis and the philosophy of mathematics. He was extremely interested in the teaching of mathematics and [1] lists 66 papers which he published on teaching mathematics at both school and university level. His 11 textbooks were, according to [1], characterised by their clear style and ingenious methods to elucidate difficult points. He was disappointed that his mathematical analysis text ... which presented a novel approach to elementary differential and integral calculus ... did not find favour. It is interesting to note that he was the first mathematical logician to hold a chair at a British University

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Goodstein.html (1 of 2) [2/16/2002 11:12:12 PM]

Goodstein

and he exercised an important influence on the development of this topic in the UK. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Goodstein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Goodstein.html (2 of 2) [2/16/2002 11:12:12 PM]

Gopel

Adolph Göpel Born: 29 Sept 1812 in Rostock, Germany Died: 7 June 1847 in Berlin, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adolph Göpel had an uncle who was the British consul in Corsica. This enabled Göpel to spend several years in Italy in his youth. He attended mathematics lectures in Pisa during 1825-26 although he was only 13 years old at the time. In 1829 Göpel entered the University of Berlin where he continued to study after taking his first degree and he was awarded a doctorate in 1835. After this he taught at the Werder Gymnasium and at the Royal Realschule. He then worked as an official in the Royal Berlin Library and had little contact with his mathematical colleagues although he was friends with Crelle for a while. Göpel's doctoral dissertation studied periodic continued fractions of the roots of integers and derived a representation of the numbers by quadratic forms. He wrote on Steiner's synthetic geometry and an important work, published after his death, continued the work of Jacobi on elliptic functions. This work was published in Crelle's Journal in 1847. W Burau in [1] writes:Göpel started from 16 theta functions in two variables ... and showed that their quotients are quadruply periodic. Of the squares of these 16 functions, four proved to be linearly independent. Göpel linked four more of these quadratics through a homogeneous fourth degree relation, later named the 'Göpel relation' which coincides with the equation of the Kummer surface. Göpel ... finally, after ingenious calculations, obtained the result that the quotients of two theta functions are solutions of the Jacobian problem for p = 2. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gopel.html (1 of 2) [2/16/2002 11:12:13 PM]

Gopel

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gopel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gopel.html (2 of 2) [2/16/2002 11:12:13 PM]

Gordan

Paul Albert Gordan Born: 27 April 1837 in Breslau, Germany (now Wroclaw, Poland) Died: 21 Dec 1912 in Erlangen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Gordan's father, David Gordan,was a merchant in Breslau, and his mother was Friedericke Friedenthal. Paul was educated in Breslau where he attended the Gymnasium, going on to study at the business school. At this stage Gordan was not heading for an academic career and he worked for several years in banks. He had, however, attended lectures by Kummer on number theory at the University of Berlin in 1855 and his interest in mathematics was strongly encouraged by N H Schellbach who acted as a private tutor to Gordan. His university career began at the University of Breslau but, as almost all German students did at this time, he undertook part of his university studies at different universities. Moving to Königsberg, Gordan studied under Jacobi, then he moved to Berlin where he began to become interested in problems concerning algebraic equations. Returning to the University of Breslau he submitted a dissertation on geodesics of spheroids in 1862. This was a fine piece of work and the dissertation, which employed methods devised by Lagrange and Jacobi, was awarded a prize by the Philosophy Faculty at Breslau. As soon as Gordan had completed his dissertation he went to visit Riemann at Göttingen. However, Riemann had caught a heavy cold which turned to tuberculosis so Gordan's visit was cut short. In 1863 Clebsch invited Gordan to come to Giessen. He lectured at Giessen, being promoted to associate professor in 1865. In 1869, while still at Giessen, Gordan married Sophie Deurer who was the daughter of the professor of law there.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gordan.html (1 of 3) [2/16/2002 11:12:15 PM]

Gordan

The first work which Gordan and Clebsch worked on in Giessen was the theory of abelian functions. They jointly wrote the treatise Theorie der Abelschen Funktionen which was published in 1866. The topic for which Gordan is most famous is invariant theory and Clebsch introduced him to this topic in 1868. For the rest of his career, although Gordan did not work exclusively on this topic, it would be fair to say that invariant theory dominated his mathematical research. For the next twenty years Gordan tried to prove the finite basis theorem conjecture for n-ary forms. He made a good start to solving this problem for n = 2 when he found a constructive proof of a finite basis for binary forms. The higher cases defeated him, however, and despite introducing more and more complicated computational techniques he failed to construct a finite basis. Gordan did not undertake the bulk of this work at Giessen, however, for he moved to Erlangen in 1874 to become professor of mathematics at the university. When Gordan was appointed Klein held the chair of mathematics at Erlangen but he moved in the following year to the Technische Hochschule at Munich. In the year 1874-75 when Gordan and Klein were together at Erlangen they undertook a joint research project examining groups of substitutions of algebraic equations. They investigated the relationship between PSL(2,5) and equations of degree five. Later Gordan went on to examine the relation between the group PSL(2,7) and equations of degree seven, then he studied the relation of the group A6 to equations of degree six. In 1888 Hilbert proved the finite basis theorem, only giving an existence proof, not one which allowed the basis to be constructed. Hilbert submitted his results to Mathematische Annalen and, since Gordan was the leading world expert on invariant theory, he was asked his opinion of the work. Gordan found Hilbert's revolutionary approach difficult to appreciate and not at all consistent with his ideas of constructive mathematics. After refereeing the paper, he sent his comments to Klein:The problem lies not with the form ... but rather much deeper. Hilbert has scorned to present his thoughts following formal rules, he thinks it suffices that no one contradict his proof ... he is content to think that the importance and correctness of his propositions suffice. ... for a comprehensive work for the Annalen this is insufficient. However, Hilbert had learnt through his friend Hurwitz about Gordan's report to Klein, and he wrote to Klein in forceful terms:... I am not prepared to alter or delete anything, and regarding this paper, I say with all modesty, that this is my last word so long as no definite and irrefutable objection against my reasoning is raised. Klein was in a difficult position. Gordan was recognised as the leading world expert on invariant theory and he was also a close friend of Klein's. However Klein recognised the importance of Hilbert's work and assured him that it would appear in the Annalen without any changes whatsoever, as indeed it did. Gordan also worked on algebraic geometry and he gave simplified proofs of the transcendence of e and . One rather unsuccessful idea which he embarked on quite late in his career was to apply invariant theory to chemical valences. His work on this, however, came in for criticism from mathematicians such as Eduard Study, and chemists were totally unimpressed with the ideas too. This was rather an unfortunate episode since it resulted in Gordan, who had enjoyed a fine reputation, losing respect from his colleagues. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gordan.html (2 of 3) [2/16/2002 11:12:15 PM]

Gordan

The style of Gordan's mathematics, which lead to his difficulties with Hilbert's basis theorem, is described in [1]:The overall style of Gordan's mathematical work was algorithmic. He shied away from presenting his ideas in informal literary forms. He derived his results computationally, working directly towards the desired goal without offering explanations of the concepts that motivated the work. Emmy Noether was Gordan's only doctoral student, but one would have to say that the lack of numbers is more than made up for by the remarkable quality of that one student who would do so much to set algebra on the path it is still on today. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

A Quotation

A Poster of Paul Gordan

Mathematicians born in the same country

Other Web sites

Clark Kimberling

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gordan.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gordan.html (3 of 3) [2/16/2002 11:12:15 PM]

Gorenstein

Daniel Gorenstein Born: 1 Jan 1923 in Boston, Massachusetts, USA Died: 26 Aug 1992 in USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Daniel Gorenstein became interested in mathematics at a young age. He taught himself calculus at the age of 12 years. His secondary schooling was at Boston Latin School from which he graduated, entering Harvard University. There he worked under Saunders MacLane and became interested in finite groups, the subject he would return to after a few years studying algebraic geometry to make it his life's major work. Gorenstein graduated in 1943 and, to contribute to the war effort, he accepted a teaching position at Harvard to teach mathematics to army personnel. At the end of World War II, Gorenstein returned to Harvard, this time to undertake graduate work with Zariski. This work led to a thesis on algebraic geometry in which he introduced rings which are now named after him. After the award of his doctorate in 1950 Gorenstein accepted a post at Clark University in 1951. He remained there, except for the year 1958-59 which he spent as a visiting professor at Cornell, until he moved to North Eastern University in 1964. In 1968-69 he was a member of the Institute for Advanced Study at Princeton. After five years at North Eastern, Gorenstein accepted a professorship at Rutgers University where he remained until his death. At Rutgers, Gorenstein was chairman of the Mathematics Department from 1975 until 1981. In 1984 Rutgers appointed him as Jacqueline B Lewis Professor of Mathematics and then, in 1989, he became Director of the newly founded National Science Foundation Science Technology Center in Discrete Mathematics and Theoretical Computer Science. This Center was a joint project between Rutgers University and Princeton University with AT&T Bell Laboratories and Bell Communications research as

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gorenstein.html (1 of 4) [2/16/2002 11:12:16 PM]

Gorenstein

partners. He returned from algebraic geometry to his early research topic of finite groups in 1957, stimulated by a collaboration with Herstein. His involvement in the classification of finite simple groups began in the year 1960-61 when he attended the Group Theory Year at the University of Chicago. He wrote [3]:My first foray into simple group theory dated from the famous 1960-61 group theory year at the University of Chicago, during which Walter Feit and John Thompson settled the solvability of groups of odd order. It was there that I began a long collaboration with John Walter and met many of the leaders in the field: Brauer, Suzuki, Graham Higman, and Ito. (It was only somewhat later that I met Philip Hall and Wielandt.) Alperin was spending the year in Chicago to write his thesis with Higman, while still a graduate student at Princeton. The classification of finite simple groups involved contributions by a host of mathematicians world wide. However it was Gorenstein who took an overview of the whole project and steered it to a successful conclusion. If is for the classification of finite simple groups that his name will always be remembered, certainly the mathematical achievement of the 20th century. If Gorenstein was the man with the best overview of this achievement, then surely we can do no better than to quote his own description of events. We quote from his response to the award of a Steele prize in 1989 given in [3]:Largely under the impetus of the odd order theorem, there was an awakening interest in finite group theory. Throughout the next decade and a half a long list of gifted young mathematicians, who were to play a prominent role in the classification proof, were attracted to the field. In the United States, John Thompson had a string of outstanding graduate students: Sims, Goldschmidt, Lyons, Griess. Glauberman was a student of Bruck's at the beginning of the period and Aschbacher near the end. Ronald Solomon wrote his thesis with Feit, Seitz with Curtis, Stephen Smith with Higman at Oxford, O'Nan with me, and Shult was essentially self-taught. But the attraction was not limited to the United States. Janko in Australia. Conway in England, and Fischer in Germany, each discovering three new sporadic groups, stimulated considerable additional interest, leading to an intensification of the search for further simple groups. Tits (entering the field somewhat earlier) had deepened our understanding of the Chevalley groups and their Steinberg-Suzuki-Ree variations, Bender in Germany was to prove the fundamental strongly embedded subgroup classification theorem, and Harada was beginning his career in Japan. At the end of the period, there were a number of others: Foote from Canada working with John Thompson in Cambridge, England, Geoffrey Mason from England, coming to the United States, and writing his thesis with Fong, himself a student of Brauer, and in Germany, Timmesfeld and Stellmacher, students of Fischer, and Stroth, a student of Huppert, but writing his thesis on a problem suggested by Held, who had himself been a student of Janko. There were a great many other group theorists as well who made significant contributions to the classification proof. But it was Aschbacher's entry into the field in the early 1970s that irrevocably altered the simple group landscape. Quickly assuming a leadership role in a single minded pursuit of the full classification theorem, he was to carry the entire "team" along with him over the following decade until the proof was completed.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gorenstein.html (2 of 4) [2/16/2002 11:12:16 PM]

Gorenstein

I was fortunate indeed to have interacted in one way or another during this twenty year period with most of the mathematicians I have mentioned. Simultaneously with this burgeoning research effort, finite simple group theory was establishing a well-deserved reputation for inaccessibility because of the inordinate lengths of the papers pouring out. The 255 page proof of the odd order theorem, filling an entire issue of the Pacific Journal, had set the tone, but it was by far not the longest paper. Moreover, the techniques being developed, no matter how seemingly powerful for the problems at hand, appeared to have no applications outside finite group theory. Although there was admiration within the mathematical community for the achievements, there was also a growing feeling that finite group theorists were off on the wrong track. No mathematical theorem could require the number of pages these fellows were taking! Surely they were missing some geometric interpretation of the simple groups that would lead to a substantially shorter classification proof. The view from the inside was quite different: all the moves we were making seemed to be forced. It was not perversity on our part, but the intrinsic nature of the problem that seemed to be controlling the directions of our efforts and shaping the techniques being developed. Gorenstein's books on finite groups and the classification of finite simple groups are Finite groups (1968), Finite simple groups : an introduction to their classification (1982), The local structure of finite groups of characteristic 2 type (jointly written with Richard Lyons) (1983) and The classification of the finite simple groups (jointly written with Richard Lyons and Ronald Solomon) (1994). Gorenstein received many honours for his work. In 1972-73 he was both Guggenheim Fellow and Fulbright Research Scholar. During 1978 he was Sherman Fairchild Distinguished Scholar at the California Institute of technology. He was elected to the National Academy of Sciences (1978) and the American Academy of Arts and Sciences (1978). He also received the Steele Prize from the American Mathematical Society in 1989. There are three Steele Prizes awarded and Gorenstein received the award for expository mathematical writing at the American Mathematical Society Summer Meeting in Boulder, Colorado, USA. The citation for the award states [3]:Gorenstein was a major figure in setting the direction of the classification program. He coordinated the activities in the program, functioning as the "coach" to a team, with optimism, perseverance, and technical power. His expository articles and books ... are beautiful accounts of this fantastic intellectual adventure. His presentation of theorems and definitions, as well as the flow of argument and the evolution of ideas, is precise and generous and reaches out to the reader. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gorenstein.html (3 of 4) [2/16/2002 11:12:16 PM]

Gorenstein

Honours awarded to Daniel Gorenstein (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1985

Other Web sites

AMS (Classification of finite simple groups)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gorenstein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gorenstein.html (4 of 4) [2/16/2002 11:12:16 PM]

Gosset

William Sealey Gosset Born: 13 June 1876 in Canterbury, England Died: 16 Oct 1937 in Beaconsfield, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Gosset was educated at Winchester, then entering New College Oxford where he studied chemistry and mathematics. While there he studied under Airy. Gosset obtained a post as a chemist in the Guinness brewery in Dublin in 1899 and did important work on statistics. He invented the t-test to handle small samples for quality control in brewing. He wrote under the name "Student". Gosset discovered the form of the t distribution by a combination of mathematical and empirical work with random numbers, an early application of the Monte-Carlo method. Writing in [7], McMullen says:To many in the statistical world "Student" was regarded as a statistical advisor to Guinness's brewery, to others he appeared to be a brewer devoting his spare time to statistics. ... though there is some truth in both these ideas they miss the central point, which was the intimate connection between his statistical research and the practical problems on which he was engaged. ... "Student" did a very large quantity of ordinary routine as well as his statistical work in the brewery, and all tat in addition to consultative statistical work and to preparing his various published papers. From 1922 he got a statistical assistant at the brewery, and he slowly built up a small statistics department which he ran until 1934. Gosset certainly did not work in isolation. He corresponded with a large number of statisticians and he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gosset.html (1 of 2) [2/16/2002 11:12:18 PM]

Gosset

often visited his father in Watlington in England and on these occasions he would visit University College, London and the Rothamsted Agricultural Experiment Station. He would discuss statistical problems with Fisher, Neyman and Pearson. In 1934 Gosset had a motor accident, described in [7]:...he ran into a lamp-post on a straight road, through looking down to adjust some stuff he was carrying... In fact when confined to bed for three months after the accident he was able to concentrate on statistics. It was a year before he was recovered but he retained a limp for the remaining few years of his life. At the end of 1935 Gosset left Ireland to take charge of the new Guinness brewery in London. Despite the hard work involved in this venture he continued to publish statistics papers. McMullen, who was a personal friend, describes Gosset in [7] as follows:... he was very kindly and tolerant and absolutely devoid of malice. He rarely spoke about personal matters but when he did his opinion was well worth listening to and not in the least superficial. Article by: J J O'Connor and E F Robertson List of References (7 books/articles) A Poster of William Gosset

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1900 to 1910

Other Web sites

University of Minnesota

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gosset.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gosset.html (2 of 2) [2/16/2002 11:12:18 PM]

Goursat

Edouard Jean-Baptiste Goursat Born: 21 May 1858 in Lanzac, Lot, France Died: 25 Nov 1936 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edouard Goursat received his doctorate in 1881 from Ecole Normale Supérieure, then taught in Toulouse until 1885. The next 12 years were spent back at Ecole Normale Supérieure, then he taught analysis at the University of Paris until his retirement. His teachers included Darboux and Hermite who influenced him to work on analysis and its applications. Goursat had a lifelong association with Emile Picard. Julia, who was Goursat's student, later collaborated with him. The Cauchy-Goursat theorem states the integral of a function round a simple closed contour is zero if the function is analytic inside the contour. Cauchy had established the theorem with the added condition that the derivative of the function was continuous. In 1891 he wrote Leçons sur l'intégration des équations aux dérivées partielles du premier ordre. Goursat's best known work is Cours d'analyse mathématique (1900-10) which introduced many new analysis concepts. Goursat became a member of the French Academy of Science in 1919. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) A Poster of Edouard Goursat

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1900 to 1910

Other Web sites

Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Goursat.html (1 of 2) [2/16/2002 11:12:20 PM]

Goursat

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Goursat.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Goursat.html (2 of 2) [2/16/2002 11:12:20 PM]

Govindasvami

Govindasvami Born: about 800 in India Died: about 860 in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Govindasvami (or Govindasvamin) was an Indian mathematical astronomer whose most famous treatise was a commentary on the Mahabhaskariya of Bhaskara I. Bhaskara I wrote the Mahabhaskariya in about 600 A. D. It is an eight chapter work on Indian mathematical astronomy and includes topics which were fairly standard for such works at this time. It discussed topics such as the longitudes of the planets, conjunctions of the planets with each other and with bright stars, eclipses of the sun and the moon, risings and settings, and the lunar crescent. Govindasvami wrote the Bhasya in about 830 which was a commentary on the Mahabhaskariya. In Govindasvami's commentary there appear many examples of using a place-value Sanskrit system of numerals. One of the most interesting aspects of the commentary, however, is Govindasvami's construction of a sine table. Indian mathematicians and astronomers constructed sine table with great precision. They were used to calculate the positions of the planets as accurately as possible so had to be computed with high degrees of accuracy. Govindasvami considered the sexagesimal fractional parts of the twenty-four tabular sine differences from the Aryabhatiya. These lead to more correct sine values at intervals of 90 /24 = 3 45 '. In the commentary Govindasvami found certain other empirical rules relating to computations of sine differences in the argumental range of 60 to 90 degrees. Both of the references [1] and [2] are concerned with the sine tables in Govindasvami's work. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Govindasvami.html (1 of 2) [2/16/2002 11:12:21 PM]

Govindasvami

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Govindasvami.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Govindasvami.html (2 of 2) [2/16/2002 11:12:21 PM]

Graffe

Karl Heinrich Gräffe Born: 7 Nov 1799 in Brunswick, Germany Died: 2 Dec 1873 in Zurich, Switzerland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karl Gräffe worked as a jeweller in Hamburg from 1813 to 1816 and he thought that he might become a goldsmith. However a lot of hard work saw him manage to pass the entrance examinations of the Carolineum at Brunswick. In 1824 Gräffe went to Göttingen where he attended lectures by Gauss and Thibaut. While in Göttingen, Gräffe wrote a prize winning dissertation. He became a lecturer in Zurich, becoming professor there in 1860. Gräffe is best remembered for his method of numerical solution of algebraic equations, developed to answer a prize question of the Berlin Academy of Sciences. It is particularly suitable for methods developed for using computers to solve mathematical problems. This method is today called the Dandelin-Gräffe method after the two mathematicians who independently investigated it. The history of the Dandelin-Gräffe method is discussed in [3] and [4]. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Graffe.html (1 of 2) [2/16/2002 11:12:22 PM]

Graffe

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Graffe.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Graffe.html (2 of 2) [2/16/2002 11:12:22 PM]

Gram

Jorgen Pedersen Gram Born: 27 June 1850 in Nustrup (18 km W of Haderslev), Denmark Died: 29 April 1916 in Copenhagen, Denmark

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jorgen Gram's father, Peder Jorgensen Gram, was a farmer and his mother was Marie Magdalene Aakjaer. After completing two parts of his primary school education, Jorgen entered the Ribe Katedralskole secondary school in 1862. He graduated in 1868 and began his university education. In 1873 Gram graduated with a Master's degree in mathematics. This degree was of a higher level than the present British/American Master's degree and more on a par with today's British/American Ph.D. Gram had published his first important mathematics paper before he had graduated. This was a work on modern algebra which appeared first in Tidsskrift for Mathematik but, in 1874, Gram published a fuller account of the same material in French. This fuller version appeared in Mathematische Annalen under the title Sur quelques théorèmes fondamentaux de l'algebre moderne. This work provided a simple and natural framework for invariant theory. In 1875 Gram was appointed as an assistant in the Hafnia Insurance Company. Around the same time he began working on a mathematical model of forest management. We give more details below of this research interest which Gram followed for many years. His career in the Hafnia Insurance Company progressed well and soon he was promoted to a more senior role in the company. However his work for the insurance company led him back into mathematical research. He began working on probability and numerical analysis, two topics whose practical applications in his day to day work in calculating insurance made their study important to him. Gram's mathematical career was always a balance between pure mathematics and very practical applications of the subject. We have already mentioned the very practical applications to forestry which he continued to study at this time, and his work on probability and numerical analysis involved both the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gram.html (1 of 3) [2/16/2002 11:12:23 PM]

Gram

theory and its application to very practical situations. He published a paper on these topics On series expansions determined by the methods of least squares and for this work he was awarded the degree of Doctor of Science in 1879. This degree, equivalent to today's British D.Sc., is at a higher level than a doctorate. Gram later published this work in the Journal für Mathematik and it proved to be of fundamental importance in the development of the theory of integral equations. The year 1879 was important for Gram in another context, for on 30 September of that year, he married Dorthe Marie Sorensen who was the daughter of a blacksmith. Also in 1879 Gram published the second of his four papers on forestry which all appeared in Danish Forestry Journals. The first had appeared in 1876, a year after he began this research, and it presented a mathematical model for maximising the profit in managing a forest. The paper was written in Danish and failed to be noticed outside his own country. In fact Gram never gained the international recognition for this work which he deserved but German researchers, unaware of Gram's contributions, published their own results on the same problems. There is little doubt that their work was far less satisfactory than that of Gram's yet because it was readily accessible it was the Germans and not Gram who gained international acclaim. Gram extended his work on forestry in later papers. After the 1879 paper he published two further papers in 1883 and 1889. These papers included developments of his forestry model which he was able to make in the light of experiments on trees which he carried out over a number of years. He work in this area was later widely used. Gram's work on probability and numerical analysis led him in a natural way to study abstract problems in number theory. In 1884 he won the Gold Medal of the Videnskabernes Society for his paper Investigations of the number of primes less than a given number which he published in the journal of the Society. Gram had corresponded with Meissel on this topic and in 1885 Meissel travelled to Denmark and met with Gram. It was 1885 that Meissel published his work on the number of primes less than 109 so the two had much to discuss on the topic. Gram also worked on the Riemann zeta function. Zeuthen writes in [1]:His brilliance and scientific training together with his practical skills made his contributions to pure and applied mathematics very significant. Although he continued to work for the Hafnia Insurance Company in more and more senior roles, Gram founded his own insurance company, the Skjold Insurance Company, in 1884. He was the director of this company from its founding until 1910. From 1895 until 1910 Gram was also an executive of the Hafnia Insurance Company. On 9 April 1895 Gram's wife Dorthe died. Gram married again just over one year later on 15 May 1896, his second wife being Emma Birgitte Hansen. From 1910 until his death in 1916 Gram was Chairman of the Danish Insurance Council. It was through his work for the insurance companies that Gram became close to another Danish mathematician, Thorvald Thiele, who worked as an actuary. Despite not teaching mathematics in a university and as a consequence never having any students, Gram still managed to influence the next generation of Danish mathematicians in a very positive way. He often lectured in the Danish Mathematical Society, he was an editor of Tidsskrift for Mathematik from 1883 to 1889, and he also reviewed papers written in Danish for the Jahrbuch über die Fortschritte der Mathematik. Gram received honours for his mathematical contributions despite being essentially an amateur mathematician. The Videnskabernes Society had awarded him their Gold Medal in 1884 before he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gram.html (2 of 3) [2/16/2002 11:12:23 PM]

Gram

became a member, but in 1888 he was honoured with election to the Society. He frequently attended meetings of the Society and published in the Society's journals. For many years he was the Treasurer of the Videnskabernes Society. Gram is best remembered for the Gram-Schmidt orthogonalisation process which constructs an orthogonal set of from an independent one. He was not however the first to use this method. The process seems to be a result of Laplace and it was essentially used by Cauchy in 1836. Gram met his death in a rather strange and very sad way. He was on his way to a meeting of the Videnskabernes Society when he as struck and killed by a bicycle. He was sixty-five years old when he met his death in this tragic accident. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gram.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gram.html (3 of 3) [2/16/2002 11:12:23 PM]

Grandi

Luigi Guido Grandi Born: 1 Oct 1671 in Cremona, Italy Died: 4 July 1742 in Pisa, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Guido Grandi was educated first at the Jesuit college in Cremona. He became a member of the Order of the Camaldolese in 1687. This Order, an offshoot of the Benedictine Order, was founded about 1012 at Camaldoli near Arezzo, Italy. The monastery and the hermitage formed one unit in this Order and Grandi studied at one of these units in Ferrara. Then, in 1693, he went to another monastery of the Camaldolese Order in Rome. The following year, Grandi became a teacher of philosophy and theology at the Camaldolese monastery in Florence. Up to this time he had shown little interest in mathematics but now his thoughts turned in that direction. However he continued to teach philosophy being appointed professor in Rome in 1700, then going to Pisa again as professor of philosophy. Grandi's first mathematical appointment came in 1707 when he became mathematician to the Grand Duke of Tuscany, Cosimo III de' Medici. In 1709 he visited England and clearly impressed the English scientists since he was elected a Fellow of the Royal Society. In 1714 Grandi was appointed Professor of Mathematics at the University of Pisa. Grandi was the author of a number of works on geometry in which he considered the analogies of the circle and equilateral hyperbola. He also considered curves of double curvature on the sphere and the quadrature of parts of a spherical surface. In 1701 Grandi discussed the conical loxodrome, the curve that cuts the generators of a cone of revolution in a constant angle. He studied the curve the Witch of Agnesi in 1703. In fact his work of 1703

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grandi.html (1 of 3) [2/16/2002 11:12:25 PM]

Grandi

is important in introducing Leibniz's calculus into Italy. In 1728 Grandi published Flores geometrici a work in which he defines the clelie curve. He named the curve after Countess Clelia Borromeo and dedicated his book to her. If the longitude and colatitude of a point P on a sphere is denoted by and and if P moves so that = m , where m is a constant, then the locus of P is a clelie. Grandi also applied the term "clelies" to the curves determined by certain trigonometric equations involving the sine function a sin

= b sin m

a sin = a - b sin m Grandi also worked on hydraulics and was involved with a number of projects such as ones to drain the Chiana Valley and the Pontine Marshes. He also published a number of works on mechanics and astronomy. His practical work on mechanics included experimenting with a steam engine. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles) Mathematicians born in the same country Cross-references to Famous Curves

1. Witch of Agnesi 2. Rhodenea curves

Other references in MacTutor

Chronology: 1720 to 1740

Honours awarded to Guido Grandi (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1709

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grandi.html (2 of 3) [2/16/2002 11:12:25 PM]

Grandi

JOC/EFR February 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Grandi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grandi.html (3 of 3) [2/16/2002 11:12:25 PM]

Granville

Evelyn Boyd Granville Born: 1 May 1924 in Washington D.C., USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Evelyn Boyd Granville's family name was Boyd, so she grew up as Evelyn Boyd. The name Granville, by which she is now known, is the name she took after her second marriage but, for the sake of simplicity, we shall refer to her during this article as Granville even from her childhood days. Evelyn Boyd Granville's father was William Boyd who had various jobs including that of a janitor, chauffeur, and a messenger. Evelyn's mother was Julia Boyd; she had been a secretary before her marriage but gave up work to bring up her family. The Great Depression began in 1929 when Granville was five years old, and by 1932 one quarter of the workers in the United States were unemployed. Granville's father worked selling vegetables from a lorry during the Great Depression and, although the family were poor, they always had food and a home. William and Julia Boyd separated while Granville was still young and, together with her elder sister who was about eighteen months older, she was brought up in the African American community in Washington, D.C by her mother. Julia Boyd's sister also played a big part in Granville's upbringing and, being more academically inclined that Granville's mother, she strongly influenced and encouraged Granville in that direction. After separating from William Boyd, Julia returned to work to support her family earning a living as a maid. Eventually she worked for the Bureau of Engraving and Printing in Washington as a currency and stamp examiner. Julia's sister, having failed to get a teaching post, also got a job with the same organisation. Granville wrote [3]:As a child growing up in the thirties in Washington, D.C., I was aware that segregation placed many limitations on Negroes, ... However, daily one came in contact with Negroes who had made a place for themselves in society; we heard about and read about individuals http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Granville.html (1 of 5) [2/16/2002 11:12:27 PM]

Granville

whose achievements were contributing to the good of all people. These individuals, men and women, served as our role models; we looked up to them and we set out goals to be like them. We accepted education as the means to rise above the limitations that a prejudiced society endeavoured to place upon us. Granville attended elementary school, junior high school, and high school in Washington D.C. She was happy at school and was an outstanding pupil. From this time on she aspired to a career as a teacher [1]:I saw black women - attractive, well dressed women - teaching school, and I wanted to be a teacher because that's all I saw. I was not aware of any other profession. The high school which she attended was Dunbar High School. It was an academically oriented school for black students which aimed to send their pupils to the top universities and there Granville was strongly encouraged by two of her mathematics teachers Ulysses Basset and Mary Cromwell. While at Dunbar High School she decided that she wanted to continue her studies at Smith College after graduating but she fully realised that her mother was not in a position to support her financially through College [1]:I did not receive a scholarship the first year at [Smith College], and I was told later that they didn't see how in the world a poor child as I could afford to go there. ... the first year, my aunt helped my mother. Of course after the first year I got scholarships. I lived in a co-op house, worked during the summers, and I was able to [pay]. It was not a financial burden after the first year. In fact both Granville's mother and aunt gave her $500 to finance her studies for a year before she won the scholarships which helped fund the remainder of her time at Smith College. The summer work which she refers to in the above quote was at the at National Bureau of Standards. On entering Smith College in 1941 Granville studied French as well as mathematics but, although she enjoyed the language, did not find French literature to her liking and soon concentrated on mathematics, theoretical physics and astronomy [3]:I was fascinated by the study of astronomy and at one point I toyed with the idea of switching my major to this subject. If I had known then that in the not too distant future the United States would launch its space program, and astronomers would be in great demand in the planning of space missions, I might have become an astronomer instead of a mathematician. Among her teachers at Smith College was Neal McCoy who was particularly supportive of women mathematicians, perhaps in part because his own sister was a mathematician. Granville graduated with distinction in 1945 and was awarded a scholarship from the Smith Student Aid Society of Smith College to undertake studies for her doctorate. Both the University of Michigan and Yale University offered her a place but only Yale was able to provide the additional financial support she required. Entering Yale in the autumn of 1945, she began research in functional analysis under Hille's supervision. She wrote a doctoral thesis On Laguerre Series in the Complex Domain and in 1949, together with Marjorie Lee Browne who graduated from the University of Michigan in the same year, she became the first black American woman to be awarded a Ph.D. in mathematics. After completing her Ph.D. from Yale, Granville spent a postdoctoral year at the New York University Institute of Mathematics working on differential equations with Fritz John. Rather sadly, neither Hille nor John encouraged her to submit her research for publication. During this year she also taught as a part-time instructor in the mathematics department of New York University. After applying unsuccessfully for a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Granville.html (2 of 5) [2/16/2002 11:12:27 PM]

Granville

teaching post at the Polytechnic Institute of Brooklyn, she accepted an offer of an associate professorship at Fisk University in Nashville, taking up the post in 1950. Murray writes [1]:In the final analysis, however, Granville - who wanted to become a teacher since she was a little girl - was unable to accept the highly restrictive terms under which black women could hold academic posts in the early 1950s. As she considered her options, it was natural for her to think about the possibility of government employment. ... In the spring of 1952, Granville decided to seek a government job and return to Washington, D.C. The job she was offered at the National Bureau of Standards gave her twice her previous academic salary so Granville [3]:The work entailed consulting with ordinance engineers and scientists on the mathematical analysis of problems related to the development of missile fuses. ... I met several mathematicians who were employed ... as computer programmers. At that time the development of electronic computers was in its infancy. The application of computers to scientific studies interested me very much, which led to my giving serious consideration to an offer of employment from International Business Machines Corporation. In December 1955 Granville left the National Bureau of Standards and she began work for IBM in January of the following year. At first she worked in Washington writing programs for the IBM 650 computer, then in 1957 she moved to New York City to take up a post as a consultant on numerical analysis at the New York City Data Processing Center of the Service Bureau Corporation, which was part of IBM. When the United States space programme began to move rapidly forward, NASA contracted IBM to write software for them. Granville was happy to return to Washington D.C. as one of a team of IBM mathematicians [3]:I can say without a doubt that this was the most interesting job of my lifetime - to be a member of a group responsible for writing computer programs to track the paths of vehicles in space. In November 1960 Granville married (but still did not take the name of Granville which was her second husband's name) and moved to Los Angeles where she continued her work on orbit calculations for the space programme at the Space Technology Laboratories. In the 1967 Granville's marriage broke up and she returned to the academic world, accepting a teaching post at California State University in Los Angeles. Her job involved undergraduate teaching and she taught both numerical analysis and computer programming. Another role was in mathematical education and she was involved in the mathematical education of those training to be elementary school teachers. This interest in mathematical education led to her involvement with the Miller Mathematics Improvement Program and as part of this program she taught mathematics for two hours each day at an elementary school in Los Angeles during session 1968-69. Out of this experience came her joint publication with Jason Frand Theory and Applications of Mathematics for Teachers (1975). The book was well received and adopted in many schools. Three years later a second edition was published but fashions change in teaching mathematics and soon after this the book ceased to be relevant to current courses. Granville had married Edward V Granville in 1970, and of course only at that time did she take the name "Granville" which we have used throughout this article. She retired from California State University in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Granville.html (3 of 5) [2/16/2002 11:12:27 PM]

Granville

1984 [3]:My husband was born and raised in East Texas and planned to return to the area when he retired from his business. I often accompanied him on visits to Texas and , after making several trips, I was convinced that a move to a rural setting in East Texas would be a welcome change from the Los Angeles metropolis. We found an ideal setting in a 16-acre parcel with a house and a four-acre lake near Tyler, Texas. She taught at Texas College from 1985 to 1988, teaching on a newly instigated computer science course. Still Granville did not want to leave the academic world and she taught at the University of Texas at Tyler, where she held the Sam A Lindsey Chair, and retired in 1997. Granville gave her views on the current problems of teaching mathematics in American schools in a lecture at Yale University. We give some quotes from that talk:I believe that math is in grave danger of joining Latin and Greek on the heap of subjects which were once deemed essential but are now, at least in America, regarded as relics of an obsolete, intellectual tradition ... ... math must not be taught as a series of disconnected, meaningless technical procedures from dull and empty textbooks. We teach that there is only one way to solve a problem, but we should let children explore various techniques. ... But we're not training teachers to provide this new approach. ... children end up crippled in mathematics at an early age. Then, when they get to the college level, they are unable to handle college classes. It's tragic because almost every academic area requires some exposure to mathematics. Make children learn how to add, subtract, multiply, and divide, and they won't need calculators. How do you teach the beauty of mathematics, how do teach them to ... solve problems, to acquaint them with various strategies of problem solving so they can take these skills into any level of mathematics? That's the dilemma we face. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Other Web sites

Agnes Scott College

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Granville.html (4 of 5) [2/16/2002 11:12:27 PM]

Granville

Glossary index

Quotations index

Mathematicians of the day JOC/EFR October 2001

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Granville.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Granville.html (5 of 5) [2/16/2002 11:12:27 PM]

Grassmann

Hermann Günter Grassmann Born: 15 April 1809 in Stettin, Prussia (now Szczecin, Poland) Died: 26 Sept 1877 in Stettin, Germany (now Szczecin, Poland)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hermann Grassmann is chiefly remembered for his development of a general calculus for vectors. Grassmann taught at the Gymnasium in Stettin from 1831 until his death except for two years (1834-1836) when he taught in Berlin. Grassmann's most important work is Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844) developed the idea of an algebra in which the symbols representing geometric entities such as points, lines and planes, are manipulated using certain rules. He represented subspaces of a space by coordinates leading to point mapping of an algebraic manifold now called the Grassmannian. Grassmann invented what is now called Exterior Algebra. This was joined to Hamilton's quaternions by Clifford in 1878. Clifford replaced Grassmann's rules ep

ep = 0 and ep

eq = - eq

ep for p q

by the rules ep ep = 1 and ep eq = - eq ep for p q. Clifford algebras are used today in the theory of quadratic forms and in relativistic quantum mechanics and they appear together with Grassmann's exterior algebra in differential geometry. See [26]. Grassmann's methods were slow to be adopted but eventually they inspired the work of Elie Cartan and have since been used in studying differential forms and their application to analysis and geometry. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grassmann.html (1 of 2) [2/16/2002 11:12:29 PM]

Grassmann

Grassmann wrote on many other subjects, for example electricity, colour, acoustics, linguistics and botany. At the age of 53 he became disappointed with the lack of interest in his mathematical ideas so he turned to Sanskrit studies, another of his interests. His Sanskrit dictionary is still widely used. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (38 books/articles)

A Quotation

A Poster of Hermann Grassmann

Mathematicians born in the same country

Cross-references to History Topics

1. An overview of the history of mathematics 2. Abstract linear spaces

Other references in MacTutor

Chronology: 1840 to 1850

Other Web sites

1. Tasmania, Australia 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Grassmann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grassmann.html (2 of 2) [2/16/2002 11:12:29 PM]

Grave

Dmitry Aleksandrovich Grave Born: 6 Sept 1863 in Kirillov, Novgorod gubernia, Russia Died: 19 Dec 1939 in Kiev, Ukraine Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dmitry Grave was educated at the University of St Petersburg where he studied under Chebyshev and his pupils Korkin, Zolotarev and Markov. Grave began research while a student, graduating with his doctorate in 1896. He had obtained his masters degree in 1889 and, in that year, began teaching at the University of St Petersburg. For his Master's Degree Grave studied Jacobi's methods for the three body problem, a topic suggested by Korkin. His doctorate was on map projections, again a topic proposed by Korkin, the degree being awarded in 1896. The work, on equal area plane projections of the sphere, built on ideas of Euler, Lagrange and Chebyshev. Grave became professor at Kharkov in 1897 and, from 1902, he was appointed professor at the University of Kiev, where he remained for the rest of his life. Grave is considered as the founder of the Kiev school of algebra which was to become the centre for algebra in the USSR. At Kiev Grave studied algebra and number theory. In particular he worked on Galois theory, ideals and equations of the fifth degree. Among his pupils were O J Schmidt, N G Chebotaryov, B N Delone and A M Ostrowski. The Revolution of 1917 had some major effects on the development of mathematics in Russia and the Ukraine. One effect was that mathematics in the Ukraine was required to be more practical and algebra did not fit into this applied mathematics and technology dominated scene. Grave had to discontinue his famous Kiev algebra seminar in the 1920s, give up teaching and research in algebra, and move to applied mathematics topics. It would not be before the 1950s, well after Grave's death, that Kiev would again play a major role in algebra research. Grave chaired the Applied Mathematics Commission of the Academy of Sciences of the Ukraine in the 1920s. After Grave stopped work on algrbra, he began to study mechanics and applied mathematics, but he never completely gave up algebra. During the 1930s there were further changes to the Soviet educational system, and there was a fair amount of reorganisation. The Institute of Mathematics of the Academy of Sciences was founded in Kiev in 1934 and Grave served as the first director of the Institute from its foundation until his death in 1939. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grave.html (1 of 2) [2/16/2002 11:12:30 PM]

Grave

His work at Institute of Mathematics was in addition to his chir at Kiev University which he continued to hold. Among the many books that Grave wrote were Theory of Finite Groups (1910) and A Course in Algebraic Analysis (1932). He also studied the history of algebraic analysis. Among the honours that were given to him was election to the Academy of Sciences of the Ukraine in 1919, election to the Shevchenko Scientific Society in 1923 and election to the Academy of Sciences of the USSR in 1929. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Grave.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grave.html (2 of 2) [2/16/2002 11:12:30 PM]

Green

George Green Born: July 1793 in Sneinton, Nottingham, England Died: 31 May 1841 in Sneinton, Nottingham, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Green's father, also called George Green, was a baker in Nottingham. After serving his apprenticeship to a baker in Nottingham George Green Senior had married Sarah Butler (the mathematician's mother) in 1791 and Sarah's father had helped George Green Senior to buy his own bakery in Wheatsheaf Yard in Nottingham. George and Sarah had one son and one daughter. The son was George Green, the mathematician and, although the date of his birth is unknown, he was baptised on 14 July 1793. His sister Ann was born two years later and the family were reasonably prosperous with the bakery business being successful. In September 1800 there were riots in England when food prices were high and people did not have enough to eat. The corn dealers and bakers were blamed for keeping back food until the prices rose even higher and crowds of people broke into bakers and tried to steal food. George Green's bakery was attacked and the he wrote a letter to the mayor asking for protection (the spelling and capital letters follow Green's letter):Sept 1th 1800 To the Right Worshipfull the Mayor and his Bretheren haveing all my Windows Broke Last night and being much thretend tonight with more mischief being done to me such as entering my House therefore Gentlemen I most Humbly crave your protection such as the Berer of this can explain to you from your Humble servant George Green. Young George Green went to Robert Goodacre's school in 1801, the year after the riots. He was only eight years old when he began his schooling and only nine when he left the school in midsummer 1802. His only schooling, therefore, consisted of four terms but, despite the short time he spent there, since Robert Goodacre's was the best and most expensive school in Nottingham, George was taught much in the four terms. George's sister Ann went on to marry William Tomlin and he was to write on George Green's life after George died. Of his time at Robert Goodacre's school, Tomlin writes:... he pursued with undeviating constancy the same as in his mature years an intense application to mathematics. ... his profound knowledge in mathematics soon exceeded that of Robert Goodacre. It is hard to see quite why Green became interested in mathematics at this age or, for that matter, whether he had access to mathematical works of any type. However, in 1802, Green left school as a nine year old boy to work in his father's bakery business. He probably learnt a little of Latin, Greek and French at school but it is hard to see how even a bright eight year old boy in a good school could learn more than

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Green.html (1 of 6) [2/16/2002 11:12:32 PM]

Green

the briefest of introductions to these subjects. George Green Senior ran a successful bakery business and he was able to buy several houses in Nottingham. In 1807 he bought a plot of land at Sneinton, just outside Nottingham. The sale was made by auction at 26 February 1807 and in the advertisement for the sale the plot is described as:A freehold estate, tythe-free, and the Land Tax redeemed, situate at Snenton, near Nottingham; consisting of excellent pasture land, and affording for Building upon one of the first Situations in the Kingdom, commanding most extensive Views of the River Trent, Trent Vale, Clifton Woods, Colwick, Belvoir Castle, and a rich and highly cultivated Country, diversified by numberless other picturesque objects... Having bought the land, he built on it a brick wind corn-mill, clearly a useful thing for a baker to own. The mill stood 16 metres high and was one of the first two brick mills in the county of Nottinghamshire. Green's father employed a manager to run the mill, and Green himself worked there. In 1817 Green Senior built a family house beside the mill and Green together with his mother and father went to live there. [The mill has been restored and we strongly recommend anyone who is visiting Nottingam to pay a visit.] Green's sister Ann did not move to Sneinton as she had married William Tomlin in 1816 and they were living in a fashionable part of Nottingham. Green must have continued working on mathematics through the years that he worked at his father's mill. However, we have no knowledge whatsoever of how he could have become acquainted with the most advanced mathematics of his day, which indeed is what happened. In [3] Cannell discusses whether there were any mathematicians living in Nottingham who could have given Green access to advanced French mathematical ideas. She comes up with only one possible candidate, John Toplis who was a mathematics graduate of Queens' College Cambridge. Toplis had become unhappy with the mathematics being taught at Cambridge and had tried to influence others to learn more of the mathematics being developed in France. He translated the first volume of Laplace's Mécanique céleste into English and he published this in Nottingham in 1814. At this time he was the headmaster of the Free Grammar School in Nottingham, and he remained there until 1819 when he returned to Cambridge as Dean of Queens' College. Certainly Green could have known Toplis since, before he moved to the new family home at the mill in 1817, Green was living in Goosegate in Nottingham just one street away from the Free Grammar School where Toplis lived at that time. As Cannell writes [3]:There is no proof that John Toplis was George Green's mentor but circumstantial evidence suggests strongly that he was guiding him in the new mathematics and helping him with the French he undoubtedly acquired in order to read the works of other French mathematicians, such as Lacroix, Poisson and Biot. The manager of Green's mill was William Smith and he had a daughter Jane Smith. George Green never married Jane Smith but together they had seven children. His relationship with her must have started in 1823 or earlier as their first child was born in 1824. Certainly in 1823 Green joined the Nottingham Subscription Library which was situated in Bromley House. This was an important event in Green's scientific development as it gave him access to a few scientific works, but perhaps most importantly of all, it gave him access to the Transactions of the Royal Society of London. In this publication Green could read some of the latest mathematical work and it also reported on works published in other countries. Green studied mathematics on the top floor of the mill, entirely on his own. The years between 1823 and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Green.html (2 of 6) [2/16/2002 11:12:32 PM]

Green

1828 were not easy for Green, and certainly not the most conducive to study. As well as having a full time job in the mill, two daughters were born, the one in 1824 mentioned above, and a second in 1827. Between these two events his mother had died in 1825 and his father was to die in 1829. Yet despite the difficult circumstances and despite his flimsy mathematical background, Green published one of the most important mathematical works of all time in 1828. On 14 December 1827 he published an advertisement in the Nottingham review:In the Press, and shortly will be published, by subscription, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. By George Green. Dedicated (by permission), to his Grace the Duke of Newcastle, K.G. Price to Subscribers, 7s. 6d. The Names of Subscribers will be received at the Booksellers, and at the Library, Bromley House. The Essay was published in March 1828 and there were 51 subscribers, most of whom were members of the Nottingham Subscription Library and paid 7s. 6d. for a work of which they could hardly have understood a word. In [1] details are given of Green's preface:In the preface Green indicated that his "limited sources of information" prevented his giving a proper historical sketch of the mathematical theory of electricity, and indeed, he cites few sources. Among them are Cavendish's single-fluid theoretical study of electricity of 1771, two memoirs by Poisson of 1812 on surface electricity and three on magnetism (1821-1823), and contributions by Arago, Laplace, Fourier, Cauchy , and T Young. The preface concludes with a request that the work be read with indulgence, in view of the limitations of the author's education. Also in [1] some details are given of the contents this important publication:The Essay begins with introductory observations emphasising the central role of the potential function. Green coined the term 'potential' to denote the results obtained by adding the masses of all the particles of a system, each divided by its distance from a given point. The general properties of the potential function are subsequently developed and applied to electricity and magnetism. The formula connecting surface and volume integrals, now known as Green's theorem, was introduced in the work, as was "Green's function" the concept now extensively used in the solution of partial differential equations. The Essay may have been of great importance but this was not realised by anyone at the time of its publication. Nobody with sufficient mathematical skills to appreciate its importance had seen the work. Green carried on working his mill and, in 1829 on the death of his father, he became solely responsible for the family business. His third child, and first son, was born in 1829. Things changed however, when he made contact with Sir Edward Bromhead. Sir Edward Bromhead was one of the subscribers to the Essay and he had written immediately to Green offering to send any further papers to the Royal Society of London, The Royal Society of Edinburgh or the Cambridge Philosophical Society. Bromhead had studied mathematics at Cambridge and had been a member of the Analytic Society. In fact what may have been the last meeting of that Society had been held at Thurlby Hall on 20 December 1817 with Bromhead in the chair. Although Bromhead was not able to appreciate the high importance of Green's essay, he did realise that Green was a very good mathematician. Green took Bromhead's offer as mere politeness and did not respond until January 1830 when a friend persuaded him to follow up Bromhead's letter.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Green.html (3 of 6) [2/16/2002 11:12:32 PM]

Green

For three years Green and Bromhead met at Thurlby Hall and during this time Green wrote three further papers. Two on electricity were sent by Bromhead to the Cambridge Philosophical Society where they were published, one in 1833 and the other in 1834. The third was on hydrodynamics and this work was published by the Royal Society of Edinburgh (of which Bromhead was a Fellow) in 1836. Bromhead was a good person for Green to become friendly with since he had good contacts with mathematicians at Cambridge. Bromhead's close friends there included Charles Babbage, John Herschel and George Peacock and he suggested to Green in April 1833 that he should consider studying mathematics at Cambridge. In June 1833 Bromhead went to Cambridge for a reunion and asked Green to go with him. However Green certainly did not realise the importance of his work. He wrote to Bromhead:You were kind enough to mention a journey to Cambridge on June 24th to see your friends Herschel, Babbage and others who constitute the Chivalry of British Science. Being as yet only a beginner I think I have no right to go there and must defer the pleasure until I shall have become tolerably respectable as a man of science should that day ever arrive. However Green took Bromhead's advice, left his mill and became an undergraduate at Cambridge in October 1833 at the age of 40. He was admitted to Caius College where Bromhead had studied. He wrote to Bromhead in May 1834:I am very happy here and am I fear too much pleased with Cambridge. This takes me in some measure from those pursuits which ought to be my proper business, but I hope on my return to lay aside my freshnesses and become a regular Second Year Man. The mathematics examinations did not prove hard for Green, but the other topics such as Latin and Greek proved much harder for someone with only four terms of school education. He graduated as fourth Wrangler in 1837, the second Wrangler that year being Sylvester. The following quote appears in [14]:Green and Sylvester were the first men of the year, but Green's want of familiarity with ordinary boy's mathematics prevented him from coming to the top in a time race. After graduating he remained at Cambridge and worked on his own mathematics. In 1838 and 1839 he had two papers on hydrodynamics (in particular wave motion in canals), two papers on reflection and refraction of light and two papers on reflection and refraction of sound published by Cambridge Philosophical Society. Bishop Harvey Goodwin was an undergraduate at Cambridge during these years following Green's graduation. He wrote of Green [3]:He stood head and shoulders above all his contemporaries inside and outside the University. Goodwin is also quoted in [14]:I was twice examined by Green. He set the problem paper in two out of the three of my college examinations... He never assisted as far as I know in lectures. This might be owing to his habits of life. His manner in the examination room was gentle and pleasant. Green was elected to a Perse fellowship on 31 October 1839. This of course was possible since he did satisfy the condition of not being married and having six children, which he had at this time, was not relevant. His time at Cambridge after being elected to the fellowship was very short indeed. By May 1840 he had returned to Nottingham suffering from ill health. A few weeks later his seventh child was born. Green clearly felt that his illness was very serious and in July 1840 he wrote a will in which he states that his health was poor but no details of the illness are given. However it would appear that he still http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Green.html (4 of 6) [2/16/2002 11:12:32 PM]

Green

hoped to return to Cambridge since he describes himself as:... late of Sneinton in the County of Nottingham and now of Caius College Cambridge, Fellow of such College. His will left all his property in Nottingham to Jane while the property at Sneinton was left to his seven children. We do know that Green died in the house where Jane Smith and his seven children lived. Jane reported his death and was with Green when he died. Rather remarkably, this is the only record of Green living in the same house as Jane Smith and his children. The Nottingham Review published a short obituary on 11 June which showed they knew little of his life and less of the importance of his work:... we believe he was the son of a miller, residing near Nottingham, but having a taste for study, he applied his gifted mind to the science of mathematics, in which he made a rapid progress. In Sir Edward Ffrench Bromhead, Bart., he found a warm friend, and to his influence he owed much, while studying at Cambridge. Had his life been prolonged, he might have stood eminently high as a mathematician. Of course, Green never knew the importance of his mathematics. That was only realised after his death [1]:Only a few weeks before Green's death, William Thomson had been admitted to St Peter's College, Cambridge. In a paper by Robert Murphy published in the Transactions of the Cambridge Philosophical Society, Thomson noticed a reference to Green's Essay, although Murphy did not mention any of his other works published in that journal. Thomson was unable to find a copy of the Essay until, just after receiving his degree in January 1845, his coach, William Hopkins, gave him three copies. Sixty years later Thomson recalled his excitement and that of Liouville and Sturm, to whom he showed the work in Paris in the summer of 1845. After returning to Cambridge, Thomson was responsible for republishing the work, with an introduction (1850-54). Through Thomson, Maxwell, and others, the general mathematical theory of potential developed by an obscure, self-taught miller's son would lead to the mathematical theories of electricity underlying twentieth-century industry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (20 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1820 to 1830

Honours awarded to George Green (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Green.html (5 of 6) [2/16/2002 11:12:32 PM]

Green

Lunar features

Crater Green

Other Web sites

1. University of Nottingham (George Green's home page) 2. Green's Mill Science Centre 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Green.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Green.html (6 of 6) [2/16/2002 11:12:32 PM]

Greenhill

Alfred George Greenhill Born: 29 Nov 1847 in London, England Died: 10 Feb 1927 in London, england

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alfred Greenhill attended Christ's Hospital School and from there he went up to St John's College, Cambridge in 1866. He was placed second in the final examinations and shared the Smith's Prize with the First Wrangler, R Pendlebury. In the year of his graduation he was elected to a fellowship at St John's College. After a short time teaching at the Royal Indian Engineering College, Greenhill returned to Cambridge in 1873, becoming a fellow and lecturer at Emmanuel College. In 1876 Greenhill was appointed professor of mathematics at the Royal Military Academy at Woolwich. He held the chair there until he retired in 1908. Greenhill's work was mainly on elliptic functions. He was interested in their applications to dynamics, hydrodynamics, elasticity, and electrostatics. As might be imagined given that Greenhill spent most of his life working in a military establishment, his work was often directed towards applications to ballistics and other military applications. Love, writing in [1], explains applications of the theory of the motion of a solid in a fluid made by Greenhill in 1879:Greenhill applied this theory to give an account of the steadiness of flight conferred upon an elongated projectile by rifling. He determined the least angular velocity about its axis for which steady motion of a solid of revolution can be stable. ... This practical application of what was regarded as a recondite mathematical theory earned for him much renown at Woolwich. An important contribution Greenhill made to the theory of elasticity was his study of the greatest length http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Greenhill.html (1 of 3) [2/16/2002 11:12:33 PM]

Greenhill

that a cylinder can have before it bends under its own weight. His applications included computing the maximum height a tree can grow. As well as his original research, Greenhill wrote several excellent texts and encyclopaedia articles. Among these books were Differential in integral calculus (1886), The applications of elliptic functions (1892) and Treatise on hydrostatics (1894). Greenhill's approach to mathematics is summed up by Love [1]:The chief characteristics of Greenhill's work were a desire for concrete realisation of abstract theories and the direction of investigation to the solution of definite problems. Hence he valued applications of analysis above the analysis itself, and was led to work out minutely the details of multitudes of special cases. He was above all things a problem solver, but, to interest him, a problem had to be a real problem about material things and the ways in which they behave. Love [1] describes Greenhill's character and interests outside mathematics:... he was keenly interested in antiquities, and had a profound knowledge of the antiquities of London. ... he had a considerable knowledge of music, and was a quite respectable performer on the organ and other instruments. He spoke French rather well, and read German easily. ... he was a sociable recluse, delighting especially in the conversation of people with tastes similar to his own. Greenhill received many honours for his work. He was elected a Fellow of the Royal Society in 1888, he won the De Morgan Medal of the London Mathematical Society in 1902 and the Royal Medal of the Royal Society in 1906. He served on the Council of the Royal Society and as President of the London Mathematical Society. He was knighted in 1908 on his retirement. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to Alfred Greenhill (Click a link below for the full list of mathematicians honoured in this way) Royal Society Royal Medal

Awarded 1906

London Maths Society President

1890 - 1892

LMS De Morgan Medal

Awarded 1902

Honorary Fellow of the Edinburgh Maths Society

Elected 1908

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Greenhill.html (2 of 3) [2/16/2002 11:12:33 PM]

Greenhill

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Greenhill.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Greenhill.html (3 of 3) [2/16/2002 11:12:33 PM]

Gregory

James Gregory Born: Nov 1638 in Drumoak (near Aberdeen), Scotland Died: Oct 1675 in Edinburgh, Scotland

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James Gregory was born in the Manse of Drumoak. This is a small parish on the river Dee, about fifteen kilometres west of Aberdeen. His father was John Gregory and his mother was Janet Anderson. John Gregory had studied at Marischal College in Aberdeen, then gone on to study theology at St Mary's College in the University of St Andrews before spending his life in the parish of Drumoak. Turnbull writes [20]:[John Gregory] was a man of courage and foresight but was not conspicuous for outstanding intellectual gifts ... James seems to have inherited his genius through his mother's side of the family. Janet Anderson's brother, Alexander Anderson, was a pupil of Viète. He acted as an editor for Viète and fully incorporated Viète's ideas into his own teaching in Paris. James was the youngest of his parents three children. He had two older brothers Alexander (the eldest) and David, and there was an age gap of ten years between James and David. James learnt mathematics first from his mother who taught him geometry. His father John Gregory died in 1651 when James was thirteen and at this stage James's education was taken over by his brother David who was about 23 at the time. James was given Euclid's Elements to study and he found this quite an easy task. He attended Grammar School and then proceeded to university, studying at Marischal College in Aberdeen. Gregory's health was poor in his youth. He suffered for about eighteen months from the quartan fever which is a fever which recurs at approximately 72-hour intervals. Once he had shaken off this problem

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.html (1 of 7) [2/16/2002 11:12:35 PM]

Gregory

his health was good, however, and he wrote some years later that the quartan fever (see for example [20]):... is a disease I am happily acquainted with, for since that time I never had the least indisposition; nevertheless that I was of a tender and sickly constitution formerly. Gregory began to study optics and the construction of telescopes. Encouraged by his brother David, he wrote a book on the topic Optica Promota. In the preface he writes:Moved by a certain youthful ardour and emboldened by the invention of the elliptic inequality, I have girded myself with these optical speculations, chief among which is the demonstration of the telescope. The reader may not understand Gregory's reference to "the elliptic inequality" which in fact refers to Kepler's discoveries. Gregory, in Optica Promota, describes the first practical reflecting telescope now called the Gregorian telescope. The book begins with 5 postulates and 37 definitions. He then gives 59 theorems on reflection and refraction of light. There follows propositions on mathematical astronomy discussing parallax, transits and elliptical orbits. Next Gregory gives details of his invention of a reflecting telescope. A primary concave parabolic mirror converges the light to one focus of a concave ellipsoidal mirror. Reflection of light rays from its surface converge to the ellipsoid's second focus which is behind the main mirror. There is a central hole in the main mirror through which the light passes and is brought to a focus by an eyepiece lens. The tube of the Gregorian telescope is thus shorter than the sum of the focal lengths of the two mirrors. His novel idea was to use both mirrors and lenses in his telescope. He showed that the combination would work more effectively than a telescope which used only mirrors or used only lenses. The book was only a theoretical description of the telescope for at this stage one had not been constructed. Gregory remarks in the book [21]:... on his lack of skill in the technique of lens and mirror making ... In 1663 Gregory went to London. There he met Collins and a lifelong friendship began. One of Gregory's aims was to have Optica Promota published and he achieved this. His other aim was to find someone who could construct a telescope to the design set out in his book. Collins advised him to seek the help of a leading optician by the name of Reive who, at Gregory's request, tried to construct a parabolic mirror. His attempt did not satisfy Gregory who decided to give up the idea of having Reive construct the instrument. However, Hooke learnt of Reive's failed attempt at making the parabolic mirror and this would lead to a successful construction of the first Gregorian telescope around ten years later. In London Gregory also met Robert Moray, president of the Royal Society, and Moray attempted to arrange a meeting between Gregory and Huygens in Paris. However, Huygens was not in Paris and the meeting did not materialise. Moray was to play a major role in Gregory's career somewhat later. In 1664 Gregory went to Italy. He visited Flanders, Rome and Paris on his journey but spent most time at the University of Padua where he worked on using infinite convergent series to find the areas of the circle and hyperbola. At Padua he worked closely with Angeli whose [20]:... teaching profoundly influenced Gregory, particularly in providing the twin keys to the calculus, the method of tangents (differentiation) and of quadratures (integration). In Padua Gregory was able to live in the house of the Professor of Philosophy who was Professor

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.html (2 of 7) [2/16/2002 11:12:35 PM]

Gregory

Caddenhead, a fellow Scot. Two works which were published by Gregory while he was in Padua are Vera circuli et hyperbolae quadratura published in 1667 and Geometriae pars universalis published right at the end of his Italian visit in 1668. Of Vera circuli et hyperbolae quadratura Dehn and Hellinger write in [5]:In this work Gregory lays down exact foundations for the infinitesimal geometry then coming into existence. It is remarkable that some decades later, at the time when analysis was in a state of revolutionary development, exactness was at a much lower standard than with Gregory, and generally with the authors writing before the discoveries of Newton and Leibniz (e.g. Huygens, Mengoli, Barrow). The work we are dealing with is of quite a different character. On the one hand, the source from which he is getting his inspiration is quite unknown to us. On the other hand we find here a singular mixture of far-reaching ideas, exact methods, incomplete deductions, and even false conclusions. The work was really trying to prove that and e are transcendental but Gregory's arguments contain a subtle error. However, this should not in any way detract from the brilliance of the work and the amazing collection of ideas which it contains such as: convergence, functionality, algebraic functions, transcendental functions, iterations etc. Before he left Padua Gregory published Geometriae pars universalis which is really [13]:... the first attempt to write a systematic text-book on what we should call the calculus. This book contained the first known proof that the method of tangents (differentiation in our modern terminology) was inverse to the method of quadratures (integration in our modern terminology). Gregory shows how to transform an integral by a change of variable and introduces the x x - 0(x) idea which is the basis of Newton's fluxions. Perhaps it is worth saying a little about how Gregory's work relates to that of Newton. By the time that Gregory published this work Newton had formed his ideas of the calculus so probably had not been influenced by Gregory. On the other hand Newton had not said anything of his ideas and so certainly these ideas could not have influenced Gregory. Essentially Newton and Gregory were working out the basic ideas of the calculus at the same time, as, of course, were other mathematicians. Gregory returned to London from Italy at about Easter 1668. He had sent a copy of Vera circuli et hyperbolae quadratura to Huygens and written a covering letter saying how he was looking forward to hearing the expert opinions of Huygens in it. Huygens did not reply but published a review of the work in July 1668. In the review he raised some objections and also claimed that he had been the first to prove some of the results. On the one hand the summer months that Gregory spent in London were profitable, particularly through his friendship with Collins. It was a time of rapid mathematical development and Gregory found that Collins, with his up-to-date knowledge of developments, was most helpful to him. On the other hand he was upset by Huygens' comments which he took to imply that Huygens was accusing him of stealing his results without acknowledgement. It was indeed unfortunate that these two great mathematicians should enter into a dispute, although having said that it is worth noting that disputes were common at this time, particularly regarding priority. Looking at the dispute with the hindsight of today's understanding of the mathematics involved we can

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.html (3 of 7) [2/16/2002 11:12:35 PM]

Gregory

say that Huygens was certainly unfair in suggesting that Gregory had stolen his results. Gregory had proved them independently and Huygens should have realised that Gregory could not have known of them. However, Huygens' main mathematical objection to Gregory's proof is a valid one. Despite there is brilliant work in this text and in [16] Scriba shows how close Gregory was to making further major discoveries. He writes [16]:Clearly [Gregory] could not see the consequences that lay concealed in his construction. But he had an unerring sense of where they would lead ... The dispute had another unfortunate consequence, namely that Gregory became much less keen to announce the methods by which he made his mathematical discoveries and, as a consequence, it was not until Turnbull examined Gregory's papers in the library in St Andrews in the 1930s that the full brilliance of Gregory's discoveries became known. We can now be certain that during the summer of 1668 Gregory was completely familiar with the series expansions of sin, cos and tan. He also established that sec x dx = log(sec x + tan x) which solved a long standing problem in the construction of nautical tables. He published the Exercitationes Geometricae as a counterattack on Huygens. Although he did not disclose his methods in the small treatise he discussed topics including various series expansions, the integral of the logarithmic function, and other related ideas. Also during his time in London in the summer of 1668 Gregory attended meetings of the Royal Society and he was elected a fellow of the Society on 11 June of that year. He presented various papers to the Society on a variety of topics including astronomy, gravitation and mechanics. We have already mentioned that Robert Moray was a member of the Royal Society with whom Gregory was friendly. Moray was a fellow Scot and a graduate of St Andrews. It is almost certain that it was through Moray that Charles II was persuaded to create the Regius Chair of Mathematics in St Andrews, principally to allow Gregory a position in which he could continue his outstanding mathematical research. Gregory arrived in St Andrews late in 1668. He was not attached to a College, as were the other professors, but given the Upper Hall of the university library as his place of work. It was the only university building which was not part of a college so was the only possible place for an unattached professor. Gregory found that St Andrews was of classical outlook where the latest mathematical work was totally unknown. In 1669, not long after arriving in St Andrews, Gregory married Mary Jamesome who was a widow. They had two daughters and one son. While in St Andrews Gregory gave two public lecture each week which were not well received:... I am often troubled with great impertinences: all persons here being ignorant of these things to admiration. However Gregory was to carry out much important mathematical and astronomical work during his six years in the Regius chair. He kept in touch with current research by corresponding with Collins. Gregory preserved all Collins' letters, writing notes of his own on the backs of Collins' letters. These are still preserved in the St Andrews University library and provide a vivid record of how one of the foremost mathematicians of his day made his discoveries.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.html (4 of 7) [2/16/2002 11:12:35 PM]

Gregory

Collins sent Barrow's book to Gregory and, within a month of receiving it, Gregory was extending the ideas in it and sending Collins results of major importance. In February 1671 he discovered Taylor's theorem (not published by Taylor until 1715), and the theorem is contained in a letter sent to Collins on 15 February 1671. The notes Gregory made in discovering this result still exist written on the back of a letter sent to Gregory on 30 January 1671 by an Edinburgh bookseller. Collins wrote back to say that Newton had found a similar result and Gregory decided to wait until Newton had published before he went into print. He still felt badly about his dispute with Huygens and he certainly did not wish to become embroiled in a similar dispute with Newton. The feather of a sea bird was to allow Gregory to make another fundamentally important scientific discovery while he worked in St Andrews. The feather became the first diffraction grating but again Gregory's respect for Newton prevented him going further with this work. He wrote:Let in the sun's rays by a small hole to a darkened house, and at the hole place a feather (the more delicate and white the better for this purpose), and it shall direct to a white wall or paper opposite to it a number of small circles and ovals (if I mistake them not) whereof one is somewhat white (to wit, the middle which is opposite the sun) and all the rest severally coloured. I would gladly hear Mr Newton's thoughts of it. The Upper Room of the library had an unbroken view to the south and was an excellent site for Gregory to set up his telescope. Gregory hung his pendulum clock on the wall beside the same window. The clock, made by Joseph Knibb of London, was purchased in 1673. Huygens patented the idea of a pendulum clock in 1656 and his work describing the theory of the pendulum was published in 1673, the year Gregory purchased his clock. In 1674 Gregory cooperated with colleagues in Paris to make simultaneous observations of an eclipse of the moon and he was able to work out the longitude for the first time. However he had already begun work on an observatory. In 1673 the university allowed Gregory to purchase instruments for the observatory, but told him he would have to make applications and organise collections for funds to build the observatory. Gregory went home to Aberdeen and took a collection outside the church doors for money to build his observatory. On 19 July 1673 Gregory wrote to Flamsteed, the Astronomer Royal, asking for advice. He then travelled to England to purchase instruments. Gregory left St Andrews for Edinburgh in 1674. His reasons for leaving again paint a sorry picture of prejudice against the brilliant mathematician. Writing after taking up his Edinburgh chair Gregory said:I was ashamed to answer, the affairs of the Observatory of St Andrews were in such a bad condition, the reason of which was, a prejudice the masters of the University did take at the mathematics, because some of their scholars, finding their courses and dictats opposed by what they had studied in the mathematics, did mock at their masters, and deride some of them publicly. After this, the servants of the colleges got orders not to wait on me at my observations: my salary was also kept back from me, and scholars of most eminent rank were violently kept from me, contrary to their own and their parents wills, the masters persuading them that their brains were not able to endure it. In Edinburgh Gregory became the first person to hold the Chair of Mathematics there. He was not to hold the chair for long, however, for he died almost exactly one year after taking up the post. It was a year in which he was still very active in research in both astronomy and mathematics. On the latter topic he had

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.html (5 of 7) [2/16/2002 11:12:35 PM]

Gregory

become interested in the problem of solving quintic equations algebraically and made some interesting discoveries on Diophantine problems. His death came suddenly. One night he was showing the moons of Jupiter to his students with his telescope when he suffered a stroke and became blind. He died a few days later at the young age of 36. Whiteside writes in [1]:For all his talent and promise of future achievement, Gregory did not live long enough to make the major discovery which would have gained him popular fame. For his reluctance to publish his "several universal methods in geometry and analysis" when he heard through Collins of Newton's own advances in calculus and infinite series, he postumously paid a heavy price ... We have mentioned in this article many of the brilliant ideas which are due to Gregory. However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood: Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor expansions more than 40 years before Taylor; he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is impressive; he appears to be the first to attempt to prove that and e are not the solution of algebraic equations; he knew how to express the sum of the nth powers of the roots of an algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could not be solved by radicals. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (22 books/articles) A Poster of James Gregory

Mathematicians born in the same country

Some pages from publications

The title page of Vera circuli et hyperbolae quadratura (1667) The title page of Geometriae pars universalis (1668)

Cross-references to History Topics

1. Mathematics in St Andrews to 1700 2. Squaring the circle 3. Pi through the ages

Other references in MacTutor

Chronology: 1650 to 1675

Honours awarded to James Gregory (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1668

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.html (6 of 7) [2/16/2002 11:12:35 PM]

Gregory

Lunar features

Crater Gregory and Catena Gregory

Other Web sites

1. The Galileo Project 2. Rouse Ball 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Gregory.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory.html (7 of 7) [2/16/2002 11:12:35 PM]

Gregory_David

David Gregory Born: 3 June 1659 in Aberdeen, Scotland Died: 10 Oct 1708 in Maidenhead, Berkshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

David Gregory was a nephew of James Gregory. He studied at Marischal College, part of the University of Aberdeen, between 1671 and 1675. Notice that he started his university education at the age of 12 years. There is however, no evidence that he took his degree. After his university studies David, still only 16 years old, visited several countries on the continent and did not return to Scotland until 1683. At the age of 24 he was appointed Professor of mathematics at the University of Edinburgh. At Edinburgh David Gregory taught Newtonian theories. He is famed for this since he was the first university teacher to teach the 'modern' theories at a time when even Cambridge was still teaching Greek natural philosophy. Gregory's lecture notes at Edinburgh were to form the basis of Maclaurin's Treatise of Practical Geometry which was published in 1745. Gregory himself published Exercitatio geometria de dimensione curvarum in 1684 while at Edinburgh. Gregory also lectured at Edinburgh on mechanics and hydrostatics. In 1690 there was political and religious unrest in Scotland and this was certainly one of the reasons that David decided to leave for England. In 1691 David was elected Savilian Professor at Oxford. Newton was a major influence in his appointment. In the same year he became Savilian Professor he was elected to be a Fellow of the Royal Society. David Gregory certainly supported Newton strongly in the Newton - Leibniz controversy arguing, as did Wallis, that Leibniz had learnt of the calculus through a letter of Collins. In 1702 he published

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory_David.html (1 of 2) [2/16/2002 11:12:37 PM]

Gregory_David

Astronomiae physicae et geometricae elementa which was a popular account of Newton's theories. David Gregory and did important work on series. He also worked on optics publishing Catoptricae et dioptricae sphericae elementa in 1695. This work describes telescopes which were a special interest of his. He also experimented with making an achromatic telescope. David Gregory took ill on a journey from Bath to London and he died in an inn in Maidenhead. Article by: J J O'Connor and E F Robertson List of References (8 books/articles) Mathematicians born in the same country Cross-references to History Topics

Pi through the ages

Other references in MacTutor

Chronology: 1700 to 1720

Honours awarded to David Gregory (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1692

Savilian Professor of Astronomy

1691

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gregory_David.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory_David.html (2 of 2) [2/16/2002 11:12:37 PM]

Gregory_Duncan

Duncan Farquharson Gregory Born: 13 April 1813 in Edinburgh, Scotland Died: 23 Feb 1844 in Edinburgh, Scotland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Duncan Gregory's great great grandfather was James Gregory and he was the youngest son of another James Gregory (1753-1821) who was professor of medicine at Edinburgh University. Duncan was educated by his mother until he was 9 years old. His liking for constructing mechanical devices was noticed at this early age. In October 1825 he entered Edinburgh Academy (where Maxwell and D'Arcy Thompson were to be educated). He showed great promise at school and, during the winter of 1827, Duncan was sent to an academy in Geneva where his mathematical talents became apparent. On his return from Geneva, Duncan became an undergraduate at Edinburgh University where he began to study advanced mathematical topics and he also conducted experiments with polarised light. In October 1833, at the age of 20, Duncan Gregory entered Trinity College, Cambridge, receiving his BA in 1838 and his MA in 1841. In October 1840 he became a Fellow of Trinity and also an assistant tutor at the College. At this time the Cambridge Mathematical Journal was starting publication and Gregory became its first editor. Many of the papers in the early parts of the Journal are written by Gregory himself. Gregory declined a chair in Toronto in 1841 due to ill health. He returned to Edinburgh, where he was an applicant for a chair, but he died there shortly afterwards in Canaan Lodge at the age of 30. His main contribution was his theory of algebra which he defined as the study of the combinations defined by the laws of operation to which they were subject. This is one of the first definitions of modern algebra. His work in this area is described in the paper On the real Nature of symbolic Algebra which Gregory published in the Transactions of the Royal Society of Edinburgh. In this work Gregory built on the foundations of Peacock but went far further towards modern algebra. Gregory, in his turn, had a major influence on Boole and it was through his influence that Boole set out on his innovative research. Two other important works by Duncan Gregory are Examples of the Processes of the Differential and Integral Calculus and A Treatise on the Application of Analysis to Solid Geometry. The first became an important text at Cambridge which, by this time, had accepted Peacock, Herschel and Babbage's Analytical Society reforms and continental methods of calculus were taught in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory_Duncan.html (1 of 2) [2/16/2002 11:12:38 PM]

Gregory_Duncan

Cambridge. The second work was unfinished at his death but completed and published the following year. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gregory_Duncan.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gregory_Duncan.html (2 of 2) [2/16/2002 11:12:38 PM]

Grosseteste

Robert Grosseteste Born: 1168 in Suffolk, England Died: 9 Oct 1253 in Buckden, Buckinghamshire, England

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Grosseteste was educated at Oxford University. He became Chancellor of Oxford University in 1215 remaining in this post until about 1221. After this he held a number of ecclesiastical positions, then from 1229 to 1235 he was a lecturer in theology to the Franciscans. He became Bishop of Lincoln in 1235 and remained in this position until his death. As Bishop of Lincoln he attended the Council of Lyon (1245) and addressed the papal congregation at Lyon in 1250. Grosseteste worked on geometry, optics and astronomy. In optics he experimented with mirrors and with lenses. He believed that experimentation must be used to verify a theory by testing its consequences. In his work De Iride he writes:This part of optics, when well understood, shows us how we may make things a very long distance off appear as if placed very close, and large near things appear very small, and how we may make small things placed at a distance appear any size we want, so that it may be possible for us to read the smallest letters at incredible distances, or to count sand, or seed, or any sort or minute objects. Grosseteste realised that the hypothetical space in which Euclid imagined his figures was the same everywhere and in every direction. He then postulated that this was true of the propagation of light. He wrote the treatise De Luce on light. In De Natura Locorum he gives a diagram which shows light being refracted by a spherical glass container full of water. Grosseteste also made Latin translations of many Greek and Arabic scientific writings. He wrote a commentary on Aristotle's Posterior Analytics and Physics and many treatises on scientific subjects

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grosseteste.html (1 of 2) [2/16/2002 11:12:40 PM]

Grosseteste

including De Generatione Stellarum, Theorica Planetarum and De astrolabio. In an astronomy text he claimed that the Milky Way was the fusion of light from many small close stars. Roger Bacon was Grosseteste's student. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Other Web sites

1. Electronic Grosseteste 2. Online Anglican resources 3. The Catholic Encyclopedia 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Grosseteste.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grosseteste.html (2 of 2) [2/16/2002 11:12:40 PM]

Grossmann

Marcel Grossmann Born: 9 April 1878 in Budapest, Hungary Died: 7 Sept 1936 in Zurich, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Marcel Grossmann attended school in Basel then studied mathematics at Zurich. In 1900 he became an assistant to the geometer W Fiedler in Zurich. Grossmann obtained his doctorate from Zurich in 1912, then became a teacher in a school in Frauenfeld, northern Switzerland, in 1901 and in Basel in 1905. He became professor of descriptive geometry at the Eidgenössische Technische Hochschule in Zurich in 1907. Grossmann was a classmate of Einstein. Einstein turned to him for help with the mathematical formulation of the theory of general relativity. Grossmann discovered the relevance of the tensor calculus of Christoffel, Ricci-Curbastro and Levi-Civita to relativity. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

General relativity

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grossmann.html (1 of 2) [2/16/2002 11:12:42 PM]

Grossmann

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Grossmann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grossmann.html (2 of 2) [2/16/2002 11:12:42 PM]

Grothendieck

Alexander Grothendieck Born: 28 March 1928 in Berlin, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alexander Grothendieck's father was Russian and he was murdered by the Nazis. Grothendieck moved to France in 1941 and later entered Montpellier University. After graduating from Montpellier he spent the year 1948-49 at the Ecole Normale Supérieure in Paris. In 1949 Grothendieck moved to the University of Nancy where he worked on functional analysis with Dieudonné. He became one of the Bourbaki group of mathematicians which included Weil, Henri Cartan and Dieudonné. He presented his doctoral thesis Produits tensoriels topologiques et espaces nucléaires. Grothendieck spent the years 1953-55 at the University of Sao Paulo and then he spent the following year at the University of Kansas. However it was during this period that his research interests changed and they moved towards topology and geometry. In fact during this period Grothendieck had been supported by the Centre National de la Recherche Scientifique, the support beginning in 1950. After leaving Kansas in 1956 he therefore returned to the Centre National de la Recherche Scientifique. However in 1959 he was offered a chair in the newly formed Institut des Hautes Etudes Scientifiques which he accepted. In [2] the next period in Grothendieck's career is described as follows:It is no exaggeration to speak of Grothendieck's years 1959-70 at the IHES as a 'Golden Age', during which a whole new school of mathematics flourished under Grothendieck's charismatic leadership. Grothendieck's Séminaire de Géométrie Algébrique established the IHES as a world centre of algebraic geometry, and him as its driving force. He received the Fields Medal in 1966. In looking back at this period, one marvels at the generosity with which Grothendieck shared his ideas with colleagues and students, the energy he and his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.html (1 of 3) [2/16/2002 11:12:45 PM]

Grothendieck

collaborators devoted to meticulous redaction, the excitement with which they set out to explore a new land. During this period Grothendieck's work provided unifying themes in geometry, number theory, topology and complex analysis. He introduced the theory of schemes in the 1960s which allowed certain of Weil's number theory conjectures to be solved. He worked on the theory of topoi which are highly relevant to mathematical logic. He gave an algebraic proof of the Riemann-Roch theorem. He provided an algebraic definition of the fundamental group of a curve. Again quoting from [2]:The mere enumeration of Grothendieck's best known contributions is overwhelming: topological tensor products and nuclear spaces, sheaf cohomology as derived functors, schemes, K-theory and Grothendieck-Riemann-Roch, the emphasis on working relative to a base, defining and constructing geometric objects via the functors they are to represent, fibred categories and descent, stacks, Grothendieck topologies (sites) and topoi, derived categories, formalisms of local and global duality (the 'six operations'), étale cohomology and the cohomological interpretation of L-functions, crystalline cohomology, 'standard conjectures', motives and the 'yoga of weights', tensor categories and motivic Galois groups. It is difficult to imagine that they all sprang from a single mind. Grothendieck was always strongly pacifist in his views and campaigned against the military built-up of the 1960s. To devote himself to this end seems to have been the chief reason that he left the IHES in 1970. He abandoned mathematics as the main focus of his energies but, in 1970-72 he held an appointment as visiting professor at the Collège de France, then a similar appointment at Orsay for 1972-73. In 1973 he accepted an appointment as professor at the University of Montpellier. He took leave during 1984-88 to direct research at the Centre National de la Recherche Scientifique. He retired at age 60 in 1988 and the publication [2] was produced to honour that 60th birthday. In contrast to his acceptance of the 1966 Fields Medal, Grothendieck declined the Crafoord Prize in 1988. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1960 to 1970

Honours awarded to Alexander Grothendieck (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1966

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.html (2 of 3) [2/16/2002 11:12:45 PM]

Grothendieck

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Grothendieck.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.html (3 of 3) [2/16/2002 11:12:45 PM]

Grunsky

Helmut Grunsky Born: 11 July 1904 in Aalen, Württemberg, Germany Died: 5 June 1986

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Helmut Grunsky's father was Heinrich Grunsky and his mother was Lydia Stahl. Helmut was brough up in Aalen where he attended high school. At this stage, although he took a great interest in mathematics, it was not the topic which he intebded to pursue at university, rather he was interested in physics and engineering. In 1922 he entered the Institute of Technology in Stuttgart where he studied physics. After three years at the Institute of Technology in Stuttgart, in 1925 he entered the Institute of Technology in Berlin. After two years study there he was awarded the degree Diplom-Ingenieur. Now at this stage he began to undertake research in mathematics at the University of Berlin with a view to a doctorate in mathematics. Grunsky worked on complex analysis for his doctorate but he took a job before submitting his thesis. In November 1930 Grunsky took a job with the journal Jahrbuch über die Fortschritte der Mathematik which was published by the Preussische Akademie der Wissenschaften. For his doctorate he was using contour integration to study different problems concerning functions which are univalent in a domain of finite connectivity. He submitted his thesis Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche to the University of Berlin in 1932 and was awarded his Dr. phil. At this stage Grunsky continued his work for the journal Jahrbuch über die Fortschritte der Mathematik while he worked on his habilitation thesis. In 1935 Grunsky married Irma Schenk; they had three children Wolfgang (born in 1936), Hiltrud (born in 1938), and Eberhard (born in 1941). In the same year that he married Grunsky became editor of the journal and, three years later he published Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen in Mathematische Zeitschrift. This was Grunsky's thirteenth paper which was written as an habilitation dissertation. As for his doctoral

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grunsky.html (1 of 3) [2/16/2002 11:12:47 PM]

Grunsky

thesis, this work again looks at applications of contour integration [2]:This paper presents a study of coefficients for functions in a domain of finite connectivity on the sphere containing the point at infinity. Now Grunsky became qualified to lecture just before the start of World War II. The war made it impossible for him to begin an academic career at this stage and he also had to leave his position as editor of the Jahrbuch über die Fortschritte der Mathematik in 1939. Difficulties at the end of the war did not allow him to enter university teaching even then so in 1945 Grunsky took a position as a high school teacher in Trossingen, Württemberg. He continued to teach mathematics at the high school until 1949 when he became a Privatdozent at the University of Tübingen. It is worth noting that due to various circumstances Grunsky did not enter university teaching until he was 45 years old. Grunsky have an invited address at the International Congress of Mathematicians held at Cambridge, Massachusetts in 1950. For the academic year 1950-51 he was Visiting Professor at Washington State College in Pullman, Washington. Returning to Germany he was appointed as Extraordinary Professor at the University of Mainz. In 1958 Grunsky moved to the University of Würzburg where he became a full Professor. In 1963-64 he spent the academic year as a visiting professor at the Middle East Technical University in Ankara, Turkey. He remained in this position at Würzburg until he retired in 1972. Following this he had some other positions, first as research consultant at Washington University, St Louis in 1973, then visiting professor at the State University of New York in Albany in 1975, finally back to Washington University, St Louis as research consultant in 1977. Grunsky published three books and 44 papers, and he supervised eight doctoral students. All are listed in [2]. Of the three book one was Lectures on the theory of functions in multiply connected domains published in 1978. It [2]:... is most closely related to Helmut Grunsky's overall activity and consists of a reworking of some of his most significant contributions to function theory, in many cases with a considerable simplification of exposition. The final book he wrote was [1] The general Stokes' theorem published in 1983. Grunsky writes that the aim of the book is to give [1]:... an intrinsic and easily comprehensible presentation of Stokes's theorem. The treatise begins with an intuitive discussion of Stokes's theorem in the plane, which is then used as a model for generalising the result to higher dimensions. Grunsky first proves Stokes's theorem for suitable k-dimensional region in Rk, and then for k-dimensional regions in Rn. He then introduces the calculus of alternating multilinear forms and gives a proof of Stokes's theorem for manifolds. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grunsky.html (2 of 3) [2/16/2002 11:12:47 PM]

Grunsky

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Grunsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Grunsky.html (3 of 3) [2/16/2002 11:12:47 PM]

Guarini

Guarino Guarini Born: 1624 in Modena, Italy Died: 1683 in Milan, Italy Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

In the age-old connections between art and mathematics - however either is defined - no one is more worthy of attention than the Italian Baroque figure of Guarino Guarini. Trained as a theologian in the small but elite order of Counter-Reformation Clerics Regular, commonly known as the Theatines and the immediate model for the Jesuit Order, Guarini was also deeply interested in mathematics following in turn the Jesuit pursuit of all the arts and especially the new discoveries that surrounded the curious-minded of the Age of Discovery. Guarini spent his novitiate in Rome where he learned at first hand what Bernini and Borromini, now recognised as great masters of Baroque architecture, were doing, and their example presumably caused him to practice architecture, then considered as a mathematical art but not strictly a member of the Quadrivium, the mathematical division of the Liberal Arts, still supreme in the world of learning. He then spent years teaching and building in Italy and Paris, all of his structures having now disappeared. In 1666 he was called from Paris by the Duke of Savoy and Prince of Piedmont to his capital Turin, to take over the design of a great dynastic chapel to house the Holy Shroud, located within the Palace but opening into the choir of the adjoining Cathedral. This was to be his masterwork - la Capella dello Sindone, sadly badly damaged by fire in 1997. He remained in Turin for the rest of his life publishing mathematical works and tutoring the ducal family, while the Sindone Chapel was finally completed after his death. Even from the beginning of the eighteenth century there was a turn against the triumphs of the Baroque, so that we now have great difficulty in understanding such a figure as Guarini and his work, which also includes his mathematical philosophy and figural meaning, or geometrical iconography. Many recent critics have muddied the waters by tending to see him as a precursor of the 'new philosophy' of the seventeenth century, including Sigfried Giedion, Rubolf Wittkower and Alberto Perez-Gomez. As can be seen from his Placita philosophica (Paris, 1665), the Modenese cleric was a reformer of Aristotelianism and a staunch supporter of the official Second Scholastic in common with both contemporary Catholic and Lutheran authorities, while at the same time open to well-considered innovations but within traditional limits. He was familiar with contemporary mathematics especially geometry, but did not adopt algebra - his encyclopedic Euclides adauctus et methodicus mathematicaeque universalis (Turin, 1671) was aimed at clerical students unable to command such a newfangled technique, or maybe he had his own reservations about it. Certainly for him, Universal Mathematics, proclaimed in the title of the Euclides adauctus, was to do with a purely contemplative usage of traditional canonics or harmonics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guarini.html (1 of 3) [2/16/2002 11:12:48 PM]

Guarini

based on Book V of Euclid. This concerned the determination of mean and extreme proportion expressed in perspectival constructions - the perspectival ladder of Brunelleschi, Alberti and Piero della Francesca, an alternate adaptation of the machine of Eratosthenes for finding two mean proportionals, also described as the mesolabium in Vitruvius, Book IX. In much of this work he is a follower of Gregory of Saint-Vincent, S. J., in the summation to infinity implicit in the perspectival ladder. The line that Guarini refuses to cross is to pass from Universal Mathematics to mathesis, a term which for him seemed to represent the rising physico-mathematics of the Cartesian school. His reason for this is that ancient and medieval mathematics was not linked to causality, which was properly examined through discourse in the manner of Aristotelian physica or cosmology. He certainly believed in his Universal Mathematics, transcending all material concern to be analogous to a divina scientia, developed in the sixteenth century as a defence against scepticism, i. e. the alignment of mathematics with metaphysics or first philosophy. His notions were of course not singular, having been promulgated by Francesco Barozzi and Guiseppe Biancani, S. J. (Aristotelis loca mathematicis, Bolgna, 1615), and were espoused by John Wallis, the best English mathematician of the day, when he stated that beyond the equalities of quantity remained the similitude between qualities, i. e. proportion, which:... belongs more to quality than quantity. Guarini referred to Wallis among the few authorities for his mathematics - both were involved in summation to infinity. Guarini was adept at most applications of advanced curvature and projective techniques, if not indulging outright in the projective geometry of Desargues as his modern admirers have alleged, confusing 'projection' with 'projective geometry'. Guarini was therefore proficient in the French tradition of stone cutting using most difficult procedures, as well as gnomonics, or the study of sundials, devoting many published pages to each discipline. The key to such attitudes can be found in the Jesuit Bolognese belletrist and mathematical encyclopedist, Mario Bettini, who Guarini implicitly relied upon in much of his work. However while there are some indications of rationalising the elements of architecture more geometrico, found in his Architectura civile, Guarini accepted the traditions of Vitruvius and the High Renaissance with a bracing freedom in the decorative development of the orders, as well as a innovative appreciation of the daring of Gothic structure, exceptional at this date. Guarini wrote a great summa of philosophy which carried a treatment of light, strictly discoursive in manner, and he was reluctant to separate light from its traditional transcendental interpretation. Guarini was aware of the transformative implications of geometrical processes, especially through light, optics and projection so rhetorically advanced by Bettini. His great achievement must surely lie in the protracted if complex elaboration of mathematics and advanced architecture that was even in his own day little understood, elegant and elite answers to great challenges that have been either maligned or misinterpreted since. Article by: James McQuillan, Famagusta, TRNC, July 2000. Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guarini.html (2 of 3) [2/16/2002 11:12:48 PM]

Guarini

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Guarini.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guarini.html (3 of 3) [2/16/2002 11:12:48 PM]

Guccia

Giovanni Battista Guccia Born: 21 Oct 1855 in Palermo, Italy Died: 29 Oct 1914 in Palermo, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giovanni Guccia's father was Giuseppe Maria Guccia and his mother was Chiara Cipponeri. Giovanni was born in Sicily into an important wealthy Sicilian family. His father was related to the Marquis of Ganzaria and no expense was spared to give Giovanni a good education. He was educated in Palermo where he not only learnt academic subjects, but he also became an expert horseman and excelled in sports in general. Guccia began his studies at the university in Palermo but he later undertook research under Cremona in Rome. While a student at Rome, Guccia attended the French Association for the Advancement of Science which was held at Rheims. The paper he read at the meeting in Rheims was on certain rational surfaces and his presentation was highly praised by Sylvester. This was encouragement indeed for the young mathematician beginning his research career. In 1880 Guccia submitted his doctoral dissertation on a class of surfaces representable point by point on a plane to the University of Rome and after successfully defending his thesis he returned to his native town of Palermo. Appointed to the university there, he began to work out an ambitious research programme. In 1884 he set up a mathematical society in Palermo and, being from a wealthy family, he was able to provide all the necessary resources to have the project rapidly become successful. Speziali writes in that, for the new society, Guccia provided [1]:... the meeting place, a library and all necessary funds. His generous offer was favourably received, and on 2 March 1884 the society's provisional statutes were signed by twenty-seven members. The goal was to stimulate the study of higher mathematics by means of original communications presented by the members of the society on the different branches of analysis and geometry, as well as on rational mechanics, mathematical physics, geodesy, and astronomy. The publication for the new society was the Rendiconti del Circolo matematico di Palermo. Guccia himself had four articles appear in the first volume of this publication, the first on Cremona transformations and a generalisation of a theorem due to Hirst, while the second was on a generalisation of a theorem due to Max Noether. This first volume appeared in four parts: July 1885, September 1886, December 1886, and September 1887. The completed volume was presented by Bertrand to the Académie des Sciences in Paris on 7 November 1887, stating that it was a publication of remarkably high

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guccia.html (1 of 2) [2/16/2002 11:12:49 PM]

Guccia

quality. When Guccia's society passed a statute allowing foreign members in February 1888, it had rapidly reached its goal of becoming a top quality international society with a leading mathematical publication. In 1889 Guccia was appointed to the chair of geometry at Palermo which he held for the rest of his life. He set up a mathematical publishing house in Palermo in 1893 adding further publications to the Rendiconti del Circolo matematico di Palermo. Guccia became editor of all these publications. As we have indicated above, Guccia's work was on geometry, in particular Cremona transformations, classification of curves and projective properties of curves. His results published in volume one of the Rendiconti del Circolo matematico di Palermo were extended by Corrado Segre in 1888 and Castelnuovo in 1897. Speziali writes in [1]:Although the majority of Guccia's publications are very short, they all contain original ideas and new relations profitably used by other geometers. This is particularly true of his publications on projective involutions, which laid the foundations for the generalisations of Federico Enriques and Francesco Severi. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Guccia.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guccia.html (2 of 2) [2/16/2002 11:12:49 PM]

Gudermann

Christoph Gudermann Born: 25 March 1798 in Vienenburg (near Hildesheim), Germany Died: 25 Sept 1852 in Münster, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Christoph Gudermann's father was a school teacher. Christoph was academically able in his schooling so, as many able boys at this time did, it was intended that he should train to became a priest. He attended the University of Göttingen but there, among the whole range of subjects he studied, he became interested in mathematics. Gudermann then followed in his father's footsteps by becoming a school teacher. He obtained a post as a teacher of mathematics in a secondary school in Kleve in 1823 and taught there until 1832 when he was appointed to the Theological and Philosophical Academy in Münster. His first appointment in the Academy in Münster was as an extraordinary professor, but later he was promoted to ordinary professor of mathematics there. Gudermann worked almost exclusively on spherical geometry and special functions but he is not remembered for any original mathematical results in these areas. This is not to say that he did not do useful original research but just that he suffered the fate that many mathematicians have suffered, namely that a comprehensive theory was developed later which meant that his contributions fused into the theory. His own contributions tended to be a whole series of special cases (although this could not have been obvious at the time) which were forgotten later when the general results which included them were found. He did write a book on spherical geometry and [1]:In the introduction he pointed out that a plane was a special case of a spherical surface, that is a sphere with infinite radius. For this reason and because of its constant curvature there exits many similarities between spherical geometry and plane geometry; yet at the same time Gudermann considered more interesting the study of cases where the similarity no longer holds. In his more extensive work on the theory of special functions Gudermann published several papers beginning in 1830 which extended work which was developed by Euler, Landen, Legendre, Abel and Jacobi. He summarised his contributions in two monographs published in 1833 and 1844. A third monograph, which he promised to write on the topic, was not written because of his early death. Gudermann is best known, however, as the teacher of Weierstrass between 1839 and 1841 while Weierstrass worked for his secondary school teacher's certificate. Gudermann, at this time, was particularly interested in the theory of elliptic functions and in the expansion of functions by power http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gudermann.html (1 of 2) [2/16/2002 11:12:50 PM]

Gudermann

series. In particular his use of power series in the study of the hyperbolic functions is of importance. This was to influence Not only did Gudermann undertake research in the theory of elliptic functions but he was one of the first to teach the topic. Weierstrass spent the academic year 1839-1840 taking Gudermann's course on elliptic functions. Much of the article [3] is devoted to studying Gudermann's work and how it influenced the direction that Weierstrass's research would take, in particular it played an important role in Weierstrass's habilitation thesis. Manning writes in [3]:The concepts on which Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Münster from 1841 through 1842, while still under the influence of Gudermann. The transformation of his conception of an analytic function from a differentiable function to a function expandable into a convergent power series was made during this early period of Weierstrass's mathematical activity. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Gudermann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gudermann.html (2 of 2) [2/16/2002 11:12:50 PM]

Guenther

Adam Wilhelm Siegmund Guenther Born: 6 Feb 1848 in Nuremburg, Germany Died: 3 Feb 1923 in Munich, Germany Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adam Guenther's father was Ludwig Leonard Guenther, who was a Nuremburg businessman, and his mother was Johanna Weiser. Guenther began his university studies of mathematics and physics in 1865. It was common at this time for students in Germany to study at several different universities, and indeed Guenther carried this out with enthusiam for he studied at the universities of Erlangen, Heidelberg, Leipzig, Berlin and Göttingen. While he was undertaking research for his doctorate political events intervened to interrupt his studies. In 1870 the political situation between France and Prussia was deteriorating. The popularity of Napoleon III, the French emperor, was declining in France and he thought a war with Prussia might change his political fortunes since his advisers having told him that the French Army could defeat Prussia. Bismarck, the Prussian chancellor, saw a war with France as an opportunity to unite the South German states. With both sides feeling that a war was to their advantage, the Franco-Prussian War became inevitable. On 14 July, Bismarck sent a telegram aimed at infuriating the French government. He succeeded, for on the 19 July France declared war on Prussia. Guenther participated in the war which ended in the defeat of France in 1871. At the end of the war Guenther returned to complete his doctoral dissertation and the degree was awarded by Erlangen for Studien zur theoretischen Photometrie in 1872. He also took his examinations to qualify as a teacher of mathematics and physics and, in 1872, he began teaching mathematics at a school in Weissenburg, Bavaria. In the following year he submitted a habilitation thesis to Erlangen: Darstellung der Näherungswerte der Kettenbrüche in independenter Form. With his qualification to teach in universities, Guenther became a docent at Munich Polytechnicum in 1874. There he taught mathematics, then two years later he was appointed professor of mathematics and physics at the Gymnasium in Ansbach also in Bavaria. After holding this position for ten years, Guenther moved back to Munich in 1886 when he was appointed as professor of geograghy at the Technische Hochschule there. He held this post until he retired in 1920 but, in addition, he was rector of the school from 1911 to 1914. The legislature of the German Empire became known as the Reichstag from 1871. This revived the name of the legislature of the Holy Roman Empire which was used from the 12th century to 1806. Guenther served on the Reichstag from 1878 to 1884 as a member of the Liberal Party, then he served on the Bavarian Landtag, the state parliament of Bavaria, from 1884 to 1899 and again from 1907 to 1918 again

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guenther.html (1 of 2) [2/16/2002 11:12:51 PM]

Guenther

as a Liberal. During World War I he headed the Bavarian flying weather service. Guenther's contributions to mathematics include a treatise on the theory of determinants (1875), hyperbolic functions (1881), and the parabolic logarithm and parabolic trigonometry (1882). He also wrote [1]:... numerous books and journal articles [which] encompass both pure mathematics and its history and physics physics, geophysics, meteorology, geography, and astronomy. The individual works on the history of science, worth reading even today, bear witness to a thorough study, a remarkable knowledge of the relevant secondary literature, and a superior descriptive ability. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Guenther.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guenther.html (2 of 2) [2/16/2002 11:12:51 PM]

Guinand

Andrew Paul Guinand Born: 3 March 1912 in Renmark, South Australia, Australia Died: 22 March 1987 in Peterborough, Ontario, Canada Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Andrew Guinand, always known as Andy, went to school at St Peter's College, Adelaide from 1924 to 1929. He then entered St Mark's College of the University of Adelaide in 1930 to study mathematics, graduating in 1933. He was a great sportsman in his university days, described in [1] as follows:... it is recorded that he was a proficient gymnast, he rowed with the Torrens Rowing Club, and he competed in bicycle races with the South Australia Amateur Cycling Association. In 1934 Guinand won a Rhodes Scholarship to attend the University of Oxford in England. This was the typical route for the top Australian academics at that time, and Guinand studied at Oxford for his doctorate under Titchmarsh's supervision. On of the examiners for his thesis was Hardy and [1]:Andy in later years treasured a note from Hardy asking him to postpone his oral examination because he (Hardy) had been asked to play in a cricket match for the Trinity College Servant's Team. In session 1937/38 Guinand studied at Göttingen, then in 1939/40 at Princeton in the United States. In 1940 he joined the Royal Canadian Air Force, returned to England and was a navigator on many missions. When he was stationed 70 km from Oxford he would ride there on his bicycle to continue his mathematical research. After being an assistant at Cambridge, he became a lecturer at the Royal Military College of Science in 1947. He was promoted to Associate Professor of Mathematics before returning, in 1955, to a chair at the University of New England at Armidale which lies on the valley slopes of Dumaresq Creek in the New England Range in North Western Australia. During his two years at Armidale he was Head of Department, then he left to take up a post in Edmonton, Canada at the University of Alberta. His next appointment was to the University of Saskatchewan in 1960, then in 1964 he became the first chairman of the mathematics department at Trent University in Peterborough in south-eastern Ontario, Canada. Trent University, 115 km east-north-east of Toronto, had been founded in 1963. Guinand worked on summation formulas and prime numbers, the Riemann zeta function, general Fourier type transformations, geometry and some papers on a variety of topics such as computing, air navigation, calculus of variations, the binomial theorem, determinants and special functions. In [1] W N Everitt writes:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guinand.html (1 of 2) [2/16/2002 11:12:52 PM]

Guinand

As a student of Titchmarsh in Oxford in the years immediately before the second world war it was natural that Guinand's research interests should be directed into the field of Fourier analysis and the Riemann zeta function. ... [In an important paper in 1948] the main application of the general result yields a deep-seated connection between the distribution of the prime numbers and the location of the zeros of the Riemann zeta function on (or near to it if the Riemann hypothesis is false) the critical line in the complex plane... Guinand was convinced that these results could lead to more information about the Riemann zeta function, and he was disappointed that he was not able to advance further in this area and that others did not take up the possibility themselves. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Guinand.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guinand.html (2 of 2) [2/16/2002 11:12:52 PM]

Guldin

Paul Guldin Born: 12 June 1577 in St Gall (now Sankt Gallen), Switzerland Died: 3 Nov 1643 in Graz, Austria

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Guldin was named Habakkuk Guldin at his birth. He became a goldsmith and worked at that during his teens. Although of Jewish descent his parents were Protestants but Guldin became a convert to Catholicism at the age of 20 and joined the Jesuit Order. At this point he changed his name to Paul. In 1609 he was sent to the Jesuit Collegio Romano in Rome where he studied under Clavius. After this he taught at Jesuit Colleges in Rome and Graz. He was also professor of mathematics at Vienna from 1623 until 1637 when he returned to Graz. In the middle of his years in Vienna he spent some time at the Silesian principality of Sagan. Guldin corresponded with Kepler, but on religious topics not mathematics or astronomy. Guldin's most important work is in 4 volumes. In Volume 1 centres of gravity are considered, in particular he discusses the centre of gravity of the Earth. Volume 2 contains Guldin's Theorem If a plane figure is rotated about an axis in its plane then the volume of the solid body formed is equal to the product of the area with the distance travelled by the centre of gravity. Volume 3 contains work on cones, cylinders and solids of revolution. Article by: J J O'Connor and E F Robertson List of References (3 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guldin.html (1 of 2) [2/16/2002 11:12:54 PM]

Guldin

Mathematicians born in the same country Other references in MacTutor

Chronology: 1600 to 1625

Other Web sites

1. Jesuit Scientists 2. The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Guldin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Guldin.html (2 of 2) [2/16/2002 11:12:54 PM]

Gunter

Edmund Gunter Born: 1581 in Hertfordshire, England Died: 10 Dec 1626 in London, England Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edmund Gunter attended Westminster School, then entered Christ Church, Oxford in 1599. He graduated in 1603 but he remained at Oxford until 1615 when he received a divinity degree. Gunter was ordained and became Rector of St George's Church in Southwark in 1615. He held this Church position until his death. In addition to being Rector of St George's Church, Gunter became professor of astronomy at Gresham College London in 1619, also holding this post until his death. A colleague and friend of Briggs, Gunter published seven figure tables of logarithms of sines and tangents in 1620 in Canon Triangulorum, or Table of Artificial Sines and Tangents. The words cosine and cotangent are due to him. He made a mechanical device, Gunter's scale, to multiply numbers based on the logs using a single scale and a pair of dividers. It was called the gunter by seamen and was an important step in the development of the slide rule. Gunter published his description in 1624 in Description and Use of the Sector, the Crosse-staffe and other Instruments. It is worth noting that in this work Gunter uses the contractions sin for sine and tan for tangent in his drawing of his scale although not in the text of the book. He also invented Gunter's chain which was 22 yards long with 100 links. It was used for surveying and the unit of area called an acre is ten square chains. Gunter also did important work on navigation, publishing New Projection of the Sphere in 1623. He also studied magnetic declination and was the first to observe the secular variation. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

The trigonometric functions

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gunter.html (1 of 2) [2/16/2002 11:12:55 PM]

Gunter

Other references in MacTutor

Chronology: 1600 to 1625

Other Web sites

1. The Galileo Project 2. G Pastori (Slide rule history) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Gunter.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Gunter.html (2 of 2) [2/16/2002 11:12:55 PM]

Haar

Alfréd Haar Born: 11 Oct 1885 in Budapest, Hungary Died: 16 March 1933 in Szeged, Hungary

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

In 1903, in his final year in school, Alfréd Haar won first prize in the Eötvös contest in mathematics. Haar travelled to Germany in 1904 to study at Göttingen and there he studied under Hilbert's supervision, obtaining his doctorate in 1909 with a dissertation entitled Zur Theorie der orthogonalen Funktionensysteme. Haar then taught at Göttingen until 1912 when he returned to Hungary and held chairs at the university in Kolozsvár (which is now Cluj in Romania), Budapest University and Szeged University. In fact after Word War I Kolozsvár was no longer in Hungary, so the University there had to move within Hungarian borders and it moved to Szeged, where there had previously been no university. Haar, together with Riesz, rapidly made a major mathematical centre from the new university. Haar worked in analysis. His doctoral thesis studied orthogonal systems of functions. Later he went on to study partial differential equations. He also wrote on Chebyshev approximations and linear inequalities. Between 1917 and 1919 he worked on the variational calculus. Haar is best remembered for his work on analysis on groups. In 1932 he introduced a measure on groups, now called the Haar measure, which allows an analogue of Lebesgue integrals to be defined on locally compact topological groups. It was used by von Neumann, by Pontryagin in 1934 and Weil in 1940 to set up an abstract theory of commutative harmonic analysis. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Haar.html (1 of 2) [2/16/2002 11:12:57 PM]

Haar

Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Alfréd Haar

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1930 to 1940

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Haar.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Haar.html (2 of 2) [2/16/2002 11:12:57 PM]

Hachette

Jean Nicolas Pierre Hachette Born: 6 May 1769 in Mézières, Ardennes, France Died: 16 Jan 1834 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean Nicolas Pierre Hachette's father was Jean Pierre Hachette, a bookseller by trade, while his mother was Marie Adrienne Gilson. He attended the Collège de Charleville, then went on to attend courses at the Ecole Royale du Génie of Mézières where he was taught by, among others, C Ferry and Monge. This first contact with Monge would be one of many throughout his career, and even at this young age Hachette greatly impressed him. In 1785, when Hachette was sixteen years old, he entered the University of Rheims. He graduated in 1787 after two years of study and then from 1788 he was employed by his old school, the Ecole Royale du Génie of Mézières, as a draftsman and technician. Monge had set up a descriptive geometry course at the Ecole Royale du Génie which Ferry was teaching when Hachette was appointed to the staff. Monge, who had alternated between Mézières and Paris for a number of years, had finally left Mézières in December 1784 but the courses which he had set up were still being taught. Hachette assisted Ferry in teaching the descriptive geometry course. Of course 1789 was an eventful year in French history with the storming of the Bastille on 14 July 1789 marking the start of the French Revolution. This was to change the course of Hachette's life, particularly since he was a strong supporter of the Revolution. At first Hachette continued with his post in Mézières, then in 1792 he entered a competition for professor of hydrography at Collioure and Port Vendres. He won the competition was was appointed to the chair. However, political events were gathering pace. France had declared war on Austria and Prussia on 20 April 1792. French defeats led to unrest in France and, on 10 August 1792, there was further revolutions

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hachette.html (1 of 3) [2/16/2002 11:12:59 PM]

Hachette

by the people with nobles and clergy murdered during September. On 21 September the monarchy was abolished in France and a republic was declared. Ferry was elected to be a deputy on the National Convention in 1873 and Hachette returned to the Ecole Royale du Génie of Mézières to take over his position teaching mathematics. During this period Hachette showed his strong support for the Revolution. He was [1]:... active in the political life of his native city, and at the Ecole du Génie he sought rapid training of officers qualified for the revolutionary army and to remove teachers and students whose patriotism seemed doubtful to him. This period in his life was short, however, for in the following year the Committee of Public Safety in Paris required his services and he went there immediately. He advised on various technological tasks, such as manufacture of weapons and military applications. In March 1794 the National Convention set up a body whose task it was to establish the Ecole Centrale des Travaux Publics (soon to become the Ecole Polytechnique). Hachette took an active role in setting up the Ecole using his experience at Mézières to good effect. He was appointed as an assistant professor in descriptive geometry at the School in November 1794. The School began to operate from June 1795 but Hachette had been teaching before that at the short-lived Ecole Normale de l'An III from January to May of that year as Monge's assistant. In 1799 Hachette was promoted to full professor at the Ecole Polytechnique and continued to serve the School both as a teacher and an organiser. The students at the School were served well as Hachette encouraged his best students with research projects, raising the level and reputation of the School. He was an editor of the Journal de l'Ecole Polytechnique and then in 1804 created a new publication for the School setting up the Correspondance sur l'Ecole Polytechnique. Hachette edited this publication until he was forced out of the School in 1816. The book [2] discusses these two publications, and Hachette's role in them. Another educational establishment, the Ecole Normale, was set up to train secondary school teachers and from 1810 Hachette taught there and at the Faculty of Sciences in Paris in addition to the Ecole Polytechnique. However, in 1816 the Government reorganised the Ecole Polytechnique and Hachette found himself excluded from the school. It is a little difficult to understand this, as is Louis XVIII refusing to confirm Hachette's election to the Académie des Sciences in 1823. The usual reason for such decisions would be political and so it must be that Hachette was out of favour. Since his political activity seems confined to Mézières in 1793, and in other ways he seems to have remained in favour in Paris, speculation about his exclusion seems pointless. Hachette worked on descriptive geometry, collected work by Monge and edited Monge's Géométrie descriptive which was published in 1799. He also published on a wide range of topics from his own major works on geometry, to works on applied mechanics including the theory of machines. His work on machines includes much in the area of applied mechanics, but he was also interested in applied hydrodynamics and steam engines. In fact he published interesting work on the history of steam engines. The final area which interested Hachette was physics. Again his interests in this topic were wide, including work on optics, electricity, magnetism and scientific instruments. His contribution is summed up in [1] as follows:Although not a scientist of the first rank, Hachette nevertheless contributed to the progress of French science at the beginning of the nineteenth century by his efforts to increase the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hachette.html (2 of 3) [2/16/2002 11:12:59 PM]

Hachette

prestige of the Ecole Polytechnique and by making Monge's work widely known, especially in descriptive and analytic geometry and in the theory of machines. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country Honours awarded to Jean Nicolas Pierre Hachette (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Jean Hachette (15th Arrondissement)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hachette.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hachette.html (3 of 3) [2/16/2002 11:12:59 PM]

Hadamard

Jacques Salomon Hadamard Born: 8 Dec 1865 in Versailles, France Died: 17 Oct 1963 in Paris, France

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jacques Hadamard's father, Amédée Hadamard, married Claire Marie Jeanne Picard on 6 June 1864. Amédée Hadamard, who was of a Jewish background, was a teacher who taught several subjects such as classics, grammar, history and geography while Jacques' mother taught piano giving private lessons in their home. At the time that Jacques was born Amédée was teaching at the Lycée Impérial in Versailles but the family moved to Paris when Jacques was three years old when his father took up a position at the Lycée Charlemagne. This was an unfortunate time for a child to be growing up in Paris. The Franco-German War which began on 19 July 1870 went badly for France and on 19 September 1870 the Germans began a siege of Paris. This was a desperate time for the inhabitants of the town who killed their horses, cats and dogs for food. Hadamard's family, like many others, eat elephant meat to survive. Paris surrendered on 28 January 1871 and The Treaty of Frankfurt, signed on 10 May 1871, was an humiliation for France. Between the surrender and the signing of the treaty there was essentially a civil war in Paris and the Hadamard's house was burnt down. The war was not the only cause of sadness for the Hadamards. Jacques' young sister Jeanne died in 1870 before the siege of Paris and another sister Suzanne, who was born in 1871, died in 1874. Jacques began his schooling at the Lycée Charlemagne where his father taught. In his first few years at school he was good at all subjects except mathematics. He excelled in particular in Greek and Latin. He wrote in 1936:... in arithmetic, until the fifth grade, I was last or nearly last.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hadamard.html (1 of 6) [2/16/2002 11:13:01 PM]

Hadamard

He was not accurate in this statement for although at first it is true he was weak in arithmetic, by the fifth class he was placed second in his class at the Lycée. By this time (1875) he was winning prizes in many subjects in the Concours Général, the national competition for school pupils. It was a good mathematics teacher who turned him towards mathematics and science, when he was in this fifth class. In 1875 Hadamard's father, having acquired a poor reputation as a teacher, was transferred to the Lycée Louis-le-Grand and Jacques attended this school from 1876. In 1882 he graduated Bachleier ès lettres et ès sciences then, in the following year, he received his Baccalauréat ès sciences. He was awarded first prize in algebra and first prize in mechanics in the Concours Général of 1883. In 1884 Hadamard took the entrance examinations for Ecole Polytechnique and Ecole Normale Supérieure; he was placed first in both examinations. He chose the Ecole Normale Supérieure, where he soon made friends with his fellow students including Duhem and Painlevé. Among his teachers were Jules Tannery, Hermite, Darboux, Appell, Goursat and Emile Picard. Already at this stage he began to undertake research, investigating the problem of finding an estimate for the determinant generated by coefficients of a power series. He graduated from the Ecole Normale Supérieure on 30 October 1888. While working on his doctorate he was a school teacher. At first he was attached to the Lycée de Caen but without teaching duties. From June 1889 he taught at the Lycée Saint-Louis and then from September 1890 at the Lycée Buffon where he taught for three years. Although his research went extremely well, his teaching was less appreciated probably because he demanded more of his pupils than their abilities allowed. His one great success was teaching Fréchet, and the two corresponded over a period of about nine years. Hadamard obtained his doctorate in 1892 for a thesis on functions defined by Taylor series. This work on functions of a complex variable was one of the first to examine the general theory of analytic functions, in particular his thesis contained the first general work on singularities. In the same year Hadamard received the Grand Prix des Sciences Mathématique for his paper Determination of the number of primes less than a given number. The topic proposed for the prize, concerning filling gaps in Riemann's work on zeta functions, had been put forward by Hermite with his friend Stieltjes in mind. Stieltjes had claimed in 1885 to have proved the Riemann hypothesis but had never published his "proof" and, after the prize topic was announced in 1890, Stieltjes discovered a gap in his "proof" which he was unable to fill. He never submitted an entry for the prize but Hadamard, between the time his thesis was submitted and his oral examination, realised that his results could be applied to zeta functions. His paper on entire functions and zeta functions was awarded first prize. The year 1892 was significant for Hadamard in addition to the academic achievements described above. In June of that year he married Louise-Anna Trénel who was, like Hadamard, of a Jewish background. They had known each other from childhood and shared a love of music. They moved to Bordeaux the following year when Hadamard was appointed as a lecturer at the University. If he had failed to come up to scratch as a teacher at the Lycée Buffon, this was far from the case now for he impressed everyone with both his research and teaching skills. On 1 February 1896 he was appointed as Professor of Astronomy and Rational Mechanics at Bordeaux. The four years which he spent in Bordeaux were not only busy ones with his family life, with two sons Pierre and Etienne being born during this time, but they were also extremely productive ones for http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hadamard.html (2 of 6) [2/16/2002 11:13:01 PM]

Hadamard

Hadamard's research. He published 29 papers during these four years but they are remarkable more for their depth and the range of the topics which they covered rather than their number. Perhaps his most important result proved during this time was the prime number theorem which he proved in 1896. This states that:The number of primes n tends to as n/loge n. This theorem was conjectured in the 18th century, but it was not proved until 1896, when Hadamard and (independently) Charles de la Vallée Poussin, used complex analysis. The proof had been outlined by Riemann in 1851, but the necessary tools had not been developed at that time. This problem was one of the major motivations for the development of complex analysis from 1851 to 1896 when Riemann's outlined proof was finally completed. Solving this important open problem was not Hadamard's only remarkable contribution of 1896. In the same year he published a paper on properties of dynamic trajectories which won the Bordin Prize of the Academy of Sciences. The topic proposed for the prize had been one on geodesics and Hadamard's work in studying the trajectories of point masses on a surface led to certain non-linear differential equations whose solution also gave properties of geodesics. His work was therefore a major contribution to both geometry and to dynamics. Another result which Hadamard published during his time in Bordeaux was his famous determinant inequality of 1893. Matrices whose determinants satisfied equality in the relation are today called Hadamard matrices and are important in the theory of integral equations, coding theory and other areas. Hadamard first became involved in politics during his time in Bordeaux. Alfred Dreyfus, a relation of Hadamard's wife, came from Alsace. Born into a Jewish family, Dreyfus embarked on a military career. In 1894, when he was in the War Ministry, he was accused of selling military secrets to the Germans and he was sentenced to life imprisonment. Although his trial had been highly irregular the anti-Semitic views of many people made the verdict popular. Forged documents and cover-ups soon showed that the legal process had been suspect. At first Hadamard, like many people, assumed that Dreyfus was guilty. However after moving to Paris in 1897 he began to discover how evidence against Dreyfus had been forged. He became a leading member of those trying to correct the injustice. Painlevé described a conversation he had with Hadamard on the matter in 1897 (see for examnple [5]):For almost an hour, [Hadamard] tried to convince me of Dreyfus's innocence, and at the end, faced with my disbelief, he did his best to make me understand the intrinsic value of his arguments and his complete lack of passion and sentimentality... he based his belief in his innocence on the facts. In 1898 the novelist Emile Zola wrote an open letter accusing the army of covering up its mistaken conviction of Dreyfus. The case split France into two opposing camps leading to issues far beyond the guilt or innocence of Dreyfus. There were demands to bring Zola to justice, anti-Semitic riots broke out, and there was a petition demanding that Dreyfus be retried. Zola was sentenced to a year in prison and fined 3,000 francs. By 1899 there had been a confession to the forgeries, followed by a suicide, and Dreyfus was retried, again found guilty, but pardoned. Hadamard took an active part in clearing Dreyfus's name which finally happened on 22 July 1906, when Dreyfus was reinstated and decorated with the Legion of Honour. Laurent Schwartz wrote (see for example [26]):It is the Dreyfus Affair which was in the sense of defence of justice the great affair of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hadamard.html (3 of 6) [2/16/2002 11:13:01 PM]

Hadamard

[Hadamard's] life. From the moment when he understood the enormity of the injustice perpetrated against a man in the name of reason of state, and the consequences which anti-Semitism could have, he devoted himself passionately to the review of the trial. This affair marked his life. Long before the Dreyfus Affair had ended Hadamard had, as we have indicated, moved from Bordeaux to Paris. In 1897 he resigned his chair in Bordeaux to take up lesser posts, one in the Faculty of Science of the Sorbonne and one at the Collège de France. Soon after arriving in Paris in October 1897, he published the first volume of Leçons de Géométrie Elémentaire . This volume on two dimensional geometry appeared in 1898, and was followed by a second volume on three dimensional geometry in 1901. This work had been requested by Darboux and it went on to have a major influence on the teaching of mathematics in France. Hadamard received the Prix Poncelet in 1898 for his research achievements over the preceding ten years. His research turned more towards mathematical physics from the time he took up the posts in Paris, yet he always argued strongly that he was a mathematician, not a physicist. He believed that it was the rigour of his work which made it mathematics. In particular he worked on the partial differential equations of mathematical physics producing results of outstanding importance. His famous 1898 work on geodesics on surfaces of negative curvature laid the foundations of symbolic dynamics. Among the topics he considered were elasticity, geometrical optics, hydrodynamics and boundary value problems. He introduced the concept of a well-posed initial value and boundary value problem. During Hadamard's first five years in Paris another three children were born, first another son Mathieu and then two daughters Cécile and Jacqueline. He continued to receive prizes for his research and he was further honoured in 1906 with election as President of the French Mathematical Society. In 1909 he was appointed to the chair of mechanics at the Collège de France. In the following year he published Leçons sur le calcul des variations which helped lay the foundations of functional analysis (he introduced the word functional). Then in 1912 he was appointed as professor of analysis at the Ecole Polytechnique where he succeed Jordan. Poincaré had strongly supported Hadamard for this chair but, within a few months, he died at the tragically young age of 58. Hadamard then undertook the hugely difficult task of surveying Poincaré's work and by the end of the summer of 1912 he had produced two major articles. As Paul Lévy wrote:One had to be Hadamard to dare to undertake the exposition of all of [Poincaré's] immense work which dealt with so many different areas, and to finish it in one summer. Near the end of 1912 Hadamard was elected to the Academy of Sciences to succeed Poincaré. He wrote later that his many years of "pure joy" from the time of his marriage came to an end in 1916. It was World War I which led to a great tragedy for Hadamard with his two older sons being killed in action. Both were killed at Verdun and Hadamard was lecturing in Rome when Pierre was killed. He left before receiving the news which he did not discover until arriving back in Paris despite the best efforts of Fano, Volterra's wife and others to get news to him. Etienne was killed near Verdun about two months later. Hadamard knew only one way to push the pain of these tragedies away enough to allow him to keep going, and that was to throw himself even more vigorously into mathematics. He was appointed to Appell's chair of analysis at the Ecole Centrale in 1920 but retained his positions in the Ecole Polytechnique and the Collège de France. During the years between his appointment and 1933 he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hadamard.html (4 of 6) [2/16/2002 11:13:01 PM]

Hadamard

travelled widely visiting the United States twice, Spin, Czechoslovakia, Italy, Switzerland, Brazil, Argentina, and Egypt. He continued to produce books and papers of the highest quality, publishing perhaps his most famous text Lectures on Cauchy's problem in linear partial differential equations in 1922. The book was based on a lecture course he had given at Yale University in the United States. He also took up new topics, writing several papers on probability theory, in particular on Markov chains. He also published many articles on mathematical education and education in general. Between the wars Hadamard's politics moved towards the left, mainly in response to the Nazis rise to power in 1933. After the start of World War II, when France fell in 1940, Hadamard and his family escaped to the United States where he was appointed to a visiting position at Columbia University. However, he failed to find a permanent post in America and in 1944 received the terrible news that his third son Mathieu had been killed in the war. Hadamard left the United States soon after and spent a year in England before returning to Paris as soon as was possible after the end of the war. After the War he became an active peace campaigner and it required the strong support of mathematicians in the USA to allow him to enter the country for the International Congress in Cambridge, Massachusetts in 1950. He was made honorary president of the Congress. One further tragedy was to hit Hadamard before his death. In 1962, when he was 96 years old, his grandson Etienne was killed in a mountaineering accident. This seemed to finally kill Hadamard's spirit and he did not leave his house after this, almost waiting to die. There is no way that an article of this length can even indicate the range of Hadamard's mathematical contributions. As well as about 300 scientific papers and books Hadamard also wrote books for a wider audience. His book The psychology of invention in the mathematical field (1945) is a wonderful work about mathematics. We should also, however, indicate Hadamard's style of teaching. At the conference to celebrate the centenary of his birth one of his students said he had been taught by:... a teacher who was active, alive, whose reasoning combined exactness and dynamism. Thus the lecture became a struggle and an adventure. Without rigour suffering, the importance of intuition was restored to us, and the better students were delighted. Laurent Schwartz spoke about Hadamard at this ceremony held to celebrate the centenary of Hadamard's birth:I believe that he had a fantastic influence on his time, and that all living analysts were shaped by him, directly or indirectly. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (27 books/articles)

Some Quotations (4)

A Poster of Jacques Hadamard

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hadamard.html (5 of 6) [2/16/2002 11:13:01 PM]

Hadamard

Cross-references to History Topics

1. Topology enters mathematics 2. Prime numbers

Other references in MacTutor

1. Prime Number Theorem 2. Chronology: 1890 to 1900

Honours awarded to Jacques Hadamard (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1932

Other Web sites

1. The Prime Pages (The Prime Number Theorem) 2. Steve Finch (Hadamard-de la Vallée Poussin or Meissel-Mertens constants) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hadamard.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hadamard.html (6 of 6) [2/16/2002 11:13:01 PM]

Hadley

John Hadley Born: 16 April 1682 in Enfield Chase (near East Barnet, now in London), Hertfordshire, England Died: 14 Feb 1744 in Barnet, Hertfordshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Hadley became a Fellow of the Royal Society in 1716. He built the first Newtonian reflecting telescope in 1721. It had a 6 inch mirror and proved very effective. He built a Gregorian reflector in 1726. It was due to him that reflecting telescopes of sufficient accuracy and power to be useful in astronomy were developed. In 1730 he invented a quadrant which measured the altitude of the Sun or of a star. In 1731 Hadley showed his new quadrant to the Royal Society. In 1734 he showed his new bubble-level to the Royal Society. It was used to determine position at sea. Edmund Stone wrote Mr Hadley tells us, that upon trial of one of these instruments, three observations made at sea of the distance between two stars with a brass octant of this kind differed from Mr Flamsteed's at land, only about a minute. Hadley's design evolved into that of a sextant. A magnifying glass was added to read the scale, a telescopic sight was added with cross-wires to divide the field of view. The arc was extended from an octant to a sextant and a stout handle was added at the back of the instrument. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) A Poster of John Hadley

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hadley.html (1 of 2) [2/16/2002 11:13:02 PM]

Hadley

Cross-references to History Topics

English attack on the Longitude Problem

Honours awarded to John Hadley (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1717

Lunar features

Rima Hadley

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hadley.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hadley.html (2 of 2) [2/16/2002 11:13:02 PM]

Hahn

Hans Hahn Born: 27 Sept 1879 in Vienna, Austria Died: 24 July 1934 in Vienna, Austria

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hans Hahn was a student at the Technische Hochschule in Vienna. There he formed a close friendship with three other students of mathematics, Paul Ehrenfest, Heinrich Tietze and Herglotz. They were known as the 'inseparable four'. He also studied in Strasbourg, Munich and Göttingen. He was appointed to the teaching staff in Vienna in 1905 and he became professor of mathematics there in 1921. In session 1905-06 Hahn substituted for Otto Stolz at Innsbruck. Hahn was a pioneer in set theory and functional analysis and is best remembered for the Hahn-Banach theorem. He also made important contributions to the calculus of variations, developing ideas of Weierstrass. During the 1920s Hahn, together with Frank and von Mises, was a member of the Vienna Circle of Logical Positivists, a discussion group of gifted scientists and philosophers who met regularly in Vienna. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Hans Hahn

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hahn.html (1 of 2) [2/16/2002 11:13:04 PM]

Hahn

Honours awarded to Hans Hahn (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Hahn

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hahn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hahn.html (2 of 2) [2/16/2002 11:13:04 PM]

Hajek

Jaroslav Hájek Born: 4 Feb 1926 in Podebrady, Bohemia Died: 10 June 1974 in Prague, Czechoslovakia

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jaroslav Hájek was brought up in Podebrady, the town of his birth, which was a spa town well known as a place where people could go to recover from heart problems. He attended primary school there but when he entered a Gymnasium it was in Prague. Of course World War II began while Hájek was studying at the Gymnasium in Prague. After the German armies took Prague, Hájek was forced to work for the German armament industry. He completed his disrupted secondary education around the time that the war ended and, after graduating from the Gymnasium, he entered the Czech Technical University to begin a course of statistical engineering. In 1947, although only two years into his undergraduate course, Hájek began teaching as an Assistant Lecturer in the Mathematics Institute at the Czech Technical University where he studied. He received his Diplom in Statistical Engineering in June 1949, the same year as his first publication appeared. This first publication by Hájek was on the subject of sampling surveys. The editors of [1] write:He was among the pioneers of unequal probability sampling. The name "Hájek predictor" now labels his contributions to the use of auxiliary data in estimating population means. Hájek's secondary school education had been disrupted by the war but this was not the end of the disruption to his career for after receiving his Diplom he had to spend two years doing military service. It was 1951 before he was free to begin his postgraduate research studies which he undertook at the Mathematical Institute of the Czechoslovak Academy of Sciences. In 1954 he was awarded his doctorate and he continued to work in the Mathematical Institute as a Senior Research Worker. The year he received his doctorate, Hájek married Alzbeta Galambos, who was known as Betty. This was a happy time for Hájek as he and his wife enjoyed bringing up two daughters born in the first five years of their marriage. For twelve year Hájek worked as a Senior Research Worker in the Mathematical http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hajek.html (1 of 3) [2/16/2002 11:13:06 PM]

Hajek

Institute of the Czechoslovak Academy of Sciences. It was a productive period during which he wrote 20 papers and two books: The theory of Probability Sampling with Applications to Sample Surveys and Probability in Science and Engineering which he wrote jointly with V Dupac. Although Hájek had not written an habilitation thesis and so did not hold a university post, he gained a strong international reputation for himself during his twelve years at the Mathematical Institute [1]:During these years Hájek established himself as a foremost international authority in several different fields of statistics, particularly in nonparametric methods and their asymptotic theory. His reputation led to an increasing number of invitations to international statistical events, to service on editorial boards of international journals, and to longer stays in foreign universities as a Visiting Professor. One important such visit was to the University of California at Berkeley which he made in 1961-62. There Lucien Le Cam was the Chairman of the Department and, despite the pressures of that role preventing him from doing as much work with Hájek as he would have liked, it was still a very useful visit. In 1962 Hájek wrote Asymptotically most powerful rank order tests which deals with contiguity. Hájek writes in that paper:The notion of contiguity had been developed independently by Le Cam and the present author. Hájek developed the property of sequences of pairs of probability measures from ideas due to de la Vallée Poussin. Both Hájek and Le Cam used the concept but the name 'contiguity' is due to Le Cam. As well as Berkeley, Hájek also made long visits to Michigan State University in East Lansing and to Florida State University in Tallahassee. Back in Prague Hájek submitted his habilitation dissertation on statistical problems in stochastic processes and was awarded the qualification. This led to an association with the Charles University of Prague where he was appointed to the Chair of Probability and Statistics in 1964 [1]:From the moment Hájek assumed the Chair, he directed great efforts towards reforming both the content and the method of teaching, which were then, admittedly, somewhat old-fashioned and obsolete; and he succeeded. in the course of the years, he brought together a group of graduate students whom he tutored and oriented in such a superb manner that the majority of them have become renowned researchers in various fields of statistical theory. In 1967 Hájek published (jointly with Z Sidak) Theory of rank tests but it was a work which had in fact been written four years before in 1963. Their methods use three lemmas of Le Cam in order to treat rank statistics under local alternatives and they established the efficiency of rank tests. Hájek's health was poor for many years as he suffered from a kidney disease. By 1970 his condition was deteriorating steadily but he bravely continued his work. He proposed an International Statistical Conference for Prague and set about inviting the main speakers. The conference Prague Symposium on Asymptotic Statistics took place in 1973. Hájek realised that the conference would be his last opportunity to meet many of his friends and colleagues and indeed it proved to be the case as he died from the kidney disease several months later. We should mention two further books by Hájek. These are A course in nonparametric statistics (1969) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hajek.html (2 of 3) [2/16/2002 11:13:06 PM]

Hajek

and Sampling from a finite population which was published in 1981, seven years after his death. Article by: J J O'Connor and E F Robertson List of References (10 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hajek.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hajek.html (3 of 3) [2/16/2002 11:13:06 PM]

Hall

Philip Hall Born: 11 April 1904 in Hampstead, London, England Died: 30 Dec 1982 in Cambridge, Cambridgeshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Philip Hall went up to King's College Cambridge in 1922. His interest in group theory came from Burnside's book. He offered parts of that book for examination in the Tripos. He graduated in 1925. Hall worked as a research assistant of Karl Pearson and his first published papers are in the theory of correlation. Returning to Cambridge in 1927 to take up a Fellowship at King's he made an important discovery in group theory, generalising the Sylow thoerems for finite soluble groups to prove what are now called Hall's theorems. In 1932 Hall wrote what is perhaps his most famous paper A contribution to the theory of groups of prime power order. It is a beautiful paper which is one of the fundamental sources of modern group theory. During the war he made an important contribution with his work at the Code and Cypher School at Bletchley Park. In 1957 Hall gave a series of lectures on nilpotent groups which have had great influence ever since. His work on infinite groups comes in papers of 1952, 1959 and 1961. The ideas of these papers continue to be one of the main areas of group theory research. Hall received many honours for his work. He was elected to the Royal Society in 1942, then he was awarded its Sylvester Medal in 1961:... in recognition of his distinguished researches in algebra. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hall.html (1 of 2) [2/16/2002 11:13:08 PM]

Hall

He was President of the London Mathematical Society during the period 1955-57, and he was awarded its De Morgan Medal in 1965. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Philip Hall

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1930 to 1940

Honours awarded to Philip Hall (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1942

Royal Society Sylvester Medal

Awarded 1961

London Maths Society President

1955 - 1957

LMS De Morgan Medal

Awarded 1965

LMS Berwick Prize winner

1958

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hall.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hall.html (2 of 2) [2/16/2002 11:13:08 PM]

Hall_Marshall

Marshall Hall Jr Born: 17 Sept 1910 in St Louis, Missouri, USA Died: 4 July 1990 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Marshall Hall showed talent for mathematics at a young age when he constructed a seven-place table of logarithms for the positive integers up to 1000. He entered Yale as an undergraduate, graduating with a B.A. in 1932. He continued graduate studies, spending the year 1932-33 at Cambridge in England. At Cambridge he was taught by several mathematicians who were to have an important influence on him such as Philip Hall, Harold Davenport and G H Hardy. Hall returned to Yale where he was awarded his doctorate in 1936 for his thesis An Isomorphism Between Linear Recurring Sequences and Algebraic Rings which was supervised by Oystein Ore. After spending the year 1936-37 at Princeton, Hall was to return to Yale. In 1941 however, he joined Naval Intelligence and was involved, as were many other mathematicians, deciphering coded messages. This work was little known about at the time but it has since emerged how significant this work proved to be. Hall's work in this area remained covered by the Official Secrets Act and when I [EFR] was in a group talking to him in 1981 about such things, he made it clear how seriously he continued to take his signing of the Act. After World War II, Hall returned to Yale where he continued to teach until 1946. That year he accepted an appointment as an associate professor at Ohio State University. He was promoted to full professor at Ohio and remained there until 1959 when he accepted a post at California Institute of Technology at Pasadena. During his time at Pasadena, Hall spent leave at Oxford in 1977, at Technion, Haifa in 1980 and at the University at Santa Barbara in 1984. In 1985 he accepted a post at Emory University in Atlanta. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hall_Marshall.html (1 of 2) [2/16/2002 11:13:09 PM]

Hall_Marshall

Hall is best known as a group theorist, perhaps because of his famous book Theory of Groups (1959) from which several generations of group theorists have learnt the subject. However he has written many papers of fundamental importance in the subject as well as an extremely important paper in 1943 on projective planes. Perhaps his best known result in group theory is his solution of the Burnside problem for groups of exponent 6. He showed that a finitely generated group in which the order of every element divides 6 must be finite. As well as the results on finite projective planes, Hall did other work of fundamental importance in the area of combinatorics, in particular on block designs. He wrote another classic text Combinatorial Theory in 1967. Among the honours Hall received were two Guggenheim Fellowships and membership of the American Academy of Arts and Sciences. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

A Quotation

Mathematicians born in the same country Other references in MacTutor

1. Chronology: 1940 to 1950 2. Chronology: 1950 to 1960 3. Chronology: 1960 to 1970

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hall_Marshall.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hall_Marshall.html (2 of 2) [2/16/2002 11:13:09 PM]

Halley

Edmond Halley Born: 8 Nov 1656 in Haggerston, Shoreditch (near London), England Died: 14 Jan 1742 in Greenwich (near London), England

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edmond (or Edmund) Halley's father was also called Edmond (or Edmund) Halley. He came from a Derbyshire family and was a wealthy soap-maker in London at a time when the use of soap was spreading throughout Europe. There is some confusion over both the date and year of Halley's birth. The confusion over the date is simply due to the change in calendar (29 October by the calendar of his time). The confusion over the year is less easy to decide, but we give 1656 which Halley himself claimed as the year of his birth. Halley's father last much in the great fire of London, which was in the year in which Halley was ten years old. His father still could afford a good education for his son and Halley was tutored privately at home before being sent to St Paul's School. It was at St Paul's School that Halley showed his talents to the full, being [16]:... equally distinguished in classics and mathematics, [he] rose to be captain of the school at fifteen, constructed dials, observed the change in the variation of the compass, and studied the heavens so closely that it was remarked by Moxon the globe maker 'that if a star were displaced in the globe he would presently find it out'. So Halley entered Queen's College Oxford in 1673, when he was seventeen years old, already an expert astronomer with a fine collection of instruments purchased for him by his father. He began working with Flamsteed in 1675, the Astronomer Royal, assisting him with observations both at Oxford and at Greenwich. Flamsteed, in a paper of 1675 published in the Philosophical Transactions of the Royal Society, remarked:Edmond Halley, a talented young man of Oxford, was present at these observations and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halley.html (1 of 6) [2/16/2002 11:13:11 PM]

Halley

assisted carefully with many of them. Halley made important observations at Oxford, including an occultation of Mars by the Moon on 21 August 1676, which he published in the Philosophical Transactions of the Royal Society. It is a little unclear what happened to Halley's undergraduate career, but what is certain is that he gave up his studies in 1676 and sailed to St Helena in the southern hemisphere in November of that year. The most likely explanation is that with the opening of the Royal Observatory at Greenwich in 1675, Flamsteed undertook the task of mapping the northern hemisphere stars and Halley decided to complement this programme with undertaking a similar task for the southern hemisphere. Such a task could not be undertaken without financial support, and indeed Halley obtained such support from his father and from no less person than King Charles II who provided a letter asking the East India Company to take Halley and a colleague to St Helena (the southern-most territory under British rule). Other important men also supported the venture, including Brouncker who was president of the Royal Society and Jonas Moore who had been a major influence in the founding of the Royal Observatory. The weather in St Helena proved less good for astronomical observations than Halley had hoped, but despite this his eighteen month spell on the island resulted in his cataloguing 341 southern hemisphere stars and discovered a star cluster in Centaurus. During the voyage [16]:... he improved the sextant, collected a number of valuable facts relative to the ocean and atmosphere, noted the equatorial retardation of the pendulum, and made on St Helena, on 7 November 1677, the first complete observation of a transit of Mercury. He proposed using transits of Mercury (and even better of Venus) to determine the distance of the Sun and therefore the scale of the solar system using Kepler's third law. Halley returned to England in 1678 and published his catalogue of southern hemisphere stars. Despite not having graduated from Oxford he found himself with the reputation of one of the leading astronomers. Honours quickly came his way. He became a graduate of the University of Oxford on 3 December 1678 without taking the degree examinations, the degree being conferred on the command of King Charles II. He was also elected a member of the Royal Society on 30 November 1678 becoming, at the age of 22, one of its youngest ever Fellows. In 1679 the Royal Society sent Halley to Danzig to arbitrate in a dispute between Hooke and Hevelius. Hooke claimed that Hevelius's observations, made without telescopic sights, could not be accurate. Hevelius at this time was 68 years old and must have been somewhat dismayed to find that a 23 year old man had been sent to judge him. However, Halley was [1]:... a man of great natural diplomacy... and after two months checking the observations Hevelius was making, he declared them to be accurate. The fame and recognition which Halley achieved so quickly did nothing to endear him to Flamsteed who, despite his praise for Halley in his student days, soon turned against him. Having the Astronomer Royal as an enemy is not the best recommendation for a young astronomer, even one as famous as Halley, who would soon pay the price. Halley did not seek a teaching post at this stage, preferring the freedom to travel and undertake research without commitments. In 1680 he set out on a European tour with a school friend, Robert Nelson. Halley observed a comet while near Calais and travelled to Paris where, together with Cassini, he made further http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halley.html (2 of 6) [2/16/2002 11:13:11 PM]

Halley

observations in an attempt to determine its orbit. Much of 1681 Halley spent in Italy. Back in England in the following year Halley married Mary Tooke, while his father remarried (Halley's mother having died ten years earlier). Not only did marriage bring financial responsibilities to Halley, but his father's marriage seems to have been a total disaster and as a consequence of this support from his father soon dried up. Further personal problems followed, for in March 1684 his father vanished and was found dead five weeks later. Halley had to administer his father's personal estate and he became involved in family, property and legal matters which are described fully in [12]. Just before his father disappeared, Halley had been involved in an exciting piece of research. He had shown that Kepler's third law implied the inverse square law of attraction and presented the results at a meeting of the Royal Society on 24 January 1684 . Wren, Hooke and Halley then discussed whether it could be shown that the inverse square law implies elliptical orbits for the planets, but failed to come up with a proof. Halley's work on these problems was disrupted during the following weeks by the difficulties surrounding his father's disappearance and death, but by August 1682 Halley was pursuing the problem further by visiting Newton in Cambridge. There he discovered that Newton had already achieved a proof of this and of other highly significant results but did not seem to be going to publish them. Chapman writes in [11]:... Halley ... had the genius to recognise the even greater mathematical genius of Newton, to urge him to write the Principia Mathematica, and then pay for the costs of publication out of his own pocket because the Royal Society was currently broke ... Glaisher, in an address delivered in Cambridge in 1888, spoke of the role which Halley played in getting Newton's Principia published:... but for Halley the Principia would not have existed. ... He paid all the expenses, he corrected the proofs, he laid aside his own work in order to press forward to the utmost the printing. All his letters show the most intense devotion to the work. By now Halley was certainly not a rich man and although in the end his financial outlay which allowed the Principia to be published was reimbursed from the sales, he now sought an academic post. In 1691 he applied for the vacant Savilian Chair of Astronomy at Oxford. Given his outstanding research in astronomy, one would have expected him to be appointed to this chair but Flamsteed was strongly against the appointment. Flamsteed was not well disposed towards Newton particularly since he felt that Newton had not given sufficient credit to observations made by the Royal Observatory in his theory of the moon. Halley's close association with Newton lowered him still further in Flamsteed's eyes. However, the argument that Flamsteed used against Halley was one which he undoubtedly believed in sincerely, writing to Oxford that Halley would [7]:... corrupt the youth of the university. Flamsteed was quite right in believing that Halley's view of Christianity was at odds with the standard view of that time which required a literal belief in the Bible. Newton also complained to Halley about the fact that Halley doubted the scientific correctness of the biblical story of the creation. Despite Halley http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halley.html (3 of 6) [2/16/2002 11:13:11 PM]

Halley

vigorously claiming that his beliefs were conventional, David Gregory was appointed to the chair. The lack of an academic post did not hold Halley back in his scientific work. Indeed he worked for the Royal Society in various roles, being editor of the Philosophical Transactions from 1685 to 1693. He published frequently important results through the Society's publications. In 1686 Halley published a map of the world showing the prevailing winds over the oceans. It has the distinction of being the first meteorological chart to be published. Another innovative piece of work was the mortality tables for the city of Breslau which he published in 1693. It was one of the earliest works to relate mortality and age in a population and was highly influential in the future production of actuarial tables in life insurance. From around 1695 Halley made a careful study of the orbits of comets. Newton favoured comets having parabolic orbits, but Halley believed that elliptical orbits might exist. Using his theory of cometary orbits he calculated that the comet of 1682 (now called Halley's comet) was periodic and was the same object as the comet of 1531, and 1607. He later also identified this comet with one which appeared in 1305, 1380, and 1456. In 1705 he published his prediction that it would return in 76 years, claiming that it would appear in December 1758. It was not an easy calculation for Halley had to take into account the perturbations to the orbit produced by Jupiter. Although Halley had been dead for fifteen years by 1758, he achieved lasting fame when the comet was observed on 25 December 1758 (very slightly later than Halley expected). Newton became Warden of the Royal Mint in London in 1696 and he used his influence to have Halley appointed as deputy controller of the mint at Chester in the same year. It was a post he held for two years before it was abolished. After leaving the mint at Chester, Halley was given the command of a warship, the Paramore Pink, by William III. This was not as strange as it sounds, for Halley had been working on determining the longitude using variation of the compass and this was the main purpose of the voyage, although he was also required by William III to [16]:... attempt the discovery of what land lies to the south of the western ocean. He sailed from Portsmouth in November 1698 but problems with his crew forced him to return, having reached Barbados. In September 1699 he sailed again making a thorough exploration of the Atlantic shores. After his return in September 1700, Halley published charts of the variation of the compass, giving the first charts with lines of equal declination plotted. Back on the Paramore Pink in 1701, Halley investigated the tides and coasts of southern England. Further journeys followed, for Queen Anne sent him to inspect the harbours around the Adriatic, and another journey saw him travel to Trieste to advise on fortifications. Halley was appointed Savilian professor of geometry at Oxford in 1704 following the death of Wallis. This certainly did not please Flamsteed who had written (see for example [13]):Dr Wallis is dead - Mr Halley expects his place - who now talks, swears and drinks brandy like a sea captain. Halley's inaugural lecture proved a great success. It was described by Thomas Hearne (see [5]):Mr Halley made his inaugural speech on Wednesday May 24, which very much pleased the generality of the University. After some compliments to the university, he proceeded to the original and progress of geometry, and gave an account of the most celebrated of the ancient and modern geometricians. Of those of our English nation he spoke in particular of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halley.html (4 of 6) [2/16/2002 11:13:11 PM]

Halley

Sir Henry Savile; but his greatest encomiums were upon Dr Wallis and Mr Newton ... This lecture is described in [24] as:... of abiding interest from the mathematics standpoint. In 1710, using Ptolemy's catalogue, Halley deduced that the stars must have small motions of their own and he was able to detected this proper motion in three stars. This achievement is described in [1] as his:... most notable achievement in stellar astronomy ... Halley played an active role in the events and controversies of his time. He supported Newton in his controversy with Leibniz over who invented the calculus, serving as secretary of a committee set up by the Royal Society to resolve the dispute. Halley did much to calm disputes, but also seemed to go out of his way to make his dispute with Flamsteed worse. In 1712 he arranged with Newton to publish Flamsteed's observations long before they were complete. To make matters worse, Halley wrote a preface, without Flamsteed's knowledge, in which [11]:... he attacked Flamsteed for sluggishness, secretiveness, and lack of public spirit. In 1720 he succeeded Flamsteed as Astronomer Royal, a position he was to hold for 21 years despite being 64 years old when appointed. Flamsteed's widow was so angry that she had all her husband's instruments from the Royal Observatory sold so that Halley would not have the use of them. At the Greenwich Royal Observatory Halley used the first transit instrument and devised a method for determining longitude at sea by means of lunar observations. He observed the Moon through one complete 18-year saros. Earlier observations of the Moon were made only at conjunction or at opposition to the Sun and it was these earlier observations on which Newton's lunar theory had been based. However, Halley has been criticised for his work as Astronomer Royal. Some claim that he made valueless observations which were no more accurate than those of Flamsteed. It has also been claimed that Halley's observations were carelessly carried out. For example in [16]:Halley took no account of fractional parts of seconds of time, and considered 10" of arc 'as the utmost attainable limit of accuracy'. His clocks were besides ill-regulated, and his system of registration unmethodical. In [22], however, Ronan argues that the criticisms are unfair. He lists in that article Halley's achievements as Astronomer Royal. Halley's other activities included studying archaeology, geophysics, the history of astronomy, and the solution of polynomial equations. He was an integral part of the English scientific community at the height of its creativity. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (25 books/articles)

A Quotation

A Poster of Edmond Halley

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halley.html (5 of 6) [2/16/2002 11:13:11 PM]

Halley

Cross-references to History Topics

1. Orbits and gravitation 2. English attack on the Longitude Problem

Other references in MacTutor

Chronology: 1675 to 1700

Honours awarded to Edmond Halley (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1678

Lunar features

Crater Halley

Planetary features

Crater Halley on Mars

Savilian Professor of Geometry

1704

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Halley.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halley.html (6 of 6) [2/16/2002 11:13:11 PM]

Halmos

Paul Richard Halmos Born: 3 March 1916 in Budapest, Hungary

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Halmos's mother died when Paul was six months old. Paul's father was a successful physician in Budapest who had the rather remarkable foresight to realise the problems that were going to befall Europe. So in 1924 Paul's father emigrated to Chicago in the United States, leaving Paul and his two elder brothers in Budapest. There they were looked after by the physician who took over his father's practice. After five years in Chicago, Paul's father became an American citizen and, at that time, brought Paul from Hungary to join him in Chicago. He attended high school in Chicago but rather remarkably he missed out four years schooling in the process. Halmos says that there was some confusion since in Hungary four years of primary schooling were followed by eight years of secondary schooling. He had completed seven of these twelve years but Halmos said ([1] or [2]):I hinted to the school authorities that I had completed three years of secondary school, and I was believed. ... a year and a half later, at the age of fifteen, I graduated from high school. While still 15 he entered the University of Illinois to study chemical engineering. His age was not a problem, he said ([1] or [2]):I was tall for my age and cocky. I pretended to be older and got along fine. After one year he changed to mathematics and philosophy but did not particularly shine at mathematics ([1] or [2]):I was a routine calculus student - I think I got B's. I did not understand about limits. I doubt that they taught it. ... But I was good at integrating and differentiating things in a mechanical sense. Somehow I like it. I kept fooling around with it. Despite being so young when he entered his undergraduate course and despite changing from chemical http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halmos.html (1 of 4) [2/16/2002 11:13:13 PM]

Halmos

engineering to mathematics and philosophy he still completed the four year degree in three years graduating in 1934. He began graduate studies at the University of Illinois at Urbana-Champaign, still with philosophy as his main subject, mathematics as his minor subject. It was not until the end of the academic year 1935-36 that Halmos made the move from philosophy to mathematics. After thinking that algebra was the right subject for him, he quickly changed to analysis and studied for his Ph.D. under Doob. This was awarded in 1938 for his thesis on measure-theoretic probability Invariants of Certain Stochastic Transformation: The Mathematical Theory of Gambling Systems. Jobs were not so easy to come by ([1] or [2]):I typed 120 letters of application, mailed them out, and got two answers, both no. The University of Illinois took pity on me and kept me on as an instructor. So in 38-39 I had a job, but I kept applying. In February 1939 Halmos was successful in obtaining a post at Reed College in Oregon. He accepted but in April his friend Warren Ambrose was offered a scholarship at the Institute for Advanced Study in Princeton. Halmos wrote ([1] or [2]):That made me mad. I wanted to go, too! I resigned my job, making the department head, whom I had never met, very unhappy, of course. I ... went to my father and asked to borrow a thousand dollars ... I wrote to Veblen and asked if I could become a member of the Institute for Advanced Study even though I had no fellowship. ... I moved to Princeton. After six months Halmos was offered a fellowship, and in his second year at Princeton he became von Neumann's assistant. Ambrose writes in [1]:This was wonderful for Paul because he ... idolised von Neumann ... This seemed to have been the first time in Paul's career when he received what he deserved and I think it must have been one of the happiest times in his life. Halmos said of von Neumann:... his speed, plus depth, plus insight, plus inspiration turned me on. A debt that Halmos owes to von Neumann is that one of his lecture courses inspired Halmos's first book. In 1942 Halmos published Finite Dimensional Vector Spaces which was to bring him instant fame as an outstanding writer of mathematics. After leaving the Institute for Advanced Study, Halmos was appointed to Syracuse University, New York. While in Syracuse he took part in teaching soldiers in the Army's Specialized Training Program. At the end of World War II Halmos decided it was time for a change and, in 1946, he became an assistant professor at the University of Chicago. In 1961 Halmos moved to the University of Michigan. In 1968-69 he served for one year as chairman of the mathematics department of the University of Hawaii. At the end of the year he accepted a professorship at Indiana University. He remained at Indiana until 1985 when he moved to Santa Clara. G L Alexanderson writes in [1]:In early 1984 I received a telephone call from Paul Halmos ... during which he said, among other things, that he would like to be someplace with more sunny days. The Bloomington winter seemed long. ... I responded that ... I would think about it. When I had, I called him and asked him whether he might consider Santa Clara... I raised the question of his joining us at Santa Clara with some hesitation because, though we may have good weather, Santa http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halmos.html (2 of 4) [2/16/2002 11:13:13 PM]

Halmos

Clara is not the kind of institution at which Paul had spent his career. Halmos is known for both his outstanding contributions to operator theory, ergodic theory, functional analysis, in particular Hilbert spaces, and for his series of exceptionally well written textbooks. These include Finite dimensional vector spaces (1942), Measure theory (1950), Introduction to Hilbert space and theory of spectral multiplicity (1951), Lectures on ergodic theory (1956), Entropy in ergodic theory (1959), Naive set theory (1960), Algebraic logic (1962), A Hilbert space problem book (1967) and Lectures on Boolean algebras (1974). In 1983 he received the Steele Prize for exposition from the American Mathematical Society. The citation read:The award for a book or substantial survey or research-expository paper is made to Paul R Halmos for his many graduate texts in mathematics, dealing with finite dimensional vector spaces, measure theory, ergodic theory and Hilbert space. Many of these books were the first systematic presentations of their subjects in English. Their felicitous style and content has had a vast influence on the teaching of mathematics in North America. His articles on how to write, talk and publish mathematics have helped all mathematicians to communicate their ideas and results more effectively. Halmos has received many other awards for his writing and teaching. For example, in 1993, he received a Distinguished Teacher award from the Mathematical Association of America. J B Conway writes in [1] about Halmos's contributions to operator theory:... Paul has a number of papers and theorems that anyone would be proud to call his own. But the thing that has always struck me about his work is the extraordinary number of topics and problems that are dominant themes in the current research of today and that have their origin in his work. Over the years Paul has demonstrated an uncanny ability to extract crucial properties from a given mathematical entity and lay it open before his colleagues in such a manner that there is a universal inclination to look and explore further. Halmos has been a frequent visitor to Scotland. He attends regularly the four-yearly St Andrews Colloquium. I [EFR] first met him at the 1972 St Andrews Colloquium and fully agree with Alastair Gillespie's comments in [1]:These Colloquia are just the sort of things that Halmos relishes in - a happy mixture of expository mathematics and recreation - a mathematical holiday, in fact. Halmos spent part of his 1973 sabbatical leave in Edinburgh and has been elected a Fellow of the Royal Society of Edinburgh. He has also been awarded an honorary D.Sc. from the University of St Andrews. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

Some Quotations (18)

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halmos.html (3 of 4) [2/16/2002 11:13:13 PM]

Halmos

Honours awarded to Paul Halmos (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh Other Web sites

Bellevue College, USA

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Halmos.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halmos.html (4 of 4) [2/16/2002 11:13:13 PM]

Halphen

George Henri Halphen Born: 30 Oct 1844 in Rouen, France Died: 23 May 1889 in Versailles, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George-Henri Halphen's father died in 1848 when George-Henri was less than four years old. Shortly after this his mother moved from Rouen to Paris where George-Henri was brought up. He was educated at the Lycée Saint-Louis which he left in 1862 to enter the Ecole Polytechnique. Political events determined the course of the next few years for Halphen and work for his doctorate would have to wait until after the Franco-German war. By July 1870 Napoleon III, the French emperor, was trying to improve his popularity. Thinking that there is nothing like a war to get people behind you, and being advised that France could win against Prussia, Napoleon was keen to start a war. Otto von Bismarck, the Prussian chancellor, saw a war as an excellent opportunity to unite the German states. Bismarck sent a provocative message to France and, as he had hoped, they declared war on 19 July. Halphen served in the French army in the conflict. It soon became obvious that Napoleon III had been badly advised and the French were no match for the Prussian forces. The French forces were defeated at the Battle of Sedan and, on 2 September, 83,000 French troops surrendered. Two weeks later the Germans besieged Paris which surrendered on 28 January 1871. It was a war in which France had been humiliated, and the terms of the treaty which ended the war reflected this. Halphen, however, had served his country with great distinction. In 1872, after leaving the army, Halphen married the daughter of Henri Aron. They had seven children, three daughters and four sons. Also in 1872 Halphen was appointed as répétiteur at the Ecole Polytechnique and he was soon making major contributions. The first result which brought him to the attention of mathematicians world-wide was his solution in 1873 of a problem of Chasles [1]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halphen.html (1 of 3) [2/16/2002 11:13:15 PM]

Halphen

Given a family of conics depending on a parameter, how many of them will satisfy a given side condition? Chasles had found a formula for this but his proof was faulty. Halphen showed that Chasles was essentially correct, but that restrictions on the kinds of singularity were necessary. Halphen's solution was ingenious ... Halphen took a different view on the problems of enumeration from his contemporaries. He defined the concepts of proper and improper solutions to an enumerative problem involving conics. Then a particular number associated with a problem about conics has enumerative significance when it counts the number of proper solutions. In fact Halphen was well ahead of his time in the ideas which he brought to these problems. This did not mean, however, that his ideas were accepted by everyone around. Halphen and Schubert engaged in a heated debate on whether an enumerative formula should be allowed to count degenerate solutions along with the nondegenerate solutions. This was, in the end, simply a special case of an old argument: is a mathematical theory important because of its external applications or because of its internal beauty? Next Halphen classified singular points of algebraic closed curves thus extending the work of Riemann. He was led to extend results due to Max Noether which, in turn, had him examine projective transformations which fix certain differential equations. A characterisation of such invariant differential equations appeared in Halphen's doctoral dissertation On differential invariants which he presented in 1878. Poincaré writes in [4] that:... the theory of differential invariants is to the theory of curvature as projective geometry is to elementary geometry. Halphen made major contributions to linear differential equations and algebraic space curves. He examined problems in the areas of systems of lines, classification of space curves, enumerative geometry of plane conics, singular points of plane curves, projective geometry and differential equations, elliptic functions, and assorted questions in analysis. He gave a formula for the number of conics in a 1-dimensional system which properly satisfy a codimension 1 condition, and also a proof of his formula for the number of conics which properly satisfy five independent conditions. This last result appeared in a paper Halphen published in the Proceedings of the London Mathematical Society in 1878. He received great honours and prizes for his work on these topics. For example, in 1880 he won the Grand Prix of the Académie des Sciences for his work on linear differential equations. Then, in 1882, he won the Steiner Prize from the Berlin Academy of Sciences for his work on algebraic curves. In 1884 Halphen was made an examinateur at the Ecole Polytechnique, then two years later he was elected to the Académie des Sciences. Sadly he died in 1889 at age 44 when at the height of his creative powers. A major figure in his time, much of Halphen's work was in areas which have fallen out of favour. Other work such as that on linear differential equations was overtaken by Lie group methods. Bernkopf writes in [1]:The amount and quality of Halphen's work is impressive, especially considering that his mathematically creative life covered only seventeen years. Why, then, is his name so little known? ... he worked in analytic and differential geometry, a subject so unfashionable today as to be almost extinct. Perhaps with its inevitable revival, analytic geometry will restore

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halphen.html (2 of 3) [2/16/2002 11:13:15 PM]

Halphen

Halphen to the eminence he earned. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Halphen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halphen.html (3 of 3) [2/16/2002 11:13:15 PM]

Halsted

George Bruce Halsted Born: 23 Nov 1853 in Newark, New Jersey, USA Died: 16 March 1922 in New York, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Halsted was a student at the University of Princeton from where he was awarded his A.B. in 1875 and, three years later, his A.M. He was Sylvester's first student at Johns Hopkins University where he studied for his doctorate which was awarded in 1879. He also studied with Borchardt in Berlin during his doctoral studies. Halsted was given outstanding references by Sylvester to present to Borchardt. Returning to the United States, Halsted was appointed a tutor at Princeton, a post he held between 1879-81, then from 1881 he was an instructor in postgraduate mathematics at Princeton until 1884. In that year he accepted a professorship at the University of Texas, a post he held for nineteen years. From 1903 he held posts at St John's College, Annapolis, Maryland (1903), then Kenyon College, Gambier, Ohio (1903-1906), and Colorado State College of Education, Greeley (1906-1914). Writing in [1], H S Tropp says of Halsted:In the period when American mathematics had few distinguished names, the eccentric and sometimes spectacular Halsted established himself as an internationally known scholar, creative teacher and promoter and popularizer of mathematics. There are a number of anecdotes showing Halsted's eccentric nature given in [2]. Halsted gave commentaries on the work of Lobachevsky, Bolyai, Saccheri and Poincaré and made translations of their works into English. He also studied the foundations of geometry which was his own mathematical speciality. His other main interest was in mathematical education and, as a mathematics educator, he criticised the careless way that mathematics was presented in the textbooks of the time. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halsted.html (1 of 2) [2/16/2002 11:13:17 PM]

Halsted

After he retired, Halsted completed the Princeton University Biographical Questionnaire. In this he wrote:I am working as an electrician as there is nothing in cultivating vacant lots. This seems to indicate that he felt badly treated by the mathematical community. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Halsted.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Halsted.html (2 of 2) [2/16/2002 11:13:17 PM]

Hamill

Christine Mary Hamill Born: 24 July 1923 in London, England Died: 24 March 1956 in Ibadan, Oyo State, Nigeria Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Christine Hamill studied first at St Paul's Girls' School, then at Perse School for Girls. She was awarded a Caroline Turle to study at Newnham College, Cambridge in 1942. At Cambridge Hamill was very successful, becoming a Wrangler in 1945 and achieving a distinction in Part III the following year. She continued to study at Cambridge working for her doctorate. In 1948 she was awarded a Newnham research fellowship and she was awarded her doctorate in 1950. The year she received her doctorate, Hamill was offered an assistant lectureship at the University of Sheffield which she accepted. She was to spend four years at Sheffield being promoted to a Lecturer in Mathematics in 1952. J A Todd supervised her research work at Cambridge and in [2] he describes the work of her doctoral dissertation:This work contains a detailed study of the finite primitive collineation groups which contain homologies of period two. Starting with an analysis of the geometrical configuration formed by the centres and the invariant primes of the homologies, she was able, by a very thorough and careful investigation, to obtain, for each of the groups, the distribution of the operations in conjugate sets, and to make the nature of these operations clear. Hamill published three papers based on her dissertation in 1948, 1951 and 1953. These papers describe groups of order 576, 6531840 and 348364800 respectively. Todd [2] assesses the importance of these papers:The groups concerned are of interest from various points of view, and the detailed results contained in her papers contain something of permanent value. In 1954 Hamill accepted a post as lecturer in the University of Ibadan in Nigeria. She had already earned a high reputation as a teacher both at Cambridge and at Sheffield and was said to have great talent at getting the best from weaker students. In Ibadan she quickly began to show the same lecturing talents giving lectures of great clarity. After four terms in Ibadan, Hamill contracted poliomyelitis and her death was rapid occurring only two days after she became ill. Todd remarks [2]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamill.html (1 of 2) [2/16/2002 11:13:18 PM]

Hamill

... she will be remembered for her natural and innate friendliness, for her complete sincerity, and for her strength of character, fortified by a firm Christian faith, and a sincere acceptance of all that that implied. She died a few months before the day on which she was to have been married. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hamill.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamill.html (2 of 2) [2/16/2002 11:13:18 PM]

Hamilton

Sir William Rowan Hamilton Born: 4 Aug 1805 in Dublin, Ireland Died: 2 Sept 1865 in Dublin, Ireland

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Rowan Hamilton's father, Archibald Hamilton, did not have time to teach William as he was often away in England pursuing legal business. Archibald Hamilton had not had a university education and it is thought that Hamilton's genius came from his mother, Sarah Hutton. By the age of five, William had already learned Latin, Greek, and Hebrew. He was taught these subjects by his uncle, the Rev James Hamilton, who William lived with in Trim for many years. James was a fine teacher. William soon mastered additional languages but a turning point came in his life at the age of 12 when he met the American Zerah Colburn. Colburn could perform amazing mental arithmetical feats and Hamilton joined in competitions of arithmetical ability with him. It appears that losing to Colburn sparked Hamilton's interest in mathematics. Hamilton's introduction to mathematics came at the age of 13 when he studied Clairaut's Algebra, a task made somewhat easier as Hamilton was fluent in French by this time. At age 15 he started studying the works of Newton and Laplace. In 1822 Hamilton found an error in Laplace's Méchanique céleste and, as a result of this, he came to the attention of John Brinkley, the Astronomer Royal of Ireland, who said: This young man, I do not say will be, but is, the first mathematician of his age. Hamilton entered Trinity College, Dublin at the age of 18 and in his first year he obtained an 'optime' in Classics, a distinction only awarded once in 20 years. He achieved this merit despite spending most of his time living with his Cousin Arthur at Trim and therefore not attending all of his lectures. In August 1824, Uncle James took Hamilton to Summerhill to meet the Disney family. It was at this point that William first met their daughter Catherine and immediately fell hopelessly in love with her. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html (1 of 7) [2/16/2002 11:13:20 PM]

Hamilton

Unfortunately, as he had three years left at Trinity College, Hamilton was not in a position to propose marriage. However Hamilton was making remarkable progress for an undergraduate and submitted his first paper to the Royal Irish Academy before the end of 1824, which was entitled On Caustics. The following February, Catherine's mother informed William that her daughter was to marry a clergyman, who was fifteen years her senior. He was affluent and could offer more to Catherine than Hamilton. In his next set of exams William was given a 'bene' instead of the usual 'valde bene' due to the fact that he was so distraught at losing Catherine. He became ill and at one point he even considered suicide. In this period he turned to poetry, which was a habit that he pursued for the rest of his life in times of anguish. In 1826 Hamilton received an 'optime' in both science and Classics, which was unheard of, while in his final year as an undergraduate he presented a memoir Theory of Systems of Rays to the Royal Irish Academy. It is in this paper that Hamilton introduced the characteristic function for optics. Hamilton's finals examiner, Boyton, persuaded him to apply for the post of Astronomer Royal at Dunsink observatory even although there had already been six applicants, one of whom was George Biddell Airy. Later in 1827 the board appointed Hamilton Professor of Astronomy at Trinity College while he was still an undergraduate aged twenty-one years. This appointment brought a great deal of controversy as Hamilton did not have much experience in observing. His predecessor, Professor Brinkley, who had become a bishop, did not think that it had been the correct decision for Hamilton to accept the post and implied that it would have been prudent for him to have waited for a fellowship. It turned out that Hamilton had made an poor choice as he lost interest in astronomy and spend all time on mathematics. Before beginning his duties in this prestigious position, Hamilton toured England and Scotland (from where the Hamilton family originated). He met the poet Wordsworth and they became friends. One of Hamilton's sisters Eliza wrote poetry too and when Wordsworth came to Dunsink to visit, it was her poems that he liked rather than Hamilton's. The two men had long debates over science versus poetry. Hamilton liked to compare the two, suggesting that mathematical language was as artistic as poetry. However, Wordsworth disagreed saying that [4]:Science applied only to material uses of life waged war with and wished to extinguish imagination. Wordsworth had to tell Hamilton quite forcibly that his talents were in science rather than poetry:You send me showers of verses which I receive with much pleasure ... yet have we fears that this employment may seduce you from the path of science. ... Again I do venture to submit to your consideration, whether the poetical parts of your nature would not find a field more favourable to their nature in the regions of prose, not because those regions are humbler, but because they may be gracefully and profitably trod, with footsteps less careful and in measures less elaborate. Hamilton took on a pupil by the name of Adare. They were a bad influence on each other as Adare's eyesight started to present problems as he was doing too much observing, while at the same time Hamilton became ill due to overwork. They decided to take a trip to Armagh by way of a holiday and visit another astronomer Romney Robinson. It was on this occasion that Hamilton met Lady Campbell, who was to become one of his favourite confidants. William also took the opportunity to visit Catherine, as she was living relatively nearby, which she then reciprocated by coming to the observatory. Hamilton was so nervous in her presence that he broke the eyepiece of the telescope whilst trying to give her a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html (2 of 7) [2/16/2002 11:13:20 PM]

Hamilton

demonstration. This episode inspired another interval of misery and poem writing. In July 1830 Hamilton and his sister Eliza visited Wordsworth and it was around this time that he started to think seriously about getting married. He considered Ellen de Vere, and he told Wordsworth that he [4]:... admired her mind ... but he did not mention love. He did, however, bombard her with poetry and was about to propose marriage when she happened to say [5] that she could ... not live happily anywhere but at Curragh. Hamilton thought this was her way of discouraging him tactfully and so he ceased to pursue her. However he was proved to be mistaken as she married the following year and did leave Curragh! Fortunately, one good thing transpired from the event as Hamilton became firm friends with Ellen's brother Aubrey although a dispute about religion in 1851 made them go their separate ways. Catherine aside, Hamilton seemed quite fickle when it came to relationships with women. Perhaps this was because he thought that he ought to marry and so, if he could not have Catherine, then it did not really matter who he married. In the end he married Helen Maria Bayly who lived just across the fields from the observatory. William told Aubrey that she was "not at all brilliant" and, unfortunately, the marriage was fated from the start. They spent their honeymoon at Bayly Farm and Hamilton worked on his third supplement to his Theory of Systems of Rays for the duration. Then at the observatory Helen did not have much of an idea of housekeeping and was so often ill that the household became extremely disorganised. In the years to come she spent most of her time away from the observatory as she was looking after her ailing mother or was indisposed herself. In 1832 Hamilton published this third supplement to Theory of Systems of Rays which is essentially a treatise on the characteristic function applied to optics. Near the end of the work he applied the characteristic function to study Fresnel's wave surface. From this he predicted conical refraction and asked the Professor of Physics at Trinity College, Humphrey Lloyd, to try to verify his theoretical prediction experimentally. This Lloyd did two months later and this theoretical prediction brought great fame to Hamilton. However, it also led to controversy with MacCullagh, who had come very close to the theoretical discovery himself but, he was forced to admit, had failed to take the last step. On 4 November 1833 Hamilton read a paper to the Royal Irish Academy expressing complex numbers as algebraic couples, or ordered pairs of real numbers. He used algebra in treating dynamics in On a General Method in Dynamics in 1834. In this paper Hamilton gave his first statement of the characteristic function applied to dynamics and wrote a second paper on the topic the following year. Hankins writes in [1]:These papers are difficult to read. Hamilton presented his arguments with great economy, as usual, and his approach was entirely different from that now commonly presented in textbooks describing the method. In the two essays on dynamics Hamilton first applied the characteristic function V to dynamics just as he had in optics, the characteristic function being the action of the system in moving from its initial to its final point in configuration space. By his law of varying action he made the initial and final coordinates the independent variables of the characteristic function. For conservative systems, the total energy H was constant along any real path but varied if the initial and final points were varied, and so the characteristic function in dynamics became a function of the 6n http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html (3 of 7) [2/16/2002 11:13:20 PM]

Hamilton

coordinates of initial and final position (for n particles) and the Hamiltonian H. The year 1834 was the one in which Hamilton and Helen had a son, William Edwin. Helen then left Dunsink for nine months leaving Hamilton to fight the loneliness by throwing himself into his work even more. In 1835 Hamilton published Algebra as the Science of Pure Time which were inspired by his study of Kant and presented to a meeting of the British Association for the Advancement of Science. This second paper on algebraic couples identified them with steps in time and he referred to the couples as 'time steps'. Hamilton was knighted in 1835 and that year his second son, Archibald Henry, was born but the next few years did not bring him much happiness. After the discovery of algebraic couples, he tried to extend the theory to triplets, and this became an obsession that plagued him for many years. The following autumn he went to Bristol for a meeting of the British Association, and Helen took the children with her to Bayly Farm for ten months. His cousin Arthur died, and not long after Helen returned from her mother's she went away again to England this time leaving the children behind after the birth of a daughter, Helen Eliza Amelia. At this point, William became depressed and started to have problems with alcohol so his sister came back to live at Dunsink. Helen returned in 1842 when Hamilton was so preoccupied with the triplets that even his children were aware of it. Every morning they would inquire [25]:Well, Papa can you multiply triplets? but he had to admit that he could still only add and subtract them. On 16 October 1843 (a Monday) Hamilton was walking in along the Royal Canal with his wife to preside at a Council meeting of the Royal Irish Academy. Although his wife talked to him now and again Hamilton hardly heard, for the discovery of the quaternions, the first noncommutative algebra to be studied, was taking shape in his mind:And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth. He could not resist the impulse to carve the formulae for the quaternions i2 = j2 = k2 = i j k = -1. in the stone of Brougham Bridge as he and his wife passed it. Hamilton felt this discovery would revolutionise mathematical physics and he spent the rest of his life working on quaternions. He wrote [25]:I still must assert that this discovery appears to me to be as important for the middle of the nineteenth century as the discovery of fluxions [the calculus] was for the close of the seventeenth. Shortly after Hamilton's discovery of the quaternions his personal life started to prey on his mind again. In 1845, Thomas Disney visited Hamilton at the observatory and brought Catherine with him. This must have upset William as his alcohol dependency took a turn for the worse. At a meeting of the Geological Society the following February he made an exhibition of himself through his intoxication. Macfarlane [16] writes:... at a dinner of a scientific society in Dublin he lost control of himself, and was so mortified that, on the advice of friends he resolved to abstain totally. This resolution he kept http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html (4 of 7) [2/16/2002 11:13:20 PM]

Hamilton

for two years, when ... he was taunted for sticking to water, particularly by Airy ... . He broke his good resolution, and from that time forward the craving for alcoholic stimulants clung to him. The year 1847 brought the deaths of his uncles James and Willey and the suicide of his colleague at Trinity College, James MacCullagh, which greatly disturbed him despite the fact that they had not always seen eye to eye. The following year Catherine began writing to Hamilton, which cannot have helped at this time of depression. The correspondence continued for six weeks and became more informal and personal until Catherine felt so guilty that she confessed to her husband. Hamilton wrote to Barlow and informed him that they would never hear from him again. However, Catherine wrote once more and this time attempted suicide (unsuccessfully) as her remorse was so great. She then spent the rest of her life living with her mother or siblings, although there was no official separation from Barlow. Hamilton persisted in his correspondence to Catherine, which he sent through her relatives. It is no surprise that Hamilton gave in to alcohol immediately after this, but he threw himself into his work and began writing his Lectures on Quaternions. He published Lectures on Quaternions in 1853 but he soon realised that it was not a good book from which to learn the theory of quaternions. Perhaps Hamilton's lack of skill as a teacher showed up in this work. Hamilton helped Catherine's son James to prepare for his Fellowship examinations which were on quaternions. He saw this as revenge towards Barlow as he was able to help his son in a way that his father could not. Later that year Hamilton received a pencil case from Catherine with an inscription that read [5]:From one who you must never forget, nor think unkindly of, and who would have died more contented if we had once more met. Hamilton went straight to Catherine and gave her a copy of Lectures on Quaternions. She died two weeks later. As a way of dealing with his grief, Hamilton plagued the Disney family with incessant correspondence, sometimes writing two letters a day. Lady Campbell was another sufferer of the burden of mail, as only she and the Disneys knew of his love for Catherine. On the other hand, Helen must have always suspected that she did not take first place in her husband's heart, a notion that must have been strengthened in 1855 when she found a letter from Dora Disney (Catherine's sister-in-law). This led to an argument, although the only consequence was that Dora had her letters addressed by her husband, they did not stop altogether. Determined to produce a work of lasting quality, Hamilton began to write another book Elements of Quaternions which he estimated would be 400 pages long and take 2 years to write. The title suggests that Hamilton modelled his work on Euclid's Elements and indeed this was the case. The book ended up double its intended length and took seven years to write. In fact the final chapter was incomplete when he died and the book was finally published with a preface by his son William Edwin Hamilton. Not everyone found Hamilton's quaternions the answer to everything they had been looking for. Thomson wrote: Quaternions came from Hamilton after his really good work had been done, and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way. Cayley compared the quaternions with a pocket map [25]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html (5 of 7) [2/16/2002 11:13:20 PM]

Hamilton

... which contained everything but had to be unfolded into another form before it could be understood. Hamilton died from a severe attack of gout shortly after receiving the news that he had been elected the first foreign member of the National Academy of Sciences of the USA. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (31 books/articles)

Some Quotations (4)

A Poster of William Rowan Hamilton

Mathematicians born in the same country

Some pages from publications

A page from a notebook showing the multiplication of quaternions.

Cross-references to History Topics

1. The four colour theorem 2. The fundamental theorem of algebra 3. Orbits and gravitation 4. Mathematical games and recreations 5. Squaring the circle 6. Memory, mental arithmetic and mathematics 7. General relativity 8. An overview of the history of mathematics 9. Abstract linear spaces

Other references in MacTutor

1. Plaque at Brougham Bridge commemorating where Hamilton discovered the Quaternions 2. Engraving of Hamilton supposedly carving it. 3. Hamilton's Icosian game 4. Chronology: 1830 to 1840 5. Chronology: 1840 to 1850 6. Chronology: 1850 to 1860 7. Chronology: 1860 to 1870

Honours awarded to William Rowan Hamilton (Click a link below for the full list of mathematicians honoured in this way) Royal Society Royal Medal

Awarded 1835

Lunar features

Crater Hamilton

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html (6 of 7) [2/16/2002 11:13:20 PM]

Hamilton

Other Web sites

1. David R Wilkins 2. European Mathematical Society (Hamilton's papers) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hamilton.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton.html (7 of 7) [2/16/2002 11:13:20 PM]

Hamilton_William

William Hamilton Born: 8 March 1788 in Glasgow, Scotland Died: 6 May 1856 in Edinburgh, Scotland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Hamilton was educated at Edinburgh and Oxford but, due to the unpopularity of Scots at Oxford, he did not receive a fellowship and returned to Edinburgh. He became the 9th Baronet after a law suit in 1861. In 1821 Hamilton was appointed professor of civil history at Edinburgh University, where, in 1836, he became professor of logic and metaphysics. Hamilton was one of the first in a series of British logicians to create the algebra of logic and introduced the 'quantification of the predicate'. Boole, De Morgan and Venn followed him. However Hamilton helped begin this development and his work, although not of great depth, influenced Boole to produce a much more sophisticated system. Hamilton stimulated an interest in metaphysics and introduced Kant and other German philosophers to the British public. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton_William.html (1 of 2) [2/16/2002 11:13:21 PM]

Hamilton_William

Other Web sites

1. Internet Encyclopedia of Philosophy 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hamilton_William.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamilton_William.html (2 of 2) [2/16/2002 11:13:21 PM]

Hamming

Richard Wesley Hamming Born: 11 Feb 1915 in Chicago, Illinois, USA Died: 7 Jan 1998 in Monterey, California, USA

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Richard Hamming entered the University of Chicago receiving his B.S. in 1937. He then went to the University of Nebraska where he was awarded his M.A. in 1939 and then he received his Ph.D. in mathematics in 1942 from the University of Illinois at Urbana-Champaign. His doctoral dissertation Some Problems in the Boundary Value Theory of Linear Differential Equations was supervised by Waldemar Trjitzinsky. In 1945 Hamming joined the Manhattan Project, a U.S. government research project to produce an atomic bomb. It was called the Manhattan Project because the first research had been done at Columbia University in Manhattan, however Hamming worked on the project at Los Alamos. After the end of World War II, Hamming joined the Bell Telephone Laboratories in 1946. While there he was able to work with both Shannon and Tukey. He was to continue to work for Bell Telephones until 1976 when he accepted a chair of computer science at the Naval Postgraduate School at Monterey, California. Hamming is best known for his work on error- detecting and error- correcting codes. His fundamental paper on this topic appeared in 1950 and with this he started a new subject within information theory. Hamming codes are of fundamental importance in coding theory and are of practical use in computer design. Work in codes is related to packing problems and error- correcting codes due to Hamming led to the solution of a packing problem for matrices over finite fields. In 1956 Hamming worked on the early computer, the IBM 650. His work here led to the development of a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamming.html (1 of 3) [2/16/2002 11:13:23 PM]

Hamming

programming language which has evolved into the high-level computer languages used to program computers today. Hamming also worked on numerical analysis, integrating differential equations, and the Hamming spectral window which is much used in computation for smoothing data before Fourier analysing it. His major works include Numerical Methods for Scientists and Engineers (1962), Introduction to applied numerical analysis (1971), Digital filters (1977), Coding and information theory (1980), Methods of mathematics applied to calculus, probability, and statistics (1985), Introduction to applied numerical analysis (1989), The Art of Probability for Scientists and Engineers (1991) and The Art of Doing Science and Engineering : Learning to Learn (1997). Hamming has received many awards for his pioneering work. In 1968 he was made a fellow of the Institute of Electrical and Electronics Engineers and awarded the Turing Prize from the Association for Computing Machinery. The Institute of Electrical and Electronics Engineers awarded Hamming the Emanuel R Piore Award in 1979 and a medal in 1988:For exceptional contributions to information sciences and systems. The IEEE have named this medal "the Hamming Medal" in his honour. Further honours included being elected a member of the National Academy of Engineering in 1980 and receiving the Harold Pender Award from the University of Pennsylvania in 1981. In 1996, in Munich, Hamming received the prestigious $130,000 Eduard Rheim Award for Achievement in Technology for his work on error correcting codes. On Hamming's death Richard Franke of the Naval Postgraduate School at Monterey wrote:He will be long remembered for his keen insights into many facets of science and computation. I'll also long long remember him for his red plaid sport coat and his bad jokes. James F Kaiser, in a brief obituary of Hamming, writes:We will all miss his engaging mind and his penetrating insight into matters scientific, engineering, and of everyday living. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles)

Some Quotations (5)

Mathematicians born in the same country Other references in MacTutor

Chronology: 1950 to 1960

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamming.html (2 of 3) [2/16/2002 11:13:23 PM]

Hamming

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hamming.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hamming.html (3 of 3) [2/16/2002 11:13:23 PM]

Hankel

Hermann Hankel Born: 14 Feb 1839 in Halle, Germany Died: 29 Aug 1873 in Schramberg (near Tübingen), Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hermann Hankel's father was Wilhelm Gottlieb Hankel who was a physicist at Halle at the time Hermann was born. Hermann began his education in Halle but, in 1849 Wilhelm was appointed to the chair of physics at Leipzig so the family moved to Leipzig where Hermann attended the Nicolai Gymnasium. At the gymnasium he [1]:... improved his Greek by reading the ancient mathematicians in the original. In 1857 Hankel entered the University of Leipzig where he studied mathematics with Möbius and physics with his own father. Following the tradition in Germany at that time Hankel did not complete his studies at one university, but moved to several different universities during the course of his studies. From Leipzig he went to Göttingen in 1860 where he became a student of Riemann and then, in the following year, he worked with Weierstrass and Kronecker in Berlin. He received his doctorate for a thesis Uber eine besondere Classe der symmetrischen Determinanten in 1862. Hankel's habilitation was accepted in 1863 and he began teaching at Leipzig where he was appointed extraordinary professor in 1867. The appointment as extraordinary professor had been in the spring but by the autumn of the same year Hankel was at Erlangen to take up an appointment as ordinary professor. He married Marie Dippe in Erlangen but again he would move fairly soon, accepting the chair at Tübingen in 1869. He worked on the theory of complex numbers, the theory of functions and the history of mathematics. His work on complex analysis, however, is not considered of the first rank and in [8] he is included with those who contributed but whose:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hankel.html (1 of 3) [2/16/2002 11:13:26 PM]

Hankel

... influence on the foundations of complex analysis was not as essential as that of those mathematicians discussed in more detail[Riemann, Weierstrass, Hurwitz, Bieberbach ...] Hankel made a systematic study of the rules of arithmetic with his Prinzip der Permanenz der formalen Gesetze (1867), see [7]. He wrote another important work which was also published in 1867 Theorie der complexen Zahlensysteme which did much to make Grassmann's ideas better known. This work [1]:... constitutes a lengthy presentation of much of what was then known of the real, complex, and hypercomplex number systems. Beginning with a revised statement of George Peacock's principle of permanence of formal laws, he developed complex numbers as well as such higher algebraic systems as Möbius' barycentric calculus, some of Hermann Grassmann's algebras, and W R Hamilton's quaternions. Hankel was the first to recognise the significance of Grassmann's long-neglected writings ... Hankel looked at Riemann's integration theory and restated it in terms of measure theoretic concepts. This, and other work he did in this area, constitutes progress towards our current integration theories. He is remembered for the Hankel transformation which occurs in the study of functions which depend only on the distance from the origin. He also studied functions, now named Hankel functions or Bessel functions of the third kind, in a series of papers which appeared in Mathematische Annalen. His historical writings are rather hard to evaluate since they contain many errors, yet they are filled with brilliant insight. In the same way that he saw the importance of Grassmann's work, Hankel also must have considerable credit for seeing the importance of Bolzano's work on infinite series. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles)

Some Quotations (2)

Mathematicians born in the same country Cross-references to History Topics

1. The fundamental theorem of algebra 2. Abstract linear spaces

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hankel.html (2 of 3) [2/16/2002 11:13:26 PM]

Hankel

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hankel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hankel.html (3 of 3) [2/16/2002 11:13:26 PM]

Hardy

Godfrey Harold Hardy Born: 7 Feb 1877 in Cranleigh, Surrey, England Died: 1 Dec 1947 in Cambridge, Cambridgeshire, England

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

G H Hardy's father, Isaac Hardy, was bursar and an art master at Cranleigh school. His mother had been a teacher at Lincoln Teacher's Training School. Both parents were highly intelligent with some mathematical skills but, coming from poor families, had not been able to have a university education. Hardy (he was always known as Hardy except to one or two close friends who called him Harold) attended Cranleigh school up to the age of twelve with great success [4]:His parents knew he was prodigiously clever, and so did he. He came top of his class in all subjects. But, as a result of coming top of his class, he had to go in front of the school to receive prizes: and that he could not bear. Hardy did not appear to have the passion for mathematics that many mathematicians experience when young. Hardy himself writes in [3]:I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively. Indeed he did win a scholarship to Winchester College in 1889, entering the College the following year. Winchester was the best school in England for mathematical training yet, despite admitting later in life that he had been well-educated there, Hardy disliked everything about the school other than the academic training he received. Like all public schools it was a rough place for a frail, shy boy like Hardy. It is significant that although he did have a passion for ball games in general and cricket in particular, he was never coached in sport at Winchester. Somehow he failed to take part fully in the non-academic activities. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hardy.html (1 of 6) [2/16/2002 11:13:28 PM]

Hardy

While at Winchester Hardy won an open scholarship to Trinity College Cambridge, which he entered in 1896. At Cambridge Hardy was assigned to the most famous coach R R Webb. He quickly realised that the point of the training was simply to achieve the best possible marks in the examinations by learning all the tricks of the trade. He was shocked to discover that Webb was not interested in the subject of mathematics, only in the tricks of examinations. Briefly Hardy thought he might change topics and study history instead. However, he managed to change his coach to A E H Love. Hardy expresses his gratitude to Love in [3]:My eyes were first opened by Professor Love, who first taught me a few terms and gave me my first serious conception of analysis. But the great debt which I owe to him was his advice to read Jordan's Cours d'analyse; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant. Hardy was placed as fourth wrangler in the tripos of 1898, a result which continued to annoy him for, despite feeling that the system was very silly, he still felt that he should have come out on top. Hardy was elected a fellow of Trinity in 1900 then, in 1901, he was awarded a Smith's prize jointly with J H Jeans 'with unspecified relative merit'. The next period of Hardy's career was up to 1911 when, as Burkill writes in [1], he:... wrote many papers on the convergence of series and integrals and allied topics. Although this work established his reputation as an analyst, his greatest service to mathematics in this early period was A course of pure mathematics (1908). This work was the first rigorous English exposition of number, function, limit, and so on, adapted to the undergraduate, and thus it transformed university teaching. This was a period of which Hardy wrote himself [3]:I wrote a great deal... but very little of any importance; there are not more than four of five papers which I can still remember with some satisfaction. It is worth noting at this point that Hardy was a remarkably honest man, and in particular he was very honest about his own abilities, strengths and weaknesses. A major change in Hardy's work came about in 1911 when he began his collaboration with J E Littlewood which was to last 35 years. Then in early 1913 he received Ramanujan's first letter from India which was to start his second major collaboration. By the time World War I started in 1914, Ramanujan was in Cambridge and this eased for Hardy what was to be a very difficult period. Littlewood left Cambridge for war service in the Royal Artillery. Hardy volunteered for war service but was rejected on medical grounds. However Hardy's views on the war left him at odds with most of his colleagues at Cambridge. He had great respect for Germany [4]:... he had a strong feeling for Germany. Germany had, after all, been the great educating force of the nineteenth century. To Eastern Europe, to Russia, to the United States, it was the German universities which had taught the meaning of research. ... in most respects the German culture, including its social welfare, appeared to him higher than his own. ... Hardy, like Russell ... did not believe that the war should have been fought. Further, with his ingrained distrust of English politicians, he thought the balance of wrong was on the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hardy.html (2 of 6) [2/16/2002 11:13:28 PM]

Hardy

English side. Deeply unhappy at Cambridge, Hardy took the opportunity to leave in 1919 when he was appointed as Savilian professor of geometry at Oxford. The was in many ways the years when he was happiest and also the years when he produced his best mathematics in the collaboration with Littlewood. This collaboration was achieved during this time with Littlewood at Cambridge and with Hardy at Oxford, making joint research a quite difficult logistical exercise. As Hardy wrote in [3]:I was at my best at a little past forty, when I was a professor at Oxford. Despite his background and the positions he held, Hardy preferred the poor and disadvantaged to those he called the 'large bottomed' who included [4]:... the confident, booming, imperialist bourgeois English. The designation included most bishops, headmasters, judges, and all politicians, with the single exception of Lloyd George. He had chosen not to live in the best rooms while at Cambridge, and Hilbert was so concerned that Hardy was not being properly treated that he wrote to the Master of the College pointing out that the best mathematician in England should have the best rooms. However, Hardy did not think that way. He held a trade union office for two years (1924-26) as President of the Association of Scientific Workers. At a time when it seemed difficult to do so, Hardy liked equally both the United States and Russia. He spent the academic year 1928-29 at Princeton in an exchange with Veblen, who spent the year in Oxford. Despite having been unhappy at Cambridge, Hardy returned to the Sadleirian chair there in 1931 when Hobson retired. Snow in [4] says that Hardy returned to Cambridge for two reasons, firstly that he still considered Cambridge the centre of English mathematics and the Sadleirian chair the foremost mathematics chair in England, and secondly, that he could keep his rooms in College at Cambridge while this was not possible at Oxford. To the unmarried Hardy, this held an attraction as he began to look toward old age. Hardy's interests covered many topics of pure mathematics - Diophantine analysis, summation of divergent series, Fourier series, the Riemann zeta function, and the distribution of primes. His long collaboration with Littlewood produced mathematics of the highest quality. It was a collaboration in which Hardy acknowledged Littlewood's greater technical mathematical skills, but at the same time Hardy brought great talents of mathematical insight and a great ability to write their work up in papers with great clarity. Even more remarkable was Hardy's collaboration with Ramanujan. Hardy instantly spotted Ramanujan's genius from a manuscript sent to him by Ramanujan from India in 1913. Two other top class mathematicians had previously failed to spot the genius. Hardy brought Ramanujan to Cambridge and they wrote 5 remarkable papers together. It was not only with Littlewood and Ramanujan that Hardy collaborated. He was a natural collaborator who also wrote joint papers with Titchmarsh, Ingham, Landau, Pólya, E M Wright, W W Rogosinski and Marcel Riesz. Hardy was a pure mathematician who hoped his mathematics could never be applied. However in 1908, near the beginning of his career, he gave a law describing how the proportions of dominant and recessive genetic traits would be propagated in a large population. Hardy considered it unimportant but it has http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hardy.html (3 of 6) [2/16/2002 11:13:28 PM]

Hardy

proved of major importance in blood group distribution. There was only one passion in Hardy's life other than mathematics and that was cricket. In fact for most of his life his day, at least during the cricket season, would consist of breakfast during which he read The Times studying the cricket scores with great interest. After breakfast he would work on his own mathematical researches from 9 o'clock till 1 o'clock. Then, after a light lunch, he would walk down to the university cricket ground to watch a game. In the late afternoon he would walk slowly back to his rooms in College. There he took dinner, which he followed with a glass of wine. When cricket was not in season, it was the Australian cricket scores he would read in The Times and he would play real tennis in the afternoons. Hardy was known for his eccentricities. He could not endure having his photograph taken and only five snapshots are known to exist. He also hated mirrors and his first action on entering any hotel room was to cover any mirror with a towel. He always played an amusing game of trying to fool God (which is also rather strange since he claimed all his life not be believe in God). For example, during a trip to Denmark he sent back a postcard claiming that he had proved the Riemann hypothesis. He reasoned that God would not allow the boat to sink on the return journey and give him the same fame that Fermat had achieved with his "last theorem". Another example of his trying to fool God was when he went to cricket matches he would take what he called his "anti-God battery". This consisted of thick sweaters, an umbrella, mathematical papers to referee, student examination scripts etc. His theory was that God would think that he expected rain to come so that he could then get on with his work. Since Hardy thought that God would then have the sun shine all day to spite him, he would be able to enjoy the cricket in perfect sunshine. As World War I had been painful for Hardy, World War II was equally so. He had remained remarkably youthful in both mind and body until 1939 when, at the age of 62, he had a heart attack. His remarkable mental powers began to leave him and sports which he had loved to participate in up till then became impossible. He was filled with anger that Europe had again entered the lunacy of war. However, Hardy had one further gift to leave to the world, namely A mathematicians apology which has inspired many towards mathematics. Hardy's book A mathematicians apology was written in 1940. It is one of the most vivid descriptions of how a mathematician thinks and the pleasure of mathematics. But the book is more, as Snow writes in [4]:A mathematicians apology is, if read with the textual attention it deserves, a book of haunting sadness. Yes, it is witty and sharp with intellectual high spirits: yes, the crystalline clarity and candour are still there: yes, it is the testament of a creative artist. But it is also, in an understated stoical fashion, a passionate lament for creative powers that used to be and that will never come again. I know nothing like it in the language: partly because most people with the literary gift to express such a lament don't come to feel it: it is very rare for a writer to realise, with the finality of truth, that he is absolutely finished. The following quotation from A mathematicians apology ([3]) gives a clear idea of Hardy's thoughts on mathematics:The mathematician's pattern's, like those of the painter's or the poet's, must be beautiful, the ideas, like the colours or the words, must fit together in a harmonious way. There is no

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hardy.html (4 of 6) [2/16/2002 11:13:28 PM]

Hardy

permanent place in the world for ugly mathematics. By the time the war ended in 1945 Hardy health was failing fast. He longed to be creative again, for that was all that really mattered to him in life, but he knew that his creativity was gone and that he became very depressed. By 1946 he could only get around by taking taxi rides, a few steps would make him short of breath. In early summer of 1947 he tried to take his own life by taking a large dose of barbiturates. He took so many, however, that he was sick and survived. Snow writes [4]:In the Evelyn nursing home, Hardy was lying in bed. As a touch of farce, he had a black eye. Vomiting from the drugs, he had hit his head on the lavatory basin. He was self-mocking. He had made a mess of it. ... He talked a little, nearly every time I saw him, about death. He wanted it. He did not fear it: what was there to fear in nothingness? His hard intellectual stoicism had come back. He would not try to kill himself again. He wasn't good at it. He was prepared to wait. With an inconsistency which might have pained him - for he ... believed in the rational to an extent that I thought irrational - he showed an intense hypochondriac curiosity about his own symptoms. Hardy received many honours for his work. He was elected a Fellow of the Royal Society in 1910, he received the Royal Medal of the Society in 1920 and Sylvester Medal of the Society in 1940:... for his important contributions to many branches of pure mathematics. He also received the Copley Medal of the Royal Society in 1947:... for his distinguished part in the development of mathematical analysis in England during the last thirty years. Hardy learnt of the award only a few weeks before his death. He was president of the London Mathematical Society from 1926 to 1928 and again from 1939 to 1941. He received the De Morgan Medal of the Society in 1929. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (20 books/articles)

Some Quotations (13)

A Poster of G H Hardy

Mathematicians born in the same country

Cross-references to History Topics

Pi through the ages

Other references in MacTutor

1. Chronology: 1900 to 1910 2. Chronology: 1910 to 1920

Honours awarded to G H Hardy (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1910

Royal Society Copley Medal

Awarded 1947

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hardy.html (5 of 6) [2/16/2002 11:13:28 PM]

Hardy

Royal Society Royal Medal

Awarded 1920

Royal Society Sylvester Medal

Awarded 1940

London Maths Society President

1926 - 1928

LMS De Morgan Medal

Awarded 1929

AMS Gibbs Lecturer

1928

Savilian Professor of Geometry

1920

Other Web sites

1. Hardy-Littlewood Constants 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Hardy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hardy.html (6 of 6) [2/16/2002 11:13:28 PM]

Hardy_Claude

Claude Hardy Born: 1598 in Le Mans, France Died: 5 April 1678 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Claude Hardy was a practising lawyer and took part in the weekly meetings of Roberval, Mersenne and others. He was also a friend of Gassendi and Mydorge. Hardy introduced Mydorge to Descartes. Hardy, in his capacity as a lawyer, worked for the Parliament in Paris from 1625. A year later he was certainly attached to the court of justice in Paris as a counsellor. In 1630 he published Examen and in 1638 Refutation. These works dealt with the problem of the duplication of the cube and Hardy pointed out a fallacy which had arisen regarding this problem. Hardy took part in many of the mathematical discussions and arguments of the time but his greatest contribution was his knowledge of Arabic and other languages which enabled him to make important Latin translations of Euclid's Data and other books. Despite the fact that he introduced Mydorge to Descartes, Hardy supported Descartes when the two had a dispute over Fermat's method of maxima and minima, a forerunner to the calculus. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Claude Hardy (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Villa Hardy (20th Arrondissement)

Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hardy_Claude.html (1 of 2) [2/16/2002 11:13:29 PM]

Hardy_Claude

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hardy_Claude.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hardy_Claude.html (2 of 2) [2/16/2002 11:13:29 PM]

Harish-Chandra

Harish-Chandra Born: 11 Oct 1923 in Kanpur, Uttar Pradesh, India Died: 16 Oct 1983 in Princeton, New Jersey, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Harish-Chandra attended school in Kanpur, then attended the University of Allahabad. Here he studied theoretical physics, this direction being the result of studying Principles of Quantum Mechanics by Dirac. He was awarded a master's degree in 1943 and then he went to Bangalore to work further on theoretical physics. After a short while Harish-Chandra went to Cambridge where he studied for his Ph.D. under Dirac's supervision. During his time in Cambridge he moved away from physics and became more interested in mathematics. While at Cambridge he attended a lecture by Pauli and pointed out a mistake in Pauli's work. The two were to become life long friends. Harish-Chandra obtained his degree in 1947 and, the same year, he went to the USA. Dirac visited Princeton for one year and Harish-Chandra worked as his assistant during this time. However he was greatly influenced by Weyl and Chevalley. The period 1950 to 1963 was his most productive and he spent these years at the Columbia University. During this time he worked on representations of semisimple Lie groups. Also during this period he had close contact with Weil. In [4] Harish-Chandra is quoted as saying that he believed that his lack of background in mathematics was in a way responsible for the novelty of his work:I have often pondered over the roles of knowledge or experience, on the one hand, and imagination or intuition, on the other, in the process of discovery. I believe that there is a certain fundamental conflict between the two, and knowledge, by advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naiveté, unburdened by conventional http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Harish-Chandra.html (1 of 2) [2/16/2002 11:13:31 PM]

Harish-Chandra

wisdom, can sometimes be a positive asset. Harish-Chandra worked at the Institute for Advanced Study at Princeton from 1963. He was appointed IBM-von Neumann Professor in 1968. He died of a heart attack at the end of a week long conference in Princeton, having earlier suffered from three heart attacks. Harish-Chandra received many awards in his career. He was a Fellow of the Royal Society and a Fellow of the National Academy of Sciences. He won the Cole prize from the American Mathematical Society in 1954 for his papers on representations of semisimple Lie algebras and groups, and particularly for his paper On some applications of the universal enveloping algebra of a semisimple Lie algebra which he had published in the Transactions of the American Mathematical Society in 1951. In 1974, he received the Ramanujan Medal from Indian National Science Academy. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles)

A Quotation

Mathematicians born in the same country Honours awarded to Harish-Chandra (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1973

AMS Colloquium Lecturer

1969

AMS Cole Prize

Awarded 1954

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Harish-Chandra.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Harish-Chandra.html (2 of 2) [2/16/2002 11:13:31 PM]

Harriot

Thomas Harriot Born: 1560 in Oxford, England Died: 2 July 1621 in London, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Harriot was a mathematician and astronomer who founded the English school of algebra. He is described in [10] by Fauvel and Goulding as:... the greatest mathematician that Oxford has produced ... yet his name has only recently become widely known, and even now his achievements are not fully appreciated by most mathematicians. We know very little of Harriot's youth. In fact all that is known is that on Friday 20 December 1577 he matriculated at the University of Oxford with an entry in the official records giving his age as seventeen, his father as a plebeian, and his birthplace Oxfordshire. It is from this record that his date of birth is deduced to be 1560 and we know that his father was a "commoner" but the very fact that Harriot was entering Oxford means that it is unlikely that he came from the poorest classes. Despite extensive searches of the Oxfordshire records, no further information concerning his birth or parentage has been found (although a number of possible relatives have been identified). As an undergraduate at Oxford, Harriot was a student at St Mary's Hall. He became friends with Richard Hakluyt and Thomas Allen, both lecturers at the university, but not at St Mary's Hall. Harriot graduated in 1580 and went to London. It is not clear exactly what he did in his first few years there but, probably from late 1583, he entered Sir Walter Raleigh's service. Hakluyt, dedicating a preface to Raleigh in February 1587, wrote (see for example [4]):Ever since you perceived that skill in the navigator's art, the chief ornament of an island kingdom, might attain its splendour amongst us if the aid of the mathematical sciences were enlisted, you have maintained in your household Thomas Harriot, a man pre-eminent in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Harriot.html (1 of 7) [2/16/2002 11:13:33 PM]

Harriot

those studies, at a most liberal salary, in order that by his aid you might acquire those noble sciences in your leisure hours ... Harriot wrote a text called Arcticon which was never published and unfortunately no copies have ever been found. This work was essentially his lecture course given at Durham House, Raleigh's lodgings in The Strand in London, where Harriot lived at this time. The lectures were given to the seamen who were being gathered by Raleigh to participate in his expeditions to the New World. Pepper describes the advances in navigational techniques made by Harriot by the time he wrote Arcticon [18]:... he solved the problem of reconciling the sun and pole star observations for determining latitude, introduced the idea of using solar amplitude to determine magnetic variation and, as well as improving methods and devices for observation of solar or stellar altitudes, he recalculated tables for the sun's declination on the basis of his own astronomical observations. ... he produced a practical numerical solution of the Mercator problem, most probably by the addition of secants ... As Roche notes in [24]:... when combined with new instruments and observational practices it is clear that Raleigh had the best navigational expertise in Europe. It was not only as a navigational instructor that Raleigh employed Harriot. He was involved with the design of the ships for Raleigh's expeditions as well as being involved in the construction of the vessels and selecting the seamen. He was Raleigh's accountant, being responsible for obtaining funding for the expeditions and keeping all the accounts. Raleigh had the captains Philip Amadas and Arthur Barlowe make an expedition to Roanoke Island off the coast of North Carolina in 1584. Although there is no direct evidence that Harriot made this voyage, Quinn in [23] argues convincingly that he was one of those making this preliminary survey. Harriot was certainly on a voyage to Virginia organised by Raleigh in 1585-86. He sailed from Plymouth on 9 April 1585 on board the Tiger and his observations of a solar eclipse on 19 April have allowed modern scientists to compute the exact position of the ship on that day. Harriot made many notes during his time in the New World, being particularly interested in the language and customs (particularly the eating habits) of the inhabitants. The object of the voyage was to colonise the New World but it was not successful in this aim. Drake was engaged in sea battles with the Spanish when he learnt that they intended to prevent the British colonists becoming established. Although Drake met up with the colonists, in June 1586 there were severe storms and there was a hurried return to England by Harriot and most of the party. Harriot, together with Drake's ships, landed at Portsmouth in July 1586 and he went immediately to Raleigh to report on the expedition. He published A Briefe and True Report of the New Found Land of Virginia in 1588, a book in which he recommends the smoking of tobacco which he himself had learnt to do in Virginia. However, he also wrote a full account of the voyage which, for some reason, he never published and, despite strenuous attempts to find a copy, seems lost. By the time Harriot had returned, Raleigh had turned his attention to Ireland. Harriot carried out surveys of the Lismore estate, which was owned by Raleigh, beginning in 1589. Nine years later he was still involved in working out the acreage of plots being leased on the estate. However the political situation was about to change and this would have dramatic implications for Harriot. Already in the 1590s there were allegations against Raleigh of atheism. The charges were against http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Harriot.html (2 of 7) [2/16/2002 11:13:33 PM]

Harriot

Raleigh's school and "the conjurer that is master thereof". Harriot felt that this was a reference to him and he discussed the allegations with John Dee (who also felt that the charges might relate to him). There is no reason to believe that Harriot (or Raleigh) were atheists but certainly they were free thinkers and Harriot's scientific approach to the world was, to say the least, viewed with great suspicion by the church. As well as problems caused by allegations, Dee and Harriot discussed scientific and mathematical matters in the 1590s. Harriot had now moved from working for Raleigh to working for Henry Percy, Duke of Northumberland. The Duke had around him a circle of friends who were scholars, many of whom held a atomistic views. Raleigh's life became so chaotic that Harriot had sought the support of a patron who could provide more stability for his scientific pursuits. In 1595 the Duke made property in Durham over to Harriot and he moved up the social ladder becoming a member of the "landed gentry". Harriot also later held estates in Cornwall and Norfolk. Not long after the Durham transaction, the Duke gave Harriot the use of one of the houses on the estate at Syon (near Kew outside London) which Harriot used both as a residence and as a scientific laboratory. We certainly know from manuscripts which survive that Harriot was engaged in deep studies of optics at Syon by 1597. Although in [4] it states that he had discovered the sine law of refraction of light before 1597, in fact we now know that the precise date of Harriot's important discovery was July 1601. As with all his other mathematical discoveries, however, Harriot did not publish his findings. It is somewhat ironical, however, that Snell (to whom the discovery of this law is now attributed) was not the first to publish the result. Snell's discovery was in 1621, about 20 years after Harriot's discovery, but the result was not published until Descartes put it in print in 1637. One of the optical problems which Harriot did study in the 1590s was Alhazen's problem. He gave a solution to Alhazen's problem which involved considering an equivalent problem, namely the problem of the maximum intercept formed between a circle and a diameter of a chord rotating about a point on a circle. The author of [8] conjectures that Harriot may have used infinitesimal techniques in demonstrating the equivalence of these problems, and certainly we know that Harriot introduced ideas later rediscovered by Barrow. Optics was not the only topic to occupy Harriot during this period. He had been asked by Raleigh in the early 1590s to apply his mathematical skills to the science of gunnery. At this time ideas of the trajectory taken by a projectile were still dominated by Aristotle's thinking. Harriot resolved the forces acting on the projectile into horizontal and vertical components. He understood that air resistance acted throughout the whole flight, and that gravity acted on the vertical component. He came very close to a vector analysis solution of the problem of finding the velocity of the projectile and, certainly by 1607, he came to the conclusion that the path of the projectile was a tilted parabola. He made one error, however, [4]:Somehow, he could not force himself to abandon the Aristotelian idea that heavier bodies fell at a faster rate than lighter ones. Other topics which Harriot began to work on before 1600 were problems of chemistry. He worked intensively on chemistry for almost exactly a year from May 1599 to May 1600 and, although his experiments were conducted with a new scientific precision, he made no discoveries of particular note. Raleigh had been a particular favourite of Elizabeth I and, when she died on 24 March 1603, it was clear

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Harriot.html (3 of 7) [2/16/2002 11:13:33 PM]

Harriot

that Raleigh's fortunes would change. Perhaps it is less clear that Harriot, by this time not so closely associated with Raleigh, would find problems too. James I became king and he quickly saw Raleigh as someone opposed to his claims to the throne. Henry Percy, the Duke of Northumberland, had taken care to put himself on a good footing with James with a letter of support for him only days before Elizabeth died. In July plots were discovered against James and Raleigh was arrested and charged with high treason. Raleigh attempted suicide but failed. He then sought Harriot's help in obtaining evidence on his behalf. Raleigh was convicted and sentenced to death by hanging. Poor Harriot was singled out in the judgement as being an atheist and an evil influence. His attempts to help Raleigh had been based on Christian principles (to which undoubtedly he adhered) but this had rather damaged Raleigh as Harriot was seen an atheist using Christian principles for convenience. Harriot was devastated and for about a year undertook no new scientific work as he tried to come to terms with what was happening. Raleigh received a last-minute reprieve from the death sentence but was imprisoned in the Tower of London. Another plot was to lead to further trouble. On 4 November 1604 Guy Fawkes and others were arrested for attempting to blow up the Houses of Parliament. Four others, including Thomas Percy, the grandson of Henry Percy, were also arrested as the main conspirators. Harriot was held on suspicion of being involved and imprisoned in the Gatehouse. He was interrogated on the charge that he had caste a horoscope of King James in an attempt to use magical powers to influence the King's future. On 27 November Henry Percy, Harriot's patron, was also put in the Tower where he remained until 1621 when he was released. No evidence seems to have been found against Harriot and, although he remained in the Gatehouse for some while writing several letters requesting his release, he was a free man probably by the end of 1604. As soon as he was released, Harriot returned to his work on optics. He now considered more complex systems and employed Christopher Tooke as a lens grinder from early 1605. His work on light now moved to the dispersion of light into colours. He began to develop a theory for the rainbow and, by 1606, Kepler had heard of the remarkable results on optics achieved by Harriot. Kepler wrote to Harriot, but the correspondence never really achieved any significant exchange of ideas. Perhaps Harriot was too wary of the difficulties that his work had nearly brought on him, or perhaps he did (as he claimed to Kepler) still intend to publish his results if his health permitted. The appearance of a comet attracted Harriot's attention and turned his scientific mind towards astronomy. He observed a comet on 17 September 1607 from Ilfracombe which would later be identified as Halley's Comet. Kepler had discovered the comet six days earlier but it would be the observations of Harriot and his friend (and student) William Lower which eventually were used by Bessel to compute its orbit. His astronomy brought back to the fore, Harriot went on to make the earliest telescopic observations in England. On 26 July 1609 at 9 p.m. he sketched the Moon which was at that time 5 days old, viewing it through a telescope with a magnification of 6. He sketched the Moon again a year later on 17 July 1610, by this time he had a telescope giving him a magnification of 10. Soon he had constructed a telescope with a magnification of 20, then by April 1611 he had a 32 magnification telescope. Harriot observed the moons of Jupiter, although since his first sighting states:My first observation of the new planets. I saw but one and that alone he must have already been aware of Galileo's discovery. As with all his scientific discoveries, Harriot did http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Harriot.html (4 of 7) [2/16/2002 11:13:33 PM]

Harriot

not publish his results. These observations of Jupiter's moons were made between 17 October 1610 and 26 February 1612. He was the first to discover sunspots, making 199 observations between 8 December 1610 and 18 January 1613. The first observation of sunspots was made while he was observing Jupiter's moons. From the data he collected he was able to deduce the period of the Sun's rotation. However, around this time his scientific work basically came to an end. He seems to have had his spirits brought low by the deaths of his friends and lost the spirit to continue research (which had brought him much trouble despite his lack of publications). Of the few pieces of work done by Harriot after 1614, one was his observation of another comet in 1618 (there were three visible comets that year and Harriot observed the third) from Syon House. In 1618 Raleigh, who had been shown the clemency of imprisonment in 1603 rather than death, was put to death. Raleigh was executed on 29 October 1618 in a public execution, with Harriot present to witness the event. However, by this time Harriot was already suffering from the cancer of the nose which eventually led to his death. The cancer seems to have started around 1613, about the time when Harriot lost interest in pushing forward his mathematical and scientific research. He consulted the top specialist in 1615 who wrote report on the consultation. He described Harriot as (see for example [4]):... a man somewhat melancholy. ... A cancerous ulcer in the left nostril eats up the septum of his nose and in proportion to its size holds the lips hard and turned upwards. It has gradually crept well into the nose. This evil the patient has suffered the last two years. Harriot would suffer this "evil" for a further three years before the cancer took his life. There are a few other major mathematical achievements due to Harriot which we should mention. He exhibited the logarithmic spiral as the stereographic projection of a loxodrome on a sphere, a projection he proved to be conformal. The loxodromes are the straight lines on the Mercator map, which Harriot computed with great precision. In fact in order to achieve this degree of precision, Harriot introduced finite-difference interpolation. There is an interesting history to a problem which has only recently been solved, yet originated with Harriot. Raleigh asked Harriot to solve certain problems regarding the stacking of cannonballs. On a manuscript dated 12 December 1591 (Sunday), Harriot set out a table to answer Raleigh's questions. He shows how, if the number of cannonballs is given, one can compute the number of cannonballs to be placed in the base of a pyramid with a triangular, square or oblong base. Raleigh posed a second question, which Harriot also answered, namely given the pyramid of cannonballs, compute the number in the pile. Harriot was too much the mathematician to stop there, however. From a study of how the cannonballs could fill space, he considered the implications for the atomic theory of matter which he believed in. Later, in his correspondence with Kepler about atomic theory, Harriot mentioned the packing problem. Kepler could not solve the problem but he believed that the densest packing of spheres would be attained if in each layer the centres of the spheres were above the centres of the holes in the layer below. This seems intuitively obvious, but resisted proof until 1998 when Thomas Hales of the University of Michigan (with the help of hours of computer generated data) finally proved the conjecture.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Harriot.html (5 of 7) [2/16/2002 11:13:33 PM]

Harriot

The one part of Harriot's work which we have not yet described is the mathematical work for which, in some ways, he is best known, namely his work on algebra. He introduced a simplified notation for algebra and his fundamental research on the theory of equations was far ahead of its time. As an example of his abilities to solve equations, even when the roots are negative or imaginary, we reproduce his solution of an equation of degree 4. The example in question is in his own handwriting and reproduced in [3]. aaaa - 6aa + 135a = 1155 --------aaaa -2aa + 1 = 4aa -136a + 1156 aa - 1 = 2a - 34 33 = 2a - aa aa - 2a = -33 aa - 2a + 1 = +1 - 33 a-1 = -32 1 - a = -32 a = 1+ -32 a = 1- -32 --------aa - 1 = 34 - 2a aa + 2a = 35 aa + 2a + 1 = 1 + 35 a + 1 = 36 a = 36 - 1 = 5 ----a - 1 = 36 a = - 36 - 1 = -7 Note how expert Harriot is in completing the square, knowing that whenever he takes a square root there are two answers to consider, and treating all answers equally whether positive or negative, real or imaginary. I [EFR] have only made one slight change to Harriot's notation. Where I have used the now . standard symbol = Harriot used Harriot invented certain symbols which are used today. However, the symbols < for "less than" and > for "greater than" were not due to Harriot (as is often claimed), but were introduced by the editor of Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas - Harriot himself used different symbols. There is still scholarly debate on how much Harriot was influenced by Viète, or whether notation and ideas introduced by Viète were learnt by him from Harriot. As we have seen from the example above, Harriot did outstanding work on the solution of equations, recognising negative roots and complex roots in a way that makes his solutions look like a present day solution. He made the observation that if a, b, c are the roots of a cubic then the cubic is (x - a)(x - b)(x c) = 0. This is a major step forward in understanding which Harriot then carried forward to equations of higher degree. Although he was far ahead of his time, his work had far less influence than it should have done since, as http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Harriot.html (6 of 7) [2/16/2002 11:13:33 PM]

Harriot

we have remarked repeatedly above, he published no mathematical work in his lifetime. Even his work on algebra Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas (1631) was published 10 years after his death and was edited by people who did not fully appreciate the depth of his work. For example, it does not discuss negative solutions. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (37 books/articles) A Poster of Thomas Harriot

Mathematicians born in the same country

Cross-references to History Topics

1. Quadratic, cubic and quartic equations 2. Thomas Harriot's manuscripts 3. The fundamental theorem of algebra 4. An overview of the history of mathematics

Other references in MacTutor

Chronology: 1625 to 1650

Honours awarded to Thomas Harriot (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Harriot

Other Web sites

1. The Galileo Project 2. High Altitude Observatory 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Harriot.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Harriot.html (7 of 7) [2/16/2002 11:13:33 PM]

Hartley

Brian Hartley Born: 15 May 1939 in Accrington, Lancashire, England Died: 8 Oct 1994 in Lake District, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Brian Hartley's early years were described by Ian Stewart, one of his doctoral students. Brian was born in Accrington. Holiday jobs included working in a pie factory and a cotton mill, where he had to jump into tubs of raw cotton to pack them down. At 15, he became one of the youngest people to be awarded a scholarship to King's College, Cambridge. Arriving as a 17-year-old he planned to study chemistry but after attending research lectures given by the mathematician Philip Hall he switched. Hartley's Ph.D. thesis was completed in 1964 under Hall's supervision. He spent a year in Chicago, then a year in MIT before being appointed to the University of Warwick in 1966. Except for spending 1969 at ANU, he remained at Warwick until he was appointed to a chair in Manchester in 1977. His over 100 publications are almost all on group theory. His main topic was locally finite groups where he used his wide knowledge of finite groups in proving properties of infinite groups which were in a sense close to finite. He collaborated with many mathematicians and loved working out ideas with them on a blackboard. His only book Rings, modules and linear algebra (written with T O Hawkes) is a widely used undergraduate text. Roger Bryant, a colleague at Manchester, described Hartley as a straightforward personality. Talked directly with everyone. No airs and graces: taxi-driver, research student, whatever. However, he stuck to what he believed in.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hartley.html (1 of 2) [2/16/2002 11:13:35 PM]

Hartley

Hartley's interests outside mathematics, and there were many, are described by Ian Stewart as follows Brian was also a musician. As a teenager he learned the trumpet. He played in St Catherine's Military Band, Accrington, and Cambridge University first Orchestra; he once performed at a mathematics conference but as he had become a little rusty, part way into the piece he and his accompanist declared they had merely been tuning up. They started again, with greater success. Brian was an outdoor person. In his younger days he ran with Ron Hill, who later competed in several Olympic marathons. The Hartleys owned a series of tandems, but Mary [his wife] tended to fold her arms and sing as they went up hills, leaving Brian to do all the work. They both took up cross-country skiing, ice-skating, and horse-riding. The whole family enjoyed camping. Above all, Brian and Mary delighted in hill walking, which occupied most weekends. In fact Brian died walking in the hills that he loved. There was a mountain in northern England that Brian always wanted to climb and he decided to try it before he got too old. He did make the top but collapsed on the way down. The picture of Brian Hartley was taken in August 1993 at the Groups Galway / St Andrews Conference, held in Galway Ireland. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Brian Hartley

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hartley.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hartley.html (2 of 2) [2/16/2002 11:13:35 PM]

Hartree

Douglas Rayner Hartree Born: 27 March 1897 in Cambridge, Cambridgeshire, England Died: 12 Feb 1958 in Cambridge, Cambridgeshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Douglas Hartree's school education was in Cambridge and Petersfield. He entered St John's College Cambridge in 1915 but World War I interrupted his studies and he joined a team studying anti-aircraft gunnery. He returned to Cambridge after the war and graduated in 1921 but, perhaps because of his interrupted studies, he only obtained a Second Class degree in Natural Sciences. However he went on to obtain a doctorate in 1926 and, after being a Fellow of St John's College (1924-27) and Christ's College (1928-29) he was appointed professor of applied mathematics at Manchester. Hartree held this chair from 1929 to 1937 when he moved to the chair of theoretical physics. After undertaking work with the Ministry of Supply during World War II, he was appointed Plummer Professor of Mathematical Physics at Cambridge. He held this post until his death. Hartree was basically a theoretical physicist, and he developed powerful methods in numerical analysis. His initial interest in numerical methods arose from his work on anti-aircraft gunnery in 1916-18. However Niels Bohr gave a lecture course in Cambridge in 1921 and Hartree was much influenced, working on applications of numerical methods for integrating differential equations to calculate atomic wave functions. Hartree learn of a differential analyzer being developed by Vannevar Bush in the USA. This machine, first proposed by Lord Kelvin (William Thomson), performs integration with a wheel rolling on a rotating disk. Hartree visited Boston to learn about the workings of the differential analyzer, then returned to Cambridge and built his own. The differential analyzer was soon to be replaced by electronic computers and when John Eckert set up http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hartree.html (1 of 2) [2/16/2002 11:13:36 PM]

Hartree

ENIAC, Hartree was asked to go to the USA to advise on its use. He showed how to use ENIAC to calculate trajectories of projectiles. In his inaugural address on his appointment to the chair in Cambridge in 1946 he said It may well be that the high-speed digital computer will have as great an influence on civilization as the advent of nuclear power. In addition to applying numerical methods to ballistics Hartree applied them to the physics of the atmosphere and to hydrodynamics. He wrote a number of important books including Numerical analysis in 1952. In this book he says Anyone intending to undertake a serious piece of calculation should realise that adequate checking against mistakes is an essential part of any satisfactory numerical process. No one, and no machine, is infallible, and it may fairly be said that the ideal to aim at is not to avoid mistakes entirely, but to find all mistakes that are made, and so free the work from any unidentified mistakes. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

A picture of ENIAC

Honours awarded to Douglas Hartree (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1932

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hartree.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hartree.html (2 of 2) [2/16/2002 11:13:36 PM]

Hasse

Helmut Hasse Born: 25 Aug 1898 in Kassel, Germany Died: 26 Dec 1979 in Ahrensburg (near Hamburg), Germany

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Helmut Hasse's father was a judge. His mother was born in Milwaukee, Wisconsin, USA but lived in Kassel from the age of five. Helmut's education was at various secondary schools near to Kassel until in 1913, when he was 15 years of age, his father was appointed to an important position in Berlin and the family moved there. Helmut studied for two years at the Fichte-Gymnasium in Berlin before volunteering for naval service during World War I. In the academic year 1917/18 Hasse was stationed at Kiel on his naval duties and he was able to attend the lectures of Otto Toeplitz. On leaving the navy he entered the University of Göttingen. His teachers there included Landau, Hilbert, Emmy Noether and Hecke. In fact he was most influenced by Hecke despite the fact that Hecke left Göttingen to take up an appointment in Hamburg only a few months after Hasse arrived in Göttingen. It might be supposed that Hasse would have followed Hecke to Hamburg but he did not take this route, going to study under Hensel at Marburg in 1920. Hensel's work on p-adic numbers was to have a major influence on the direction of Hasse's research. In October 1920 Hasse discovered the 'local-global' principle which shows that a quadratic form that represents 0 non-trivially over the p-adic numbers for each prime p, and over the real numbers, represents 0 non-trivially over the rationals. The importance of this result, now known as the Hasse principle, is that both the representability of a number by a given form and whether two forms are equivalent can be decided using only local information. The 'local-global' principle and its applications form an important part of his doctoral thesis Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen of 1921 and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hasse.html (1 of 4) [2/16/2002 11:13:38 PM]

Hasse

also of his habilitation thesis Über die Aquivalenz quadratischer Formen im Körper der rationalen Zahlen. In 1922 Hasse was appointed a lecturer at the University of Kiel, then three years later he was appointed professor at Halle. During his time at Kiel, Hasse kept in close contact with the mathematicians at Hamburg including Artin, Hecke, Ostrowski and Schreier. He extended Heinrich Weber's work on class field theory writing several important papers and starting work on his famous report on class field theory which included the contributions of Kronecker, Heinrich Weber, Hilbert, Furtwängler and Takagi. As H M Edwards says in [1]:... [the text], like any good exposition, contained a great deal of Hasse's own reworking of the material. At Halle Hasse obtained fundamental results on the structure of central simple algebras over local fields. In 1930 Hensel retired from Marburg and Hasse was appointed to fill his chair. While in Marburg he began joint work with Brauer and Emmy Noether on simple algebras, culminating in the complete determination of what is today called the Brauer group of an algebraic number field. He also started work on elliptic curves and, with Baer, on topological fields. At this time he obtained a result that is particularly associated with his name, when (inspired by Mordell and Davenport) he proved the analogue of the Riemann Hypothesis for zeta functions of elliptic curves. The year 1933 was to be significant for all of Germany and for Hasse in particular. In that year the Nazis came to power and its effect on mathematics in Germany was profound. MacLane, who was at Göttingen at this time writes in [2]:On April 7, 1933, a new law ... summarily dismissed all those who were Jewish ...The effect on the Mathematical Institute [at Göttingen] was drastic. ... All told, in 1933 eighteen mathematicians left or were driven out from the faculty at the Mathematical Institute in Göttingen. When Weyl resigned from his professorship in Göttingen, Hasse received an offer as his successor. As Edwards says in [1]:... [Hasse] appeared to be potentially acceptable to the Nazis and yet was a mathematician of the highest calibre. Following the advice of Weyl, Hasse decided to accept the offer, although (see [4] and [10]) he met with fierce opposition from the fundamentalist Nazi functionaries within the Mathematics Institute and the University. In 1934 he was appointed to Göttingen. It is hard to understand exactly what Hasse's views were in the middle of this political mess. Edwards sums up the position in [1] as he saw it:Hasse's political views and his relations with the Nazi government are not easily categorised. On the one hand, his relations with his teacher Hensel, who was unambiguously Jewish by Nazi standards, were extremely close, right up to Hensel's death in 1941 ... One of his most important papers was a collaboration with Emmy Noether and Richard Brauer, both Jewish, published in 1932 in honour of Hensel's 70th birthday. ... Hasse did not compromise his mathematics for political reasons ... he struggled against Nazi functionaries who tried (sometimes successfully) to subvert mathematics to political http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hasse.html (2 of 4) [2/16/2002 11:13:38 PM]

Hasse

doctrine ... On the other hand, he made no secret of his strong nationalistic views and his approval of many of Hitler's policies. At one point, Hasse hoped to increase his influence with the authorities and so he applied for membership in the Nazi Party. But one of Hasse's antecendents was a Jew and, therefore, membership was not granted. Officially his application was put on hold till after the war. See [2] and [10]. From 1939 until 1945 Hasse was on war leave from Göttingen and he returned to naval duty, working in Berlin on problems in ballistics. He returned to Göttingen where, in September 1945 he was dismissed from his post by the British occupation forces. His right to teach was terminated and he refused the offer of a research only position, moving to Berlin in 1946 when he took up a research post at Berlin Academy. H W Leopoldt writes in [5]:When he resumed teaching in 1948 in Berlin he attracted a large audience. In his first official lecture he compared aesthetic principles working in music and in number theory. Most of his examples regarding music he took from the late piano sonatas of Beethoven, which he - always an ardent piano player - intimately studied during those years. I had just begun to study mathematics, and this lecture made a lasting impression on me, in fact, it decided the further course of my studies. In May 1949, Hasse was appointed professor at the Humboldt University in East Berlin. His work on determining arithmetical properties of abelian number fields Über die Klassenzahl abelscher Zahlkörper was published at this time. At about the same time his textbook Zahlentheorie appeared which contained, for the first time, a systematic introduction to algebraic number theory based on the local method. It was later translated into English. In 1950 Hasse was appointed to Hamburg where he continued to teach until he retired in 1966. His influence is summed up in [1] where Edwards writes:One of the most important mathematicians of the twentieth century, Helmut Hasse was a man whose accomplishments spanned research, mathematical exposition, teaching and editorial work. This reference to editorial work is made because Hasse was editor of Crelle's Journal for 50 years. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) A Poster of Helmut Hasse

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1920 to 1930

Other Web sites

Clark Kimberling

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hasse.html (3 of 4) [2/16/2002 11:13:38 PM]

Hasse

History Topics Time lines Glossary index

Societies, honours, etc.

Famous curves

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR December 1996 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Hasse.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hasse.html (4 of 4) [2/16/2002 11:13:38 PM]

Hausdorff

Felix Hausdorff Born: 8 Nov 1868 in Breslau, Germany (now Wroclaw, Poland) Died: 26 Jan 1942 in Bonn, Germany

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Felix Hausdorff graduated from Leipzig in 1891 and then taught there until 1910 when he went to Bonn. Within a year of his appointment to Leipzig he was offered a post at Göttingen but, rather surprisingly given Göttingen's reputation, he turned it down. Hausdorff worked at Bonn until 1935 when he was forced to retire by the Nazi regime. Although as early as 1932 he sensed the oncoming calamity of Nazism he made no attempt to emigrate while it was still possible. As a Jew his position became more and more difficult. In 1941 he was scheduled to go to an internment camp but managed to avoid being sent. However by 1942 he could no longer avoid being sent to the internment camp and, together with his wife and his wife's sister, he committed suicide. Hausdorff's main work was in topology and set theory. He introduced the concept of a partially ordered set and from 1906 to 1909 he proved a series of results on ordered sets. In 1907 he introduced special types of ordinals in an attempt to prove Cantor's continuum hypothesis. He also posed a generalisation of the continuum hypothesis by asking if 2 to the power n was equal to n+1. Hausdorff proved further results on the cardinality of Borel sets in 1916. Building on work by Fréchet and others, he created a theory of topological and metric spaces with Grundzüge der Mengenlehre (1914). Earlier results on topology fitted naturally into the framework set up by Hausdorff. In 1919 he introduced the notion of Hausdorff dimension, which was a real number lying between the topological dimension of an object and 3. It is used to study objects such as Koch's curve. He also

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hausdorff.html (1 of 2) [2/16/2002 11:13:40 PM]

Hausdorff

introduced the Hausdorff measure and the term 'metric space' is due to him. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) A Poster of Felix Hausdorff

Mathematicians born in the same country

Cross-references to History Topics

Topology enters mathematics

Other references in MacTutor

1. Curves with non-integral dimension 2. Chronology: 1910 to 1920

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hausdorff.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hausdorff.html (2 of 2) [2/16/2002 11:13:40 PM]

Hawking

Stephen William Hawking Born: 8 Jan 1942 in Oxford, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stephen Hawking's parents lived in London where his father was undertaking research into medicine. However, London was a dangerous place during World War II and Stephen's mother was sent to the safer town of Oxford where Stephen was born. The family were soon back together living in Highgate, north London, where Stephen began his schooling. In 1950 Stephen's father moved to the Institute for Medical Research in Mill Hill. The family moved to St Albans so that the journey to Mill Hill was easier. Stephen attended St Albans High School for Girls (which took boys up to the age of 10). When he was older he attended St Albans school but his father wanted him to take the scholarship examination to go to Westminster public school. However Stephen was ill at the time of the examinations and remained at St Albans school which he had attended from the age of 11. Stephen writes in [2]:I got an education there that was as good as, if not better than, that I would have had at Westminster. I have never found that my lack of social graces has been a hindrance. Hawking wanted to specialise in mathematics in his last couple of years at school where his mathematics teacher had inspired him to study the subject. However Hawking's father was strongly against the idea and Hawking was persuaded to make chemistry his main school subject. Part of his father's reasoning was that he wanted Hawking to go to University College, Oxford, the College he himself had attended, and that College had no mathematics fellow. In March 1959 Hawking took the scholarship examinations with the aim of studying natural sciences at Oxford. He was awarded a scholarship, despite feeling that he had performed badly, and at University College he specialised in physics in his natural sciences degree. He only just made a First Class degree in 1962 and in [1] he explains how the attitude of the time worked against him:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hawking.html (1 of 4) [2/16/2002 11:13:42 PM]

Hawking

The prevailing attitude at Oxford at that time was very anti-work. You were supposed to be brilliant without effort, or accept your limitations and get a fourth-class degree. To work hard to get a better class of degree was regarded as the mark of a grey man - the worst epithet in the Oxford vocabulary. From Oxford, Hawking moved to Cambridge to take up research in general relativity and cosmology, a difficult area for someone with only a little mathematical background. Hawking had noticed that he was becoming rather clumsy during his last year at Oxford and, when he returned home for Christmas 1962 at the end of his first term at Cambridge, his mother persuaded him to see a doctor. In early 1963 he spent two weeks having tests in hospital and motor neurone disease (Lou Gehrig's disease) was diagnosed. His condition deteriorated quickly and the doctors predicted that he would not live long enough to complete his doctorate. However Hawking writes:... although there was a cloud hanging over my future, I found to my surprise that I was enjoying life in the present more than I had before. I began to make progress with my research... The reason that his research progressed was that he met a girl he wanted to marry and realised he had to complete his doctorate to get a job so:... I therefore started working for the first time in my life. To my surprise I found I liked it. After completing his doctorate in 1966 Hawking was awarded a fellowship at Gonville and Caius College, Cambridge. At first his position was that of Research Fellow, but later he became a Professorial Fellow at Gonville and Caius College. In 1973 he left the Institute of Astronomy and joined to the Department of Applied Mathematics and Theoretical Physics at Cambridge. He became Professor of Gravitational Physics at Cambridge in 1977. In 1979 Hawking was appointed Lucasian Professor of Mathematics at Cambridge. The man born 300 years to the day after Galileo died now held Newton's chair at Cambridge. Between 1965 and 1970 Hawking worked on singularities in the theory of general relativity devising new mathematical techniques to study this area of cosmology. Much of his work in this area was done in collaboration with Roger Penrose who, at that time, was at Birkbeck College, London. From 1970 Hawking began to apply his previous ideas to the study of black holes. Continuing this work on black holes, Hawking discovered in 1970 a remarkable property. Using quantum theory and general relativity he was able to show that black holes can emit radiation. His success with proving this made him work from that time on combining the theory of general relativity with quantum theory. In 1971 Hawking investigated the creation of the Universe and predicted that, following the big bang, many objects as heavy as 109 tons but only the size of a proton would be created. These mini black holes have large gravitational attraction governed by general relativity, while the laws of quantum mechanics would apply to objects that small. Another remarkable achievement of Hawking's using these techniques was his no boundary proposal made in 1983 with Jim Hartle of Santa Barbara. Hawking explains that this would mean:... that both time and space are finite in extent, but they don't have any boundary or edge. ... there would be no singularities, and the laws of science would hold everywhere, including at the beginning of the universe. In 1982 Hawking decided to write a popular book on cosmology. By 1984 he had produced a first draft http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hawking.html (2 of 4) [2/16/2002 11:13:42 PM]

Hawking

of A Brief History of Time. However Hawking was to suffer a further illness:I was in Geneva, at CERN, the big particle accelerator, in the summer of 1985. ... I caught pneumonia and was rushed to hospital. The hospital in Geneva suggested to my wife that it was not worth keeping the life support machine on. But she was having none of that. I was flown back to Addenbrooke's Hospital in Cambridge, where a surgeon called Roger Grey carried out a tracheotomy. That operation saved my life but took away my voice. Hawking was given a computer system to enable him to have an electronic voice. It was with these difficulties that he revised the draft of A Brief History of Time which was published in 1988. The book broke sales records in a way that it would have been hard to predict. By May 1995 it had been in The Sunday Times best-sellers list for 237 weeks breaking the previous record of 184 weeks. This feat is recorded in the 1998 Guinness Book of Records. Also recorded there is the fact that the paperback edition was published on 6 April 1995 and reached number one in the best sellers in 3 days. By April 1993 there had been 40 hardback editions of A Brief History of Time in the United States and 39 hardback editions in the UK. Of course Hawking has received, and continues to receive, a large number of honours. He was elected a Fellow of The Royal Society in 1974, being one of its youngest fellows. He was awarded the CBE in 1982, and was made a Companion of Honour in 1989. Hawking has also received many foreign awards and prizes and was elected a Member of the National Academy of Sciences of the United States. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles)

Some Quotations (3)

Mathematicians born in the same country Honours awarded to Stephen Hawking (Click a link below for the full list of mathematicians honoured in this way) Lucasian Professor of Mathematics Other Web sites

1980 1. Bob Bruen 2. Stephen Hawking's home page 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hawking.html (3 of 4) [2/16/2002 11:13:42 PM]

Hawking

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hawking.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hawking.html (4 of 4) [2/16/2002 11:13:42 PM]

Heath

Thomas Little Heath Born: 5 Oct 1861 in Barnetby le Wold, Lincoln, England Died: 16 March 1940 in Ashtead, Surrey, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Heath went up to Trinity College Cambridge in 1879. He was awarded a first class degree in both mathematics and classics. He took the Civil Service examination in 1884 and obtained the top mark. He became a clerk in the Treasury and was quickly promoted. he left the Treasury in 1919 for the National Debt Office, holding a post there until he retired in 1926. However Heath had two separate careers, one as a civil servant, the other as one of the leading world experts on the history of mathematics. He was a specialist in the history of Greek mathematics. He wrote articles on 'Pappus' and 'Porisms' for Encyclopaedia Britannica while still an undergraduate. In his first year at the Treasury he wrote an essay on Diophantus and this won him a Cambridge Fellowship. Cayley recommended its publication by Cambridge University Press. In 1896 he published Apollonius of Perga, in 1897 Archimedes and in 1908 the three volume work on Euclid. Heath's translation of Euclid has since become the standard English version of the text. Heath was elected a Fellow of the Royal Society in 1912. Perhaps his most famous work History of Greek Mathematics appeared in 1921. Article by: J J O'Connor and E F Robertson List of References (4 books/articles)

Some Quotations (2)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heath.html (1 of 2) [2/16/2002 11:13:43 PM]

Heath

Mathematicians born in the same country Cross-references to History Topics

1. Squaring the circle 2. Doubling the cube 3. Trisecting an angle 4. How do we know about Greek mathematicians? 5. How do we know about Greek mathematics?

Honours awarded to Thomas L Heath (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1912

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heath.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heath.html (2 of 2) [2/16/2002 11:13:43 PM]

Heaviside

Oliver Heaviside Born: 18 May 1850 in Camden Town, London, England Died: 3 Feb 1925 in Torquay, Devon, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oliver Heaviside caught scarlet fever when he was a young child and this affected his hearing. This was to have a major effect on his life making his childhood unhappy with relations between himself and other children difficult. However his school results were rather good and in 1865 he was placed fifth from 500 pupils. Academic subjects seemed to hold little attraction for Heaviside however and at age 16 he left school. Perhaps he was more disillusioned with school than with learning since he continued to study after leaving school, in particular he learnt Morse code, studied electricity and studied further languages in particular Danish and German. He was aiming at a career as a telegrapher and in this he was advised and helped by his uncle Charles Wheatstone (the piece of electrical apparatus the Wheatstone bridge is named after him). In 1868 Heaviside went to Denmark and became a telegrapher. He progressed quickly in his profession and returned to England in 1871 to take up a post in Newcastle upon Tyne in the office of Great Northern Telegraph Company which dealt with overseas traffic. Heaviside became increasingly deaf but he worked on his own researches into electricity. While still working as chief operator in Newcastle he began to publish papers on electricity, the first in 1872 and then the second in 1873 was of sufficient interest to Maxwell that he mentioned the results in the second edition of his Treatise on Electricity and Magnetism. Maxwell's treatise fascinated Heaviside and he gave up his job as a telegrapher and devoted his time to the study of the work. He later wrote:I saw that it was great, greater, and greatest, with prodigious possibilities in its power. I

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heaviside.html (1 of 4) [2/16/2002 11:13:45 PM]

Heaviside

was determined to master the book... It took me several years before I could understand as much as I possible could. Then I set Maxwell aside and followed my own course. And I progressed much more quickly. Although his interest and understanding of this work was deep, Heaviside was not interested in rigour. His poorest subject at school had been the study of Euclid, a topic in which the emphasis was on rigorous proof, an idea strongly disliked by Heaviside who later wrote:It is shocking that young people should be addling their brains over mere logical subtleties, trying to understand the proof of one obvious fact in terms of something equally ... obvious. Despite this hatred of rigour, Heaviside was able to greatly simplify Maxwell's 20 equations in 20 variables, replacing them by two equations in two variables. Today we call these 'Maxwell's equations' forgetting that they are in fact 'Heaviside's equations'. FitzGerald wrote:Maxwell's treatise is cumbered with the debris of his brilliant lines of assault, of his entrenched camps, of his battles. Oliver Heaviside has cleared these away, has opened up a direct route, has made a broad road, and has explored a considerable trace of country. Heaviside results on electromagnetism, impressive as they were, were overshadowed by the important methods in vector analysis which he developed in his investigations. His operational calculus, developed between 1880 and 1887, caused much controversy however. Heaviside introduced his operational calculus to enable him to solve the ordinary differential equations which came out of the theory of electrical circuits. He replaced the differential operator d/dx by a variable p transforming a differential equation into an algebraic equation. The solution of the algebraic equation could be transformed back using conversion tables to give the solution of the original differential equation. Although highly successful in obtaining the answer, the correctness of Heaviside's calculus was not proved until Bromwich's work. Burnside rejected one of Heaviside's papers on the operational calculus, which he had submitted to the Proceedings of the Royal Society, on the grounds that it contained errors of substance and had irredeemable inadequacies in proof. Tait championed quaternions against the vector methods of Heaviside and Gibbs and sent frequent letters to Nature attacking Heaviside's methods. Heaviside went on to achieved further advances in knowledge, again receiving less than his just deserts. In 1887 Preece, a GPO technical expert, wrote a paper on clear telephone circuits. His paper is in error and Heaviside pointed this out in Electromagnetic induction and its propagation published in the Electrician on 3 June 1887. Heaviside, never one to avoid controversy, wrote:Sir W Thomson's theory of the submarine cable is a splendid thing. .. Mr Preece is much to be congratulated upon having assisted at the experiments upon which (so he tells us) Sir W Thomson based his theory; he should therefore have an unusually complete knowledge of it. But the theory of the eminent scientist does not resemble that of the eminent scienticulist, save remotely. In this paper Heaviside gave, for the first time, the conditions necessary to transmit a signal without distortion. His idea for an induction coil to increase induction was never likely to be taken up by the GPO while Preece was in charge of research proposals. Heaviside dropped the idea but it was patented in 1904 http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heaviside.html (2 of 4) [2/16/2002 11:13:45 PM]

Heaviside

in the United States. Michael Pupin of Columbia University and George Campbell of AT&T both read Heaviside's papers about using induction coils at intervals along the telephone line. Both Campbell and Pupin applied for a patent which was awarded to Pupin in 1904. Not all went badly for Heaviside however. Thomson, giving his inaugural address in 1889 as President of the Institute of Electrical Engineers, described Heaviside as an authority. Lodge wrote to Nature describing Heaviside as a man whose profound researches into electro-magnetic waves have penetrated further than anyone yet understands. Heaviside was elected a Fellow of the Royal Society in 1891, perhaps the greatest honour he received. Whittaker rated Heaviside's operational calculus as one of the three most important discoveries of the late 19th Century. In 1902 Heaviside predicted that there was an conducting layer in the atmosphere which allowed radio waves to follow the Earth's curvature. This layer in the atmosphere, the Heaviside layer, is named after him. Its existance was proved in 1923 when radio pulses were transmitted vertically upward and the returning pulses from the reflecting layer were received. It would be a mistake to think that the honours that Heaviside received gave him happiness in the last part of his life. On the contrary he seemed to become more and more bitter as the years went by. In 1909 Heaviside moved to Torquay where he showed increasing evidence of a persecution complex. His neighbours related stories of Heaviside as a strange and embittered hermit who replaced his furniture with granite blocks which stood about in the bare rooms like the furnishings of some Neolithic giant. Through those fantastic rooms he wandered, growing dirtier and dirtier, and more and more unkempt - with one exception. His nails were always exquisitely manicured, and painted a glistening cherry pink. Perhaps Heaviside has become more widely known due to the Andrew Lloyd Webber song Journey to the Heaviside Layer in the musical Cats, based on the poems of T S Eliot. Up up up past the Russell hotel Up up up to the Heaviside layer ... although it is doubtful if many people understand the greatness and significance of the achievements of this sad misunderstood genius. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (22 books/articles)

Some Quotations (5)

A Poster of Oliver Heaviside

Mathematicians born in the same country

Other references in MacTutor

An article giving more details of his life in Newton Abbott

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heaviside.html (3 of 4) [2/16/2002 11:13:45 PM]

Heaviside

Honours awarded to Oliver Heaviside (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1891

Lunar features

Crater Heaviside

Planetary features

Crater Heaviside on Mars

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heaviside.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heaviside.html (4 of 4) [2/16/2002 11:13:45 PM]

Heawood

Percy John Heawood Born: 8 Sept 1861 in Newport, Shropshire, England Died: 24 Jan 1955 in Durham, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Percy Heawood attended Queen Elizabeth's Grammar School in Ipswich being awarded an Open Scholarship to study at Oxford in 1880. There Heawood was most influenced by Henry Smith and he went on to be a Wrangler in 1883 (the year in which Henry Smith died). Heawood was awarded a Junior Mathematical Scholarship in 1882 and a Senior Mathematical Scholarship in 1886. In 1886 he was also awarded the Lady Herschell Prize. In 1887 Heawood was appointed Lecturer in Mathematics at Durham Colleges (later Durham University). In fact Heawood worked at Durham University all his life being appointed to the Chair of Mathematics there in 1911. He did not retire until 1939 when he was 78 years of age but still went on to enjoy 16 years of retirement. Heawood spent 60 years of his life working on the four colour theorem. He published his first paper on the topic Map colour theorems in 1890 when he pointed out the mistake in Kempe's proof. In the same paper he proves that five colours suffice. He also proved in this paper that for a surface of connectivity h, a map requires at most [7/2 + ((24h - 23)1/2)/2] colours. Another result from the same paper considered empires, countries which have a number of colonies which must all be coloured with the same colour. Heawood proved that if an empire contains at most r disjoint portions then on a surface of connectivity h, a map can be coloured with at most (6r + 1 + (24h +(6r + 1)2 - 72)1/2)/2

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heawood.html (1 of 3) [2/16/2002 11:13:47 PM]

Heawood

colours. He gave a map with 12 countries, each in two disjoint parts, which required 12 colours. Heawood wrote on the four colour problem again in 1897, 1932, 1936, 1943, 1944 and his final paper on the topic in 1949 was given the same title Map colour theorems as his first paper. Other topics which Heawood wrote on were continued fractions, approximation theory, and quadratic residues. He also wrote five papers and 23 notes for the Mathematical Gazette on a variety of mathematical topics but perhaps more on geometry than any other topic. Dirac, in [1], describes Heawood's appearance and character in the following words:In his appearance, manners and habits of thought, Heawood was an extravagantly unusual man. He had an immense moustache and a meagre, slightly stooping figure. He usually wore an Inverness cape of strange pattern and manifest antiquity, and carried an ancient handbag. His walk was delicate and hasty, and he was often accompanied by a dog, which was admitted to his lectures. ... His transparent sincerity, piety and goodness of heart, and his eccentricity and extraordinary blend of naiveté and shrewdness secured for him not only the fascinated interest, but also the regard and respect of his colleagues. Heawood had one passion outside mathematics and university life. In 1928 Durham Castle was found to be insecure with the foundations moving on the cliff on which it was built. Very large amounts of money were required to save the castle and the University of Durham failed in its attempt to raise the necessary money. Heawood however refused to give up and for years he worked almost on his own as Secretary of the Durham Castle Restoration Fund to raise the cash. Without Heawood's efforts Durham Castle would not be standing today. He succeeded against all the odds to raise the money to ensure that the foundations were permanently secured. He received the O.B.E. in 1939 for this work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Percy J Heawood

Mathematicians born in the same country

Cross-references to History Topics

The four colour theorem

Other references in MacTutor

Chronology: 1890 to 1900

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heawood.html (2 of 3) [2/16/2002 11:13:47 PM]

Heawood

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heawood.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heawood.html (3 of 3) [2/16/2002 11:13:47 PM]

Hecht

Daniel Friedrich Hecht Born: 8 July 1777 in Sosa (near Eibenstock), Saxony (now Germany) Died: 13 March 1833 in Freiberg, Saxony (now Germany) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Little is known of Daniel Hecht before the age of 26 when he enrolled at the Bergakademie at Freiberg, Saxony. It is reasonable to assume that before this Hecht worked in the mining industry in some capacity. After his studies Hecht took up the position of a mine manager, then he became a teacher at the Freiberger Berschule. Hecht was appointed second professor of mathematics at the Freiberg Berakademie in 1816, becoming the first professor there in 1826. At first he taught pure and applied mathematics but later he taught only mechanics and mining machinery. He is best known for the high school texts which he wrote. These widely used school texts were on mathematics, geometry and surveying. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hecht.html (1 of 2) [2/16/2002 11:13:48 PM]

Hecht

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hecht.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hecht.html (2 of 2) [2/16/2002 11:13:48 PM]

Hecke

Erich Hecke Born: 20 Sept 1887 in Buk, Posen, Germany (now Poznan, Poland) Died: 13 Feb 1947 in Copenhagen, Denmark

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Erich Hecke's father, Heinrich Hecke, was an architect. Erich attended primary school at Buk before going to Posen for his secondary school studies. He graduated from secondary school in 1905 and in that year he entered the University of Breslau. Following the tradition in Germany at that time Hecke did not complete his studies at a single university, but moved to several different universities during the course of his studies. After Breslau he worked under Landau at Berlin and then from there he went to Göttingen where he worked under Hilbert. Hecke was awarded his doctorate by Göttingen in 1910 for a dissertation Zur Theorie der Modulfunktionen von zwei Variablen und ihrer Anwendund auf die Zahlentheorie which had been supervised by Hilbert. Hecke remained at Göttingen where he was appointed as an assistant to Hilbert and Klein. He submitted his habilitation to the University of Göttingen in 1912 and earned the right to teach there, which indeed he did as a Privatdozent. In 1915 Hecke was appointed to a chair at the University of Basel but, three years later, he returned to a chair of mathematics at Göttingen. One might have imagined that Hecke would have been happy to stay at Göttingen but, having been there for only one year, he accepted the chair of mathematics at Hamburg in 1919. There was then a rather strange episode regarding chairs of mathematics at both Berlin and Göttingen. Carathéodory, who went from Göttingen to Berlin in 1918 left in the following year and, at the request of the Greek government, he took up a post at the University of Smyrna. This left Berlin trying to fill Carathéodory's chair and Göttingen trying to fill Hecke's chair both in 1919. The two universities drew

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hecke.html (1 of 3) [2/16/2002 11:13:50 PM]

Hecke

up very similar lists of possible mathematicians to fill the posts. Brouwer and Weyl topped the list for the chair at both universities but both turned down the offers they received from each of Göttingen and Berlin. Courant then accepted the offer of Hecke's chair at Göttingen while Berlin, having been refused by their third choice Herglotz, tried to entice Hecke to leave Hamburg and accept the chair at Berlin. Hecke, however, was happy with his new post at Hamburg and turned down the offer from Berlin. He would remain at Hamburg from the rest of his career. The paper [5] is a published version of a talk given by Schoeneberg at a conference organised in Hamburg to celebrate the 100th anniversary of Hecke's birthday in 1987. Schoeneberg describes Hecke's contributions to a number of topics which he lists as follows: Hilbert modular functions, Dedekind zeta functions, arithmetical notions and methods, elliptic modular forms of level N, algebraic functions, Dirichlet series with functional equation, Hecke-operators Tn, and physics where he made contributions to the kinetic theory of gases. Hecke's best work was in analytic number theory where he continued work of Riemann, Dedekind and Heinrich Weber. Complex multiplication and modular forms had been treated in the 19th century by Kronecker and Heinrich Weber, who discovered their link with class field theory. For his doctoral work Hilbert suggested to Hecke that he extend Kronecker's ideas to curves of genus 2. Although Hecke achieved important results following this line of investigation, he considered that his attempts had been unsuccessful. However it was highly successful in the sense that the results which Hecke obtained were to lead him to further major discoveries. Hecke's contributions in number theory are discussed in detail in [2]. He studied the Riemann zeta function, and its extension to arbitrary number fields, discovering important results. He used these results to simplify theorems in class field theory. He then introduced the new concept of "Grossencharakter" and the corresponding L-series, to which he extended the properties of analytic continuation he had proved for the zeta functions in 1917. He discovered the decomposition laws for the divisors of discriminants for class fields. Probably Hecke's most important work was in 1936 with his discovery of the properties of the algebra of Hecke operators and of the Euler products associated with them. He [1]:... discovered new connections between prime numbers and analytic functions and new rules for the representation of natural numbers through positive integral quadratic forms of an even number of variables. Hecke married and had a son who tragically died young. Hecke himself died of cancer before reaching his 60th birthday. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hecke.html (2 of 3) [2/16/2002 11:13:50 PM]

Hecke

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hecke.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hecke.html (3 of 3) [2/16/2002 11:13:50 PM]

Hedrick

Earle Raymond Hedrick Born: 27 Sept 1876 in Union City, Indiana, USA Died: 3 Feb 1943 in Los Angeles, California, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Earle Hedrick attended High School in Ann Arbor from 1891 to 1892 preparing to enter the University of Michigan. He studied there from 1892 to 1896, graduating with a A.B. He then spent a year as a teacher of mathematics at the High School in Sheboygan, Wisconsin before beginning graduate studies at Harvard University in 1897. At Harvard his studies were directed by Bôcher and Osgood among others and he was awarded his Master's degree in 1898. While at Harvard he wrote his first paper which was on three dimensional determinants. He received a fellowship to study abroad and he spent the sessions 1899-00 and 1900-01 at the University of Göttingen in Germany. There he profited greatly by attending lectures by Hilbert, Klein and other exceptional mathematicians. He was awarded a doctorate by Göttingen in February 1901 for a dissertation, supervised by Hilbert, Über den analytischen Charakter der Lösungen von Differentialgleichungen (On the analytic character of solutions of differential equations). Harvard awarded Hedrick a scholarship for a third year to study at the Ecole Normale Supérieure in Paris and there he spent part of 1901 in contact with Goursat, Emile Picard, Hadamard, Appell and Jules Tannery. This strengthen his interests in differential equations, the calculus of variations, and functions of a real variable which he would work on for the rest of his life. It also led to Hedrick translating Goursat's Cours d'Analyse into English which provided an important text for students. After returning to the United States, Hedrick was appointed an instructor in mathematics at Yale University, a post he held from 1901 to 1903. In 1903 he was appointed professor of mathematics at the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hedrick.html (1 of 3) [2/16/2002 11:13:51 PM]

Hedrick

University of Missouri and remained there until 1920 when he was appointed professor of mathematics, and head of department, at the University of California at Los Angeles. In 1937 he became the provost of the University of California. In addition to his research in pure mathematics, Hedrick was also interested in applications of mathematics and he wrote papers on a generalised form of Hooke's law and the transmission of heat in boilers. He became an active member of the Society for the Promotion of Electrical Engineering, the American Society of Mechanical Engineering and the American Institute of Electrical Engineers. Hedrick's editorial work, however, was extraordinary. He was editor of the American Mathematical Monthly from 1913 to 1915, editor-in-chief of the Bulletin of the American Mathematical Society from 1921 to 1937, he was editor of 34 volumes in the Engineering Science Series and 35 volumes in the Series of Mathematical Texts. In addition to his editorial work for the American Mathematical Society he was vice-president in 1916 and president in 1929-30. Archibald relates that his main hobby is carrying out:... long-continued experiments in crossing varieties of flowers to produce new types. Hedrick is buried in Glendale City, Los Angeles county, California in the Forest Lawn Memorial Park which is a cemetery famous for its elaborate statues and reproductions of famous shrines and works of art. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Earle Hedrick (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1929 - 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hedrick.html (2 of 3) [2/16/2002 11:13:51 PM]

Hedrick

JOC/EFR October 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hedrick.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hedrick.html (3 of 3) [2/16/2002 11:13:51 PM]

Heegaard

Poul Heegaard Born: 2 Nov 1871 in Copenhagen, Denmark Died: 7 Feb 1948 in Oslo, Norway

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Poul Heegaard's name occurs frequently (quite often misspelled as Heegard or even Hegard) in the area of three-manifolds where 'Heegaard decompositions' and the associated 'Heegaard diagrams' remain important tools 100 years after they first occurred in Heegaard's 1898 Copenhagen University dissertation. Even though the dissertation is in Danish, it quickly became internationally well known, mainly because it also contains a counter-example to the version of Poincaré duality published by Poincaré shortly before. This counter-example sent Poincaré back to the drawing board and thereby contributed to a clarification of some basic notions of algebraic topology. Another important Heegaard contribution is his 1907 survey article (with Max Dehn) Analysis Situs where the authors set forth the foundations of combinatorial topology. This enables them to give the first rigorous proof of the classification of compact surfaces. After his 1893 M.Sc. degree from Copenhagen University, Poul Heegaard had received a stipend for a year's study abroad. He first went to Paris and it has often been assumed that his interest in Poincaré's work started there. However, in his recently (1997) discovered autobiographical notes (see references), Heegaard indicates that he never met Poincaré and he laments that his visit to Paris was mathematically very disappointing. Therefore, after one semester he moved to Göttingen where his contact with Felix Klein became very influential for his future work. In particular, his interest in topology came from an attempt to study http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heegaard.html (1 of 2) [2/16/2002 11:13:53 PM]

Heegaard

algebraic functions of two complex variables by means of generalized (four dimensional) Riemann surfaces. The study of three dimensional manifolds mentioned above really is there because he has to give up on the four dimensional case. After his dissertation, Heegaard taught at various military schools in the Copenhagen area for more than 10 years. In 1910, he accepted a chair at Copenhagen University, but seven years later he resigned, quoting a heavy work load and disagreements with colleagues as his reasons. Shortly after his resignation, Heegaard received an offer from the University of Kristiania (now Oslo) in Norway. Here, he became a cofounder of the Norwegian Mathematical Society, and a very popular teacher until he retired in 1941. An extensive bibliography for Poul Heegaard is contained in [2]. Article by: H J Munkholm, Odense Denmark Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1900 to 1910

Other Web sites

1. Theseus 2. Odense, Denmark (Including a transcript of reference [1]) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heegaard.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heegaard.html (2 of 2) [2/16/2002 11:13:53 PM]

Heilbronn

Hans Arnold Heilbronn Born: 8 Oct 1908 in Berlin, Germany Died: 28 April 1975 in Toronto, Canada

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hans Heilbronn was born into a middle class Jewish-German family. When he was six years old he entered the Realgymnasium Berlin-Schmargenhof. This school, which he attended for twelve years from 1914 to 1926, provided him with a good background in science and modern languages. During his school years he was attracted towards science and mathematics but when he began his university studies in 1926 he still had not made any definite decision to specialise in mathematics. He entered the University of Berlin taking courses from physics, chemistry and mathematics but soon he was specialising more and more in mathematics. As was the custom of German students at this time, Heilbronn moved from one university to another as he progressed through his studies. From Berlin he moved to Freiburg, then he arrived at Göttingen which was the leading mathematics research centre in the world at that time. There Heilbronn began to undertake research in number theory, his studies being directed by Edmund Landau. By all accounts Heilbronn seems to have participated in the typical student life of that time. In [3] it is noted that:It was only rebels who abhorred duelling - Heilbronn carried a duelling scar from his Göttingen days throughout his life. In 1930 Heilbronn was appointed as Landau's assistant and he continued to work for his doctorate. The problem which Heilbronn worked on for his doctorate was related to a conjecture made by Bertrand in 1845. Bertrand conjectured that there was always at least one prime between x and 2x for x > 1. Chebyshev proved Bertrand's conjecture in 1850 and then in 1930 Hoheisel proved that there exists a t < 1 such that for all large x, there is a prime p between x and x + xt. Heilbronn found a simpler proof to that http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heilbronn.html (1 of 5) [2/16/2002 11:13:55 PM]

Heilbronn

given by Hoheisel and he also proved a stronger result by giving a smaller value of t. In his thesis Heilbronn also applied his result to primes in an arithmetic progression and to estimates of the sum of the Möbius function. This thesis earned Heilbronn his doctorate in 1933 and by the end of that year he had six publications, some of them joint publications with Landau. These joint publications looked at the Tauberian theorems, giving simpler proof and also giving applications of the theorems. The authors of [3] describe Heilbronn's relations with Landau:By all accounts Landau was a rather formidable man and a demanding task master. Heilbronn seems, however, to have managed very well. Landau came to have a very high opinion of him and, judging by letters he wrote on his behalf in 1933, considered him his star pupil. On the other hand, those years with Landau were formative ones for Heilbronn. Throughout his life his principal research interest remained with number theory, largely analytic. In the shorter term, Landau also clearly exerted an influence on Heilbronn's mathematical style ... It was not only Landau who formed a high opinion of Heilbronn. Davenport visited Göttingen for the summer months of 1933 and was very impressed by him. This was a piece of luck for Heilbronn who realised that he was in an increasingly difficult position in Germany. On 30 January 1933 Hitler had come to power and on 7 April 1933 the Civil Service Law had been passed which provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Heilbronn was dismissed from his position as an Assistant at Göttingen and fled to England, arriving in Cambridge with enough money to support him for around six months. Of course Heilbronn was not the only German scientist in this position and an Academic Assistance Council had been set up in Britain to deal with such cases. However, it did not have the necessary funds to help support the academics who were arriving in Britain but nevertheless Heilbronn wrote to them for help explaining:There is no possibility for me to continue scientific work in Germany. The Academic Assistance Council had already been informed about Heilbronn before he fled from Germany since Davenport had written to them in August 1933, after returning from Göttingen, and Hardy had also written in October of that year. Mordell arranged accommodation for Heilbronn in Manchester and asked the Academic Assistance Council if they could provide a small salary. However, the Academic Assistance Council had no funds. Even when Hassé, the Head of Department in Bristol, wrote saying that he had partial support offered by the Jewish community around Bristol to support Heilbronn, the Academic Assistance Council could not make up the difference needed. Members of the Mathematics department in Bristol then agreed to help out and an invitation went to Heilbronn in the middle of December offering him a temporary post at Bristol. Heilbronn gladly accepted the offer from the University of Bristol and he spent about eighteen months there. This was a very productive time for Heilbronn who published what turned out to be his most famous result during this time. While at Göttingen, Heilbronn had begun a collaboration with E H Linfoot and two papers which Heilbronn wrote soon after arriving in Bristol came out of this

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heilbronn.html (2 of 5) [2/16/2002 11:13:55 PM]

Heilbronn

collaboration. One paper appears under Heilbronn's name alone, the other under the joint authorship of Heilbronn and Linfoot. The first of the two papers proved a conjecture of Gauss on imaginary quadratic number fields using ideas of Hecke, Deuring and Mordell. Heilbronn proved the conjecture which asserts that the class number of the quadratic number field Q( -d) tends to infinity as d tends to infinity. This was generalised by Siegel and Brauer. In the joint paper with Linfoot, Heilbronn proved that there are at most ten quadratic number fields Q( -d) of class number one. Nine such fields, namely those with d = 3, 4, 7, 8, 11, 19, 43, 67, and 163, had essentially been known since Euler's time. It is worth noting that Heilbronn and Linfoot remained friends throughout their lives but their interests diverged with Linfoot's interests turning from number theory to research in optics and astronomy. Of course these important results helped Heilbronn obtain further temporary posts in Britain. He moved from Bristol to Manchester when Mordell offered him a two year scholarship near the end of 1934. His stay in Manchester was short, however, since he was offered the Bevan Fellowship in Trinity College, Cambridge in May 1935. He accepted this offer, mainly made through Hardy's efforts, with pleasure and moved to Cambridge. There he bought a house and his parents and sister came over from Germany to live with him. Patrick du Val give a good description of Heilbronn at this time:I met him when he first came to Cambridge ... He was I think rather shy, and not easy to get to know quickly. My mother lived in Cambridge then, and he went to tea with her once while I was abroad ...; she wrote to me quoting Dr Johnson, "His intellect is as exalted as his stature; I am half afraid of him, yet he is no less amiable than formidable". This seems to me to give a very good impression of him in those days. After he got his parents and sister out of Germany, and settled them in a house in Chesterton Road, we became very friendly; I was often at their house, and they used to come to ours. His mother was very musical, collected a circle of refugee musicians, and used often to give musical parties; she herself was an excellent pianist. I was at many of these parties; they rather bored Hans, who had no interest in music, but he was present at many of them, as a very correct and attentive host. After Heilbronn moved to Cambridge he began a collaboration with Davenport which started in 1936 and lasted until Davenport's death in 1969, with Heilbronn publishing further joint papers with Davenport up to 1971. While at Cambridge he published four papers with Davenport, and one paper on his own, on Waring's problem. These deal with writing integers as sums of fourth powers, on sums of two cubes and a square, and on the sum of a prime and a k-th power. Heilbronn also published results on the Epstein zeta-function showing that the Riemann Hypothesis fails for this zeta function. In April 1939 Heilbronn applied for British citizenship but he was a few days too late to enable the papers to be processed before the start of World War II. As a contribution to the war effort Heilbronn organised the Trinity College A.R.P. Fire Service. However, he was interned the Isle of Man in 1940 as an enemy alien, something which he greatly resented since nobody could have been more opposed to the Nazis than he was. The Academic Assistance Council, Hardy and other made strong representations on his behalf and he was released after many months of detention. He then served with the British forces with the Signals Corps and Military Intelligence until autumn of 1945. The Bevan Fellowship at Trinity would have ended in 1940 and Trinity had already made arrangements

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heilbronn.html (3 of 5) [2/16/2002 11:13:55 PM]

Heilbronn

to continue to pay his salary. However his internment and subsequent military service put an end to this plan. He returned to Bristol in 1946 when offered a Readership there. In 1949 he was promoted to Professor and he became Head of the Department of Mathematics. While at Bristol he worked on the Euclidean algorithm (norm function) in a general setting having already proved while at Cambridge that there are only finitely many real quadratic fields with a Euclidean algorithm. He examined the existence of a norm function for a broad class of cyclic fields. With Davenport he showed that any real indefinite diagonal quadratic form, in 5 or more variables, takes arbitarily small values for nonzero integral arguments. For his outstanding contribution to mathematics, he was elected a Fellow of the Royal Society in 1951. During this period as Head of Department at Bristol Heilbronn was still, according to du Val:... rather stiff and formal in external manner ... I have a feeling he wanted to be popular, but found it uphill work. Heilbronn was very interested in sport, especially rowing and tennis, and while at Bristol he coached the University Boat Club. In many ways Heilbronn was very successful at Bristol, building up a strong department. The authors of [3] write:... many other members of the department, notably the number theoreticians, were stimulated by Heilbronn and helped by his active interest in their research. He was always accessible. His manner was most direct and his intellectual honesty did not allow him to pass over in politeness whatever he felt was below standard. But although some may have found him a bit frightening at first acquaintance, he was a kind and modest man, who entirely lacked pomposity or aggressiveness. The situation in British universities began to worry Heilbronn. The Robbins Report proposed opening up university education to a large proportion of young people. The required rapid expansion greatly worried Heilbronn who felt that it was inevitable that such a rapid expansion would bring staff into universities who were below standard. He argued strongly against the proposals both at national level and within his own university. Realising that it was a battle he could never win, he resigned in 1964 and moved to North America without having a job. Not long before resigning, Heilbronn married Dorothy Greaves who was a widow whom he had known through a shared interest in bridge since his days in Cambridge. After a short stay at the California Institute of Technology at Pasadena as guests of Olga Taussky-Todd, Heilbronn and his wife moved to Toronto where he was appointed to a chair of mathematics at Toronto University. Heilbronn became a Canadian citizen in 1970 and spent the rest of his life in Canada, taking an active part national role in mathematics there. He attended the International Congress of Mathematicians in Nice in 1970 as a Canadian delegate and presented the official invitation to the delegates to attend the 1974 Congress in Vancouver. However, Heilbronn was not able to put as much effort into organising the 1974 Congress as he would have liked since he suffered a heart attack in November 1973. He made a good recovery but had to lead a quieter life. He died while undergoing an operation to fit a pacemaker to his heart. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heilbronn.html (4 of 5) [2/16/2002 11:13:55 PM]

Heilbronn

List of References (6 books/articles) Mathematicians born in the same country Honours awarded to Hans Heilbronn (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1951

London Maths Society President

1959 - 1961

Other Web sites

Heilbronn Triangle Constants

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heilbronn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heilbronn.html (5 of 5) [2/16/2002 11:13:55 PM]

Heine

Heinrich Eduard Heine Born: 16 March 1821 in Berlin, Germany Died: 21 Oct 1881 in Halle, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eduard Heine is important for his contributions to analysis. He went to lectures by Gauss and was taught by Dirichlet. Heine worked on Legendre polynomials, Lamé functions and Bessel functions. He is best remembered for the Heine-Borel theorem:a subset of the reals is compact if and only if it is closed and bounded. Heine also formulated the concept of uniform continuity. The paper [2] analyses letters written by Heine and found in 1988 in the Institut Henri Poincaré. The second part of this paper covers the history of the Heine-Borel theorem and is summarised in the following review:The last half of the paper is devoted to a more systematic account of the gradual discovery and formulation of the so-called Heine-Borel theorem. It begins with the implicit use of the theorem in various proofs of the theorem stating that a continuous function on a closed, bounded interval is uniformly continuous. The first proof of this theorem was given by Dirichlet in his lectures of 1862 (published 1904) before Heine proved it in 1872. Dugac shows that Dirichlet used the idea of a covering and a finite subcovering more explicitly than Heine. This idea was also used by Weierstrass and Pincherle. Borel formulated his theorem for countable coverings in 1895 and Schönflies and Lebesgue generalized it to any type of covering in 1900 and 1898 (published 1904), respectively. Dugac shows that the story is in fact much more complicated and includes names such as Cousin, Thomae, Young, Vieillefond, Lindelöf. The priority questions are nicely illustrated with quotes from the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heine.html (1 of 2) [2/16/2002 11:13:57 PM]

Heine

correspondence between Lebesgue and Borel and other letters. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) A Poster of Eduard Heine

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1870 to 1880

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heine.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heine.html (2 of 2) [2/16/2002 11:13:57 PM]

Heisenberg

Werner Karl Heisenberg Born: 5 Dec 1901 in Würzburg, Germany Died: 1 Feb 1976 in Munich, Germany

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Werner Heisenberg's father was August Heisenberg and his mother was Anna Wecklein. At the time that Werner was born his father was about to progress from being a school teacher of classical languages to being appointed as a Privatdozent at the University of Würzburg. Anna's father, Nikolaus Wecklein, was the headmaster of the Maximilians Gymnasium in Munich and it was while August Heisenberg was a trainee teacher at that school that he had met Anna. August and Anna were married in May 1899. Werner had an older brother Erwin, born in March 1900, who was therefore nearly two years older than his brother. August Heisenberg was [3]:... a rather stiff, tightly controlled, authoritarian figure. He was an Evangelical Lutheran and his wife Anna had converted from being a Roman Catholic to make sure there were no religious problems with their marriage. August and Anna, however, were only religious for the sake of convention. A Christian belief was expected of people of their status so for them it was a social necessity. In private, however, they expressed their lack of religious beliefs, and in particular they brought up their children to follow Christian ethics but showed total disbelief in the historical side of Christianity. In September 1906, shortly before his fifth birthday, Werner enrolled in a primary school in Würzburg. He spent three years at that school but then in 1909 his father was appointed Professor of Middle and Modern Greek at the University of Munich. In June 1910, a few months after his father took up the professorship, Werner and the rest of the family moved to Munich. There he attended the Elisabethenschule school from September, spending only one year at this school before entering the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heisenberg.html (1 of 6) [2/16/2002 11:13:59 PM]

Heisenberg

Maximilians Gymnasium in Munich. This of course was the school where his grandfather was the headmaster. In 1914 World War I began and the Gymnasium was occupied by troops. Lessons were arranged in different buildings and as a result of the disruption Heisenberg undertook much independent studying which probably had a beneficial effect on his education. His best subjects were mathematics, physics and religion but his record throughout his school career was excellent all round. In fact his mathematical abilities were such that in 1917 he tutored a family friend who was at university in calculus. During this period he belonged to a paramilitary organisation which operated in the Gymnasium with the intention of preparing the young men for later military service. Heisenberg also worked on farms as his contribution to another voluntary organisation which sent the boys to help in the fields in spring and summer. This work took him away from home for the first time in 1918 when he was sent to work on a dairy farm in Upper Bavaria. It was a time of great hardship with long hours of labour made worse since there was insufficient food. He spent his spare time playing chess, which he did to a very high standard, and also read mathematics texts he had taken with him. In fact by this time he had become interested in number theory and he read Kronrcker's work and tried to find a proof of Fermat's Last Theorem. After the war ended in 1918 the situation in Germany became unstable with different factions trying to take power by force. Heisenberg took part in the military suppression of the Bavarian Soviet forces but, although it was a very serious business, the young men probably almost treated it almost as a game. He later wrote [4]:I was a boy of 17 and I considered it a kind of adventure. it was like playing cops and robbers ... In the Gymnasium Heisenberg led a youth movement and he later led a movement within the Young Bavarian League. In 1920 he took his Abitur examination and was one of two pupils entered from the Maximilians Gymnasium for a Bavarian wide competition for a scholarship from the Maximilianeum Foundation. Eleven scholarships were available and Heisenberg just made it by coming in eleventh place. His examination results in mathematics and physics were classed as extraordinary, but his essay on "tragedy as poetic art" was much less impressive. He declined the offer of free accommodation from the Foundation, preferring to live with his parents. In the period between taking his Abitur examination and entering the University of Munich Heisenberg went off hiking with his youth group. He nearly died of typhoid which he contracted after spending the night in a castle which had been used as a military hospital. He recovered, despite the problems of obtaining suitable food, in time to begin his university studies. During the summer of 1920 Heisenberg was, as he had been for some time, intending to study pure mathematics at university. He had read Weyl and also Bachmann's text which gave a complete survey of number theory and this was to be his intended research topic for his doctorate. He approached Ferdinand von Lindemann to see if he would be his research supervisor. Had the interview with Lindemann been a success then Heisenberg might today be known as an outstanding number theorist. However, the interview did not go well, almost certainly since Lindemann was only two years off retiring and had only agreed to see Heisenberg as a favour to his father who was a friend and colleague. Following this Heisenberg had an interview with Sommerfeld who happily

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heisenberg.html (2 of 6) [2/16/2002 11:13:59 PM]

Heisenberg

accepted him as a student. With his fellow student Pauli, Heisenberg began to study theoretical physics under Sommerfeld in October 1920. At first he was cautious, taking mostly mathematics classes and making sure that he could revert to mathematics if the theoretical physics went badly. He avoided courses by Lindemann, however, so his mathematical interests moved from number theory to geometry. Soon his confidence in theoretical physics was such that by the second semester he was taking all of Sommerfeld's courses. He also took courses in experimental physics, which were compulsory, and he began to plan to undertake research in relativity. However Pauli, who was at that time working on his major survey of the theory of relativity, advised him against doing research in that topic. On atomic structure, however, Pauli explained, much needed to be done since theory and experiment did not agree. In [6] Heisenberg wrote of his early days at university:My first two years at Munich University were spent in two quite different worlds: among my friends of the youth movement and in the abstract realm of theoretical physics. Both worlds were so filled with intense activity that I was often in a state of great agitation, the more so as I found it rather difficult to shuttle between the two. In June 1922 he attended lectures by Niels Bohr in Göttingen. Returning to Munich, Sommerfeld gave him a problem in hydrodynamics to keep him busy while he (Sommerfeld) spent session 1922-23 in the United States. Heisenberg presented preliminary results on the problem on turbulence at a conference in Innsbruck before going again to Göttingen to study with Born, Franck, and Hilbert while his supervisor was away. There he worked with Born on atomic theory, writing a joint paper with him on helium. His doctoral dissertation, presented to Munich in 1923, was on turbulence in fluid streams. After taking his doctorate Heisenberg went on a trip to Finland then, in October 1923, he returned to Göttingen as Born's assistant. In March 1924 he visited Niels Bohr at the Institute for Theoretical Physics in Copenhagen where he met Einstein for the first time. Returning again to Göttingen he delivered his habilitation lecture on 28 July 1924 and qualified to teach in German universities. Heisenberg later wrote:I learned optimism from Sommerfeld, mathematics at Göttingen, and physics from Bohr. From September 1924 until May 1925 he worked, with the support of a Rockefeller grant, with Niels Bohr at the University of Copenhagen, returning for the summer of 1925 to Göttingen. Heisenberg invented matrix mechanics, the first version of quantum mechanics, in 1925. He did not invent these concepts as a matrix algebra, however, rather he focused attention on a set of quantised probability amplitudes. These amplitudes formed a non-commutative algebra. It was Max Born and Pascual Jordan in Göttingen who recognised this non-commutative algebra to be a matrix algebra. Matrix mechanics was further developed in a three author paper by Heisenberg, Born and Jordan published in 1926. In May 1926 Heisenberg was appointed Lecturer in Theoretical Physics in Copenhagen where he worked with Niels Bohr. In 1927 Heisenberg was appointed to a chair at the University of Leipzig and he delivered his inaugural lecture on 1 February 1928. He was to hold this post until, in 1941, he was made director of the Kaiser Wilhelm Institute for Physics in Berlin. In 1932 he was awarded the Nobel Prize in physics for:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heisenberg.html (3 of 6) [2/16/2002 11:13:59 PM]

Heisenberg

The creation of quantum mechanics, the application of which has led, among other things, to the discovery of the allotropic forms of hydrogen. In the presentation speech H Pleijel said:Heisenberg ... viewed his problem, from the very beginning, from so broad an angle that it took care of systems of electrons, atoms, and molecules. According to Heisenberg one must start from such physical quantities as permit of direct observation, and the task consists of finding the laws which link these quantities together. The quantities first of all to be considered are the frequencies and intensities of the lines in the spectra of atoms and molecules. Heisenberg now considered the combination of all the oscillations of such a spectrum as one system, for the mathematical handling of which, he set out certain symbolical rules of calculation. It had formerly been determined already that certain kinds of motions within the atom must be viewed as independent from one another to a certain degree, in the same way that a specific difference is made in classical mechanics between parallel motion and rotational motion. It should be mentioned in this connection that in order to explain the properties of a spectrum it had been necessary to assume self-rotation of the positive nuclei and the electrons. These different kinds of motion for atoms and molecules produce different systems in Heisenberg's quantum mechanics. As the fundamental factor of Heisenberg's theory can be put forward the rule set out by him with reference to the relationship between the position coordinate and the velocity of an electron, by which rule Planck's constant is introduced into the quantum-mechanics calculations as a determining factor. ... Heisenberg's quantum mechanics has been applied by himself and others to the study of the properties of the spectra of atoms and molecules, and has yielded results which agree with experimental research. It can be said that Heisenberg's quantum mechanics has made possible a systemization of spectra of atoms. It should also be mentioned that Heisenberg, when he applied his theory to molecules consisting of two similar atoms, found among other things that the hydrogen molecule must exist in two different forms which should appear in some given ratio to each other. This prediction of Heisenberg's was later also experimentally confirmed. Heisenberg is perhaps best known for the Uncertainty Principle, discovered in 1927, which states that determining the position and momentum of a particle necessarily contains errors the product of which cannot be less than the quantum constant h. These errors are negligible in general but become critical when studying the very small such as the atom. It was in 1927 that Heisenberg attended the Solvay Conference in Brussels. He wrote in 1969:To those of us who participated in the development of atomic theory, the five years following the Solvay Conference in Brussels in 1927 looked so wonderful that we often spoke of them as the golden age of atomic physics. The great obstacles that had occupied all our efforts in the preceding years had been cleared out of the way, the gate to an entirely new field, the quantum mechanics of the atomic shells stood wide open, and fresh fruits seemed ready for the picking. Heisenberg published The Physical Principles of Quantum Theory in 1928. In 1929 he went on a lecture tour to the United States, Japan, and India. In the 1930s Heisenberg and Pauli used a quantised realisation of space in their lattice calculations. Heisenberg hoped this mathematical property would lead to a fundamental property of nature with a 'fundamental length' as one of the constants of nature.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heisenberg.html (4 of 6) [2/16/2002 11:13:59 PM]

Heisenberg

In 1932 Heisenberg wrote a three part paper which describes the modern picture of the nucleus of an atom. He treated the structure of the various nuclear components discussing their binding energies and their stability. These papers opened the way for others to apply quantum theory to the atomic nucleus. In 1935 the Nazis brought in a law whereby professors over 65 had to retire. Sommerfeld was 66 and he had already indicated that he wanted Heisenberg to succeed him. It was an appointment which Heisenberg badly wanted and in 1935 Sommerfeld again indicated that he wanted Heisenberg to fill his chair. However this was the period when the Nazis wanted "German mathematics" to replace "Jewish mathematics" and "German physics" to replace "Jewish physics". Relativity and quantum theory were classed as "Jewish" and as a consequence Heisenberg's appointment to Munich was blocked. Although he was in no way Jewish, Heisenberg was subjected to frequent attacks in the press describing him to be of "Jewish style". In 1937 Heisenberg married Elisabeth Schumacher. He met her through his music which was important to him throughout his life. An excellent pianist, Heisenberg met Elisabeth Schumacher at a concert in which he was performing at the house of a published friend. Elizabeth was only 22 when they met, Heisenberg was 35. They were married on 29 April 1937, less than three months after they first met. Heisenberg had been asked to take up the appointment at Munich in March but had asked for the date to be delayed until August because of his wedding. It was agreed that he should take up the appointment on 1 August. He and his wife arrived in Munich in July but his appointment was blocked by the Nazis. During the Second World War Heisenberg headed the unsuccessful German nuclear weapons project. He worked with Otto Hahn, one of the discoverers of nuclear fission, on the development of a nuclear reactor but failed to develop an effective program for nuclear weapons. Whether this was because of lack of resources or a lack of a desire to put nuclear weapons in the hands of the Nazis, it is unclear. After the war he was interned in Britain with other leading German scientists. However he returned to Germany in 1946 when he was appointed director of the Max Planck Institute for Physics and Astrophysics at Göttingen. In the winter of 1955-1956 he gave the Gifford Lectures "On physics and philosophy" at the University of St Andrews. When the Max Planck Institute moved to Munich in 1958 Heisenberg continued as its director. He held this post until he resigned in 1970. He was also interested in the philosophy of physics and wrote Physics and Philosophy (1962) and Physics and Beyond (1971). Heisenberg received many honours for his remarkable contributions in addition to the Nobel Prize for Physics. He was elected a Fellow of the Royal Society of London, and was a member of the academies of Göttingen, Bavaria, Saxony, Prussia, Sweden, Rumania, Norway, Spain, The Netherlands, Rome, the Akademie der Naturforscher Leopoldina, the Accademia dei Lincei, and the American Academy of Sciences. Among the prizes he received was the Copernicus prize. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles)

Some Quotations (4)

A Poster of Werner Heisenberg

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heisenberg.html (5 of 6) [2/16/2002 11:13:59 PM]

Heisenberg

Cross-references to History Topics

The quantum age begins

Honours awarded to Werner Heisenberg (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1932

Fellow of the Royal Society

Elected 1955

Other Web sites

1. Nobel prizes site (A biography of Heisenberg and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heisenberg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heisenberg.html (6 of 6) [2/16/2002 11:13:59 PM]

Hellinger

Ernst David Hellinger Born: 30 Sept 1883 in Striegau (now Strzegom, Poland), Silesia, Germany Died: 28 March 1950 in Chicago, Illinois, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernst Hellinger's parents were Emil and Julie Hellinger and the fact that the family was Jewish meant that he would have major problems after the Nazis came to power. Hanna, a sister of Ernst, later became Hanna Meissner and wrote the article [3] about her brother. Ernst grew up in Breslau where he attended school, graduating from the Gymnasium there in 1902. It was at the Breslau Gymnasium that Hellinger first became fascinated in mathematics, and this was the result of an excellent mathematics teacher at the school. Hellinger entered the University of Heidelberg but, following the tradition in Germany at that time, he did not complete his studies at a single university but moved to several different universities during the course of his studies. His second university was Breslau, so he returned to the town where he was brought up, and then in 1904 he followed his friend Max Born to Göttingen. Hellinger would keep in touch with Born and developments in quantum mechanics for much of his life. In Göttingen Hellinger was a student of Hilbert and, not long after he began his studies there, he was joined by Courant and Toeplitz who had been his fellow students at Breslau. Hellinger was awarded his doctorate by the University of Göttingen in 1907 for a thesis entitled Die Orthogonalinvarianten quadratischer Formen von unendlichvielen Variablen. He introduced a new type of integral, the Hellinger integral in his doctoral thesis and, jointly with Hilbert, he produced the important Hilbert-Hellinger theory of forms. Then from 1907 to 1909 he was an assistant at Göttingen and, during this time, he [1]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hellinger.html (1 of 4) [2/16/2002 11:14:01 PM]

Hellinger

... edited Hilbert's lecture notes and Felix Klein's influential Elementarmathematik vom höheren Standpunkte aus (Berlin, 1925) which was translated into English (New York, 1932). From Göttingen Hellinger went to Marburg where he was a Privatdozent from 1909 to 1914. He was then appointed to a chair at the new university of Frankfurt am Main. He was not the only appointment to Frankfurt in 1914 for Szasz was appointed a Privatdozent in that year, later to be promoted to professor. Bieberbach was also at Frankfurt in this early period, although he left after World War I to take up a chair at Berlin. Of course 1914 marked the start of World War I and Hellinger was involved in war service. However, after the end of the war, there were a number of further important appointments to Frankfurt which built up an impressive mathematics department there. Epstein was appointed in 1919, Dehn in 1921 and Siegel in 1922. Others such as Toeplitz were frequent visitors to the Frankfurt Mathematics Seminar. This Seminar is described in [2] and [6] which both concentrate on the period from 1922 to the difficult years of the 1930s. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. However, there was an exemption clause which exempted non-Aryans who had fought for Germany in World War I. Hellinger certainly qualified under this clause and this allowed him to keep his lecturing post in Frankfurt in 1933. Hellinger, however, was forced to retire in 1936 because by this stage the rules that non-Aryans who served in World War I were allowed to keep their posts was being ignored after decisions at the Nuremberg party congress in the autumn of 1935. Hellinger continued to live in Frankfurt. On the Kristallnacht (so called because of the broken glass in the streets on the following morning), the 9-10 November 1938, 91 Jews were murdered, hundreds were seriously injured, and thousands were subjected to horrifying experiences. Thousands of Jewish businesses were burnt down together with over 150 synagogues. The Gestapo arrested 30,000 well-off Jews and a condition of their release was that they emigrate. The Gestapo did not arrest Hellinger that night because there was nowhere else to put prisoners but [6]:he refused to flee ... because he wanted to stay and see just how far beyond the traditional standards of justice and ethics the authorities would go in his case. On 13 November 1938 he found out how far the authorities would go. He was arrested, first taken to the Festhalle and then put into Dachau concentration camp. By this time his sister, Hanna Meissner, was in the United States and she describes in [3] how Siegel wrote to her to say that Hellinger had been sent to the concentration camp. Fortunately friends were able to arrange a temporary job for Hellinger at Northwestern University at Evanston in the United States. He was released from the Dachau camp after six weeks on condition that he emigrate immediately. Siegel writes [6]:I saw Hellinger in Frankfurt a few days after his release. He looked emaciated from the utterly insufficient diet at the camp, but maintained a strong will to live as a result of his impending emigration. He refused to discuss his horrifying experiences and was never able to forget the humiliation that was done him. He emigrated to the United States in late February 1939.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hellinger.html (2 of 4) [2/16/2002 11:14:01 PM]

Hellinger

Hellinger's position at Evanston throughout the war was precarious with a series of one-year appointments but he acquired American citizenship in 1944 and worked at Evanston until 1949 when he retired. Of course after just a few years work he would not have received a pension to enable him to live and he had constant financial worries.After retirement he accepted a post at Illinois Institute of Technology but he fell ill with cancer in November 1949 and died a few months later. At Frankfurt, Hellinger had continued his mathematical work on the spectral theory of Jacobi forms and continued fractions. He did important work on Stieltjes' moment problem. With Toeplitz he wrote a monumental survey of the literature on integral equations up to 1923 for Klein's Encyclopadie der Mathematischen Wissenschaften. This article, Integralgleichungen und Gleichungen mit unendlichvielen Unbekannten, was published in the Encyclopedia in 1927 and then as a separate article the following year. It is now considered one of the great classics and has been reprinted several times. Hellinger also worked on the history of mathematics and, while at Frankfurt, he wrote an important paper on James Gregory. Hellinger was a gifted teacher who concerned himself deeply with the students who he taught [6]:He always took the welfare of the students to heart even outside his lectures and study groups. The letter [7] is written by two of Hellinger's students and praises highly his teaching prowess:At Northwestern he was one of the most liked professors in the Mathematics Department and was recognised by all for the excellence, skill, and understandability of his lectures and especially his proofs. He had an odd sense of humour but a strong sense of loyalty and some equally strong dislikes. He had a gift for making friends. In [7] a story shows his sense of humour:Shortly after his arrival at Northwestern, one of the professors in describing Northwest's mathematics program to him remarked that in the honours course Felix Klein's 'Elementary mathematics from an advanced standpoint' was used as a text and "perhaps Hellinger was familiar with it". At this Hellinger ... replied "familiar with it, I wrote it!". Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) A Poster of Ernst Hellinger

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hellinger.html (3 of 4) [2/16/2002 11:14:01 PM]

Hellinger

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hellinger.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hellinger.html (4 of 4) [2/16/2002 11:14:01 PM]

Helly

Eduard Helly Born: 1 June 1884 in Vienna, Austria Died: 28 Nov 1943 in Chicago, Illinois, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eduard Helly came from a Jewish family in Vienna. He studied at the University of Vienna and was awarded his doctorate in 1907 after writing a thesis under the direction of Wirtinger and Mertens. His thesis was on Fredholm equations. Wirtinger arranged a scholarship for Helly so that he could continue his studies at Göttingen and he went there after graduating from Vienna. At Göttingen Helly studied under Hilbert, Klein, Minkowski and Runge in 1907-8. Returning to Vienna in 1908 he had no university post, but supported himself in a number of different ways. He taught in a Gymnasium, gave private tuition, and wrote solution manuals for a series of standard textbooks. During this period, he undertook research on functional analysis and proved the Hahn-Banach theorem in 1912, fifteen years before Hahn published essentially the same proof and 20 years before Banach gave his new setting. We discuss this more fully later in this article. On the outbreak of World War I, Helly enlisted in the army. While serving as a lieutenant in September 1915 he was shot. The bullet went through his lung and did damage to his health from which he never recovered throughout the rest of his life. After being shot he was captured by the Russians. After this he spent years in hospitals and prisoner of war camps in Siberia. One might have expected that the end of World War 1 in 1918 would have led to Helly's release but by this time the Russian armies were fighting each other and escape was impossible. Even after leaving Russia it was a long route back to Vienna for Helly who travelled through Japan, the Far East, Egypt and the Middle East before reaching home in 1920. Helly was married in 1921 to Elise Bloch who also had a doctorate in mathematics from Vienna. In the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Helly.html (1 of 3) [2/16/2002 11:14:03 PM]

Helly

same year he submitted his habilitation thesis and received the right to teach. He was appointed to Vienna in 1921 but to an unpaid post. His wife believed that he was denied a professorship [4]:... partly because Helly was Jewish and also because Hahn thought a younger person should be preferred. Helly was forced to earn a living working in a bank, then as an actuary when the bank collapsed in 1929. He then found a job with an insurance company but even this difficult life became worse in 1938. Hitler himself had entered Austria on 12 March 1938 with the German army, and a Nazi government had been set up there. Helly was dismissed from his post because he was a Jew. He fled from Austria to save himself and his family, emigrating to the United States in 1938. Life remained difficult for Helly and his family in the United States. At first he resorted to giving private tuition as he had done in Vienna many years before. His first break occurred in 1939 when Einstein supported him for a position in Paterson Junior College in New Jersey. With support from so eminent a person as Einstein, he was successful and received the position. Two years later, in 1941, he moved to Monmouth Junior College, also in New Jersey. The following year saw both Helly and his wife employed as mathematicians by the Signal Corps in Chicago. Helly again took up familiar work since he was preparing mathematics training manuals, while his wife taught mathematics. It was during his time with the Signal Corps in Chicago that he suffered his first heart attack. This was still as a direct result of the damage which he had suffered when shot during World War I. He recovered from the heart attack and things began to look up when he was offered a chair of mathematics as Illinois Institute of Technology. Sadly he died after a second heart attack shortly after. He is remembered for Helly's theorem, published in 1923, which states that if there are given n convex subsets of a d-dimensional euclidean space with n d+1 and if each collection of d + 1 of the subsets has a point in common then there is a common point of the n subsets. However, as we noted above, in a paper he published in 1912 there are a number of important results. First there is Helly's selection principle which says that given a sequence of functions of bounded variation which are of uniform bounded variation and uniformly bounded at a point, then there exists a subsequence which converges to a function of bounded variation. There are other results in the paper which should have given Helly a much higher profile in the world of mathematics than he has achieved. One is the fact that he gives the Hahn-Banach theorem for the space C[a, b], while he is providing a simpler proof of a theorem which Riesz had published the previous year. He also gives the uniform boundedness principle for linear functionals, the Banach-Steinhaus theorem. As the authors of [4] write:Had Helly succeeded in staying in the mainstream of mathematics, as an academician who published and participated in seminars, he would have undoubtedly have capitalised on his earlier contributions. He not only might have seen to it that proper credit should be ascribed, but it is likely that he would have extended his results further. ... In most careers there are some disappointments and failures, but Helly's career derailed early, and life never gave him a chance to get back on the right track. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Helly.html (2 of 3) [2/16/2002 11:14:03 PM]

Helly

Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Eduard Helly

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Helly.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Helly.html (3 of 3) [2/16/2002 11:14:03 PM]

Helmholtz

Hermann Ludwig Ferdinand von Helmholtz Born: 31 Aug 1821 in Potsdam, Germany Died: 8 Sept 1894 in Berlin, Germany

Click the picture above to see nine larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hermann von Helmholtz's father was August Ferdinand Julius Helmholtz while his mother was Caroline Penn. Hermann was the eldest of his parents four children. His childhood had a strong influence on both his character and his later career. In particular the views on philosophy held by his father restricted Helmholtz's own views. Ferdinand Helmholtz had served in the Prussian army in the fight against Napoleon. Despite having a good university education in philology and philosophy, he became a teacher at Potsdam Gymnasium. It was a poorly paid job and Hermann was brought up in financially difficult circumstances. Ferdinand was an artistic man and his influence meant that Hermann grew up to have a strong love of music and painting. Caroline Helmholtz was the daughter of an artillery officer. From her Hermann inherited [1]:... the placidity and reserve which marked his character in later life. Hermann attended Potsdam Gymnasium where his father taught philology and classical literature. His interests at school were mainly in physics and he would have liked to have studied that subject at university. The financial position of the family, however, meant that he could only study at university if he received a scholarship. Such financial support was only available for particular topics and Hermann's father persuaded him that he should study medicine which was supported by the government. In 1837 Helmholtz was awarded a government grant to enable him to study medicine at the Royal Friedrich-Wilhelm Institute of Medicine and Surgery in Berlin. He did not receive the money without strings attached, however, and he had to sign a document promising to work for ten years as a doctor in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Helmholtz.html (1 of 5) [2/16/2002 11:14:05 PM]

Helmholtz

the Prussian army after graduating. In 1838 he began his studies in Berlin. Although he was officially studying at the Institute of Medicine and Surgery, being in Berlin he had the opportunity of attending courses at the University. He took this chance, attending lectures in chemistry and physiology. Given Helmholtz's contributions to mathematics later in his career it would be reasonable to have expected him to have taken mathematics courses at the University of Berlin at this time. However he did not, rather he studied mathematics on his own, reading works by Laplace, Biot and Daniel Bernoulli. He also read philosophy works at this time, particularly the works of Kant. His research career began in 1841 when he began work on his dissertation. He rejected the direction which physiology had been taking which had been based on vital forces which were not physical in nature. Helmholtz strongly argued for founding physiology completely on the principles of physics and chemistry. Helmholtz graduated from the Medical Institute in Berlin in 1843 and was assigned to a military regiment at Potsdam, but spent all his spare time doing research. His work still concentrated, as we remarked above, on showing that muscle force was derived from chemical and physical principles. If some vital force were present, he argued, then perpetual motion would become possible. In 1847 he published his ideas in a very important paper Uber die Erhaltung der Kraft which studied the mathematical principles behind the conservation of energy. Helmholtz argued in favour of the conservation of energy using both philosophical arguments and physical arguments. He based many ideas on the earlier works by Sadi Carnot, Clapeyron, Joule and others. That philosophical arguments came right up front in this work was typical of all of Helmholtz's contributions. He argued that physical scientists had to conduct experiments to find general laws. Then theoretical argument (quoting from the paper): ... endeavours to ascertain the unknown causes of processes from their visible effects; it seeks to comprehend them according to the laws of causality. ... Theoretical natural science must, therefore, if it is not to rest content with a partial view of the nature of things, take a position in harmony with the present conception of the nature of simple forces and the consequences of this conception. Its task will be completed when the reduction of phenomena to simple forces is completed, and when it can at the same time be proved that the reduction given is the only one possible which the phenomena will permit. He showed that the assumption that work could not continually be produced from nothing led to the conservation of kinetic energy. This principle he then applied to a variety of different situations. He demonstrated that in various situations where energy appears to be lost, it is in fact converted into heat energy. This happens in collisions, expanding gases, muscle contraction, and other situations. The paper looks at a broad number of applications including electrostatics, electrostatics, galvanic phenomena and electrodynamics. The paper is an important contribution and it was quickly seen as such. In fact it played a large role in Helmholtz's career for the following year he was released from his obligation to serve as an army doctor so that he could accept the vacant chair of physiology at Königsberg. He married Olga von Velten on 26 August 1849 and settled down to an academic career. On one hand his career progressed rapidly in Königsberg. He published important work on physiological optics and physiological acoustics. His received great acclaim for his invention of the ophthalmoscope in 1851 and rapidly gained a strong international reputation. In 1852 he published important work on physiological optics with his theory of colour vision. However, experiments which he carried out at this http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Helmholtz.html (2 of 5) [2/16/2002 11:14:05 PM]

Helmholtz

time led him to reject Newton's theory of colour. The paper was rightly criticised by Grassmann and Maxwell. Helmholtz was always prepared to admit his mistakes and indeed he did just this three years later when he published new experimental results showing these of his 1852 paper to be incorrect. A visit to Britain in 1853 saw him form an important friendship with William Thomson. However, on the other hand, there were problems in Königsberg. Franz Neumann, the professor of physics in Königsberg was involved in disputes concerning priority with Helmholtz and the cold weather in Königsberg had a bad effect on his wife's delicate health. He requested a move and, in 1855, was appointed to the vacant chair of anatomy and physiology in Bonn. In 1856 he published the first volume of his Handbook of physiological optics, then in 1858 he published his important paper in Crelle's Journal on the motion of a perfect fluid. Helmholtz's paper Uber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen began by decomposing the motion of a perfect fluid into translation, rotation and deformation. Helmholtz defined vortex lines as lines coinciding with the local direction of the axis of rotation of the fluid, and vortex tubes as bundles of vortex lines through an infinitesimal element of area. Helmholtz showed that the vortex tubes had to close up and also that the particles in a vortex tube at any given instant would remain in the tube indefinitely so no matter how much the tube was distorted it would retain its shape. Helmholtz was aware of the topological ideas in his paper, particularly the fact that the region outside a vortex tube was multiply connected which led him to consider many-valued potential functions. He described his theoretical conclusions regarding two circular vortex rings with a common axis of symmetry in the following way:If they both have the same direction of rotation they will proceed in the same sense, and the ring in front will enlarge itself and move slower, while the second one will shrink and move faster, if the velocities of translation are not too different, the second will finally reach the first and pass through it. Then the same game will be repeated with the other ring, so the ring will pass alternately one through the other. This paper, highly rigorous in its mathematical approach, did not attract much attention at the time but its impact on the future work by Tait and Thomson was very marked. For details of the impact of this work, particularly Helmholtz's results on vortices, see the article Topology and Scottish mathematical physics. Before the publication of this paper Helmholtz had become unhappy with his new position in Bonn. Part of the problem seemed to revolve round the fact that the chair involved anatomy and complaints were made to the Minister of Education that his lectures on this topic were incompetent. Helmholtz reacted strongly to these criticisms which, he felt, were made by traditionalists who did not understand his new mechanical approach to the subject. It was a somewhat strange position for Helmholtz to be in for he had a very strong reputation as a leading world scientist. When he was offered the chair in Heidelberg in 1857, he did not accept at once however. When further sweeteners were put forward in 1858 to entice him to accept, such as the promise of setting up a new Physiology Institute, Helmholtz agreed. Helmholtz suffered some personal problems. His father died in 1858, then at the end of 1859 his wife, whose health had never been good, died. He was left to bring up two young children and within eighteen months he married again. On 16 May 1861 Helmholtz married Anna von Mohl, the daughter of another professor at Heidelberg [1]:Anna, by whom Helmholtz later had three children, was an attractive, sophisticated woman http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Helmholtz.html (3 of 5) [2/16/2002 11:14:05 PM]

Helmholtz

considerably younger than her husband. The marriage opened a period of broader social contacts for Helmholtz. Some of his most important work was carried out while he held this post in Heidelberg. He studied mathematical physics and acoustics producing a major study in 1862 which looked at musical theory and the perception of sound. In mathematical appendices he advocated the use of Fourier series. In 1843 Ohm had stated the fundamental principle of physiological acoustics, concerned with the way in which one hears combination tones. Helmholtz explained the origin of music on the basis of his fundamental physiological hypotheses. He formulated a resonance theory of hearing which provided a physiological explanation of Ohm's principle. His contributions to the theory of music are discussed fully in [8]. From around 1866 Helmholtz began to move away from physiology and move more towards physics. When the chair of physics in Berlin became vacant in 1870 he indicated his interest in the position. Kirchhoff was the other main candidate and because he was considered a superior teacher to Helmholtz he was offered the post. However, when Kirchhoff decided not to accept Helmholtz was in a strong position. He was able to negotiate a high salary as well as having Prussia agree to build a new physics institute under Helmholtz control in Berlin. In 1871 he took up this post. Helmholtz had begun to investigate the properties of non-Euclidean space around the time his interests were turning towards physics in 1867. Bernardo in [9] writes:In the second half of the 19th century, scientists and philosophers were involved in a heated discussion on the principles of geometry and on the validity of so-called non-Euclidean geometry. ... Helmholtz's research on the subject began between 1867 and 1868. Moving from the observation that our geometric faculties depend on the existence, in nature, of rigid bodies, he presumed he had given a proof that Euclidean geometry was the only one compatible with these bodies, maintaining, at the same time, the empirical, not a priori, origin of geometry. In 1869, after Beltrami's letter ... he realized he had made a mistake: the empirical concept of a rigid body and mathematics alone were not enough to characterize Euclidean geometry. The following year, fully sharing the mathematical itinerary that, through Gauss, Riemann, Lobachevsky and Beltrami, led to the creation of the new geometry, he proposed to spread this knowledge among philosophers while at the same time criticizing the Kantian system. This marked the beginning of a heated philosophical discussion that led Helmholtz in 1878 to try to appease the criticisms of the Kantian a priori. A major topic which occupied Helmholtz after his appointment to Berlin was electrodynamics. He discussed with Weber the compatibility of Weber's electrodynamics with the principle of the conservation of energy. In fact the argument was heated and lasted throughout the 1870s. It was an argument which neither really won and the 1880s saw Maxwell's theory accepted. Helmholtz attempted to give a mechanical foundation to thermodynamics, and he also tried to derive Maxwell's electromagnetic field equations from the least action principle. R Steven Turner writes in [1]:Helmholtz devoted his life to seeking the great unifying principles underlying nature. His career began with one such principle, that of energy, and concluded with another, that of least action. No less than the idealistic generation before him, he longed to understand the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Helmholtz.html (4 of 5) [2/16/2002 11:14:05 PM]

Helmholtz

ultimate, subjective sources of knowledge. That longing found expression in his determination to understand the role of the sense organs, as mediators of experience, in the synthesis of knowledge. To this continuity with the past Helmholtz and his generation brought two new elements, a profound distaste for metaphysics and an undeviating reliance on mathematics and mechanism. Helmholtz owed the scope and depth characteristic of his greatest work largely to the mathematical and experimental expertise which he brought to science. ... Helmholtz was the last great scholar whose work, in the tradition of Leibniz, embraced all the sciences, as well as philosophy and the fine arts. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (32 books/articles)

A Quotation

Mathematicians born in the same country Cross-references to History Topics

Topology and Scottish mathematical physics

Honours awarded to Hermann von Helmholtz (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1860

Royal Society Copley Medal

Awarded 1873

Planetary features

Crater Cayley on Mars

Other Web sites

West Chester University

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Helmholtz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Helmholtz.html (5 of 5) [2/16/2002 11:14:05 PM]

Heng

Zhang Heng Born: 78 in Nan-yang, China Died: 139 in China

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Heng was a mathematician, astronomer and geographer. He became chief astrologer and minister under the Emperor An'ti of China. In the year 123 he corrected the calendar to bring it into line with the seasons. In 132 Heng invented the first seismoscope for measuring earthquakes. Heng's device was in the shape of a cylinder with eight dragon heads around the top, each with a ball in its mouth. Around the bottom were eight frogs, each directly under a dragon head. When an earthquake occurred, a ball fell out of a the dragon's mouth into a frog's mouth, making a noise. Heng was the first person in China to construct a rotating celestial globe. He wrote about this in his work Hun-i chu where he describes his version of the universe as follows:The sky is like a hen's egg, and is as round as a crossbow pellet; the Earth is like the yolk of the egg, lying alone at the centre. The sky is large and the Earth small. In another work Lin Hsien, he describes the stars:North and south of the equator there are 124 groups which are always brightly shining. 320 stars can be named. There are in all 2500, not including those which the sailors observe. Of the very small stars there are 11520. Heng also proposed

= 10 or approximately 3.162.

Article by: J J O'Connor and E F Robertson A Reference (One book/article)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heng.html (1 of 2) [2/16/2002 11:14:07 PM]

Heng

A Poster of Heng

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heng.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heng.html (2 of 2) [2/16/2002 11:14:07 PM]

Henrici

Olaus Magnus Friedrich Erdmann Henrici Born: 1840 in Meldorf, Holstein (now Germany) Died: 10 Aug 1918 in Chandler's Ford, Hampshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

While an engineering apprentice Olaus Henrici's talents were recognised by Clebsch and he persuaded Hesse to take Henrici on as a Ph.D. student at Heidelberg. Henrici then went to Berlin and studied under Weierstrass and Kronecker. He came to England in 1865 and Hesse introduced him to Sylvester. Sylvester in turn introduced he to Hirst who helped him to a chair at University College London in 1870. From 1884 he held a chair at Bedford College. Henrici introduced graphical statics into Bedford and at the Central Technical College he introduced a Mechanics Laboratory and a Harmonic Analyser. He was elected to the Royal Society in 1874. He was awarded an honorary degree by St Andrews in 1884. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Honours awarded to Olaus Henrici (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1874

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Henrici.html (1 of 2) [2/16/2002 11:14:08 PM]

Henrici

London Maths Society President

1882 - 1884

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Henrici.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Henrici.html (2 of 2) [2/16/2002 11:14:08 PM]

Hensel

Kurt Hensel Born: 29 Dec 1861 in Königsberg, Prussia (now Kaliningrad, Russia) Died: 1 June 1941 in Marburg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kurt Hensel was born in the East Prussia in the city then called Königsberg. He studied mathematics in Berlin and Bonn. Among his teachers were Lipschitz, Weierstrass, Kirchhoff, Helmholtz and especially Kronecker. He devoted several years to the editing of Kronecker's collected works. His work followed that of Kronecker in the development of arithmetic in algebraic number fields. In 1897 the Weierstrass method of power-series development for algebraic functions led him to the invention of the p-adic numbers. Hensel was interested in the exact power of a prime which divides the discriminant of an algebraic number field. The p-adic numbers can be regarded as a completion of the rational numbers in a different way from the usual completion which leads to the real numbers. His invention led to the development of the concept of a field with valuation which has had a great influence on later mathematics. He was able to use his methods to prove many results in the theory of quadratic forms and number theory. It was not until 1921 that the potential of the p-adic numbers was demonstrated by Hasse when he formulated the local-global principle. He showed, at least for quadratic forms, that an equation has a rational solution if and only if it has a solution in the p-adic numbers for each prime p and a solution in the reals. Hensel was a professor at the University of Marburg until his retirement in 1930. From 1901 he was editor of the prestigious and influencial Crelle's Journal.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hensel.html (1 of 2) [2/16/2002 11:14:10 PM]

Hensel

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) A Poster of Kurt Hensel

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1890 to 1900

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hensel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hensel.html (2 of 2) [2/16/2002 11:14:10 PM]

Heraclides

Heraclides of Pontus Born: 387 BC in Heraclea Pontica (now Eregli, Turkey) Died: 312 BC in Heraclea Pontica

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Heraclides of Pontus has achieved fame for a long time as the first to propose that the sun was the centre of the solar system but this has been shown to be due to a misinterpretation of what he wrote. We do have some details of Heraclides' life. His father was named Euthyphron, a wealthy man of high status from Heraclea Pontica, who was descended from one of the original founders of this Greek city on the south coast of the Black Sea. Heraclides attended the Academy in Athens and was left in charge of it during Plato's third visit to Sicily in 360 BC. Although in some sense he was a pupil of Plato, he also studied with Aristotle and with Speusippus who was Plato's successor as head of the Academy. When Speusippus died in 339 BC there was an election for the new leader despite Xenocrates having been chosen to head the Academy by Speusippus. It was a close battle between Xenocrates, Menedemus of Pyrrha and Heraclides Ponticus but Xenocrates triumphed by just a few votes. At this point Heraclides left the Athens and returned to Heraclea Pontica. Stories told of his death are not really believable yet they must at least point to the type of person that Heraclides was. It is said that Heraclea Pontica suffered a famine and Heraclides bribed the messengers sent to the Delphic oracle to say that the city would be saved if Heraclides was given a gold crown and made a hero after his death. The story relates that Heraclides died while being presented with the golden crown. Perhaps Heraclides had the last laugh here for indeed he did become a hero after is death based on a false interpretation of his writing. For example Heath [2] writes:Heraclides of Pontus, Plato's famous pupil, is known on clear evidence to have discovered

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heraclides.html (1 of 3) [2/16/2002 11:14:12 PM]

Heraclides

that Venus and Mercury revolve round the sun like satellites. He may have come to the same conclusion about the superior planets but this is not certain... The misunderstanding comes from a commentary by Calcidius in the fifth century AD on Plato's Timaeus. This reads (in Neugebauer's translation in [3]):Heraclides Ponticus, when describing the circle of Venus as well as that of the sun, and giving the two circles one centre and one mean motion, showed how Venus is sometimes above, sometimes below the sun. T H Martin, in 1849, pointed out the significance of the passage saying that Venus is sometimes above, sometimes below the sun clearly means that Heraclides believed that it was in orbit round the sun. Schiaparelli accepted Martin's argument and went further to claim that Heraclides must have proposed the theory that the sun revolves round the earth, but the planets revolve round the sun. This theory, first proposed by Tycho Brahe at the end of the 16th century, was never as far as we know put forward by a Greek astronomer. Neugebauer [3] shows clearly that the passage indicating that Venus is sometimes above, sometimes below the sun, means that it is sometimes ahead of the sun, sometimes behind it. Toomer in [1] agrees completely with Neugebauer's interpretation but a still more amazing interpretation of Heraclides had been proposed by van der Waerden. Rather than basing his argument on Calcidius's words, van der Waerden interpreted a diagram in Calcidius to mean that the sun, Venus and the earth all revolve round a common centre. The article [6] by van der Waerden was an attempt by him to defend his hypothesis despite the new interpretation of the situation by Neugebauer. It is an unconvincing article and it seems to only repeat van der Waerden's earlier hypothesis without making any attempt to counter the rather simple and totally convincing argument by Neugebauer. Heraclides does, however, still have a claim to fame for an astronomical hypothesis. A number of sources quote his belief that the earth is at the centre of the universe but that it rotates on its axis once a day. Certainly he is the first known to have held this view and deserves great credit for it. We do know a little more concerning Heraclides. He is said to have (see [1]):... dressed richly, was very fat and stately, and was nicknamed "stately and magnificent" by the Athenians. This nickname is a Greek pun based on the word Pompikos (meaning stately and magnificent) replacing Pontikos (meaning from Pontus). Apart from his writings on astronomy, Heraclides wrote on many of the usual topics that a leading philosopher of his day would have written on. These topics included ethics, literature, rhetoric, history, politics and music. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heraclides.html (2 of 3) [2/16/2002 11:14:12 PM]

Heraclides

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heraclides.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heraclides.html (3 of 3) [2/16/2002 11:14:12 PM]

Herbrand

Jacques Herbrand Born: 12 Feb 1908 in Paris, France Died: 27 July 1931 in La Bérarde, Isère, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jacques Herbrand entered the Ecole Normale Supérieure at the age of 17. This was quite exceptional at that time. For his doctoral thesis he studied mathematical logic which was a surprising choice, given the lack of interest in that topic in France in this period. His doctoral thesis was approved in April 1929 and in October of that year Jacques joined the army for his military service. After his spell in the army, Herbrand was awarded a Rockefeller fellowship to allow him to study at various places in Europe. His first period, until May 1931, was spent at the University of Berlin where he worked with von Neumann. From Berlin, Herbrand went to Hamburg where he spent the month of June working with Artin. His final visit was to Göttingen where he spent the month of July 1931 studying with Emmy Noether. After leaving Göttingen, Herbrand decided on a holiday in the Alps before his intended return to France. However he was never to complete his plans for he died in a mountaineering accident in the Alps only a few days after his holiday began. His death at the age of 23 in one of the tragic losses to mathematics. It is incredible how much Herbrand achieved in the short time he had to undertake mathematical research. He made contributions to mathematical logic where Herbrand's theorem on the theory of quantifiers appears in his doctoral thesis. See [5] for discussion of a gap which was found in Herbrand's proof in 1963. Herbrand's theorem establishes a link between quantification theory and sentential logic which is important in that it gives a method to test a formula in quantification theory by successively testing formulas for sentential validity. Since testing for sentential validity is a mechanical process, Herbrand's http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herbrand.html (1 of 2) [2/16/2002 11:14:13 PM]

Herbrand

theorem is today of major importance in software developed for theorem proving by computer. Herbrand also worked on field theory considering abelian extensions of algebraic number fields. In the few months on which he worked on this topic, Herbrand published ten papers. These papers simplify proofs of results by Kronecker, Heinrich Weber, Hilbert, Takagi and Artin. Herbrand also generalised some of the results by these workers in class field theory as well as proving some important new theorems of his own. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other Web sites

Clark Kimberling

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Herbrand.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herbrand.html (2 of 2) [2/16/2002 11:14:13 PM]

Herigone

Pierre Hérigone Born: 1580 in France Died: 1643 in Paris, France Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Hérigone was of Basque origin. Little is known of his life except that he taught for most of it in Paris. Hérigone's only important work is the 6 volume Cursus mathematicus which is a compendium of elementary mathematics written in French and Latin. It introduced a complete system of mathematical and logical notation, yet none is used today. We know that Hérigone served on a number of committees and took a full part in the mathematical life of Paris. One committee which he served on was set up to judge whether Morin's scheme for determining longitude from the Moon's motion was practical. The committee members, in addition to Hérigone, were Etienne Pascal, Mydorge, Beaugrand, J C Boulenger and L de la Porte. Hérigone, and the rest of the committee, became involved in a dispute with Morin. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Longitude and the Académie Royale 2. The trigonometric functions

Honours awarded to Pierre Hérigone (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Herigonius and Rima Herigonius

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herigone.html (1 of 2) [2/16/2002 11:14:15 PM]

Herigone

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Herigone.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herigone.html (2 of 2) [2/16/2002 11:14:15 PM]

Hermann

Jakob Hermann Born: 16 July 1678 in Basel, Switzerland Died: 11 July 1733 in Basel, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jakob Hermann, who was a distant relative of Euler, studied theology and mathematics at Basel. He studied mathematics under Jacob Bernoulli at Basel, receiving his first degree in 1695. In 1701 he became a member of the Berlin academy, his election being very much due to support from Leibniz. In 1707 Hermann was appointed, on Leibniz's recommendation, to the chair of mathematics at he University of Padua. He held this post in Italy until 1713 when he went to Germany. He spent 11 years in Frankfurt-an-der-Oder, but in 1724 he went to Russia and held a chair of higher mathematics in the St Petersburg Academy. While in St Petersburg Hermann gave tuition in mathematics to Peter the Great's grandson, Peter Aleksevevich. Peter Aleksevevich was crowned Peter II in 1727 at the age of 12 but died in 1730. In 1731 Hermann returned to Basel to the chair of ethics and natural law. Hermann worked in mechanics and studied the 'inverse problem' where one has to determine the orbit from a knowledge of the law of force. He proposed the term 'phoronomia' in 1716 for the topic which was sometimes called 'rational mechanics' and now called 'theoretical mechanics'. In [4] Hermann's contribution to dynamics is described in these terms:He belonged to the Basel school headed by the Bernoullis and, sharing their methodology, made important contributions to the analytical treatment of dynamics. However, he also

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hermann.html (1 of 2) [2/16/2002 11:14:16 PM]

Hermann

leaned towards Newton and, on many occasions, preferred to deal with dynamical problems in terms of geometry. In 1733 Hermann was elected to the Académie Royal des Sciences in Paris. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Honours awarded to Jakob Hermann (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Hermann

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hermann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hermann.html (2 of 2) [2/16/2002 11:14:16 PM]

Hermann_of_Reichenau

Hermann of Reichenau Born: 18 July 1013 in Altshausen, Germany Died: 24 Sept 1054 in Altshausen, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hermann of Reichenau is also called Hermann the Lame or Hermann Contractus. Hermann entered the Cloister School at Reichenau in September 1020. He became a monk at the Benedictine Monastery at Reichenau in 1043, becoming Abbot of the Monastery. Hermann is called 'the Lame' or 'Contractus' for very good reason. He was extremely disabled, having only limited movement and limited ability to speak. Despite these disabilities he was a key figure in the transmission of Arabic mathematics, astronomy and scientific instruments from Arabic sources into central Europe. In other words he published in Latin much scientific work which before this time had been only available in Arabic. One would expect, from this description, that he would be an Arabic speaker but it is thought almost certain that he could not read Arabic. Hermann introduced three important instruments into central Europe, knowledge of which came from Arabic Spain. He introduced the astrolabe, a portable sundial and a quadrant with a cursor. His works include De Mensura Astrolabii and De Utilitatibus Astrolabii (some parts of these works may not have been written by Hermann). Hermann's contributions to mathematics include a treatise dealing with multiplication and division, although this book is written entirely with Roman numerals. He also wrote on a complicated game based on Pythagorean number theory which was derived from Boethius. As well as his scientific work, Hermann wrote poetry and hymns. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hermann_of_Reichenau.html (1 of 2) [2/16/2002 11:14:17 PM]

Hermann_of_Reichenau

Mathematicians born in the same country Other references in MacTutor

Chronology: 900 to 1100

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hermann_of_Reichenau.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hermann_of_Reichenau.html (2 of 2) [2/16/2002 11:14:17 PM]

Hermite

Charles Hermite Born: 24 Dec 1822 in Dieuze, Lorraine, France Died: 14 Jan 1901 in Paris, France

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Hermite's father was Ferdinand Hermite and his mother was Madeleine Lallemand. Ferdinand Hermite was a trained engineer and he worked in this capacity in a salt mine near Dieuse. After he married Madeleine he joined in the draper's trade in which her family were involved. However he was an artistic man who always wanted to pursue art as a career. He had his wife look after the draper's business and he took up art. Charles was the sixth of his parents seven children and when he was about seven years old his parents left Dieuse and went to live in Nancy to where the business had moved. Education was not a high priority for Charles's parents but despite not taking too much personal interest in their children's education, nevertheless they did provide them with good schooling. Charles was something of a worry to his parents for he had a defect in his right foot which meant that he moved around only with difficulty. It was clear that this would present him with problems in finding a career. However he had a happy disposition and bore his disability with a cheerful smile. Charles attended the Collège de Nancy, then went to Paris where he attended the Collège Henri. In 1840-41 he studied at the Collège Louis-le-Grand where some fifteen years earlier Galois had studied. In fact he was taught mathematics there by Louis Richard who had taught Galois. In some ways Hermite was similar to Galois for he preferred to read papers by Euler, Gauss and Lagrange rather than work for his formal examinations. If Hermite neglected the studies that he should have concentrated on, he was showing remarkable research ability publishing two papers while at Louis-le-Grand. Also like Galois he was attracted by the problem of solving algebraic equations and one of the two papers attempted to show that the quintic

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hermite.html (1 of 5) [2/16/2002 11:14:19 PM]

Hermite

cannot be solved in radicals. That he was unfamiliar with Galois's contributions, despite being at the same school, is not at all surprising since the mathematical community were completely unaware of them at this time. However he might reasonably have known of the contributions of Ruffini and Abel to this question, but apparently he did not. Again like Galois, Hermite wanted to study at the Ecole Polytechnique and he took a year preparing for the examinations. He was tutored by Catalan in 1841-42 and certainly Hermite fared better than Galois had done for he passed. However it was not a glorious pass for he only attained sixty-eighth place in the ordered list. After one year at the Ecole Polytechnique Hermite was refused the right to continue his studies because of his disability. Clearly this was an unfair decision and some important people were prepared to take up his case and fight for him to have the right to continue as a student at the Ecole Polytechnique. The decision was reversed so that he could continue his studies but strict conditions were imposed. Hermite did not find these conditions acceptable and decided that he would not graduate from the Ecole Polytechnique. Hermite made friends with important mathematicians at this time and frequently visited Joseph Bertrand. On a personal note this was highly significant for he would marry Joseph Bertrand's sister. More significantly from a mathematical point of view he began corresponding with Jacobi and, despite not shining in his formal education, he was already producing research which was ranking as a leading world-class mathematician. The letters he exchanged with Jacobi show that Hermite had discovered some differential equations satisfied by theta-functions and he was using Fourier series to study them. He had found general solutions to the equations in terms of theta-functions. Hermite may have still been an undergraduate but it is likely that his ideas from around 1843 helped Liouville to his important 1844 results which include the result now known as Liouville's theorem. After spending five years working towards his degree he took and passed the examinations for the baccalauréat and licence which he was awarded in 1847. In the following year he was appointed to the Ecole Polytechnique, the institution which had tried to prevent him continuing his studies some four years earlier; he was appointed répétiteur and admissions examiner. Hermite made important contributions to number theory and algebra, orthogonal polynomials, and elliptic functions. He discovered his most significant mathematical results over the ten years following his appointment to the Ecole Polytechnique. In 1848 he proved that doubly periodic functions can be represented as quotients of periodic entire functions. In 1849 Hermite submitted a memoir to the Académie des Sciences which applied Cauchy's residue techniques to doubly periodic functions. Sturm and Cauchy gave a good report on this memoir in 1851 but a priority dispute with Liouville seems to have prevented its publication. Another topic on which Hermite worked and made important contributions was the theory of quadratic forms. This led him to study invariant theory and he found a reciprocity law relating to binary forms. With his understanding of quadratic forms and invariant theory he created a theory of transformations in 1855. His results on this topic provided connections between number theory, theta functions, and the transformations of abelian functions. On 14 July 1856 Hermite was elected to the Académie des Sciences. However, despite this achievement,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hermite.html (2 of 5) [2/16/2002 11:14:19 PM]

Hermite

1856 was a bad year for Hermite for he contracted smallpox. It was Cauchy who, with his strong religious conviction, helped Hermite through the crisis. This had a profound effect on Hermite who, under Cauchy's influence, turned to the Roman Catholic religion. Cauchy was also a very staunch royalist and Hermite was influenced by him to also became a royalist. We made comparisons with Galois earlier on in this article, but with royalist views, Hermite was now completely opposed to the views which the staunch republican Galois had held. The next mathematical result by Hermite which we must mention is one for which he is rightly famous. Although an algebraic equation of the fifth degree cannot be solved in radicals, a result which was proved by Ruffini and Abel, Hermite showed in 1858 that an algebraic equation of the fifth degree could be solved using elliptic functions. He applied these results to number theory, in particular to class number relations of quadratic forms. In 1862 Hermite was appointed maître de conference at the Ecole Polytechnique, a position which had been specially created for him. In the following year he became an examiner there. The year 1869 saw him become a professor when he succeeded Duhamel as professor of analysis both at the Ecole Polytechnique and at the Sorbonne. Hermite resigned his chair at the Ecole Polytechnique in 1876 but continued to hold the chair at the Sorbonne until he retired in 1897. In the 1890s Hermite became much less interested in the new results found by the mathematicians of the next generation. The 1870s saw Hermite return to problems which had interested him earlier in his career such as problems concerning approximation and interpolation. In 1873 Hermite published the first proof that e is a transcendental number. This is another result for which he is rightly famous. Using method's similar to those of Hermite, Lindemann established in 1882 that was also transcendental. Many historians of science regret that Hermite, despite doing most of the hard work, failed to use it to prove the result on which would have brought him fame outside the world of mathematics. Hermite is now best known for a number of mathematical entities that bear his name: Hermite polynomials, Hermite's differential equation, Hermite's formula of interpolation and Hermitian matrices. For Hermite certain areas of mathematics were much more interesting than other areas. Hadamard, who unlike his teacher Hermite worked in all areas of mathematics, spoke of Hermite's dislike for geometry:[Hermite] had a kind of positive hatred of geometry and once curiously reproached me with having made a geometrical memoir. Hermite's great love was for analysis and, not surprisingly, he had a great respect for Weierstrass. When Mittag-Leffler arrived in Paris to study with him, Hermite greeted him warmly but said:You have made a mistake, sir, you should follow Weierstrass's course in Berlin. He is the master of us all. Poincaré is almost certainly the best known of Hermite's students. He once suggested that Hermite's mind did not proceed in logical fashion. He wrote:But to call Hermite a logician! Nothing can appear to me more contrary to the truth. Methods always seemed to be born in his mind in some mysterious way. Hadamard like Poincaré was very interested in the way that mathematics was discovered. He also had this to say about the way that Hermite made his discoveries:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hermite.html (3 of 5) [2/16/2002 11:14:19 PM]

Hermite

Hermite used to observe [that biology] may be a most useful study even for mathematicians, as hidden and eventually fruitful analogies may appear between processes in both kinds of studies. Hadamard had great respect for Hermite as a teacher. He said:I do not think that those who never listened to him can realise how magnificent Hermite's teaching was, overflowing with enthusiasm for science, which seemed to come to life in his voice and whose beauty he never failed to communicate to us, since he felt it so much himself to the very depth of his being. [Hermite] was making a deep impression on us, not only with his methods and those of Weierstrass, but also with his enthusiasm and love of science; in our brief but fruitful conversations, Hermite loved to direct to me remarks such as: "He who strays from the paths traced by providence crashes." These were the words of a profoundly religious man, but an atheist like me understood them very well, especially when he added at other times: "In mathematics , our role is more of servant than of master." It goes without saying that gradually, as years and my scientific work unfolded, I came to understand more and more deeply the aptness and scope of his words. Cross, reviewing [11] where 125 letters from Hermite to Mittag-Leffler are reproduced, writes:So there stands revealed one of the most engaging and influential men in Parisian and French mathematics in the second half of the 19th century, one might even say the central character for the period in which he published, 1842-1901. What radiates from the text is [Hermite's] humility, his Catholicism, his concern for his (very extended) family, his willingness to fight for colleagues whose merit he discerns, and his devotion to family, merit, and principle rather than simple influence. In terms of his family life Hermite had married Louise Bertrand, Joseph Bertrand's sister. One of their two daughters married Emile Picard. Struik writes:Hermite lived a retired life, with his family. His working hours were devoted to mathematical research and teaching. His outlook on mathematics was realistic in the Platonic sense: a mathematician, like a naturalist, discovers an outside world, in his case a world of ideas. Hermite, therefore, disliked Cantor's world, in which a new mathematical world was created. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (16 books/articles)

Some Quotations (4)

A Poster of Charles Hermite

Mathematicians born in the same country

Cross-references to History Topics

Abstract linear spaces

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hermite.html (4 of 5) [2/16/2002 11:14:19 PM]

Hermite

Other references in MacTutor

1. Chronology: 1840 to 1850 2. Chronology: 1870 to 1880

Honours awarded to Charles Hermite (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1873

Lunar features

Crater Hermite

Paris street names

Rue Charles Hermite and Square Charles Hermite (18th Arrondissement) 1. Hermite's Constants

Other Web sites

2. The Catholic Encyclopedia 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hermite.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hermite.html (5 of 5) [2/16/2002 11:14:19 PM]

Heron

Heron of Alexandria Born: about 10 in (possibly) Alexandria, Egypt Died: about 75 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sometimes called Hero, Heron of Alexandria was an important geometer and worker in mechanics. Perhaps the first comment worth making is how common the name Heron was around this time and it is a difficult problem in the history of mathematics to identify which references to Heron are to the mathematician described in this article and which are to others of the same name. There are additional problems of identification which we discuss below. A major difficulty regarding Heron was to establish the date at which he lived. There were two main schools of thought on this, one believing that he lived around 150 BC and the second believing that he lived around 250 AD. The first of these was based mainly on the fact that Heron does not quote from any work later than Archimedes. The second was based on an argument which purported to show that he lived later that Ptolemy, and, since Pappus refers to Heron, before Pappus. Both of these arguments have been shown to be wrong. There was a third date proposed which was based on the belief that Heron was a contemporary of Columella. Columella was a Roman soldier and farmer who wrote extensively on agriculture and similar subjects, hoping to foster in people a love for farming and a liking for the simple life. Columella, in a text written in about 62 AD [5]:... gave measurements of plane figures which agree with the formulas used by Heron, notably those for the equilateral triangle, the regular hexagon (in this case not only the formula but the actual figures agree with Heron's) and the segment of a circle which is less than a semicircle ... However, most historians believed that both Columella and Heron were using an earlier source and claimed that the similarity did not prove any dependence. We now know that those who believed that Heron lived around the time of Columella were in fact correct, for Neugebauer in 1938 discovered that Heron referred to a recent eclipse in one of his works which, from the information given by Heron, he was able to identify with one which took place in Alexandria at 23.00 hours on 13 March 62. From Heron's writings it is reasonable to deduce that he taught at the Museum in Alexandria. His works look like lecture notes from courses he must have given there on mathematics, physics, pneumatics, and mechanics. Some are clearly textbooks while others are perhaps drafts of lecture notes not yet worked into final form for a student textbook. Pappus describes the contribution of Heron in Book VIII of his Mathematical Collection. Pappus writes (see for example [8]):http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heron.html (1 of 5) [2/16/2002 11:14:21 PM]

Heron

The mechanicians of Heron's school say that mechanics can be divided into a theoretical and a manual part; the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of work in metals, architecture, carpentering and painting and anything involving skill with the hands. ... the ancients also describe as mechanicians the wonder-workers, of whom some work by means of pneumatics, as Heron in his Pneumatica, some by using strings and ropes, thinking to imitate the movements of living things, as Heron in his Automata and Balancings, ... or by using water to tell the time, as Heron in his Hydria, which appears to have affinities with the science of sundials. A large number of works by Heron have survived, although the authorship of some is disputed. We will discuss some of the disagreements in our list of Heron's works below. The works fall into several categories, technical works, mechanical works and mathematical works. The surviving works are: On the dioptra dealing with theodolites and surveying. It contains a chapter on astronomy giving a method to find the distance between Alexandria and Rome using the difference between local times at which an eclipse of the moon is observed at each cities. The fact that Ptolemy does not appear to have known of this method led historians to mistakenly believe Heron lived after Ptolemy; The pneumatica in two books studying mechanical devices worked by air, steam or water pressure. It is described in more detail below; The automaton theatre describing a puppet theatre worked by strings, drums and weights; Belopoeica describing how to construct engines of war. It has some similarities with work by Philon and also work by Vitruvius who was a Roman architect and engineer who lived in the 1st century BC; The cheirobalistra about catapults is thought to be part of a dictionary of catapults but was almost certainly not written by Heron; Mechanica in three books written for architects and described in more detail below; Metrica which gives methods of measurement. We give more details below; Definitiones contains 133 definitions of geometrical terms beginning with points, lines etc. In [15] Knorr argues convincingly that this work is in fact due to Diophantus; Geometria seems to be a different version of the first chapter of the Metrica based entirely on examples. Although based on Heron's work it is not thought to be written by him; Stereometrica measures three-dimensional objects and is at least in part based on the second chapter of the Metrica again based on examples. Again it is though to be based on Heron's work but greatly changed by many later editors; Mensurae measures a whole variety of different objects and is connected with parts of Stereometrica and Metrica although it must be mainly the work of a later author; Catoprica deals with mirrors and is attributed by some historians to Ptolemy although most now seem to believe that this is a genuine work of Heron. In this work, Heron states that vision results from light rays emitted by the eyes. He believes that these rays travel with infinite velocity. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heron.html (2 of 5) [2/16/2002 11:14:21 PM]

Heron

Let us examine some of Heron's work in a little more depth. Book I of his treatise Metrica deals with areas of triangles, quadrilaterals, regular polygons of between 3 and 12 sides, surfaces of cones, cylinders, prisms, pyramids, spheres etc. A method, known to the Babylonians 2000 years before, is also given for approximating the square root of a number. Heron gives this in the following form (see for example [5]):Since 720 has not its side rational, we can obtain its side within a very small difference as follows. Since the next succeeding square number is 729, which has 27 for its side, divide 720 by 27. This gives 26 2/3. Add 27 to this, making 53 2/3, and take half this or 26 5/6. The side of 720 will therefore be very nearly 26 5/6. In fact, if we multiply 26 5/6 by itself, the product is 720 1/36, so the difference in the square is 1/36. If we desire to make the difference smaller still than 1/36, we shall take 720 1/36 instead of 729 (or rather we should take 26 5/6 instead of 27), and by proceeding in the same way we shall find the resulting difference much less than 1/36. Heron also proves his famous formula in Book I of the Metrica : if A is the area of a triangle with sides a, b and c and s = (a + b + c)/2 then A2 = s (s - a)(s b)(s - c). In Book II of Metrica, Heron considers the measurement of volumes of various three dimensional figures such as spheres, cylinders, cones, prisms, pyramids etc. His preface is interesting, partly because knowledge of the work of Archimedes does not seem to be as widely known as one might expect (see for example [5]):After the measurement of surfaces, rectilinear or not, it is proper to proceed to solid bodies, the surfaces of which we have already measured in the preceding book, surfaces plane and spherical, conical and cylindrical, and irregular surfaces as well. The methods of dealing with these solids are, in view of their surprising character, referred to Archimedes by certain writers who give the traditional account of their origin. But whether they belong to Archimedes or another, it is necessary to give a sketch of these results as well. Book III of Metrica deals with dividing areas and volumes according to a given ratio. This was a problem which Euclid investigated in his work On divisions of figures and Heron's Book III has a lot in common with the work of Euclid. Also in Book III, Heron gives a method to find the cube root of a number. In particular Heron finds the cube root of 100 and the authors of [9] give a general formula for the cube root of N which Heron seems to have used in his calculation: a + b d/(b d + aD)(b - a), where a3 < N < b3, d = N - a3, D = b3 - N. In [9] it is remarked that this is a very accurate formula, but, unless a Byzantine copyist is to be blamed for an error, they conclude that Heron might have borrowed this accurate formula without understanding how to use it in general. The Pneumatica is a strange work which is written in two book, the first with 43 chapters and the second with 37 chapters. Heron begins with a theoretical consideration of pressure in fluids. Some of this theory is right but, not surprisingly, some is quite wrong. Then there follows a description of a whole collection of what might best be described as mechanical toys for children [1]:Trick jars that give out wine or water separately or in constant proportions, singing birds

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heron.html (3 of 5) [2/16/2002 11:14:21 PM]

Heron

and sounding trumpets, puppets that move when a fire is lit on an altar, animals that drink when they are offered water ... Although all this seems very trivial for a scientist to be involved with, it would appear that Heron is using these toys as a vehicle for teaching physics to his students. It seems to be an attempt to make scientific theories relevant to everyday items that students of the time would be familiar with. There is, rather remarkably, descriptions of over 100 machines such as a fire engine, a wind organ, a coin-operated machine, and a steam-powered engine called an aeolipile. Heron's aeolipile, which has much in common with a jet engine, is described in [2] as follows:The aeolipile was a hollow sphere mounted so that it could turn on a pair of hollow tubes that provided steam to the sphere from a cauldron. The steam escaped from the sphere from one or more bent tubes projecting from its equator, causing the sphere to revolve. The aeolipile is the first known device to transform steam into rotary motion. Heron wrote a number of important treatises on mechanics. They give methods of lifting heavy weights and describe simple mechanical machines. In particular the Mechanica is based quite closely on ideas due to Archimedes. Book I examines how to construct three dimensional shapes in a given proportion to a given shape. It also examines the theory of motion, certain statics problems, and the theory of the balance. In Book II Heron discusses lifting heavy objects with a lever, a pulley, a wedge, or a screw. There is a discussion on centres of gravity of plane figures. Book III examines methods of transporting objects by such means as sledges, the use of cranes, and looks at wine presses. Other works have been attributed to Heron, and for some of these we have fragments, for others there are only references. The works for which fragments survive include one on Water clocks in four books, and Commentary on Euclid's Elements which must have covered at least the first eight books of the Elements. Works by Heron which are referred to, but no trace survives, include Camarica or On vaultings which is mentioned by Eutocius and Zygia or On balancing mentioned by Pappus. Also in the Fihrist, a tenth century survey of Islamic culture, a work by Heron on how to use an astrolabe is mentioned. Finally it is interesting to look at the opinions that various writers have expressed as to the quality and importance of Heron. Neugebauer writes [7]:The decipherment of the mathematical cuneiform texts made it clear that much of the "Heronic" type of Greek mathematics is simply the last phase of the Babylonian mathematical tradition which extends over 1800 years. Some have considered Heron to be an ignorant artisan who copied the contents of his books without understanding what he wrote. This in particular has been levelled against the Pneumatica but Drachmann, writing in [1], says:... to me the free flowing, rather discursive style suggests a man well versed in his subject who is giving a quick summary to an audience that knows, or who might be expected to know, a good deal about it. Some scholars have approved of Heron's practical skills as a surveyor but claimed that his knowledge of science was negligible. However, Mahony writes in [1]:In the light of recent scholarship, he now appears as a well-educated and often ingenious

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heron.html (4 of 5) [2/16/2002 11:14:21 PM]

Heron

applied mathematician, as well as a vital link in a continuous tradition of practical mathematics from the Babylonians, through the Arabs, to Renaissance Europe. Finally Heath writes in [5]:The practical utility of Heron's manuals being so great, it was natural that they should have great vogue, and equally natural that the most popular of them at any rate should be re-edited, altered and added to by later writers; this was inevitable with books which, like the "Elements" of Euclid, were in regular use in Greek, Byzantine, Roman, and Arabian education for centuries. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (19 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Greek Astronomy 2. Doubling the cube

Other references in MacTutor

1. Minimal paths 2. Chronology: 1AD to 500

Other Web sites

1. Kevin Brown (Heron's triangle formula) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heron.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heron.html (5 of 5) [2/16/2002 11:14:21 PM]

Herschel

John Frederick William Herschel Born: 7 March 1792 in Slough, Buckinghamshire, England Died: 11 May 1871 in Hawkhurst, Kent, England

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Herschel was the son of William Herschel, the astronomer who discovered Uranus. His mother Mary Pitt was the daughter of a wealthy merchant. She was 38 years old when she married William Herschel, her first husband and her only child having both died. When John Herschel was born in 1792 William was 55 years old and Mary was 42. In addition to his parents, his aunt Caroline Herschel was another important figure in John Herschel's upbringing. William Herschel was not only a leading astronomer but he also had a great talent for music. Both he and his sister Caroline Herschel were extremely musical, and they had both used their musical talents to help augment their income after first arriving in England. John was brought up in Observatory House, with its 40 foot telescope, where music, science and religion were dominant. Caroline Herschel had left her brother's home when he married, but she continued to come to Observatory House every day to help William reduce his data and she proved an outstanding teacher to John, carrying out experiments in physics and chemistry with the young boy. Schooling did present some problems for John. After studying at Dr Gretton's School in Hitcham, he was sent to Eton College when he was eight years old but he was bullied by the other boys. He was removed from the school by his mother after a few months. In addition to schooling at Clewer and Hitcham, John was tutored at home by Mr Rogers, a private mathematics tutor, to prepare him for university. He entered St John's College Cambridge in 1809. As an undergraduate Herschel made friends with Peacock and Babbage. In 1812 the three undergraduates founded the Analytical Society which had as its aim the introduction of Continental methods of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herschel.html (1 of 6) [2/16/2002 11:14:23 PM]

Herschel

mathematical analysis into English universities. It is fair here to say that the aim was to bring Continental mathematical theory into English universities since the Scottish universities were at this time more progressive. We should also say that the Analytical Society was not the first move towards Continental mathematics in England, for Woodhouse who was one of Herschel's lecturers at Cambridge, had written a fine book which took the Leibniz approach to the calculus rather than Newton's approach. Yet the mathematical syllabus at Cambridge reflected none of the theories of d'Alembert, Leibniz (in particular Euler's development of this approach), or of the more algebraic approach of Lagrange. Herschel, together with Peacock, translated Lacroix's Traité du calcul différentiel et du calcul intégral which examined these different approaches to the calculus. Herschel did not offer these three approaches with equal recommendation for he believed that the algebraic approach of Lagrange was the right one. However the Analytical Society did not survive for very long. Herschel graduated in 1813 taking first place in the final examination. Peacock came second to Herschel while Babbage had withdrawn mainly because he could not compete with Herschel and he was not prepared to enter a competition which he knew that he could not win. Following his graduation Herschel became first Smith's prizeman and was elected a fellow of St John's College. Also in 1813 he was elected as a fellow of the Royal Society of London, having published a mathematics paper On a remarkable application of Cotes's theorem in the Transactions of the Royal Society. Despite the demise of the Analytical Society Herschel continued to work on mathematics. He studied algebras and published papers on trigonometrical series. Among many mathematical works he published a two volume book in 1820 on examples of applications of finite differences. His mathematical work continued up to 1820 (and his interest in mathematics was evident in many later works on other topics) but even before 1820 his interest in other subjects had begun to take him away from research in mathematics. It is interesting to think that it was in some ways due to Herschel's remarkable all round abilities that he failed to make an advance of the depth that he was clearly capable of in any of the subjects that he studied. In all of them he could have been the person that we remember today as the world leader of his time. Yet the fact that he contributed to so many areas meant that he had usually moved on to something else when he might have been consolidating his work in a single area. Certainly his mathematical work, important as it was, never had the influence which he might have been able to have achieved had he continued to work in the area. Perhaps most surprising of all was the decision that Herschel made after graduating. He decided to enter the legal profession, much against the advice of his father who wanted him to join the Church, and he went to London in February 1814 to begin training. It was not long before he found that it was not right for him and he wrote to Babbage in September 1814 saying that he would stick it out since it was his chosen career. However, after 18 months, he gave up his legal training and returned to Cambridge as a tutor and examiner in mathematics. Herschel spent a holiday with his father during the summer of 1816. He seems to have decided during this holiday to turn to astronomy, almost certainly influenced by the fact that at 78 years of age his father's health was failing and there was nobody else to continue his father's work. He wrote on 10 October 1816 to Babbage (see C A Ronan's article in [5]):I shall go to Cambridge on Monday where I mean to stay just enough time to pay my bills, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herschel.html (2 of 6) [2/16/2002 11:14:23 PM]

Herschel

pack up my books and bid a long - perhaps a last farewell to the University. ... I am going under my father's directions, to take up the series of his observations where he has left them (for he has now pretty well given over regular observing) and continuing his scrutiny of the heavens with powerful telescopes ... Indeed John Herschel began to undertake work in astronomy from this time although he also studied other topics. Even before his first astronomy paper was published, Herschel published details of his chemical and photography experiments in 1819 which, 20 years later, would prove of fundamental importance in the development of photography. He was very much involved with the founding of the Astronomical Society in 1820 and he was elected vice-president at the second meeting of the Society when the officers of the Society were elected. Herschel's great versatility is shown by the fact that in 1821, having recently become involved in astronomy and chemistry, he was awarded the Copley Medal of the Royal Society of London for his work on mathematical analysis. Babbage and Herschel remained close friends and the two travelled together to Italy and Switzerland in 1821. They shared a love of climbing mountains but their exploits here were not simply for pleasure as Herschel continually learnt about science from his environment. Other trips abroad by Herschel included one in 1822 and another with Babbage in 1824. These trips included visits to other scientist, for example while in Paris Herschel and Babbage had discussed topics of common interest with Arago, Laplace and Biot. On the 1824 trip Herschel had visited Fraunhofer and had met with Fox Talbot who was visiting Fraunhofer at the time. He also visited Pfaff, who was working on a German translation of his father's work, and his aunt Caroline Herschel who had returned to Hanover after the death of her brother (John Herschel's father) in 1822. In fact 1822 was the year in which John Herschel published his first paper on astronomy, a relatively minor work on a new method to calculate eclipses of the moon. His first major publication in astronomy was a catalogue of double stars which he published in the Transactions of the Royal Society in 1824 and for which he received honours. The Paris Academy awarded him its Lalande Prize in 1825 and the Astronomical Society awarded him its Gold Medal the following year. The work on double stars had been undertaken as a continuation of his father's work which attempted to measure the parallax of a star. The parallax is the apparent change in position of a relatively nearby star against the background of very distant stars due to the change in position of the earth in its orbit around the sun. John Herschel published an important paper on this topic in 1826 but did not succeed in determining the parallax of any star. He continued to work on double stars until 1833, in particular he developed methods to determine the orbits of those double stars which orbited a common centre of gravity and for this work he received the Royal Medal of the Royal Society in 1833. Bessel was the first to demonstrate the parallax of a star, publishing his results in 1840. In 1830 Herschel published his famous treatise Discourse on Natural Philosophy the importance of which is beautifully described by Faraday in writing to Herschel:... when your work on the study of Natural Philosophy came out, I read it, as others did with delight. I took it as a school book for philosophers, and feel that it has made me a better reasoner and even experimenter, and has altogether heightened my character, and made me, if I may be permitted to say so, a better philosopher. Herschel's involvement with the Royal Society had important influences on his career. He had been

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herschel.html (3 of 6) [2/16/2002 11:14:23 PM]

Herschel

elected Secretary of the Society in 1824 (although he resigned in 1827) and, in 1831, he was proposed by Babbage for President of the Royal Society. This was part of a battle that was going on in the Society between reformers and traditionalists, Herschel being the champion of the reformers. In a slightly embarrassing episode for Herschel, he failed narrowly to be elected, the traditionalists winning the day. He had been elected as President of the Astronomical Society in 1827 and he was knighted in 1831 so he received great recognition despite his disappointment with the Royal Society. The Royal Society episode may have been the main reasons why Herschel decided to make a long visit to the Cape of Good Hope in South Africa. Also it became possible at this time because of the death of his mother Mary Herschel in 1832. The Royal Observatory had been completed at the Cape of Good Hope in 1828 with the scientific aim to catalogue interesting astronomical objects which could not be observed from the northern hemisphere. Herschel sailed for South Africa in 1833, taking with him his own 20 foot refractor telescope. He travelled with his family, having married Margaret Brodie Stewart four years earlier, and their ship reached South Africa in January 1834. Among the many important scientific advances made by Herschel in South Africa was his observations of Halley's comet on its 1835 return. He recognised that the comet was being subjected to major forces other than gravitation and he was able to calculate that the force was one repelling it from the sun. This could in some sense be said to constitute the discovery of the solar wind which is indeed the reason for the repulsive force discovered by Herschel. He also made the important discovery that gas was evaporating from the comet. The geologist Lyell saw the irony in the fact that Herschel had done this work in South Africa because he failed to be elected President of the Royal Society. He wrote (see for example [5]):Fancy exchanging Herschel at the Cape for Herschel as President of the Royal Society, which he so narrowly missed being, and I voted for him too! I hope to be forgiven for that. In 1838 Herschel returned to England not having had much time to reduce the many observations that he had made while there. His intention was to work on these as soon as he returned but he was kept so busy with other events that this task was not completed until 1847. Despite the delay, the publication was an event of great importance and Herschel received his second Copley Medal from the Royal Society for this work. The events which prevented him from working on his observations after his return were partly non-scientific such as his investment to a baronetcy in June 1838 which he accepted with considerable reluctance. The main distractions were, however, scientific ones. On 22 January 1839 Herschel heard of Daguerre's work on photography from a casual remark in a letter written by Beaufort to Margaret his wife. Without knowing any details, Herschel was able to take photographs himself within a few days. Fox Talbot visited him on the 1 February to discuss their separate experiments in photography. In his article in [5] Roberts quotes from a letter by Herschel's wife:I happen to remember well the visit to Slough of Mr Fox Talbot, who came to show Herschel his beautiful little pictures of Ferns and Laces taken by his new process. - when something was said about the difficulty of fixing the pictures, Herschel said "Let me have this one for a few minutes" and after a short time he returned and gave the picture to Mr Fox Talbot saying "I think you'll find that fixed" - this was the beginning of the hyposulphite plan of fixing. Indeed Herschel was able to achieve this remarkably rapid breakthrough due to the work that he had

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herschel.html (4 of 6) [2/16/2002 11:14:23 PM]

Herschel

conducted and published in 1819 on chemical processes related to photography. Herschel went on to publish a number of further papers on photography, in 1839, 1840 and 1842. In fact many people have wondered why Herschel himself never made the steps which would have led to his being recognised as the inventor of photography. There was a period of around 20 years from the time of his 1819 work when he might have made the breakthrough but again it was probably due to his wide ranging talents that he failed to do so. Not only did his talents take him into a wide range of other activities but his great skill as an artist meant that he had less need to invent photography than most others! In 1850 Herschel made a rather strange decision as to the future direction of his career. He had turned down entering parliament as a Cambridge University member and he had also turned down a proposal that he become president of the Royal Society. He did become rector of Marischal College in Aberdeen in 1842 and served as president of the British Association at Cambridge in 1845. In 1850, however, he accepted the post of Master of the Mint. He thus became head of the Mint at a very difficult time, with a major reform already agreed but its implementation not begun. It was not a job which Herschel found to his liking. There were difficulties in dealing with staff, difficulties in dealing with the Treasury, but perhaps most significantly of all to Herschel, he no longer could pursue his scientific interests. Herschel was a scientist by nature, not a business man. After difficult years, frequently separated from his family by the demands of the job which he tried his best to do well, he resigned. His health had suffered through the stresses of the post and it must have come as a great relief to him to be free of the problems. He essentially retired to Collingwood at this stage in his career being 63 years of age. To say he retired needs of course to be qualified with the remark that throughout his life he never held any paid scientific post. He certainly did not retire from scientific work but, free of the huge demands of the job at the Mint for which he had received a fair salary, he was able to complete many tasks which he and his father had started but had never reached completion. As Tait wrote (see for example [9]):Every day of Herschel's long and happy life added its share to his scientific services. Although he made no major breakthroughs for which he is remembered today he was considered by many of his contemporaries as the leading scientist of his day. Biot considered him the natural successor to Laplace when he died in 1827. In the obituary [12] it is remarked:In John Frederick William Herschel British science has sustained a loss greater than any which it has suffered since the death of Newton, and one not likely to be replaced... The article [12] then goes on to say that of even higher value than his research:... was the influence of his teaching and example in wakening the public to the power and beauty of science, and stimulating and guiding its pursuit. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herschel.html (5 of 6) [2/16/2002 11:14:23 PM]

Herschel

Cross-references to History Topics

1. Orbits and gravitation 2. Mathematical discovery of planets 3. A visit to James Clerk Maxwell's house

Other references in MacTutor

Chronology: 1810 to 1820

Honours awarded to John Herschel (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1813

Royal Society Copley Medal

Awarded 1821 and 1847

Royal Society Royal Medal

Awarded 1833, 1836 and 1840

Royal Society Bakerian lecturer

1823

Fellow of the Royal Society of Edinburgh Lunar features

Crater J Herschel

Planetary features

Crater Herschel on Mars

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Herschel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herschel.html (6 of 6) [2/16/2002 11:14:23 PM]

Herschel_Caroline

Caroline Lucretia Herschel Born: 16 March 1750 in Hannover, Hanover (now Germany) Died: 9 Jan 1848 in Hannover, Hanover (now Germany)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Caroline Herschel was the daughter of Isaac Herschel and Anna Ilse Moritzen. She was sister of William Herschel and the aunt of John Herschel. Caroline's father Isaac was an oboist in the Hanovarian Foot Guards and rose to become the bandmaster. Although a man with no formal education, he tried hard to give his four sons and two daughters a good education. His interests in music, philosophy and astronomy led to lively conversations in their home but Caroline's mother disapproved of learning in general and although she reluctantly accepted that her four sons should have some education, she strongly opposed her daughters doing anything other than the household chores. Caroline Herschel's four brothers were all brought up to be musicians while Caroline showed an enthusiasm for knowledge which her father tried to satisfy despite all her mother's efforts to ensure that she did nothing but household tasks. Caroline recalled that her father took her [4]:... on a clear frosty night into the street, to make me acquainted with several of the beautiful constellations, after we had been gazing at a comet which was then visible. Caroline could never have thought in her wildest dreams that one day she would make a major contribution to the study of comets. After the French occupation of Hanover in 1757, Isaac was occupied fighting the French and so was not at home. William escaped to England, where he became a music teacher, and Caroline was left under the control of her mother who sent her to learn to knit and otherwise kept her fully occupied with household chores. In 1760 Isaac returned home in poor health and Caroline essentially lived the life of a servant until he died in 1767. The death of her father seems to have made Caroline realise that she had to take some control of her own life and she took lessons in dressmaking and studied to qualify as a governess. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herschel_Caroline.html (1 of 4) [2/16/2002 11:14:25 PM]

Herschel_Caroline

However, fitting in the studies while her mother demanded so much work from her proved a great strain. In 1766 William became an organist in Bath and, in 1772, Caroline joined him there. She made this move despite strong protests from her mother who was very unhappy at effectively losing a servant. Caroline had always been very close to her brother William and, after arriving in Bath, she trained as a singer receiving lessons from her brother. William taught Caroline more than musical skills. He had studied mathematics and astronomy in his spare time at the end of a long day after many hours teaching music, reading works such as Maclaurin's Fluxions. Now he began to teach Caroline English and mathematics while he himself became more and more involved with astronomy. Caroline began giving successful singing performances [6]:As first treble in the Messiah, Judas Macabaeus, etc., she sang at Bath or Bristol sometimes five nights in the week, but declined an engagement for the Birmingham festival, having resolved to appear only where her brother conducted. In addition to her singing, Caroline helped William with his musical activities and looked after him while he spent many hours with his new hobby of constructing telescopes. Slowly Caroline turned more and more towards helping William with his astronomical activities while he continued to teach her algebra, geometry and trigonometry. In particular Caroline studied spherical trigonometry which would be important for reducing astronomical observations. However, she was not interested in mathematics for its own sake, finding only those parts which were useful in applications worth studying. Almost inevitably Caroline's role changed from looking after William to helping him with his scientific activities which soon occupied every available moment. Caroline wrote (see for example [4]):Every leisure moment was eagerly snatched at for resuming some work which was in progress, without taking time or changing dress, and many a lace ruffle ... was torn or bespattered by molten pitch. ... I was even obliged to feed him by putting the vitals by bits into his mouth; - this was once the case when at the finishing of a 7 foot mirror he had not left his hands from it for 16 hours ... Astronomy changed from a hobby for William in 1781 when he achieved fame by discovering the planet now named Uranus. King George III gave William a 200 per year salary which was less than generous but sufficient to allow him to become a full-time astronomer. Giving up their musical activities the Herschels moved to Datchet in August 1782 where they remained until June 1785 when they moved again, this time into Clay Hall, near Windsor. It was certainly not without many regrets that Caroline abandoned music and began to take an active part in astronomy. William gave her a telescope with which she began to make observations, in particular searching for comets making methodical sweeps of the sky. Caroline found much less time than she expected to make her own observations as she became fully involved helping William with his astronomical projects. By day Caroline would work on the results obtained by William while observing on the previous night. She carried out the lengthy calculations necessary to reduce William's data with remarkable accuracy. In fact only when William was away from home was Caroline able to spend much time with her own program of research. In April 1786 William and Caroline moved to a new home they called Observatory House which was in Slough and there, on 1 August 1786, Caroline discovered her first comet which was described by some as the "first lady's comet". This discovery brought Caroline a certain degree of fame and articles were written about her. In one such article she is described as (see [6]):... very little, very gentle, very modest, and very ingenuous... http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herschel_Caroline.html (2 of 4) [2/16/2002 11:14:25 PM]

Herschel_Caroline

while another describes her as:... a most excellent, kind-hearted creature... In 1787 King George III gave Caroline a 50 per year salary as assistant to William. In the following year William married Mary Pitt and Caroline's life changed markedly [9]:Initially Caroline was deeply affected by the marriage, and moved out to lodgings at Upton. She continued to support her brother's work and in making the daily walk to Observatory House, became a well-known figure. Often, with William resting after a long night of observation, the house was kept as quiet as possible during the day. Eventually the relationship between the two ladies - Mary and Caroline - warmed ... Caroline kept a diary into which she had recorded her thoughts, in particular she had recorded her great distress at the change in relationship with her brother and she also recorded her bitterness towards his wife. However, as the relationship between the two ladies improved, Caroline regretted her bitter comments against Mary and she destroyed every page of her diary over this time in her life. In total Caroline discovered eight comets between 1786 and 1797 and she then embarked on a new project of cross-referencing and correcting the star catalogue which had been produced by Flamsteed. In 1798 Caroline submitted to the Royal Society an Index to Flamsteed's Observations of the Fixed Stars together with a list of 560 stars which had been omitted. This publication marked the temporary end of her own researches which she would not begin again until 25 years later after William's death. This period of 25 years was not one which lacked interest for Caroline. She became involved with the education of John Herschel, William and Mary Herschel's son who was born in 1792. John Herschel spent long periods with his aunt during the vacations and was greatly influenced by Caroline. She saw him educated at Cambridge, make a name for himself as a mathematician, become elected to the Royal Society, join his father in research in astronomy and be awarded the Copley Medal of the Royal Society for his achievements. Caroline continued to assist William with his observations but her status had greatly improved from the housekeeper she had been in her young days. She was the guest of Maskelyne at the Royal Observatory in 1799 and a guest of members of the Royal Family at various times in 1816, 1817 and 1818. Caroline returned to Hanover after William's death in 1822. In many ways it was a bad decision, made too quickly, which she soon regretted but she was always one to keep a promise whatever the personal consequences so she would never return to England. All her energies had been directed towards helping her brother in his astronomical work during his lifetime but now she turned to help his son John Herschel. Certainly this help was not given in the same way as a personal assistant, but rather now as independent researcher producing a catalogue of nebulae to assist John in his astronomical work. She completed her catalogue of 2500 nebulae and, in 1828, the Royal Astronomical Society awarded her its gold medal for this work. Although Caroline regretted spending her last 25 years in Hanover there were many compensations. She was now a celebrity in the world of science and she was visited by many scientists including Gauss. Her nephew John Herschel, visiting Caroline in June 1832 when she was 83 years old, wrote of her (see for example [6]):She runs about the town with me, and skips up her two flights of stairs. In the morning, till

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herschel_Caroline.html (3 of 4) [2/16/2002 11:14:25 PM]

Herschel_Caroline

eleven or twelve, she is dull and weary, but as the day advances she gains life, and is quite 'fresh and funny' at ten p.m., and sings old rhymes, nay, even dances. Caroline Herschel received many honours for her scientific achievements. Together with Mary Somerville, she was elected to honorary membership of the Royal Society in 1835. They were the first honorary women members. She was also elected a member of the Royal Irish Academy in 1838 and then on her 96th birthday she received a letter [3]:His Majesty the King of Prussia, in recognition of the valuable service rendered to astronomy by you, as the fellow worker of your immortal brother, wishes to convey to you in his name the Large Gold Medal for science. On her 97th birthday Caroline [6]:... entertained the crown prince and princess with great animation for two hours, even singing to them a composition of her brother William. A minor planet was named Lucretia in 1889 in Caroline Lucretia Herschel's honour, a fitting tribute one who had contributed so much yet had so little personal ambition that she disliked praise directed towards her least it detract from her brother William. Article by: J J O'Connor and E F Robertson List of References (10 books/articles) A Poster of Caroline Herschel

Mathematicians born in the same country

Cross-references to History Topics

Mathematical discovery of planets

Honours awarded to Caroline Herschel (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater C Herschel

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Herschel_Caroline.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herschel_Caroline.html (4 of 4) [2/16/2002 11:14:25 PM]

Herstein

Israel Nathan Herstein Born: 28 March 1923 in Lublin, Poland Died: 9 Feb 1988 in Chicago, Illinois, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yitz Herstein was named Yitzchak and known as 'Yitz' by his friends. His family emigrated to Canada in 1928. He studied at Manitoba, receiving his B.A. in 1945, then at Toronto where he was awarded an M.A. the following year. Herstein moved to the University of Indiana and received a Ph.D. in 1948 for a thesis written under Zorn's supervision. Herstein worked at the University of Kansas for two years, then at Ohio State University for a year before being appointed to Chicago in 1951. There he was influenced by Abraham Albert. During this time he worked on a topic which was to be one of the main themes of his work, namely on conditions on a ring which imply commutativity. For example he worked on conditions of the type xn = x first studied by Jacobson in 1945. After posts at Pennsylvania and Cornell, he returned to Chicago in 1962 and remained there for the rest of his life. In addition to work on rings and algebras Herstein also worked on groups and fields. In particular he examined finite subgroups of a division ring. In [1] 115 publications on these topics are listed. Herstein is perhaps best known for his beautifully written algebra texts, especially the undergraduate text Topics in algebra (1964). Other algebra books included a more advanced ring theory book Noncommutative rings (1968) and a book which he worked on in the last two years of his life Abstract algebra (1986). Much of his own research is put into context in his book Rings with involution (1976). http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herstein.html (1 of 2) [2/16/2002 11:14:27 PM]

Herstein

Herstein supervised 30 research students. One said of him:He was someone of great warmth who took an intense personal interest in his students and had a knack of getting them to believe in themselves. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article)

A Quotation

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Herstein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Herstein.html (2 of 2) [2/16/2002 11:14:27 PM]

Hesse

Ludwig Otto Hesse Born: 22 April 1811 in Königsberg, Germany (now Kaliningrad, Russia) Died: 4 Aug 1874 in Munich, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Otto Hesse studied under Jacobi at Königsberg and spent a while as a teacher of physics and chemistry before he graduated from Königsberg in 1840. Hesse then became a lecturer at Königsberg and spent 16 years there publishing his work in Crelle's Journal. In 1856 he was appointed to Heidelberg and remained there until 1868 when he took up a post at Munich. Hesse's main work was in the development of the theory algebraic functions and the theory of invariants. He introduced the Hessian determinant in a paper in 1842 during an investigation of cubic and quadratic curves. His work was influenced by Steiner, particularly work he did on the geometrical interpretation of algebraic transformations. Hesse worked on some topics that Cayley was also working on and both produced a theory of homogeneous forms which they published at the same time. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hesse.html (1 of 2) [2/16/2002 11:14:28 PM]

Hesse

List of References (5 books/articles) A Poster of Otto Hesse

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1840 to 1850

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hesse.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hesse.html (2 of 2) [2/16/2002 11:14:28 PM]

Heuraet

Hendrik van Heuraet Born: 1633 in Haarlem, Netherlands Died: 1660 in (probably) Leiden, Netherlands Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hendrik van Heuraet entered the University of Leiden in 1653 intending to study medicine. However he studied mathematics as well as medicine, studying under van Schooten with fellow students Huygens and Hudde. Little is known of van Heuraet's life, all that is known that he was at Saumur, a town on the river Loire in western France, in 1658. Van Schooten had established a vigorous research school in Leiden which included van Heuraet, and this school was one of the main reasons for the rapid development of Cartesian geometry in the mid 17th century. Van Schooten edited and published a Latin translation of Descartes's La Géométrie in 1649. A second two-volume translation of the same work (1659-1661) contained appendices by de Witt, Hudde and van Heuraet. In van Heuraet's only publication he effectively computes the integral (1+y'2) dx and applies his methods to the parabola. His methods of rectification of curves became part of a more general theory by Fermat which was produced independently and at about the same time. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1650 to 1675

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heuraet.html (1 of 2) [2/16/2002 11:14:29 PM]

Heuraet

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heuraet.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heuraet.html (2 of 2) [2/16/2002 11:14:29 PM]

Heyting

Arend Heyting Born: 9 May 1898 in Amsterdam, Netherlands Died: 9 July 1980 in Lugano, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Arend Heyting's father was Johannes Heyting and his mother was Clarissa Kok. Both Arend's parents were school teachers and Johannes Heyting was particularly successful in his profession being appointed as head of a secondary school. Arend spent his school years with the intention that he would make a career in engineering. Only near the end of his schooling did his love and ability in mathematics mean that the course of his career changed and he went to university to study mathematics. Although Heyting's father was a successful school teacher, the family were still in financial problems when Heyting began his studies in 1916 at the University of Amsterdam. Both Heyting and his father earned the extra money necessary to pay for his studies by taking on private tutoring work. At the University of Amsterdam Heyting was taught by Brouwer who had a large influence on his future work. In 1922 Heyting graduated with a degree of master's standard. At this point in his career Heyting began to follow the same road as his parents by beginning a career as a secondary school teacher. He taught in two school in the town of Enschede. In the Overijssel province in eastern Netherlands, standing on the Twente Canal near the German border, this industrial town with its cotton-textile industry was not an ideal place for an academic to be living. Heyting was not well placed to make contact with colleagues in the universities, yet he spent all his free time working on his research. He received his doctorate in 1925 for a thesis written under Brouwer's supervision. His dissertation Intuitionistische axiomatieks der projektieve meetkunde (Intuitionistic axiomatics of projective geometry) was the first study of axiomatisation in constructive mathematics. When the Dutch Mathematical Association announced a prize question in 1927 they gave Heyting an ideal topic on which to compete.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heyting.html (1 of 3) [2/16/2002 11:14:31 PM]

Heyting

They asked for a formalisation of Brouwer's intuitionist theories and Heyting's outstanding essay was awarded the prize in 1928. This essay was then polished and expanded by Heyting and published in 1930. It made Heyting's name well known among those interested in the philosophy of mathematics. This work had another beneficial effect as far as Heyting was concerned for it brought him to the attention of Heinrich Scholz who held the chair of mathematical logic in Münster. Scholz made his extensive library available to Heyting, fortunate since Münster was relatively close to Enschede, and a lifelong friendship arose between the two. Heyting's academic isolation in Enschede no longer seemed the problem that it might have been. By this time Heyting had married Johanne Friederieke Nijenhuis. They couple were married in 1929 and had eleven children. After 31 years of marriage they divorced in 1960. Heyting attended the Erkenntnis symposium at Konigsberg in September 1930. There he represented intuitionism while Carnap and von Neumann represented logicism and formalism respectively. Each argued their own case and against that of the other two. Although Heyting's version of intuitionist logic differed somewhat from that of Brouwer, it is clear that one of his main aims was to make Brouwer's ideas more accessible and better known. Brouwer had presented his ideas in a deliberately non-formal, and very personal, way. There were others interested in intuitionist logic working on similar problems of formalisation at the same time as Heyting. One was Kolmogorov who corresponded with Heyting. The article [6] ([7] is the English translation) reproduces three letters which Kolmogorov sent to Heyting, the first in 1931 questions the distinction between a proposition P and the statement "P is provable". In 1934 Heyting published Intuitionism and Proof Theory [1]:... a concise and well-written survey in which the viewpoints of intuitionism and formalism are clearly described and contrasted. Heyting was appointed as a Privatdozent at the University of Amsterdam in 1936 and in the following year he was appointed as a lecturer. He spent the rest of his career at the University of Amsterdam, being promoted to professor in 1948. He held this position for twenty years until he retired in 1968. Heyting published a paper on intuitionistic algebra in 1941 and intuitionistic Hilbert spaces in the 1950's. These were ground-breaking works. Another major treatise which has presented intuitionism to both mathematicians and logicians was Intuitionism: an Introduction (1956, second edition 1966). Gilmore begins his excellent review of this book as follows:This is an introduction to intuitionistic mathematics for mature mathematicians. The reader is taken rapidly to the heart of several different branches of intuitionistic mathematics. The speed of development is achieved by condensing the proofs and by presuming familiarity with the classical counterparts to the theories discussed. The book is written as a dialogue between Class (a classical mathematician), Form (a formalist), Int (an intuitionistic mathematician), Letter (a finitistic nominalist), Prag (a pragmatist), and Sign (a significist). In the first chapter Int defends intuitionistic mathematics against the criticism of the others, asking them finally to judge for themselves. In the remaining chapters Int presents mathematics for them to judge. In these chapters Class, except for Int, is the most loquacious; he frequently compares classical results with http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heyting.html (2 of 3) [2/16/2002 11:14:31 PM]

Heyting

corresponding intuitionistic results and his questions lead Int to a more detailed discussion of some points. The device of dialogue allows abbreviation of statements without loss of clarity. The article [3] shows the major influence that Heyting has had on the study of the foundations of mathematics and in so doing shows the importance of Heyting's contributions. Franchella argues that Heyting has been the cause of two major changes of direction. Firstly, at least partly because of him, the topic has moved away from trying to answer the big problems such as "What is mathematics?". Heyting moved away from these big problems, concentrating on trying to identify formal, intuitive, and logical concepts in the study of mathematics. The second change which Franchella argues that Heyting brought about was a realisation that there exist degrees of evidence in mathematics. This is a particularly important aspect of mathematics today when computer programs are being used to verify mathematical proofs:What was specific to intuitionism, however, was the thesis that mathematics is an activity, a process of becoming, the exhaustive description of which is impossible, just as it is impossible to define once and for all its elementary concepts. We should end this biography by giving an indication of Heyting's personality. Troelstra writes in [1]:Heyting was retiring and modest, lacking all ostentation. His interests were very wide-ranging and varied: music, literature, linguistics, philosophy, astronomy, and botany; he also was fond of walking. As a teacher and lecturer he impressed his students and his international audiences at congresses with his exceptionally clear presentations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) A Poster of Arend Heyting

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Heyting.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Heyting.html (3 of 3) [2/16/2002 11:14:31 PM]

Higman

Graham Higman Born: 19 Jan 1917

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Graham Higman is the second son of the Reverend Joseph Higman. He was educated at Sutton Secondary School in Plymouth and won a natural sciences scholarship to study at Balliol College Oxford. He chose Balliol College because that was the College where his elder brother had studied but, since his elder brother had read chemistry, Graham decided he had to be different in some respect and so, despite holding a natural sciences scolarship, he chose to read mathematics. Higman's tutor was Henry Whitehead and it took a while before he realised that Higman was a natural pure mathematician since most students on a natural sciences scholarship would only take mathematics courses to support their science studies. Following a suggestion by Whitehead, Higman founded the Invariant Society (an Oxford undergraduate mathematical society). The first speaker was G H Hardy who addressed the Invariant Society on round numbers. After taking special topic courses on group theory and differential geometry, Higman received an MA. After graduating Higman continued to study for his doctorate at Oxford. His doctoral research was supervised by Henry Whitehead and he was awarded a DPhil. for his thesis The units of group-rings. In this work, among other results, he classified group rings over the rational numbers without non-trivial units. After his doctoral studies Higman spent a year at the University of Cambridge where he was strongly influenced by Philip Hall. He also met Max Newman in Cambridge and Newman's interest in the interaction between group theory and logic had a lasting influence on him. By this time World War II had started and Higman had signed up as a conscientious objector. However he did war service in the Meteorological Office from 1940 to 1946. He began this work, which did not involve using any mathematical skills or knowledge, in Lincolnshire, also spending time in Northern Ireland and Gibraltar. At the end of the war Higman decided to apply for a permanent post in the Meteorological Office but, after being asked at an interview why he had not chosen to enter the academic world, he turned down the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Higman.html (1 of 4) [2/16/2002 11:14:33 PM]

Higman

offered post and looked for an academic career. The first offer of such a post came from the University of Durham but, preferring to go to Manchester to be with Max Newman, he turned down Durham before receiving any offer from Manchester. The offer from Manchester did come and, in 1946, Higman was appointed as a lecturer in mathematics at the University of Manchester. When he first arrived at Manchester Higman worked with Max Newman. Walter Ledermann was also appointed to Manchester in 1946 and then Bernhard Neumann arrived in 1948. Higman and Neumann collaborated proving a number of very significant results which we comment on below. Despite the large amount of activity in group theory which was going on in Manchester, Higman was ambitious and began to apply for professorships. It is not surprising that at this early stage in his career he was unsuccessful, but Henry Whitehead encouraged him to return to Oxford rather than to seek a chair at a second rate place. He did so in 1955 being appointed as a Lecturer in Mathematics in Oxford and then, very soon after, he was promoted to Reader in Mathematics. In 1958 he was honoured with election as a Fellow of the Royal Society of London and, in the same year, he became a Senior Research fellow at his old Oxford College, Balliol College. Higman was appointed Waynflete Professor of Pure Mathematics at Oxford in October 1960 and, at the same time, he was elected a Fellow of Magdalen College Oxford. He held these positions until he retired in 1984. Immediately after he retired from Oxford, Higman went to the United States where he was George A Miller visiting professor at the University of Illinois for the two years from 1984 to 1986. Higman is known for his outstanding work in all aspects of the theory of groups. He published on units in group rings, the subject of his doctoral thesis, in 1940 then there was a break in his publication record during the time he worked in the Meteorological Office. His 1948 papers are on somewhat different topics, being on topological spaces and linkages. They show the influences of Henry Whitehead and, to a lesser extent, Max Newman. In 1949 Higman published one of several major pieces of work which stand as a landmark in the development of group theory. His paper Embedding theorems for groups written jointly with both Bernhard Neumann, who as we noted was a colleague of Graham Higman's at Manchester at that time, and with Hanna Neumann, introduces the now standard group construction of HNN extensions (Higman Neumann - Neumann extensions). Higman published further important papers in 1951 when he gave an example of a finitely presented group which is isomorphic to a proper factor of itself, and Higman's famous example of a finitely generated infinite simple group. Other work which he did around this time was on unrestricted free products and topological groups. Two further papers written jointly with Bernhard Neumann were Groups as groupoids with one law and On two questions of Ito. After working on finitely generated nilpotent groups and infinite simple permutation groups, Higman, together with Philip Hall, produced another of his landmark papers in 1956 On the p-length of p-soluble groups and reduction theorems for Burnside's problem. It is this paper which introduced many important ideas but the most significant result was a reduction theorem for the restricted Burnside problem which essentially reduced the problem to looking only at groups of prime power exponent. This result plays a vital part in Zelmanov's positive solution to the restricted Burnside problem in the early 1990s. Higman made other contributions to the Burnside problem with a paper on http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Higman.html (2 of 4) [2/16/2002 11:14:33 PM]

Higman

groups of exponent 5. Perhaps his most surprising result, and one of his most influential, appeared in Subgroups of finitely presented groups published in the Proceedings of the Royal Society in 1961. There he proved:Any finitely generated group can be embedded in a finitely presented group if and only if it is recursively presented. As a corollary to this theorem Higman proved the existence of a universal finitely presented group containing every finitely presented group as a subgroup. Another application in the paper produced a new example of a finitely presented group with unsolvable word problem. Higman also worked on topics such as: varieties of groups; enumerating p-groups; and Lie ring methods for finite nilpotent groups. Then in 1967 Higman became interested in the sporadic finite simple groups being discovered at this time and played an important role in constructing certain of these groups from a knowledge of their character tables. He published papers on the Higman-Sims simple group (named after D G Higman and not Graham Higman) and on Janko's group of order 50232960. I [EFR] attended a lecture course which he gave at Oxford on this topic in the 1960s. It was a very exciting course, presenting results which he had only just proved, and often, it seemed, results he worked out on the blackboard during the lectures. Together will Bill Boone, Higman worked on the word problem and together they wrote two papers on the algebraic structure of groups with soluble word problem and with soluble order problem. [A finitely generated group has soluble order problem if given any word in the generators there is an algorithm to determine its order.] In the years before he retired from Oxford in 1984, Higman gave a course on recent work on existentially closed groups. Elizabeth Scott took notes of the course and the lecture notes were published as a joint Higman-Scott publication. Higman has received many honours for his outstanding work in group theory. He was awarded the Berwick Prize from the London Mathematical Society in 1962 and the De Morgan Medal from that Society in 1974. He served the London Mathematical Society as its 52nd president from 1965 to 1967. In addition to his election as a Fellow of the Royal Society in 1958, the Royal Society awarded him its Sylvester Medal in 1979. He has received a number of honorary degrees. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Graham Higman (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Higman.html (3 of 4) [2/16/2002 11:14:33 PM]

Higman

Fellow of the Royal Society

Elected 1958

Royal Society Sylvester Medal

Awarded 1979

London Maths Society President

1965 - 1967

LMS De Morgan Medal

Awarded 1974

LMS Berwick Prize winner

1962

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2001 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Higman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Higman.html (4 of 4) [2/16/2002 11:14:33 PM]

Hilbert

David Hilbert Born: 23 Jan 1862 in Königsberg, Prussia (now Kaliningrad, Russia) Died: 14 Feb 1943 in Göttingen, Germany

Click the picture above to see eight larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

David Hilbert attended the gymnasium in his home town of Königsberg. After graduating from the gymnasium, he entered the University of Königsberg. There he went on to study under Lindemann for his doctorate which he received in 1885 for a thesis entitled Über invariante Eigenschaften specieller binärer Formen, insbesondere der Kugelfunctionen. One of Hilbert's friends there was Minkowski, who was also a doctoral student at Königsberg, and they were to strongly influence each others mathematical progress. In 1884 Hurwitz was appointed to the University of Königsberg and quickly became friends with Hilbert, a friendship which was another important factor in Hilbert's mathematical development. Hilbert was a member of staff at Königsberg from 1886 to 1895, being a Privatdozent until 1892, then as Extraordinary Professor for one year before being appointed a full professor in 1893. In 1892 Schwarz moved from Göttingen to Berlin to occupy Weierstrass's chair and Klein wanted to offer Hilbert the vacant Göttingen chair. However Klein failed to persuade his colleagues and Heinrich Weber was appointed to the chair. Klein was probably not too unhappy when Weber moved to a chair at Strasbourg three years later since on this occasion he was successful in his aim of appointing Hilbert. So, in 1895, Hilbert was appointed to the chair of mathematics at the University of Göttingen, where he continued to teach for the rest of his career. Hilbert's eminent position in the world of mathematics after 1900 meant that other institutions would have liked to tempt him to leave Göttingen and, in 1902, the University of Berlin offered Hilbert Fuchs' chair. Hilbert turned down the Berlin chair, but only after he had used the offer to bargain with Göttingen http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hilbert.html (1 of 5) [2/16/2002 11:14:35 PM]

Hilbert

and persuade them to set up a new chair to bring his friend Minkowski to Göttingen. Hilbert's first work was on invariant theory and, in 1888, he proved his famous Basis Theorem. Twenty years earlier Gordan had proved the finite basis theorem for binary forms using a highly computational approach. Attempts to generalise Gordan's work to systems with more than two variables failed since the computational difficulties were too great. Hilbert himself tried at first to follow Gordan's approach but soon realised that a new line of attack was necessary. He discovered a completely new approach which proved the finite basis theorem for any number of variables but in an entirely abstract way. Although he proved that a finite basis existed his methods did not construct such a basis. Hilbert submitted a paper proving the finite basis theorem to Mathematische Annalen. However Gordan was the expert on invariant theory for Mathematische Annalen and he found Hilbert's revolutionary approach difficult to appreciate. He refereed the paper and sent his comments to Klein:The problem lies not with the form ... but rather much deeper. Hilbert has scorned to present his thoughts following formal rules, he thinks it suffices that no one contradict his proof ... he is content to think that the importance and correctness of his propositions suffice. ... for a comprehensive work for the Annalen this is insufficient. However, Hilbert had learnt through his friend Hurwitz about Gordan's letter to Klein and Hilbert wrote himself to Klein in forceful terms:... I am not prepared to alter or delete anything, and regarding this paper, I say with all modesty, that this is my last word so long as no definite and irrefutable objection against my reasoning is raised. At the time Klein received these two letters from Hilbert and Gordan, Hilbert was an assistant lecturer while Gordan was the recognised leading world expert on invariant theory and also a close friend of Klein's. However Klein recognised the importance of Hilbert's work and assured him that it would appear in the Annalen without any changes whatsoever, as indeed it did. Hilbert expanded on his methods in a later paper, again submitted to the Mathematische Annalen and Klein, after reading the manuscript, wrote to Hilbert saying:I do not doubt that this is the most important work on general algebra that the Annalen has ever published. In 1893 while still at Königsberg Hilbert began a work Zahlbericht on algebraic number theory. The German Mathematical Society requested this major report three years after the Society was created in 1890. The Zahlbericht (1897) is a brilliant synthesis of the work of Kummer, Kronecker and Dedekind but contains a wealth of Hilbert's own ideas. The ideas of the present day subject of 'Class field theory' are all contained in this work. Rowe, in [18], describes this work as:... not really a Bericht in the conventional sense of the word, but rather a piece of original research revealing that Hilbert was no mere specialist, however gifted. ... he not only synthesized the results of prior investigations ... but also fashioned new concepts that shaped the course of research on algebraic number theory for many years to come. Hilbert's work in geometry had the greatest influence in that area after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance. He published Grundlagen der Geometrie in 1899 putting geometry in a formal axiomatic setting. The http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hilbert.html (2 of 5) [2/16/2002 11:14:35 PM]

Hilbert

book continued to appear in new editions and was a major influence in promoting the axiomatic approach to mathematics which has been one of the major characteristics of the subject throughout the 20th century. Hilbert's famous 23 Paris problems challenged (and still today challenge) mathematicians to solve fundamental questions. Hilbert's famous speech The Problems of Mathematics was delivered to the Second International Congress of Mathematicians in Paris. It was a speech full of optimism for mathematics in the coming century and he felt that open problems were the sign of vitality in the subject:The great importance of definite problems for the progress of mathematical science in general ... is undeniable. ... [for] as long as a branch of knowledge supplies a surplus of such problems, it maintains its vitality. ... every mathematician certainly shares ..the conviction that every mathematical problem is necessarily capable of strict resolution ... we hear within ourselves the constant cry: There is the problem, seek the solution. You can find it through pure thought... Hilbert's problems included the continuum hypothesis, the well ordering of the reals, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet's principle and many more. Many of the problems were solved during this century, and each time one of the problems was solved it was a major event for mathematics. Today Hilbert's name is often best remembered through the concept of Hilbert space. Irving Kaplansky, writing in [2], explains Hilbert's work which led to this concept:Hilbert's work in integral equations in about 1909 led directly to 20th-century research in functional analysis (the branch of mathematics in which functions are studied collectively). This work also established the basis for his work on infinite-dimensional space, later called Hilbert space, a concept that is useful in mathematical analysis and quantum mechanics. Making use of his results on integral equations, Hilbert contributed to the development of mathematical physics by his important memoirs on kinetic gas theory and the theory of radiations. Many have claimed that in 1915 Hilbert discovered the correct field equations for general relativity before Einstein but never claimed priority. The article [11] however, shows that this view is in error. In this paper the authors show convincingly that Hilbert submitted his article on 20 November 1915, five days before Einstein submitted his article containing the correct field equations. Einstein's article appeared on 2 December 1915 but the proofs of Hilbert's paper (dated 6 December 1915) do not contain the field equations. As the authors of [11] write:In the printed version of his paper, Hilbert added a reference to Einstein's conclusive paper and a concession to the latter's priority: "The differential equations of gravitation that result are, as it seems to me, in agreement with the magnificent theory of general relativity established by Einstein in his later papers". If Hilbert had only altered the dateline to read "submitted on 20 November 1915, revised on [any date after 2 December 1915, the date of Einstein's conclusive paper]," no later priority question would have arisen. In 1934 and 1939 two volumes of Grundlagen der Mathematik were published which were intended to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hilbert.html (3 of 5) [2/16/2002 11:14:35 PM]

Hilbert

lead to a 'proof theory', a direct check for the consistency of mathematics. Gödel's paper of 1931 showed that this aim is impossible. Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations. Hilbert's mathematical abilities were nicely summed up by Otto Blumenthal, his first student:In the analysis of mathematical talent one has to differentiate between the ability to create new concepts that generate new types of thought structures and the gift for sensing deeper connections and underlying unity. In Hilbert's case, his greatness lies in an immensely powerful insight that penetrates into the depths of a question. All of his works contain examples from far-flung fields in which only he was able to discern an interrelatedness and connection with the problem at hand. From these, the synthesis, his work of art, was ultimately created. Insofar as the creation of new ideas is concerned, I would place Minkowski higher, and of the classical great ones, Gauss, Galois, and Riemann. But when it comes to penetrating insight, only a few of the very greatest were the equal of Hilbert. Among Hilbert's students were Hermann Weyl, the famous world chess champion Lasker, and Zermelo. Hilbert received many honours. In 1905 the Hungarian Academy of Sciences gave a special citation for Hilbert. In 1930 Hilbert retired and the city of Königsberg made him an honorary citizen of the city. He gave an address which ended with six famous words showing his enthusiasm for mathematics and his life devoted to solving mathematical problems:Wir müssen wissen, wir werden wissen - We must know, we shall know. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (27 books/articles)

Some Quotations (20)

A Poster of David Hilbert

Mathematicians born in the same country

Cross-references to History Topics

1. The beginnings of set theory 2. Abstract linear spaces 3. Topology enters mathematics 4. General relativity 5. An overview of the history of mathematics

Other references in MacTutor

1. Chronology: 1890 to 1900 2. Chronology: 1900 to 1910

Honours awarded to David Hilbert (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1928

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hilbert.html (4 of 5) [2/16/2002 11:14:35 PM]

Hilbert

Lunar features

Crater Hilbert

Other Web sites

1. David Joyce (A list of Hilbert's 23 problems) 2. The text of his 1901 speech 3. Clark Kimberling 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hilbert.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hilbert.html (5 of 5) [2/16/2002 11:14:35 PM]

Hill

George William Hill Born: 3 March 1838 in New York, USA Died: 16 April 1914 in West Nyack, New York, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

In 1846 George Hill's family moved to West Nyack where George attended school. After graduating from school he studied at Rutgers University graduating in 1859. The following year he began his study of the lunar theory of Delaunay and Hansen. He was to continue this study for twelve years before he produced any publications of his own. In 1861 Hill joined the Nautical Almanac Office working in Cambridge, Massachusetts. After two years he returned to West Nyack where he worked from his home. Except for a period of 10 years from 1882 to 1892 when he worked in Washington on the theory and tables for the orbits of Jupiter and Saturn, this was to be his working pattern for the rest of his life. Writing in [4], E W Brown says:He was essentially of the type of scholar and investigator who seems to feel no need of personal contacts with others. While the few who knew him speak of the pleasure of his companionship in frequent tramps over the country surrounding Washington, he was apparently quite happy alone, whether at work or taking recreation. Hill was the first to use infinite determinants to study the orbit of the Moon (1877). His Researches in Lunar Theory appeared in 1878 in the new American Journal of Mathematics. This publication contains important new ideas on the three-body problem. He also introduced infinite determinants and other methods to give increased accuracy to his results. Brown wrote in 1915 that Hill's memoir Researches in Lunar Theory :.. of but fifty quarto pages has become fundamental for the development of celestial http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hill.html (1 of 3) [2/16/2002 11:14:37 PM]

Hill

mechanics in three different directions. ... Poincaré's remark that in it we may perceive the germ of all progress which has been made in celestial mechanics since its publication is doubtless fully justified. Newcomb persuaded Hill to develop a theory of the orbits of Jupiter and Saturn and Hill's work is a major contribution to mathematical astronomy. Hill's most important work dealt with the gravitational effects of the planets on the Moon's orbit so in this work he was considering the 4-body problem. Although he must be considered a mathematician, his mathematics was entirely based on that necessary to solve his orbits problems. He had no interest in any modern developments in other areas of mathematics. In fact Hill worked on very similar problems to Adams. In fact Adams was also led to investigate infinite determinants independently of Hill. From 1898 until 1901 Hill lectured at Columbia University, but [4], characteristically returned the salary, writing that he did not need the money and that it bothered him to look after it. Hill became a Fellow of the Royal Society (1902) receiving its Copley Medal in 1909. He was president of the American Mathematical Society from 1894 to 1896. He won the Damoiseau Prize from the Institut de France in 1898, was elected to the Royal Society of Edinburgh in 1908, elected to the academies of Belgium (1909), Christiania (1910), Sweden (1913) and others. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country Cross-references to History Topics

Orbits and gravitation

Honours awarded to George Hill (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1902

Royal Society Copley Medal

Awarded 1909

Fellow of the Royal Society of Edinburgh American Maths Society President

1895 - 1896

Lunar features

Crater Hill

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hill.html (2 of 3) [2/16/2002 11:14:37 PM]

Hill

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hill.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hill.html (3 of 3) [2/16/2002 11:14:37 PM]

Hille

Einar Carl Hille Born: 28 June 1894 in New York, USA Died: 12 Feb 1980 in La Jolla, California, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

First we should note that Einar Carl Hille was not given this name at birth. His name was Carl Einar Heuman and he was the son of Carl August Heuman, a civil engineer, and Edla Eckman. Hille's parents separated before his birth and he was brought up by his mother. The name Hille was simply a mistake for Heuman but Hille's mother adopted the name even although it came from an error and called herself Edla Hille. That was the name by which Einar Hille was always known. Hille did not meet his father until 1937. Although he was born in the United States, Hille's parents were Swedish. When Einar Hille was two years old his mother returned to Sweden and lived in Stockholm. Hille would spend the next 24 years of his life in Sweden only returning to the United States when he was 26 years old. In 1900 Einar entered the Palmsgrenska Samskola, a private school in Stockholm. When he left in 1911 it was not mathematics in which he intended to make his career, rather it was chemistry. Hille entered the University of Stockholm in 1911 with the intention of reading for a degree in chemistry. In fact he studied the subject for two years at university and he was taught by an exceptional chemist in Hans von Euler-Chelpin. He had been appointed professor of general and inorganic chemistry in 1906 and he worked on the biochemistry of sugar and phosphates and he was awarded the Nobel prize for chemistry in 1929 for his work on enzymes in the fermentation of sugar. He also helped determine the chemical structures of several vitamins. Hille made an impressive start to his career in chemistry with his first publication in 1913 being jointly with Euler-Chelpin. However, Hille decided that he did not have the necessary dexterity to make a career in a subject which involved delicate experiments. He therefore decided to give up his work with Euler-Chelpin and to study a topic which required no experimental

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hille.html (1 of 4) [2/16/2002 11:14:39 PM]

Hille

expertise at all, namely mathematics. Hille was fortunate in not only having a world class chemist to work under in Stockholm, but also world class mathematicians. He began to study under Ivar Bendixson and Helge von Koch. Hille was awarded his first degree in mathematics in 1913 and the equivalent of a Master's degree in the following year. Then Hille began working with Marcel Riesz on conformal mappings and submitted a thesis on that topic in 1916; for this he was awarded a Lic. Ph. During this time at Stockholm he was also strongly influenced by Mittag-Leffler. By this time Europe was engulfed by World War I and Hille undertook war work, serving in the Swedish army for two years as a typist. While in the army he decided to undertake research for his Ph.D. and this he did without any further supervision. He received a Ph.D. from Stockholm in 1918 for a doctoral dissertation entitled Some Problems Concerning Spherical Harmonics. In 1919 Hille was awarded the Mittag-Leffler prize for his outstanding contributions, and was given the right to teach at the University of Stockholm. He was employed in the Swedish civil service for two years starting from the end of World War I. Hille obtained a fellowship to work with Birkhoff in Harvard and he returned to the land of his birth in 1920, spending the academic year 1920-21 at Harvard where, as well as working with Birkhoff, he also studied with Kellogg. In academic year 1921-22 Hille was Benjamin Peirce Instructor at Harvard then, in 1922, he went to Princeton as an instructor. In 1923 Hille was promoted to assistant professor at Princeton, and in 1927 to associate professor. In 1933, he was appointed full professor at Yale University and four years later, in 1937, he married Kirsti Ore, who was the sister of his Yale colleague, the mathematician Oystein Ore. Kirsti Hille wrote the article [5] after the death of her husband. Kirsti and Einar had two children, both sons. Hille was appointed director of graduate studies in 1938 and he held this position, and his chair, at Yale until he retired in 1962. Hille's main work was on integral equations, differential equations, special functions, Dirichlet series and Fourier series. Later in his career his interests turned more towards functional analysis. Zund writes in [9]:Hille was one of the few mathematicians who brought to his study of functional analysis operator theory some twenty years experience in classical analysis. Moreover, he was almost unique among mathematicians in applying functional analysis to investigate classical problems, rather than simply considering abstract situations for their own sake. He is well known as an author of a number of texts. The first of these was Functional analysis and semigroups (1948). Stone, reviewing the book, writes:It is impossible in a review of this kind to give an adequate account of the wealth of material covered in the book, or to emphasize the interest of specific results. It is appropriate to note the systematic, incisive and polished character of the author's treatment. His scholarly handling of the subject will be appreciated by all users of the book, as will the clarity of his expository style. Among Hille's other texts were Analytic function theory Vol 1 (1959), Vol 2 (1964); Analysis Vol 1 (1964), Vol 2 (1966); Lectures on ordinary differential equations (1969); Methods in classical and

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hille.html (2 of 4) [2/16/2002 11:14:39 PM]

Hille

functional analysis (1972); and Ordinary differential equations in the complex domain (1976). In the preface of Methods in classical and functional analysis Hille explains both about his aims in writing the text and his view of mathematics:Modes come and go in mathematics as in most fields. During the half-century and more that I have worked in the vineyard I have heard many dire predictions for the fate of my ideas and interests. Abstraction has been in the saddle during most of the time and has ridden us mercilessly. In a modest way I have taken part in this development. I did not believe in abstraction per se, one should know what one is trying to generalize and one should show that the generalization is significant. I have tried to keep at least one foot on the ground while craning my neck to look into Heaven. What is Heaven? There are some doubts, and the more extravagant claims of the abstract mathematicians to be the sole dispenser of the true faith and the arbiters of values are received with a healthy scepticism.... This book may be regarded as part of the backlash. If the book has a thesis, it is that a functional analyst is an analyst, first and foremost, and not a degenerate species of a topologist. His problems come from analysis and his results should throw light on analysis.... It seemed to me that I could do some useful work in giving the student a historical perspective and in showing how the multitude of abstract concepts have arisen and are present in Euclidean spaces. The article [2] list 175 papers and 12 books written by Hille. The book [1] reproduces 47 selected papers of Hille, published in the years 1922-1969. The book also contains a personal account of Hille's mathematical career which he gave at the Yale Colloquium in May 1962. The same autobiographical paper appears as [4]. Hille served as president of the American Mathematical Society (1937-38) and was the Society's Colloquium lecturer in 1944. As well as editorial work with the Annals of Mathematics (1929-33) and the Transactions of the American Mathematical Society (1937-43) he received many honours including election to the National Academy of Sciences (1953) and the Royal Academy of Sciences of Stockholm. He was honoured by Sweden with their award of the Order of the North Star. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Honours awarded to Einar Carl Hille (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1947 - 1948

AMS Colloquium Lecturer

1944

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hille.html (3 of 4) [2/16/2002 11:14:39 PM]

Hille

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hille.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hille.html (4 of 4) [2/16/2002 11:14:39 PM]

Hindenburg

Carl Friedrich Hindenburg Born: 13 July 1741 in Dresden, Germany Died: 17 March 1808 in Leipzig, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Carl Friedrich Hindenburg was the son of a merchant. He did not attend school but his father arranged that he be taught privately in his home by a tutor. Hindenburg entered the University of Leipzig in 1757 but at this stage his interests were not focused on mathematics, rather he was interested in a wide range of subjects. he took courses in medicine, philosophy, Latin, Greek, physics, mathematics, and aesthetics. Christian Fürchtegott Gellert, whose whole career was spent at the University of Leipzig, had been promoted to professor there six years before Hindenburg entered the university. Gellert's lectures on poetry, rhetoric, and ethics were exceptionally popular. Gellert, who tutored Hindenburg, arranged with him that he should take on the task of accompanying a student named Schoenborn through his education. This was an important event for Hindenburg for Schoenborn's increasing interest in mathematics took Hindenburg in that direction too. As well as at Leipzig, Schoenborn studied at Göttingen and while he was there Hindenburg became a friend of Kaestner, who had himself taught at Leipzig earlier in his career. Through this Hindenburg did not neglect his own studies and he was awarded a Master's degree from the University of Leipzig in 1771 and appointed as a Privatdozent there in that year. Even before his appointment as a Privatdozent, Hindenburg had published articles but these were not in mathematics. In 1763 and 1769 he published on philology which is the study of language. His first papers on mathematics were published in 1776 when he studied series. Two years later he published his first papers on combinatorics, the topic for which he became famous. Hindenburg published a series of works on combinatorial mathematics, in particular probability, series and formulas for higher differentials. Hindenburg hoped for combinatorial operations to have the same importance as those of arithmetic, algebra and analysis but his expectations were not realised. He is recognised, however, as starting [2]:... the first scientific school of combinatorial mathematics. Although essentially forgotten now, Hindenburg's combinatorics was very fashionable 1800 although it is now clear that its importance being much overestimated. His ideas centred around the so-called polynomial theorem which was a generalisation of the binomial theorem. It would be too easy to dismiss Hindenburg's combinatorics, however, for they had some important consequences. Gudermann, best known as the teacher of Weierstrass, worked on the expansion of functions into power series and, as

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hindenburg.html (1 of 2) [2/16/2002 11:14:40 PM]

Hindenburg

shown by Manning in [4], it was Hindenburg's combinatorial analysis which was the main influence on this work. In 1781 Hindenburg was appointed as professor of philosophy in the University of Leipzig. After presenting a dissertation on water pumps, he was appointed as professor of physics in 1786. This later post was one which he continued to hold until his death over twenty years later. It was not only for his school of combinatorial analysis that Hindenburg is famous. He also made important contributions to publishing mathematics in Germany. Between 1780 and 1800 he was involved at different times with the publishing of four different journals all relating to mathematics and its applications. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hindenburg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hindenburg.html (2 of 2) [2/16/2002 11:14:40 PM]

Hipparchus

Hipparchus of Rhodes Born: 190 BC in Nicaea (now Iznik), Bithynia (now Turkey) Died: 120 BC in probably Rhodes, Greece

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Little is known of Hipparchus's life, but he is known to have been born in Nicaea in Bithynia. The town of Nicaea is now called Iznik and is situated in north-western Turkey. Founded in the 4th century BC, Nicaea lies on the eastern shore of Lake Iznik. Reasonably enough Hipparchus is often referred to as Hipparchus of Nicaea or Hipparchus of Bithynia and he is listed among the famous men of Bithynia by Strabo, the Greek geographer and historian who lived from about 64 BC to about 24 AD. There are coins from Nicaea which depict Hipparchus sitting looking at a globe and his image appears on coins minted under five different Roman emperors between 138 AD and 253 AD. This seems to firmly place Hipparchus in Nicaea and indeed Ptolemy does describe Hipparchus as observing in Bithynia, and one would naturally assume that in fact he was observing in Nicaea. However, of the observations which are said to have been made by Hipparchus, some were made in the north of the island of Rhodes and several (although only one is definitely due to Hipparchus himself) were made in Alexandria. If these are indeed as they appear we can say with certainty that Hipparchus was in Alexandria in 146 BC and in Rhodes near the end of his career in 127 BC and 126 BC. It is not too unusual to have few details of the life of a Greek mathematician, but with Hipparchus the position is a little unusual for, despite Hipparchus being a mathematician and astronomer of major importance, we have disappointingly few definite details of his work. Only one work by Hipparchus has survived, namely Commentary on Aratus and Eudoxus and this is certainly not one of his major works. It is however important in that it gives us the only source of Hipparchus's own writings. Most of the information which we have about the work of Hipparchus comes from Ptolemy's Almagest

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hipparchus.html (1 of 6) [2/16/2002 11:14:42 PM]

Hipparchus

but, as Toomer writes in [1]:... although Ptolemy obviously had studied Hipparchus's writings thoroughly and had a deep respect for his work, his main concern was not to transmit it to posterity but to use it and, where possible, improve upon it in constructing his own astronomical system. Where one might hope for more information about Hipparchus would be in the commentaries on Ptolemy's Almagest. There are two in particular by the excellent commentators Theon of Alexandria and by Pappus, but unfortunately these follow Ptolemy's text fairly closely and fail to add the expected information about Hipparchus. Since when Ptolemy refers to results of Hipparchus he does so often in an obscure way, at least he seems to assume that the reader will have access to the original writings by Hipparchus, and it is certainly surprising that neither Theon or Pappus fills in the details. One can only assume that neither of them had access to the information about Hipparchus on which we would have liked them to report. Let us first summarise the main contribution of Hipparchus and then examine them in more detail. He made an early contribution to trigonometry producing a table of chords, an early example of a trigonometric table; indeed some historians go so far as to say that trigonometry was invented by him. The purpose of this table of chords was to give a method for solving triangles which avoided solving each triangle from first principles. He also introduced the division of a circle into 360 degrees into Greece. Hipparchus calculated the length of the year to within 6.5 minutes and discovered the precession of the equinoxes. Hipparchus's value of 46" for the annual precession is good compared with the modern value of 50.26" and much better than the figure of 36" that Ptolemy was to obtain nearly 300 years later. We believe that Hipparchus's star catalogue contained about 850 stars, probably not listed in a systematic coordinate system but using various different ways to designate the position of a star. His star catalogue, probably completed in 129 BC, has been claimed to have been used by Ptolemy as the basis of his own star catalogue. However, Vogt shows clearly in his important paper [26] that by considering the Commentary on Aratus and Eudoxus and making the reasonable assumption that the data given there agreed with his star catalogue, then Ptolemy's star catalogue cannot have been produced from the positions of the stars as given by Hipparchus. This last point shows that in any detailed discussion of the achievements of Hipparchus we have to delve more deeply than just assuming that everything in the Ptolemy's Almagest which he does not claim as his own must be due to Hipparchus. This view was taken for many years but since Vogt's 1925 paper [26] there has been much research done trying to ascertain exactly what Hipparchus achieved. So major shifts have taken place in our understanding of Hipparchus, first it was assumed that his discoveries were all set out by Ptolemy, then once it was realised that this was not so there was a feeling that it would be impossible to ever have detailed knowledge of his achievements, but now we are in a third stage where it is realised that it is possible to gain a good knowledge of his work but only with much effort and research. Let us begin our detailed description of Hipparchus's achievements by looking at the only work which has survived. Hipparchus's Commentary on Aratus and Eudoxus was written in three books as a commentary on three different writings. Firstly there was a treatise by Eudoxus (unfortunately now lost) in which he named and described the constellations. Aratus wrote a poem called Phaenomena which was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hipparchus.html (2 of 6) [2/16/2002 11:14:42 PM]

Hipparchus

based on the treatise by Eudoxus and proved to be a work of great popularity. This poem has survived and we have its text. Thirdly there was commentary on Aratus by Attalus of Rhodes, written shortly before the time of Hipparchus. It is certainly unfortunate that of all of the writings of Hipparchus this was the one to survive since the three books on which Hipparchus was writing a commentary contained no mathematical astronomy. As a result of this Hipparchus chose to write at the same qualitative level in the first book and also for much of the second of his three book. However towards the end of the second book, continuing through the whole of the third book, Hipparchus gives his own account of the rising and setting of the constellations. Towards the end of Book 3 Hipparchus gives a list of bright stars always visible for the purpose of enabling the time at night to be accurately determined. As we noted above Hipparchus does not use a single consistent coordinate system to denote stellar positions, rather using a mixture of different coordinates. He uses some equatorial coordinates, although often in a rather strange way as for example saying that a star (see [1]):... occupies three degrees of Leo along its parallel circle... He has therefore divided each small circle parallel to the equator into 12 portions of 30 each and this means that the right ascension of the star referred to in the quotation is 123 . The data in the Commentary on Aratus and Eudoxus has been analysed by many authors. In particular the authors of [15] argue that Hipparchus used a mobile celestial sphere with the stars pictured on the sphere. They claim that the data was taken from on a star catalogue constructed around 140 BC based on observations accurate to a third of a degree or even better. In the earlier work [16] by the same authors, they suggest that the observations were made at a latitude of 36 15' which corresponds to that of northern Rhodes. This would tend to confirm that this work by Hipparchus was done near the end of his career. As Toomer writes in [1]:Far from being a "work of his youth", as it is frequently described, the commentary on Aratus reveals Hipparchus as one who had already compiled a large number of observations, invented methods for solving problems in spherical astronomy, and developed the highly significant idea of mathematically fixing the positions of the stars... There is of course no agreement on many of the points discussed here. For example Maeyama in [13] sees major differences between the accuracy of the data in Commentary on Aratus and Eudoxus (claimed to be written around 140 BC) and Hipparchus's star catalogue (claimed to be produced around 130 BC). Maeyama writes [13]:... Hipparchus's "Commentary" contains his own observations of the stellar positions, great in number but inaccurate in operation, despite all his ability for accurate observations. ... the observational accuracy [of] his two different epochs have nothing in common, as if they dealt with two different observers. Within an interval of 10 years everything can happen, particularly in the case of a man like Hipparchus. Those views which consider Hipparchus's astronomical activities at his two different epochs as similar are completely unfounded. Perhaps the discovery for which Hipparchus is most famous is the discovery of precession which is due to the slow change in direction of the axis of rotation of the earth. This work came from Hipparchus's attempts to calculate the length of the year with a high degree of accuracy. There are two different definitions of a 'year' for one might take the time that the sun takes to return to the same place amongst the fixed stars or one could take the length of time before the seasons repeated which is a length of time defined by considering the equinoxes. The first of these is called the sidereal year while the second is

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hipparchus.html (3 of 6) [2/16/2002 11:14:42 PM]

Hipparchus

called the tropical year. Of course the data needed by Hipparchus to calculate the length of these two different years was not something that he could find over a few years of observations. Swerdlow [19] suggests that Hipparchus calculated the length of the tropical year using Babylonian data to arrive at the value of 1/300 of a day less than 365 1/4 days. He then checked this against observations of equinoxes and solstices including his own data and those of Aristarchus in 280 BC and Meton in 432 BC. Hipparchus also calculated the length of the sidereal year, again using older Babylonian data, and arrived at the highly accurate figure of 1/144 days longer than 365 1/4 days. This gives his rate of precession of 1 per century. Hipparchus also made a careful study of the motion of the moon. There are difficult problems in such a study for there are three different periods which one could determine. There is the time taken for the moon to return to the same longitude, the time taken for it to return to the same velocity (the anomaly) and the time taken for it to return to the same latitude. In addition there is the synodic month, that is the time between successive oppositions of the sun and moon. Toomer [22] writes:For his lunar theory [Hipparchus] needed to establish the mean motions of the Moon in longitude, anomaly and latitude. The best data available to him were the Babylonian parameters. But he was not content merely to accept them: he wanted to test them empirically, and so he constructed (purely arithmetically) the eclipse period of 126007 days 1 hour, then looked in the observational material available to him for pairs of eclipses which would confirm that this was indeed an eclipse period. The observations thus played a real role, but that role was not discovery, but confirmation. In calculating the distance of the moon, Hipparchus not only made excellent use of both mathematical techniques and observational techniques but he also gave a range of values within which be calculated that the true distance must lie. Although Hipparchus's treatise On sizes and distances has not survived details given by Ptolemy, Pappus, and others allow us to reconstruct his methods and results. The reconstruction of Hipparchus's techniques is beautifully presented in [24] where the author shows that Hipparchus based his calculations on an eclipse which occurred on 14 March 190 BC. Hipparchus's calculations led him to a value for the distance to the moon of between 59 and 67 earth radii which is quite remarkable (the correct distance is 60 earth radii). The main reason for his range of values was that he was unable to determine the parallax of the sun, only managing to give an upper value. Hipparchus appears to know that 67 earth radii for the distance of the moon comes from this upper limit of solar parallax, while the lower value of 59 earth radii corresponds to the sun being at infinity. Hipparchus not only gave observational data for the moon which enabled him to compute accurately the various periods, but he developed a theoretical model of the motion of the moon based on epicycles. He showed that his model did not agree totally with observations but it seems to be Ptolemy who was the first to correct the model to take these discrepancies into account. Hipparchus was also able to give an epicycle model for the motion of the sun (which is easier), but he did not attempt to give an epicycle model for the motion of the planets. Finally let us examine the contributions which Hipparchus made to trigonometry. Heath writes in [6]:Even if he did not invent it, Hipparchus is the first person whose systematic use of trigonometry we have documentary evidence.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hipparchus.html (4 of 6) [2/16/2002 11:14:42 PM]

Hipparchus

The documentary evidence comes from Ptolemy and Theon of Alexandria who explicitly says that Hipparchus wrote a work on chords in 12 books. However, Neugebauer [7] points out that:... this number is obvious nonsense since 13 books sufficed for the whole of the "Almagest" or of Euclid's "Elements"... Toomer ([1] or [23]) reconstructs Hipparchus's table of chords, and the mathematical means by which Hipparchus calculated it. The table was based on a circle divided into 360 degrees with each degree divided into 60 minutes. The radius of the circle is then 360.60/2 = 3438 minutes and the chord function Crd of Hipparchus is related to the sine function by (Crd 2a)/2 = 3438 sin a. Toomer claims that Hipparchus defined his Crd function at 7 1/2 intervals (1/48 of the circle) and used linear interpolation to find the value at intermediate points. He then goes on to show that the table can be computed from some basic formulas which would be known to Hipparchus, one of which is the supplementary angle theorem, essentially Pythagoras's theorem, and the half-angle theorem. The only trace of Hipparchus's tables that survives is in Indian tables which are thought to have been based on that of that of Hipparchus. Toomer summarises the contributions of Hipparchus in this area when he writes in [1]:... it seems highly probable that Hipparchus was the first to construct a table of chords and thus provide a general solution for trigonometrical problems. A corollary of this is that, before Hipparchus, astronomical tables based on Greek geometrical methods did not exist. If this is so, Hipparchus was not only the founder of trigonometry but also the man who transformed Greek astronomy from a purely theoretical into a practical predictive science. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (26 books/articles) A Poster of Hipparchus Cross-references to History Topics

Mathematicians born in the same country 1. Longitude and the Académie Royale 2. Greek Astronomy 3. The trigonometric functions

Other references in MacTutor

Chronology: 500BC to 1AD

Honours awarded to Hipparchus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Hipparchus

Planetary features

Crater Hipparchus on Mars

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hipparchus.html (5 of 6) [2/16/2002 11:14:42 PM]

Hipparchus

Other Web sites

1. Kevin Brown (Some information about Hipparchus and compound statements) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hipparchus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hipparchus.html (6 of 6) [2/16/2002 11:14:42 PM]

Hippias

Hippias of Elis Born: about 460 BC in Elis, Peloponnese, Greece Died: about 400 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hippias of Elis was a statesman and philosopher who travelled from place to place taking money for his services. He lectured on poetry, grammar, history, politics, archaeology, mathematics and astronomy. Plato describes him as a vain man being both arrogant and boastful, having a wide but superficial knowledge. Heath tells us something of this character when he writes in [3]:He claimed ... to have gone once to the Olympian festival with everything that he wore made by himself, ring and sandal (engraved), oil-bottle, scraper, shoes, clothes, and a Persian girdle of expensive type; he also took poems, epics, tragedies, dithyrambs, and all sorts of prose works. As to Hippias's academic achievements, Heath writes:He was a master of the science of calculation, geometry, astronomy, 'rhythms and harmonies and correct writing'. He also had a wonderful system of mnemonics enabling him, if he once heard a string of fifty names to remember them all. A rather nice story, which says more of the Spartans than it does of Hippias, is that it was reported that he received no payment for the lectures he gave in Sparta since [3]:... the Spartans could not endure lectures on astronomy or geometry or calculation; it was only a small minority of them who could even count; what they liked was history and archaeology. Since Hippias was reported to give lectures on archaeology, he seems to have chosen the wrong topics when he lectured in Sparta! Hippias's only contribution to mathematics seems to be the quadratrix which may have been used by him for trisecting an angle and squaring the circle. The curve may be used for dividing an angle into any number of equal parts. Perhaps the highest compliment that we can pay to Hippias is to report on the arguments of certain historians of mathematics who have claimed that the Hippias who discovered the quadratrix cannot be Hippias of Elis since geometry was not far enough advanced at this time to have allowed him to make these discoveries. However, their arguments are not generally accepted and there is ample evidence to attribute the discovery of the quadratrix to Hippias of Elis. Heath [3] writes:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hippias.html (1 of 3) [2/16/2002 11:14:44 PM]

Hippias

It was probably about 420 BC that Hippias of Elis invented the curve known as the quadratrix for the purpose of trisecting any angle, and it was in the first half of the fourth century (BC) that Archytas used [it] for duplication of the cube. However this is far from certain and there is some evidence to suggest that Geminus, writing in the first century BC, had in his possession a treatise by Hippias of Elis on the quadratrix which indicated how it could be used to square the circle. If this is indeed the case then the treatise by Hippias must have been lost between this time and that of Sporus in the third century AD. Pappus wrote his major work on geometry Synagoge in 340. It is a collection of mathematical writings in eight books. Book IV contains a description of the quadratrix of Hippias. Look at the diagram of the quadratrix. ABCD is a square and BED is part of a circle, centre A radius AB. As the radius AB rotates about A to move to the position AD then the line BC moves at the same rate parallel to itself to end at AD. Then the locus of the point of intersection F of the rotating radius AB and the moving line BC is the quadratrix. Hence angle EAD/angle BAD = arc ED/arc BED = FH/AB, so, taking AB = 1, angle EAD = arc ED = FH

/2.

To divide the angle FAD in a given ratio, say p : q, then draw a point P on the line FH dividing it in the ratio p : q. Draw a line through P parallel to AD to meet the quadratrix at Q. Then AQ divides angle FAD in the ratio p : q. Pappus also gives the rather more complicated version of the construction necessary to square the circle. However, Pappus reports that Sporus had two criticisms of Hippias's method with which he agrees. The second is specifically related to the construction necessary for squaring the circle which we have not described. The first however relates to the construction of the quadratrix itself. Pappus reports that Sporus writes (see [3]):The very thing for which the construction is thought to serve is actually assumed in the hypothesis. For how is it possible, with two points starting from B, to make one of them move along a straight line to A and the other along a circumference to D in an equal time, unless you first know the ratio of the straight line AB to the circumference BED? In fact this ratio must also be that of the speeds of motion. For, if you employ speeds not definitely adjusted to this ratio, how can you make the motions end at the same moment, unless this should sometime happen by pure chance? Is not the thing thus shown to be absurd? The point here seems to be a question of what exactly Hippias is trying to show with his quadratrix. Certainly he knew perfectly well that he was not providing a ruler and compass construction for squaring the circle. Exactly what he has proved concerning squaring the circle is, as Pappus and Sporus suggest, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hippias.html (2 of 3) [2/16/2002 11:14:44 PM]

Hippias

far from clear. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

Squaring the circle

Cross-references to Famous Curves

Quadratrix of Hippias

Other references in MacTutor

Chronology: 500BC to 1AD

Other Web sites

1. Internet Encyclopedia of Philosophy 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hippias.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hippias.html (3 of 3) [2/16/2002 11:14:44 PM]

Hippocrates

Hippocrates of Chios Born: about 470 BC in Chios (now Khios), Greece Died: about 410 BC

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hippocrates of Chios taught in Athens and worked on the classical problems of squaring the circle and duplicating the cube. Little is known of his life but he is reported to have been an excellent geometer who, in other respects, was stupid and lacking in sense. Some claim that he was defrauded of a large sum of money because of his naiveté. Iamblichus [4] writes:One of the Pythagoreans [Hippocrates] lost his property, and when this misfortune befell him he was allowed to make money by teaching geometry. Heath [6] recounts two versions of this story:One version of the story is that [Hippocrates] was a merchant, but lost all his property through being captured by a pirate vessel. He then came to Athens to persecute the offenders and, during a long stay, attended lectures, finally attaining such proficiency in geometry that he tried to square the circle. Heath also recounts a different version of the story as told by Aristotle:... he allowed himself to be defrauded of a large sum by custom-house officers at Byzantium, thereby proving, in Aristotle's opinion, that, though a good geometer, he was stupid and incompetent in the business of ordinary life. The suggestion is that this 'long stay' in Athens was between about 450 BC and 430 BC. In his attempts to square the circle, Hippocrates was able to find the areas of lunes, certain crescent-shaped figures, using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii. We describe this impressive achievement more fully below. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hippocrates.html (1 of 4) [2/16/2002 11:14:46 PM]

Hippocrates

Hippocrates also showed that a cube can be doubled if two mean proportionals can be determined between a number and its double. This had a major influence on attempts to duplicate the cube, all efforts after this being directed towards the mean proportionals problem. He was the first to write an Elements of Geometry and although his work is now lost it must have contained much of what Euclid later included in Books 1 and 2 of the Elements. Proclus, the last major Greek philosopher, who lived around 450 AD wrote:Hippocrates of Chios, the discoverer of the quadrature of the lune, ... was the first of whom it is recorded that he actually compiled "Elements". Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration. Eudemus of Rhodes, who was a pupil of Aristotle, wrote History of Geometry in which he described the contribution of Hippocrates on lunes. This work has not survived but Simplicius of Cilicia, writing in around 530, had access to Eudemus's work and he quoted the passage about the lunes of Hippocrates 'word for word except for a few additions' taken from Euclid's Elements to make the description clearer. We will first quote part of the passage of Eudemus about the lunes of Hippocrates, following the historians of mathematics who have disentangled the additions from Euclid's Elements which Simplicius added. See [6] both for the translation which we give and for a discussion of which parts are due to Eudemus:The quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation of lunes to the circle, were first investigated by Hippocrates, and his exposition was thought to be correct; we will therefore deal with them at length and describe them. He started with, and laid down as the first of the theorems useful for the purpose, the proposition that similar segments of circles have the same ratio to one another as the squares on their bases. And this he proved by first showing that the squares on the diameters have the same ratio as the circles. Before continuing with the quote we should note that Hippocrates is trying to 'square a lune' by which he means to construct a square equal in area to the lune. This is precisely what the problem of 'squaring the circle' means, namely to construct a square whose area is equal to the area of the circle. Again following Heath's translation in [6]:After proving this, he proceeded to show in what way it was possible to square a lune the outer circumference of which is that of a semicircle. This he affected by circumscribing a semicircle about an isosceles right-angled triangle and a segment of a circle similar to those cut off by the sides. Then, since the segment about the base is equal to the sum of those about the sides, it follows that, when the part of the triangle above the segment about the base is added to both alike, the lune will be equal to the triangle. Therefore the lune, having been proved equal to the triangle, can be squared.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hippocrates.html (2 of 4) [2/16/2002 11:14:46 PM]

Hippocrates

To follow Hippocrates' argument here, look at the diagram. ABCD is a square and O is its centre. The two circles in the diagram are the circle with centre O through A, B, C and D, and the circle with centre D through A and C. Notice first that the segment marked 1 on AB subtends a right angle at the centre of the circle (the angle AOB) while the segment 2 on AC also subtends a right angle at the centre (the angle ADC). Therefore the segment 1 on AB and the segment 2 on AC are similar. Now segment 1/segment 2 = AB2/AC2 = 1/2 since AB2 + BC2 = AC2 by Pythagoras's theorem, and AB = BC so AC2 = 2AB2. Now since segment 2 is twice segment 1, the segment 2 is equal to the sum of the two segments marked 1. Then Hippocrates argues that the semicircle ABC with the two segments 1 removed is the triangle ABC which can be squared (it was well known how to construct a square equal to a triangle). However, if we subtract the segment 2 from the semicircle ABC we get the lune shown in the second diagram. Thus Hippocrates has proved that the lune can be squared. However, Hippocrates went further than this in studying lunes. The proof we have examined in detail is one where the outer circumference of the lune is the arc of a semicircle. He also studied the cases where the outer arc was less than that of a semicircle and also the case where the outer arc was greater than a semicircle, showing in each case that the lune could be squared. This was a remarkable achievement and a major step in attempts to square the circle. As Heath writes in [6]:... he wished to show that, if circles could not be squared by these methods, they could be employed to find the area of some figures bounded by arcs of circles, namely certain lunes, and even of the sum of a certain circle and a certain lune. There is one further remarkable achievement which historians of mathematics believe that Hippocrates achieved, although we do not have a direct proof since his works have not survived. In Hippocrates' study of lunes, as described by Eudemus, he uses the theorem that circles are to one another as the squares on their diameters. This theorem is proved by Euclid in the Elements and it is proved there by the method of exhaustion due to Eudoxus. However, Eudoxus was born within a few years of the death of Hippocrates, and so there follows the intriguing question of how Hippocrates proved this theorem. Since Eudemus seems entirely satisfied that Hippocrates does indeed have a correct proof, it seems almost certain from this circumstantial evidence that we can deduce that Hippocrates himself developed at least a variant of the method of exhaustion. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hippocrates.html (3 of 4) [2/16/2002 11:14:46 PM]

Hippocrates

Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Squaring the circle 2. Doubling the cube 3. Trisecting an angle

Other references in MacTutor

Chronology: 500BC to 1AD

Other Web sites

1. Simon Fraser University 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hippocrates.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hippocrates.html (4 of 4) [2/16/2002 11:14:46 PM]

Hironaka

Heisuke Hironaka Born: 9 April 1931 in Yamaguchi-ken, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Heisuke Hironaka attended Kyoto University. This university was founded in 1897 to train small numbers of selected students as academics. By the time Hironaka entered Kyoto University, after World War II, it had been integrated into a mass higher education system but had maintained its prestige. From Kyoto University Hironaka went to the United States where he continued his studies at Harvard. After completing his studies there, Hironaka was appointed to the staff at Harvard. In 1970 Hironaka had the distinction of being awarded a Fields Medal at the International Congress at Nice. This was for his work on algebraic varieties which we describe below. Among the many other honours he has received is the Order of Culture from Japan in 1975. Two algebraic varieties are said to be equivalent if there is a one-to-one correspondence between them with both the map and its inverse regular. Two varieties U and V are said to be birationally equivalent if they contain open sets U' and V' that are in biregular correspondence. Classical algebraic geometry studies properties of varieties which are invariant under birational transformations. Difficulties that arise as a result of the presence of singularities are avoided by using birational correspondences instead of biregular ones. The main problem in this area is to find a nonsingular algebraic variety U, that is birationally equivalent to an irreducible algebraic variety V, such that the mapping f: U but not biregular.

V is regular

Hironaka gave a general solution of this problem in any dimension in 1964. His work generalised that of Zariski who had proved the theorem concerning the resolution of singularities on an algebraic variety for dimension not exceeding 3. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hironaka.html (1 of 2) [2/16/2002 11:14:48 PM]

Hironaka

Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1960 to 1970

Honours awarded to Heisuke Hironaka (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1970

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hironaka.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hironaka.html (2 of 2) [2/16/2002 11:14:48 PM]

Hirsch

Kurt August Hirsch Born: 12 Jan 1906 in Berlin, Germany Died: 4 Nov 1986 in London, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kurt Hirsch studied at the University of Berlin where he was taught by Bieberbach, von Mises, Schmidt and Schur. Although most influenced by Schur, his doctoral dissertation was on the philosophy of mathematics. The thesis examines the 1920s dispute between Hilbert and Brouwer on the foundations of mathematics. Completed and examined in 1930 (Bieberbach was an examiner), it was not until 1933 that Kurt could afford to get it printed, so he did not receive the degree until 1933. Kurt became a journalist, writing a scientific column. However he kept up his mathematics attending a study group where he studied Emmy Noether's work and read Schreier's paper on the Jordan-Hölder theorem. Influenced by these ideas he decided to study soluble groups with the maximum condition on subgroups. Kurt's wife, who he married in 1928, was Jewish and he adopted the faith for her sake. He had little choice but to leave for England (where he had distant relatives). On arriving in England he was met by Bernhard Neumann and Hanna von Caemmerer (later to become Hanna Neumann). He had known Bernhard as a fellow student in Berlin. At this time Hanna was studying with Philip Hall in Cambridge. Kurt was introduced to Hall and he was encouraged to pursue his intention of working on soluble groups with the maximum condition on subgroups. Despite having a doctorate, Hirsch completed a second one at Cambridge in 1937 on polycyclic groups under Hall's supervision. Appointed to Leicester in 1938 he was interned as "an enemy alien" in 1940 in a POW camp on the Isle of Man. He worked there as a cook (and retained an interest in recipes all his

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hirsch.html (1 of 2) [2/16/2002 11:14:50 PM]

Hirsch

life) but was soon released and returned to Leicester. That he had interests outside mathematics is illustrated by the fact that he was Leicester County Chess Champion in 1945-46. In 1948 Hirsch moved to King's College, Newcastle (founded in 1937 as a part of the University of Durham and becoming the University of Newcastle in 1963). While at Newcastle he began translating Kurosh's The theory of groups into English, a project he was to work on for a number of years. He was a leading reformer of the mathematics syllabus at Newcastle where again he found time to win the County Chess Championship in 1950. Then in 1951 Hirsch was appointed to Queen Mary College of the University of London where he remained building up a strong algebra school. He sought Hall's advice in appointments of algebraists and attracted many research students to make a thriving group theory school. In [1] Gruenberg writes:Hirsch was a shrewd judge of people and managed to create at Queen Mary College an unusually friendly environment for students as well as staff. Everyone felt encouraged to be cooperative. Younger members of staff found him easy to work with and knew they could count on his help and protection. He gave them generously of his time with sound advice on teaching, on examining and on supervising research students. All Hirsch's publications were in group theory, in addition to the work on polycyclic groups he published on locally nilpotent groups and automorphism groups of torsion free abelian groups. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Kurt Hirsch

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hirsch.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hirsch.html (2 of 2) [2/16/2002 11:14:50 PM]

Hirst

Thomas Archer Hirst Born: 22 April 1830 in Heckmondwike, Yorkshire, England Died: 16 Feb 1891 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Hirst's father was in the wool trade. It was a prosperous family, so much so that Hirst's father retired when only thirty-one years old since at that age he inherited his own father's money. Thomas's mother also came from a family involved in the wool trade and Thomas was the youngest of his parents four sons. The family were certainly education minded and Thomas's father moved from Heckmondwike to be close to Wakefield when Thomas was five years old so that the boys could attend better schools. Thomas attended Wakefield primary school, then in 1841 he entered West Riding Proprietary School where he spent four years. He wrote later [2] that in this school:... I could obtain the most rudimentary and necessary instruction. I remember, however, that here mathematics was my favourite study ... In 1844 Thomas's father died in an accident. Thomas's mother was keen to find careers for here sons and for this reason Thomas was taken away from school when fifteen years of age and began to work as an apprentice engineer. He was sent to Halifax and his first task was to work on a survey for the proposed railway line from Halifax to Keighley. He began to write a diary at this time and it is largely for this that he is well known among mathematicians today. Many quotations from this diary, in which he describes well-known mathematicians he had met, are given in this archive. The chief surveyor of the engineering firm was John Tyndall. He was ten years older than Hirst and he was to have a major influence on the direction which Hirst's life took. They quickly became firm friends. The first important influence which Tyndall had on Hirst was to encourage him, in addition to carrying out the many tasks relating to his job as a surveyor with the engineering firm, to continue with his education by reading on his own. Hirst read many works of literature, scientific texts and mathematics http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hirst.html (1 of 4) [2/16/2002 11:14:52 PM]

Hirst

books such as Euclid's Elements, Hutton's Mathematics and Brewster's Life of Sir Isaac Newton. In February 1848 Hirst enrolled at the Halifax Mechanics Institute where he seemed for a while to be casting about seeking the subjects which interested him most. Slowly he turned to studying more and more mathematics. Tyndall left the engineering firm and, after a short spell as a teacher, went to the University of Marburg to study chemistry. Hirst visited Tyndall in Germany in August 1949 but he had to return home earlier than planned on the news that his mother had died. One result of his mother's death was that he came into quite a bit of money and, deciding that surveying was not the career for him, he formed a plan to complete his apprenticeship and then to follow his friend Tyndall to Marburg. He completed his apprenticeship on 31 August 1850 and set out with Tyndall, who was returning after the summer vacation, to travel to Marburg which they reached in October. On 2 November 1850 Hirst enrolled at the University of Marburg to study mathematics, physics and chemistry. Despite earlier moves towards mathematics, he was yet to make a decision on the topic in which he would specialise. At first he was attracted to chemistry, particularly to the lectures of Robert Bunsen. However Bunsen left Marburg in the spring of 1851 and after this Hirst's interests turned increasingly towards mathematics, sometimes to such an extent that he would ignore his other subjects. When he was close to completing his doctorate (this was a first degree in Germany for those who took the course at this time) a comment from Tyndall made him decide to carry on with his studies of mathematics and to visit other German universities such as Berlin and Göttingen. He took the necessary oral examination in the spring of 1852 and, after a brief visit back to England, he returned to Marburg to complete his dissertation for his doctorate. He submitted the thesis On conjugate diameters of the triaxial ellipsoid and was awarded his doctorate in July 1852. Leaving Marburg, Hirst travelled to Göttingen where he spent two weeks attending lectures. He met Weber and Gauss at this time. After spending some time travelling in Germany and Austria with friends he went to Berlin where he spent the winter semester. Eisenstein, who he had hoped to visit, died the day before Hirst arrived. However he did make friends with many of the mathematicians in Berlin and he continued to study mathematics concentrating on geometry. In particular he attended lectures by Dirichlet and Steiner, being strongly influenced by Steiner to undertake further research on geometry. From Berlin Hirst made the journey to Paris where he spent two months attending lectures by Liouville and Lamé. He returned to England in the middle of 1853 and was appointed to a teaching post at Queenwood College near Salisbury. He married Anna Martin in late 1854, a young English lady whom he had met while in Marburg. Unfortunately soon after they were married Anna showed signs of tuberculosis. In 1856 Hirst resigned his teaching post at Queenwood so that he could take care of his wife and they travelled to the south of France hoping that the warmer weather would cure Anna. Leaving the south of France they settled in Paris where Hirst continued with his mathematical researches, publishing two papers on which he had begun to work while at Göttingen. Sadly Anna died in July 1857. Hirst began to attend lectures again in Paris and his own researches into geometry progressed well. In August 1858 he left Paris to spend a year in Italy. He became friends with Cremona, particularly sharing his keen involvement in the Italian war of unification. He returned to England in the summer of 1859. Hirst was appointed to the University College School in 1860. The following year he was nominated for

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hirst.html (2 of 4) [2/16/2002 11:14:52 PM]

Hirst

a fellowship of the Royal Society, Boole and Sylvester being among his proposers. He was elected as a fellow in 1861. However, he resigned from his post at University College School in July 1864 so that he might concentrate on his research. Later in 1864 he became a founder member of the X-club which had as its aim the promotion of science in England. It was not the only event of importance for Hirst in 1864 for that year saw him elected to the Council of the Royal Society. He was indeed becoming an important figure in British Science, something which was confirmed by him becoming a major figure in the London Mathematical Society. At a preliminary meeting to discuss setting up a mathematical society on 7 November 1864, the name London Mathematical Society was chosen. In fact this was meant to be the first proper meeting of the Society but De Morgan was ill and could not attend. The chair at this preliminary meeting was taken by Hirst and the date for the first meeting proper was set for 16 January 1865. Hirst was elected the first Vice-President of the Society and he served it for the next twenty years as a member of Council, the Treasurer of the Society and then as its President in 1872-74. Hirst was appointed as professor of physics at University College London in 1865 and he began lecturing in October of that year. In 1866 he received further honours being elected a fellow of the Royal Astronomical Society as well as being appointed General Secretary of the British Association for the Advancement of Science. He succeeded De Morgan to the chair of mathematics at University College London in 1867. One of Hirst's students gave this description of him [7]:His presence in the classroom was striking. He was tall, and held himself erect with an almost military air. he had a long black beard and a great, bald, dome-like forehead. He was a man with whom it was impossible to imagine the most audacious student venturing to take a liberty. There was something about him that invested his unlovely subject with dignity, if not interest. Less, perhaps, than any of the other professors, did he seem to think of examinations. To him, I believe, incredible as it sounds, mathematics must have been a solemn, high pursuit: a passion, if not religion. Yet with all his aloofness of manner he could be very simple, very patient, and extremely kind. Certainly to one of his most hopeless pupils he showed himself all three. As well as promoting science in general, Hirst worked in particular for the education of women. In 1869 he gave a course of twenty-four lectures on the Elements of Geometry to the Ladies educational Association of London. The lectures were very successful both for their quality and for the large number of women who signed up for the course. He was less than happy with his heavy lecturing commitments, however, and so he decided to ask University College to give him an administrative post in exchange for the Chair of Mathematics. Hirst throughout his life always became unhappy when commitments prevented him from undertaking mathematical research. He became Assistant Registrar of University College in March 1870. In January 1871 the Association for the Improvement of Geometrical Teaching was founded and Hirst became its first president. This fitted well with his long held belief that Euclid's Elements should be replaced as the main geometry teaching text in schools. The association soon took on board the improvement of teaching of all mathematical topics in schools and was renamed the Mathematical Association. In 1873 Hirst was appointed as Director of Studies at the Royal Naval College in Greenwich. His

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hirst.html (3 of 4) [2/16/2002 11:14:52 PM]

Hirst

research had been mostly in geometry, in particular on Cremona transformations, and it was for this work that he was awarded the Royal Medal from the Royal Society in 1883. However his health, which had been less than good for a number of years became steadily worse. Kidney stones and a stomach tumour were diagnosed and, somewhat depressed by the death of his brother John, Hirst resigned from Greenwich in 1883 at the age of 53. The authors of [7] write:Finally, in 1890, he finished his memoir on the correlation of two spaces. He had worked on it for a long time and after its completion he destroyed his mathematical notebooks. Suddenly he seemed old, spending his time in watching the rapidly changing world from his clubs, his flat, and the park. By January 1892, now suffering from cancer of the prostate, he caught flu in an epidemic which hit London. He made the last entry in his diary which he had kept from the age of fifteen on 18 January 1892 and he died four weeks later. Article by: J J O'Connor and E F Robertson List of References (9 books/articles)

A Quotation

Mathematicians born in the same country Honours awarded to Thomas Hirst (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1861

Royal Society Royal Medal

Awarded 1883

London Maths Society President

1872 - 1874

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hirst.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hirst.html (4 of 4) [2/16/2002 11:14:52 PM]

Hniedenko

Hniedenko This biography is now under Gnedenko. You should be automatically forwarded to the correct page. If your browser leaves you on this page, then press HERE. JOC/EFR January 2001

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hniedenko.html [2/16/2002 11:14:52 PM]

Hobbes

Thomas Hobbes Born: 5 April 1588 in Westport, Malmesbury, Wiltshire, England Died: 4 Dec 1679 in Hardwick Hall, Derbyshire, England

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Hobbes was the author of Leviathan (1651 and was a renowned philosopher and political theorist. Educated at Oxford, where he graduated in 1608, he then journyed on the continent of Europe and visited Galileo, Mersenne, Gassendi and Roberval. Hobbes spent many periods in England and on the continent. He was in contact with Mersenne and his group on a number of these occasions, in particular the time he spent following 1640 when he feared for his safety in England. He became interested in light and wrote Tractatus opticus which was published by Mersenne . He wrote on geometry, not always correctly as De Morgan pointed out, but he loved the subject. However Wallis's Algebra he described as a scab of symbols which disfigured the page as if a hen had been scraping there. Hobbes attacked Wallis and others in Six Lessons to the Professors of Mathematics in the University of Oxford (1656). In 1660 Hobbes attacked the 'new' methods of mathematical analysis. In Dialogus Physicus, sive de Natura Aeris (1661) he attacked Boyle and those setting up the Royal Society. Wallis replied with unfair charges of disloyalty but Hobbes ended the argument with Mr. Hobbes Considered in His Loyalty, Religion, Reputation, and Manners (1662). Article by: J J O'Connor and E F Robertson List of References (21 books/articles)

Some Quotations (8)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hobbes.html (1 of 2) [2/16/2002 11:14:54 PM]

Hobbes

A Poster of Thomas Hobbes

Mathematicians born in the same country

Other Web sites

1. Internet Encyclopedia of Philosophy 2. The Galileo Project 3. Oregon 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hobbes.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hobbes.html (2 of 2) [2/16/2002 11:14:54 PM]

Hobson

Ernest William Hobson Born: 27 Oct 1856 in Derby, England Died: 19 April 1933 in Cambridge, Cambridgeshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernest Hobson was educated at Cambridge. He graduated as Senior Wrangler in 1878. He was appointed to the staff and taught there for the rest of his life. Perhaps rather surprisingly he tutored Miss Fawcett, the Senior Woman Wrangler of 1890, despite his well known strongly held views against women. Hobson published A Treatise on Trigonometry in 1891. He was introduced to modern analysis by Young and after this he began to make a real contribution to research. This research concentrated on convergence, in particular convergence of series of orthogonal functions. His book Theory of Functions of a Real Variable written in 1907 was the first English book on the measure and integration of Baire, Borel and Lebesgue. Hardy believed that this work had been of major importance in the development of pure mathematics in Britain. Probably mainly due the this particularly influential work, Hobson was elected Sadlerian professor at Cambridge in 1910. The Sadlerian chair only became vacant that year because Forsyth had been forced him to resign the chair after a scandal resulted from his love affair with the wife of C.V. Boys. Hobson was an active member of the London Mathematical Society, being president of the Society 1900-2 and receiving the De Morgan Medal of the Society in 1920. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hobson.html (1 of 2) [2/16/2002 11:14:55 PM]

Hobson

List of References (2 books/articles) A Poster of Ernest Hobson

Mathematicians born in the same country

Honours awarded to Ernest Hobson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1893

Royal Society Royal Medal

Awarded 1907

London Maths Society President

1900 - 1902

LMS De Morgan Medal

Awarded 1920

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hobson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hobson.html (2 of 2) [2/16/2002 11:14:55 PM]

Hodge

William Vallance Douglas Hodge Born: 17 June 1903 in Edinburgh, Scotland Died: 7 July 1975 in Cambridge, Cambridgeshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Hodge was a student at Edinburgh in Scotland and was taught by Whittaker and advised by him to study at Cambridge. His main interests were in algebraic geometry and differential geometry. In 1930 he applied ideas of Lefschetz to solve a problem posed by Severi. Hodge developed the relationship between geometry, analysis and topology and is best remembered for his theory of harmonic integrals. He held a chair at Cambridge from 1936 to 1970. He was one of the originators of the British Mathematical Colloquium, an annual conference which visits different British universities. He also played a major role in setting up the International Mathematical Union in 1952. He received the Royal Medal of the Royal Society of London in 1957 and was knighted in 1959. The Royal medal was awarded by the Royal Society in:... recognition of his distinguished work on algebraic geometry. He was also awarded the Copley Medal by the Royal Society in 1974:... in recognition of his pioneering work in algebraic geometry, notably in his theory of harmonic integrals. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hodge.html (1 of 2) [2/16/2002 11:14:57 PM]

Hodge

List of References (5 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1950 to 1960

Honours awarded to William Hodge (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1938

Royal Society Copley Medal

Awarded 1974

Royal Society Royal Medal

Awarded 1957

Fellow of the Royal Society of Edinburgh London Maths Society President

1947 - 1949

LMS De Morgan Medal

Awarded 1959

Honorary Fellow of the Edinburgh Maths Society

Elected 1954

LMS Berwick Prize winner

1952

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hodge.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hodge.html (2 of 2) [2/16/2002 11:14:57 PM]

Holder

Otto Ludwig Hölder Born: 22 Dec 1859 in Stuttgart, Germany Died: 29 Aug 1937 in Leipzig, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Otto Hölder worked on the convergence of Fourier series and in 1884 he discovered the inequality now named after him. He became interested in group theory through Kronecker and Klein and proved the uniqueness of the factor groups in a composition series. Hölder studied engineering at the polytechnic in Stuttgart for a year then, from 1877, he studied at the University of Berlin. At Berlin he was a fellow student of Runge and he attended lectures by Weierstrass, Kronecker and Kummer. Hölder's interest in algebra came partly through the influence of Kronecker at this time and Kronecker's liking for rigour almost certainly was to have a profound influence on Hölder's later work in algebra. Hölder presented his dissertation to the University of Tübingen in 1882. His dissertation investigates analytic functions and summation procedures by arithmetic means. After taking his doctorate Hölder went to Leipzig. Klein was there at the time but there seems to have been little interaction between the two at the time, Hölder at this time still being interested in function theory, although Klein had a strong influence on Hölder later. Hölder became a lecturer at Göttingen in 1884 and at first he worked on the convergence of Fourier series. Shortly after be began working at Göttingen he discovered the inequality now named after him. It appears that Hölder became interested in group theory while at Göttingen, through von Dyck and Klein. Hölder was offered a post in Tübingen in 1889 but unfortunately he suffered a mental collapse. The faculty at Tübingen kept their confidence in Hölder and he made a steady recovery, giving his inaugural lecture in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Holder.html (1 of 3) [2/16/2002 11:14:59 PM]

Holder

1890. He began to study the Galois theory of equations and from there he was led to study compostion series of groups. Although Hölder did not consider that he invented the notion of a factor group, the concept appears clearly for the first time in a paper of Hölder's of 1889. Hölder clarifies the concept which he claims is neither new nor difficult but is not sufficiently appreciated. Hölder proved the uniqueness of the factor groups in a composition series, the theorem now called the Jordan-Hölder theorem. With the help of group theory and Galois theory methods Hölder returned to a study of the irreducible case of the cubic in the Cardan-Tartaglia formula in 1891. Hölder made many other contributions to group theory. He searched for finite simple groups and in an 1892 paper he showed that all simple groups up to order 200 are already known. His methods use the Sylow theorems in a similar way to how the problem would be solved today. Hölder also studied groups of orders p3, pq2, pqr and p4 for p, q, r primes, publishing his results in 1893, these results again heavily rely on the use of Sylow theorems. Concepts which were introduced by Hölder include inner and outer automorphisms. In 1895 he wrote a long paper on extensions of groups. From 1900 he became interested in the geometry of the projective line and later he studied philosophical questions. In [5] van der Waerden writes:reading Hölder's papers again and again is a profound intellectual treat. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Otto Hölder

Mathematicians born in the same country

Cross-references to History Topics

The development of group theory

Other references in MacTutor

Chronology: 1880 to 1890

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Holder.html (2 of 3) [2/16/2002 11:14:59 PM]

Search Form

Holder

Glossary index

Quotations index

Mathematicians of the day JOC/EFR August 1997

Poster index Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Holder.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Holder.html (3 of 3) [2/16/2002 11:14:59 PM]

Hollerith

Herman Hollerith Born: 29 Feb 1860 in Buffalo, New York, USA Died: 17 Nov 1929 in Washington D.C., USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Herman Hollerith's parents were immigrants to the United States from Germany in 1848 after political disturbances in that country. School was not very easy for Herman despite the fact that he was clever. Ashurst recounts [4]:Herman is said to have been a bright and able child at school, but had an inability to learn spelling easily. His determined teacher made his life miserable to the extent that he used to avoid school whenever possible and run away when his teacher showed renewed effort to improve his spelling. The consequence of these school problems were that Herman was eventually taken away from school and he was tutored privately at home by the family's Lutheran minister. Hollerith entered the City College of New York in 1875 and he became an engineering graduate of the Columbia School of Mines in 1879, obtaining a distinction in his final examinations. His undergraduate record had been outstanding and one of his teachers, Professor W P Trowbridge, was so impressed that he asked Hollerith to become his assistant. So after graduating Hollerith became an assistant to Trowbridge, first at Columbia University but later he joined the US Census Bureau as a statistician when Trowbridge was appointed Chief Special Agent to the Census Bureau. This appointment was very significant because it was in solving the problems of analysing the large amounts of data generated by the 1880 US census that Hollerith was led to look for ways of manipulating data mechanically. The idea in fact came from Dr John Shaw Billings who Hollerith came in contact with in his work for the US Census Bureau. Hollerith wrote much later (see [6]):http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hollerith.html (1 of 4) [2/16/2002 11:15:01 PM]

Hollerith

One evening at Dr B's tea table he said to me 'There ought to be a machine for doing the purely mechanical work of tabulating population and similar statistics'. In a slightly different version of the same story Dr Billings was reported to have said (see for example [4]):There ought to be some mechanical way of doing this job, something on the principle of the Jacquard loom, whereby holes in a card regulate the pattern to be woven. In 1882 Hollerith joined the Massachusetts Institute of Technology where he taught mechanical engineering. At this time he investigated Billings suggestion, examining the way that the Jacquard loom worked with a view to seeing if it could be used in census work. He found that in most respects the function of the Jacquard loom was too far removed from what might be useful to the census work, however he did realise that the punched cards were an efficient way to store information. Another idea struck him one day on a train journey as he watched the ticket collector punch tickets. This was an easy way to punch information onto cards. While he worked at the Massachusetts Institute of Technology Hollerith began his first experiments. These used a paper tape, rather than cards, with pins which would go through a hole in the tape and complete an electrical contact. The idea was nearly right but the tape had drawbacks since it had to stop to allow the pin to go through the hole to make the contact. Hollerith realised that cards would provide a better solution. Hollerith did not enjoy teaching so he soon sought another job. In 1884 he obtain a post in the U.S. Patent Office in Washington, D.C. This was either good luck or a brilliant career move depending on how far sighted Hollerith was in seeing that he would be in the best possible position to make full use of skills learnt in the patent office in patenting his own inventions. In 1884 Hollerith applied for his first patent (he would receive more than 30 patents from the United States during his career and many overseas patents). He developed the early work he had done at the Massachusetts Institute of Technology on methods to convert the information on punched cards into electrical impulses. These impulses in turn would activate mechanical counters. He used at first the punch that was used for tickets on the railway to make the holes in the cards. This showed that the system worked but since the punch could only make holes near the edge of the card, the full potential was not being used. Hollerith designed punches specially made for his system, the Hollerith Electric Tabulating System. He also improved the machines which read the cards. Engineering developments improved the accuracy of the pin going through the hole in the card to make an electrical connection with mercury placed beneath. The resulting electrical current activated a mechanical counter and the amount of information which could be handled on each card rapidly increased. Hollerith's system was first tested on tabulating mortality statistics in Baltimore, New Jersey in 1887 and again in New York City. This punched card system was in use by the time of the 1890 US census but it was not the only system to be considered for use with the census. It won convincingly in competition with two other systems considered for the 1890 census showing that it could handle data more quickly. Having won, Hollerith now had to have punches and counting devices manufactured. The punches were made by Pratt and Whitney, later famed for building engines for aircraft. The punch was constructed in a similar way to a typewriter having a simple keyboard. The counting machines were made by the Western http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hollerith.html (2 of 4) [2/16/2002 11:15:01 PM]

Hollerith

Electric Company. Everything was in place by June 1890 and the first data from the census arrived in September of that year. The counting was completed by 12 December 1890 having taken about three months to process instead of the expected time of two years if counting had been done by hand. The total population of the United States in 1890 was found to be 62,622,250. Speed was not the only benefit of using Hollerith's system. It was possible to gather more data, and data such as the number of children born in a family, the number of children still alive in a family, and the number of people who spoke English were part of the 1890 census. Although Hollerith had left the academic world, he clearly was still attracted to certain aspects of it, for he wrote up the details of his tabulating systems and submitted the work for a doctorate at the Columbia School of Mines. Hollerith was awarded his doctorate in 1890. The Hollerith system was clearly a great leap forward. It saved the United States 5 million dollars for the 1890 census by completing the analysis of the data in a fraction of the time it would have taken without it and with a smaller amount of manpower than would have been necessary otherwise. The system was again used for the 1891 census in Canada, Norway and Austria and later for the 1911 UK census. Honours came to Hollerith from all sides for his outstanding invention. He was awarded the prestigious Elliot Cresson Medal by the Franklin Institute of Philadelphia in 1890. He received the Gold Medal of the Paris Exposition and the Bronze Medal of the World's Fair in 1893. He was asked to address learned societies around the world, for example he spoke to the Royal Statistical Society in London. In 1896 Hollerith founded the Tabulating Machine Company to exploit his inventions. By this time he had added a mechanism to feed the cards automatically and other automatic sorting procedures which added sophistication to the original simple mechanical counting process. His system was used again for the US census of 1900, but by this time he was asking such a high price for the use of his technology that questions began to be asked about the wisdom of using the system. Because Hollerith had a virtual monopoly he had set the price well beyond what it would have cost to count the 1900 census data by hand. The Census Bureau became a permanent institution by an Act of Congress in 1903 and it began to prepare for the 1910 census. The cost of using Hollerith's system in 1900 made them decide to develop their own system and, despite the short time and the difficulty of getting round Hollerith's patents, they were able to have more advanced machines ready in time for the 1910 census. There is a rather strange twist to this story for the engineer who was in charge of the development of the rival machines at the Census Bureau, James Powers, was strangely allowed to patent these more advanced machines in his own name. Powers was now in a strong position and in 1911, after the census, he left the Census Bureau and formed the Powers Tabulating Machine Company which was now more than a match for Hollerith's Tabulating Machine Company. A merger with another company saw Hollerith's company become the Computer Tabulating Recording Company in 1911 but the new company largely was forced out of the market for counting machines. Hollerith served as a consulting engineer with the Computer Tabulating Recording Company until he retired in 1921. The Computer Tabulating Recording Company had recovered its leading role by 1920, due not to Hollerith but to Thomas J Watson who joined the company in 1918. The company was renamed International Business Machines Corporation (IBM), in 1924. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hollerith.html (3 of 4) [2/16/2002 11:15:01 PM]

Hollerith

Although Hollerith made a very significant contribution to the development of the modern electronic computer with his punched card technology not all his ideas were similar great successes. In the 1880s, at the same time as he was developing his first punched card system, he invented a new brake system for trains. However his electrically actuated brake system lost out to the Westinghouse steam-actuated brake. Hollerith died of a heart attack in 1929, eight years after retiring. Article by: J J O'Connor and E F Robertson List of References (10 books/articles) Mathematicians born in the same country Other references in MacTutor

A picture of Hollerith's machine

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hollerith.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hollerith.html (4 of 4) [2/16/2002 11:15:01 PM]

Holmboe

Bernt Michael Holmboe Born: 23 March 1795 in Vang, Norway Died: 28 March 1850 in Christiania (now Oslo), Norway

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bernt Holmboe was the son of a minister. He graduated from the Cathedral School of Christiania, then served for a short time in a military campaign against Sweden in 1814. Holmboe was appointed assistant to the astronomer C Hansteen at Christiania in 1815. In 1818 he became a teacher at the Cathedral School of Christiania. It was at this school that he was the mathematics teacher of Abel and he helped pay for Abel's university education. In 1826 Holmboe accepted a position as lecturer at the University of Christiania, a move which some have criticised since this might have been a possible post for Abel. However Abel did not seem to feel this way as he remained firm friends with Holmboe. From 1826 to 1850 Holmboe lectured at the military academy in Christiania. In 1834 he was appointed to the chair of pure mathematics at the University. During the years 1828-30 Holmboe lectured on astronomy since Hansteen went on a geomagnetic expedition to Siberia. After Abel's death Holmboe edited Abel's complete works in 1839. Holmboe also wrote a textbook. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Holmboe.html (1 of 2) [2/16/2002 11:15:03 PM]

Holmboe

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Holmboe.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Holmboe.html (2 of 2) [2/16/2002 11:15:03 PM]

Honda

Taira Honda Born: 2 June 1932 in Fukui, Japan Died: 15 May 1975 in Osaka, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Taira Honda began his university studies in 1951 when he entered Tokyo University. He joined T Tamagawa's seminar his 1954-55 which was his final undergraduate year. After graduating in 1955 he continued to study at Tokyo University, undertaking research supervised by Tamagawa. In the autumn of 1955 an International Symposium on Algebraic Number Theory was held in Japan. This was an important event as Iyanaga relates in [1];Mathematicians like E Artin, Chevalley and A Weil came to our country for this occasion, and significant contributions to long-standing problems of the theory of complex multiplication by Goro Shimura and Yutaka Taniyama as well as by A Weil were reported at the Symposium. These circumstances surely had a strong influence on the mathematical formation of Honda in addition to the personal direction given to him by Tamagawa. Honda published his first paper Isogenies, rational points and section points of group varieties in 1960 [1]:He restudied and generalised the mathematical theories of Kummer fields and of cyclotomic fields from the standpoint of abelian varieties over algebraic number fields. Honda's next three papers all considered the problem of class numbers of algebraic number fields. Then in 1966 he published an important paper which began his study of applications of the theory of commutative formal groups to the arithmetical theory of abelian varieties. The main ideas of his doctoral thesis, which continued work which appeared in his first paper, was published in 1968. Isogeny classes of abelian varieties over finite fields appeared in Journal of the Mathematical Society of Japan in 1968. In this important paper Honda gives a complete classification of Abelian varieties up to isogeny over a finite field. The paper built on work of J Tate which had been published two years earlier http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Honda.html (1 of 2) [2/16/2002 11:15:05 PM]

Honda

and used results of Shimura and Taniyama in its proof. An important consequence is that Honda was able to give a short proof of Manin's conjecture about formal groups. Further work on the class numbers of algebraic number fields saw Honda prove that there are infinitely many real quadratic fields whose class numbers are divisible by 3 and also to classify those n for which the class number of Q(3 n) is a multiple of 3. In 1970 he published On the theory of commutative formal group. Many more papers on formal groups followed, in particular relating them to the zeta function. We have described Honda's mathematical research, which was mainly devoted to the investigation of the arithmetic properties of commutative formal groups. However we have not yet described his career following his postgraduate work. He was appointed to Osaka University in 1961 and then to Osaka City University in 1974. Shortly after that, however, his brilliant career was cut short when he took his own life. Iyanaga writes in [1] that Honda was:... a most active mathematician, beloved by his colleagues and students as well as by his family. Most unfortunately, he took his own life for some unknown reason in May 1975. Describing Honda's personality, Iyanaga writes:Always frank and friendly, he was popular among both his colleagues and students. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2001 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Honda.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Honda.html (2 of 2) [2/16/2002 11:15:05 PM]

Hooke

Robert Hooke Born: 18 July 1635 in Freshwater, Isle of Wight, England Died: 3 March 1703 in London, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Hooke went to school in Westminster where he learnt Latin and Greek, but unlike his contemporaries he never wrote in Latin. In 1653 he went to Christ College, Oxford where he won a chorister's place. At Oxford Hooke met Boyle and in 1655 he was employed by Boyle to construct his air pump. In 1660 he discovered Hooke's law of elasticity. Hooke worked on optics, simple harmonic motion and stress in stretched strings. For 30 years he was professor of geometry at Gresham College, London, being appointed there in 1665. The year 1665 was the one when Hooke first achieved worldwide scientific fame. His book Micrographia, published that year, contains beautiful pictures of objects Hooke had studied through a microscope he had made himself. The book also contains a number of fundamental biological discoveries. Pepys wrote in his diary Before I went to bed I sat up till two o'clock in my chamber reading Mr Hooke's Microscopical Observations, the most ingenious book that ever I read in my life. Hooke invented the conical pendulum and was the first person to build a Gregorian reflecting telescope. He made important astronomical observations including the fact that Jupiter revolves on its axis and his drawings of Mars were later used to determine its period of rotation. In 1666 he proposed that gravity could be measured using a pendulum. In addition to his post as professor of geometry at Gresham College, London Hooke held the post of City Surveyor. He was a very competent architect and was chief assistant to Wren in his project to rebuild London after the Great Fire of 1666. In 1672 Hooke attempted to prove that the Earth moves in an ellipse round the Sun and six years later proposed that inverse square law of gravitation to explain planetary motions. Hooke wrote to Newton in 1679 asking for his opinion:of compounding the celestiall motions of the planetts of a direct motion by the tangent (inertial motion) and an attractive motion towards the centrall body ... my supposition is that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall... http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hooke.html (1 of 3) [2/16/2002 11:15:06 PM]

Hooke

Hooke seemed unable to give a mathematical proof of his conjectures. However he claimed priority over the inverse square law and this led to a bitter dispute with Newton who, as a consequence, removed all references to Hooke from the Principia. No portrait of Hooke is known to exist. A possible reason for this is that he has been described as a lean, bent and ugly man and so he may not have been willing to sit for a painting of his portrait. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (29 books/articles)

Some Quotations (2)

Mathematicians born in the same country Cross-references to History Topics

1. Orbits and gravitation 2. English attack on the Longitude Problem 3. The rise of the calculus

Other references in MacTutor

Chronology: 1650 to 1675

Honours awarded to Robert Hooke (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1663

Lunar features

Crater Hooke

Lunar features

Crater Hooke on Mars

Other Web sites

1. The Galileo Project 2. Linda Hall Library (Drawing of the Moon) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hooke.html (2 of 3) [2/16/2002 11:15:06 PM]

Hooke

JOC/EFR February 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hooke.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hooke.html (3 of 3) [2/16/2002 11:15:06 PM]

Hopf

Heinz Hopf Born: 19 Nov 1894 in Gräbschen (near Breslau), Germany (now Wroclaw, Poland) Died: 3 June 1971 in Zollikon, Switzerland

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Heinz Hopf's father was Wilhelm Hopf and his mother was Elizabeth Kirchner. Wilhelm Hopf was from a Jewish family. He joined Heinrich Kirchner at his brewery in Breslau in 1887. Wilhelm married Elizabeth, Heinrich Kirchner's eldest daughter, in 1892 and by that time he owned the brewery firm. They had two children, the eldest Hedwig was born in 1893 while Heinz was born in the following year. Elizabeth Hopf was a Protestant and, in 1895, Wilhelm converted to his wife's religion. Heinz attended Dr Karl Mittelhaus' school from 1901 until 1904 and following this he began his studies at the König-Wilhelm Gymnasium in Breslau. He attended the Gymnasium until 1913 and it was at this school that his talent for mathematics first became clear to his teachers. In his other subjects, however, his results were less good and it is probable that he devoted too much time to sport, he was particularly fond of swimming and tennis, and not enough to his academic subjects. He left the Gymnasium with the mathematics report stating:He has shown an extraordinary gift in this topic, especially in the algebraic direction. In April 1913 Hopf entered the Silesian Friedrich Wilhelms University in Breslau to read for a degree in mathematics. There he was taught by Kneser, Schmidt, and Rudolf Sturm. He also attended lectures by Dehn and Steinitz who taught at the polytechnic in Breslau. However, his studies were interrupted by the outbreak of World War I in 1914. He immediately enlisted and for the duration of the war he fought on the Western front as a lieutenant. During a fortnight's leave from military service in 1917 Hopf went to a class by Schmidt on set theory at the University of Breslau. From that time on he knew that he wanted to undertake research in mathematics. He wrote in [13] about the influence Schmidt's lectures had on him :http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopf.html (1 of 5) [2/16/2002 11:15:08 PM]

Hopf

I was fascinated; this fascination - of the power of the method of the mapping degree - has never left me since, but has influenced major parts of my work. And when I look for the cause of this effect, I see particularly two things: firstly, Schmidt's vividness and enthusiasm in his lecture, and secondly my own increased receptiveness during a fortnight off after many years of military service. After the war Hopf returned to his studies in Breslau but after about a year he left and went to the University of Heidelberg. By this time Schmidt had left Breslau and it appears that Hopf wanted to go to Heidelberg to be with his sister who had begun her studies there in the previous year. At Heidelberg Hopf took courses in philosophy and psychology as well as attending courses by Perron and Stäckel. In 1920 Hopf went to study for his doctorate at the University of Berlin where Schmidt was now teaching. He attended several courses by Schur in Berlin and he received his doctorate in 1925 with a thesis, supervised by Schmidt, studying the topology of manifolds. Among other results, he classified simply connected Riemannian 3-manifolds of constant curvature in this thesis. It was an impressive piece of work which received the following praise from Schmidt in his report (see for example [11]):The boldness of the questions deserves as much admiration as the surprising results of the solutions. But the most beautiful thing in the thesis is the method of proving, which is, particularly rarely found in works in that area, abstract and comprehensible in every step, and which, due to the abstractness, shows equally clearly the richness of the concrete geometric imagination. Bieberbach and Schmidt examined him in mathematics, while Planck examined him in physics. Hopf went to Göttingen in 1925 where he met Emmy Noether. Her contributions would play an important part in Hopf's developing ideas. Perhaps even more significant was the fact that Aleksandrov was also spending time in Göttingen and Hopf wrote in [13]:My most important experience in Göttingen was to meet Pavel Aleksandrov. The meeting soon became friendship; not only topology, not only mathematics was discussed; it was a fortunate and also a very happy time, not restricted to Göttingen but continued on many joint journeys. During this year in Göttingen Hopf worked on his habilitation thesis which was completed by the autumn of 1926. The thesis contains a different proof of the fact just shown by Lefschetz that for any closed manifold the sum of the indices of a generic vector field is a topological invariant, namely the Euler characteristic. Aleksandrov and Hopf spent some time in 1926 in the south of France with Neugebauer. Then the two spent the academic year 1927-28 at Princeton in the United States. This was an important year in the development of topology with Aleksandrov and Hopf in Princeton and able to collaborate with Lefschetz, Veblen and Alexander. During their year in Princeton, Aleksandrov and Hopf planned a joint multi-volume work on Topology the first volume of which did not appear until 1935. This was the only one of the three intended volumes to appear since World War II prevented further collaboration on the remaining two volumes. Hopf married Anja von Mickwitz in October 1928. He was offered an assistant professorship by Princeton in December 1929 but he rejected the offer. In 1930 Weyl left his chair in the ETH in Zurich to take up a chair at Göttingen and in 1931 Hopf was approached to see if he was interested in accepting this chair. In part the offer had been prompted by a very positive recommendation which Schur had sent http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopf.html (2 of 5) [2/16/2002 11:15:08 PM]

Hopf

to Zurich:Hopf is an excellent lecturer, a mathematician of strong temperament and strong influence, a leading example in his discipline ... I cannot wish you a better colleague in respect to his manners, his education and his sympathetic nature. Hopf replied to the approach of the ETH in Zurich indicating that he would accept a formal offer:A call to Switzerland, to the beautiful city of Zurich, could indeed tempt and honour me, particularly to such a famous chair. I therefore declare that I am in principle willing to accept such an offer. However, before receiving the formal offer from Zurich, Hopf received the offer of a chair at Freiburg but he waited for the Zurich offer and accepted it. He took up his duties in Zurich in April 1931. The next few years were not easy ones for Hopf. After the Nazis came to power in Germany in 1933, Hopf's father, being Jewish, came under increasing pressure. Hopf continued to visit his parents in Breslau up until 1939. Seeing the difficulties that his father faced Hopf arranged for his parents to receive immigration papers for Switzerland. However, his father fell ill and could not travel. Hopf was able to provide refuge in Switzerland for friends who had to flee Germany under the Nazis. In particular Schur came for a while before finally going to Palestine in 1939. Hopf's own position became more difficult, however, for he was still a German citizen. Lefschetz, realising Hopf's difficulties, invited him to Princeton but Hopf refused. Then in 1943 he was told to move back to Germany or he would lose his German citizenship. Faced with this he had little choice but to quickly apply for Swiss citizenship, which was soon granted. With the end of World War II Hopf was able to help his German friends again. He did much more than this, however, for he put much energy into trying to re-establish a mathematical community in Germany. His visit to the research centre in Oberwolfach in August 1946 was part of his efforts. Soon after the Oberwolfach visit, Hopf went to the United States where he spent six months and there he renewed many old friendships. He was offered professorships by many of the most prestigious of the American universities but, after careful consideration, he decided to remain loyal to Zurich. Over the next few years he enjoyed invitations to lecture at leading international conferences, and he visited many places including Paris, Brussels, Rome and Oxford. He spent the academic year 1955-56 with his wife in the United States. Most of Hopf's work was in algebraic topology where he can be thought of as continuing Brouwer's work. He studied homotopy classes and vector fields producing a formula about the integral curvature. Hopf extended Lefschetz's fixed point formula in work which he undertook in 1928. It is in this 1928 paper that he first explicitly used homology groups. His work on the homology of manifolds, undertaken in Princeton in 1927-28, led to his definition of the intersection ring by defining a product on cycles by their intersection. This idea was later seen to be connected to cohomology. He defined what is now known as the 'Hopf invariant' in 1931. This was done in his work on maps between spheres of different dimensions which cannot be distinguished homologically so required the introduction of a new invariant. In 1939 he examined the homology of a compact Lie group. This was to attack questions posed to him by Elie Cartan. The ides which he introduced in this investigation led to him defining what is today called a Hopf algebra. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopf.html (3 of 5) [2/16/2002 11:15:08 PM]

Hopf

In the early 1940s Hopf published [11]:The paper Fundamentalgruppe und zweite Bettische Gruppe [which] is legitimately regarded to be the beginning of homological algebra. It opened the way for the definition for the homology and cohomology of a group. This step was made independently at different places shortly after the paper became known ... The honours which Hopf received are almost too numerous to list. He was President of the International Mathematical Union from 1955 until 1958. He received honorary doctorates from many universities including Princeton, Freiburg, Manchester, Sorbonne, Brussels, and Lausanne. He was awarded many prizes including the Gauss-Weber medal and the Lobachevsky award. He was elected to honorary membership of many learned societies throughout the world. Frudenthal gives this description of Hopf in [1]:Hopf was a short, vigorous man with cheerful, pleasant features. His voice was well modulated, and his speech slow and strongly articulated. his lecture style was clear and fascinating; in personal conversation he conveyed stimulating ideas. Frei and Stammbach in [11] pay this tribute to Hopf:Without doubt Heinz Hopf was one of the most distinguished mathematicians of the twentieth century. His work is closely linked with the emergence of algebraic topology; it is most decisively thanks to his early works that this area established itself as a new and important branch of mathematics. his work has influenced profoundly the evolution not only of topology but of a large part of mathematics. But Heinz Hopf was not only a gifted researcher: he was also an excellent teacher and a personality of the highest integrity. at the same time, he effervesced with charm and subtle humour. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles) A Poster of Heinz Hopf

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1920 to 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopf.html (4 of 5) [2/16/2002 11:15:08 PM]

Hopf

JOC/EFR September 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hopf.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopf.html (5 of 5) [2/16/2002 11:15:08 PM]

Hopf_Eberhard

Eberhard Frederich Ferdinand Hopf Born: 4 April 1902 in Salzburg, Austria Died: 24 July 1983

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eberhard Hopf, an Austrian mathematician who made significant contributions in topology and ergodic theory, was born in Salzburg. Most of his scientific formation, however, was in Germany, where he received a Ph.D. in Mathematics in 1926 and, in 1929, his Habilitation in Mathematical Astronomy from the University of Berlin. In 1930 Hopf received a fellowship from the Rockefeller Foundation to study classical mechanics with Birkhoff at Harvard in the United States. He arrived Cambridge, Massachusetts in October of 1930 but his official affiliation was not the Harvard Mathematics Department but, instead, the Harvard College Observatory. While in the Harvard College Observatory he worked on many mathematical and astronomical subjects including topology and ergodic theory. In particular he studied the theory of measure and invariant integrals in ergodic theory and his paper On time average theorem in dynamics which appeared in the Proceedings of the National Academy of Sciences is considered by many as the first readable paper in modern ergodic theory. Another important contribution from this period was the Wiener-Hopf equations, which he developed in collaboration with Norbert Wiener from the Massachusetts Institute of Technology. By 1960, a discrete version of these equations was being extensively used in electrical engineering and geophysics, their use continuing until the present day. Other work which he undertook during this period was on stellar atmospheres and on elliptic partial differential equations. On 14 December 1931, with the help of Norbert Wiener, Hopf joined the Department of Mathematics of the Massachusetts Institute of Technology accepting the position of Assistant Professor. Initially he had a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopf_Eberhard.html (1 of 3) [2/16/2002 11:15:10 PM]

Hopf_Eberhard

three years contract but this was subsequently extended to four years (1931 to 1936). While at MIT, Hopf did much of his work on ergodic theory which he published in papers such as Complete Transitivity and the Ergodic Principle (1932), Proof of Gibbs Hypothesis on Statistical Equilibrium (1932) and On Causality, Statistics and Probability (1934). In this 1934 paper Hopf discussed the method of arbitrary functions as a foundation for probability and many related concepts. Using these concepts Hopf was able to give a unified presentation of many results in ergodic theory that he and others had found since 1931. He also published a book Mathematical problems of radiative equilibrium in 1934 which was reprinted in 1964. In addition of being an outstanding mathematician, Hopf had the ability to illuminate the most complex subjects for his colleagues and even for non specialists. Because of this talent many discoveries and demonstrations of other mathematicians became easier to understand when described by Hopf. In 1936, at the end of the MIT contract, Hopf received an offer of full professorship in the University of Leipzig. As a result of this Hopf, with his wife Ilse, returned to Germany which, by this time, was already being ruled by the Nazi party. In Leipzig Hopf undertook research on quantic mechanics (1937), Geodesics on manifolds of negative curvature (1939), Statistik der geod (1939) and on the influence of curvature of a closed Riemannian manifold on its topology (1941). One important event from this period was the publication of the book Ergodentheorie (1937), most of which was written when Hopf was still at the Massachusetts Institute of Technology. In that book containing only 81 pages, Hopf made a precise and elegant summary of ergodic theory. In 1940 Hopf was on the list of the invited lecturers to the International Congress of Mathematicians to be held in Cambridge, Massachusetts. Because of the start of World War II, however, this Congress was cancelled. In 1942 Hopf was drafted to work in the German Aeronautical Institute. In 1944, one year before the end of World War II, Hopf was appointed to a professorship at the University of Munich. He held this post until 1947 by which time he had returned to the United States, where he presented, the definitive solution of Hurewicz's problem. On 22 February 1949 Hopf became a US citizen. He joined Indiana University as a Professor in 1949, a position he held until he retired in 1972. In 1962 he was made Research Professor of Mathematics, staying in that position until his death. An important publication from this period was An inequality for positive linear integral operators (1963) which appeared in the Journal for Mathematics and Mechanics. This paper is concerned with some extensions of Jentzsch's theorem on the existence of a positive eigenfunction for a positive integral operator. In 1971 Hopf was the American Mathematical Society Gibbs Lecturer. Coming out of this lecture was a paper Ergodic theory and the geodesic flow on surfaces of constant negative curvature which he published in the Bulletin of the American Mathematical Society. Hopf wrote in the introduction to that paper:Famous investigations on the theory of surfaces of constant negative curvature have been carried out around the turn of the century by F Klein and H Poincaré in connection with complex function theory. The theory of the geodesics in the large on such surfaces was developed later in the famous memoirs by P Koebe. This theory is purely topological. The measure-theoretical point of view became dominant in the later thirties after the advent of ergodic theory, and the papers of G A Hedlund and E Hopf on the ergodic character of the geodesic flow came into being. The present paper is an elaboration of the author's Gibbs

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopf_Eberhard.html (2 of 3) [2/16/2002 11:15:10 PM]

Hopf_Eberhard

lecture of this year [1971] and at the same time of the author's paper of 1939 on the subject, at least of its part concerning constant negative curvature. Hopf was never forgiven by many people for his moving to Germany in 1936, where the Nazi party was already in power. As a result most of his work to ergodic theory and topology was neglected or even attributed to others in the years following the end of World War II. An example of this was the dropping of Hopf's name from the discrete version of the so called Wiener-Hopf equations, which are currently referred to as "Wiener filter". In [4] Icha summarises Hopf's mathematical achievements:His interests and principal achievements were in the fields of partial and ordinarydifferential equations, calculus of variations, ergodic theory, topological dynamics, integral equations, differential geometry, complex function theory and functional analysis. Hopf's work is also of the greatest importance to the hydrodynamics, theory of turbulence and radiative transfer theory. Article by: J J O'Connor and E F Robertson based on a biography submitted by Osvaldo de Oliveira Duarte which in turn made substantial use of [3]. Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hopf_Eberhard.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopf_Eberhard.html (3 of 3) [2/16/2002 11:15:10 PM]

Hopkins

William Hopkins Born: 2 Feb 1793 in Kingston-on-Soar, Derbyshire, England Died: 13 Oct 1866 in Cambridge, Cambridgeshire, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Hopkins's father was also named William Hopkins. Now Hopkins senior was a farmer, not in the sense of working on a farm but in the sense of owning a farm and employing others to do the hard work. William junior was supposed to follow in his father's footsteps so his education was not considered important. William showed little progress, particularly when he was taken to Norfolk and instructed in the practical matters concerned with the running of a farm. Sticking with the plan that his son would become a farmer, William Hopkins senior purchased a small farm near Bury St Edmunds in Suffolk. William junior and his wife (he had by now married a Miss Braithwaite) made an unconvincing attempt to run the farm. He neither enjoyed the work nor did it bring in enough money to allow him to live a reasonable life and debts began to mount. When his wife died, Hopkins saw the chance to start a new life that he would enjoy. Hopkins sold the small farm which his father had bought him and with the money he was able to pay off his debts. In 1822, at the age of twenty-nine, Hopkins entered Peterhouse, the oldest of the Colleges of the University of Cambridge. There he studied mathematics and, particularly when one takes into account his previous poor education, he was highly successful. He graduated in 1827 placed Seventh Wrangler in the Mathematical Tripos. This means that he was seventh in the list of those gaining first class honours in that year. Just ahead of him was De Morgan, who was in the same year and placed Fourth Wrangler in the Mathematical Tripos. Hopkins married for a second time, to Caroline Boys, while he was an undergraduate at Cambridge. They had three children, one son and two daughters. After graduating Hopkins became a private tutor at Cambridge, having Tait, Thomson, Stokes, Maxwell and Todhunter among his pupils. He was highly successful being called the "senior wrangler maker". Rouse Ball relates that in 1849:[Hopkins] was able to say that he had among his pupils nearly two hundred wranglers, of whom seventeen had been senior and forty-four in one of the first three places. A little calculation shows how remarkable these figures are. By 1849 there had only been about 20 senior wranglers during the time he tutored (there was only one top student in each year) and around sixty placed in the top three. Of course once he gained a reputation as the best tutor he was able to select the best students from the first year class to tutor. Some tutors concentrated entirely on examination technique but Hopkins [2]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopkins.html (1 of 3) [2/16/2002 11:15:11 PM]

Hopkins

... was conspicuous for encouraging in his pupils a disinterested love of their studies, instead of limiting their aspirations to examination honours. Hopkins made relatively little contributions to pure mathematics, other than in his tutoring and with a two volume text Elements of Trigonometry written in 1833 and 1847 which contains interesting historical comments. However, he did make major contributions to the application of mathematics to geology. He became interested in applying mathematics to geology in 1833. This came about because of his friendship with Adam Sedgwick. Now Sedgwick had also trained as a mathematician and had been appointed professor of geology at Cambridge in 1818. Sedgwick began fieldwork at Barmouth in north Wales in 1831 where he established the order of local rocks and gave the name Cambrian to the oldest fossiliferous strata. The name was chosen because Cambria is the old name for Wales. Now Hopkins accompanied Sedgwick to Barmouth on many of these trips and through this work, which he greatly enjoyed, he began to feel that geology would benefit by being put on a firm mathematical basis. He made mathematical models which won him the Wollaston Medal of the Geological Society of London in 1850. Hopkins was President of the Geological Society of London in 1851 and 1852 (Sedgwick had been President of the Society in 1829). He was president of the British Association in 1853 when it met in Hull. In his presidential address he [2]:... referred to a series of important experiments which he had instituted at Manchester with the advice of Sir William Thomson and the assistance of Messrs Joule and Fairbairn, to determine the temperature of melting substances under great pressure. The experiments were being conducted by Hopkins in his study of the interior of the Earth. It would not be unfair to say that most of the geological theories which Hopkins proposed have turned out to be false. As Beckinsale writes [1]:... except in the popularisation of quantification and in the broader field of geophysics, Hopkins' effect on contemporary geology was frequently retrogressive rather than progressive. He was often lacking in geological insight ... As to his character we refer to [2] where Anderson writes:He was a man of marked dignity of character and most affectionate nature. He took a keen pleasure in poetry and music, had great conversational power, and his sense of natural beauty led to his taking up, not unsuccessfully, landscape painting late in life as a recreation. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopkins.html (2 of 3) [2/16/2002 11:15:11 PM]

Hopkins

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hopkins.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopkins.html (3 of 3) [2/16/2002 11:15:11 PM]

Hopkinson

John Hopkinson Born: 27 July 1849 in Manchester, England Died: 27 August 1898 in Evalona, Switzerland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Hopkinson was born the eldest of a large family of thirteen children. He was fortunate in that being brought up in Manchester he had good schooling and in 1865 entered Owens College in that city. Owens College, which went on to become the University of Manchester, was an excellent institution in which to study. After showing great abilities in mathematics Hopkinson was awarded a scholarship to allow him to continue his study of that subject at Trinity College Cambridge. He entered Cambridge in 1867 and graduated with a mathematics degree in 1871. Although his scholarship would have allowed him to continue his mathematical studies at Cambridge, Hopkinson decided to put his mathematics to practical use in engineering. There had been no chair of engineering at Owens College when Hopkinson studied there, but it is interesting to note that Osborne Reynolds was appointed to such a chair while Hopkinson was studying at Cambridge. In 1872 Hopkinson was appointed as the engineering manager of Chance Brothers and Company, a glass manufacturing company in Birmingham. There he studied the problems of efficient ways of shining lights from a lighthouse and, in particular, he recommended the use of flashing groups of lights. Dibner writes in [1]:Hopkinson's application of Maxwell's electromagnetic theories to the analysis of residual charge and displacement in electrostatic capacity led to his election as a fellow of the Royal Society in 1877. In 1878 Hopkinson founded his own electrical engineering company. Collaborating with Edward Hopkinson, one of his brothers, he applied the theory of electricity and magnetism to the development of electric motors. He used his mathematical expertise to give a general theory of alternating currents and he applied this theory to the operation of alternating current generators in parallel. Hopkinson was appointed professor of electrical engineering at King's College London in 1890. At the same time he became director of the newly founded Siemens Laboratory. When at the height of his powers Hopkinson was sadly killed in a mountaineering accident while on holiday in Switzerland. His son and two daughters died in the same accident on Mount Petite Dent de http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopkinson.html (1 of 2) [2/16/2002 11:15:13 PM]

Hopkinson

Veisivi. Although Hopkinson's life was cut short by the accident, he had already received several honours for his contributions of applying mathematics to engineering. In addition to his election to the Royal Society which we mentioned above, he was President of the Institution of Electrical Engineers on two occasions. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Hopkinson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopkinson.html (2 of 2) [2/16/2002 11:15:13 PM]

Hopper

Grace Brewster Murray Hopper Born: 9 Dec 1906 in New York, USA Died: 1 Jan 1992 in Arlington, Virginia, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Grace Hopper was born Grace Brewster Murray, the oldest of three children. Her father, Walter Murray, was an insurance broker while her mother, Mary Van Horne, had a love of mathematics which she passed on to her daughter. Both Grace's parents believed that she and her sister should have an education of the same quality as her brother. The book [2] contains a fascinating account of her childhood. It tells of summers spent with her cousins in their cottage on Lake Wentworth in Wolfeboro, New Hampshire and the games they played there such as kick-the-can, hide-and-seek and cops-and-robbers. It also describes her hobbies of needlepoint, reading and playing the piano. There were certainly signs in Grace's childhood of her fascination with machines and in [2] there is a delightful story of how, when she was seven years old, she took her alarm clock to pieces to find out how it worked. Unable to reassemble it, she took to pieces the other seven clocks she found in the house before her mother discovered what was happening. Grace was educated at two private schools for girls, namely Graham School and Schoonmakers School both in New York City. Intending to enter Vassar College in 1923 she failed a Latin examination and was required to wait another year. She spent the academic year at Hartridge School in Plainfield, New Jersey then entered Vassar College in 1924. She studied mathematics and physics at Vassar College graduating with a BA in 1928. After graduating she undertook research in mathematics at Yale University. In 1930 Grace Murray married Vincent Foster Hopper, an English teacher from New York University. A Vassar College Fellowship allowed her to study at Yale University and, also in 1930, Yale awarded her an MA. In 1931 she began teaching mathematics at Vassar College as an instructor in the Department of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopper.html (1 of 5) [2/16/2002 11:15:15 PM]

Hopper

Mathematics and she continued on the staff there until 1943, having been promoted by that time to an associate professorship. Hopper was awarded her doctorate by Yale University in 1934 for a thesis New Types of Irreducibility Criteria which was supervised by Oystein Ore. Hopper attended New York University as a Vassar Faculty Fellow in 1941. Hopper wanted to join the military as soon as the United States entered World War II. However her at 34 she was too old (and not heavy enough for her heoght) to enlist and anyway as a mathematics professor her job was considered essential to the war effort. However she was determined to join the Navy and, despite being told that she could serve her country best by remaining in her teaching post at Vassar College, she eventually persuaded the Naval Reserve to accept her in 1943 and she also persuaded Vassar College to grant her leave. After initial training at Midshipman's School, after which she was commissioned a Lieutenant, Hopper was assigned to the Bureau of Ordnance Computation Project at the Cruft Laboratories at Harvard University. From 1944 she worked with Aiken on the Harvard Mark I computer [7]:On her arrival at Cruft Laboratory she immediately encountered the Mark I computer. For her it was an attractive gadget, similar to the alarm clocks of her youth; she could hardly wait to disassemble it and figure it out. ... Hopper became the third person to program the Mark I. Aiken gave her as a first programming task immediately she arrived at Harvard which was to:Compute the coefficients of the arctan series by next Thursday. By the end of the war, Hopper was working on the Harvard Mark II computer. In 1946 Hopper ended her active duty with the Navy but remained a duty reservist. She resigned her post at Vassar College so that she could remain at Harvard where she was appointed a Research Fellow in Engineering Sciences and Applied Physics in the Computation Laboratory. She continued to work on the Mark II, then later on the Mark III computer. In 1949 Hopper joined the Eckert-Mauchly Computer Corporation as a Senior Mathematician and there she worked with John Eckert and John Mauchly on the UNIVAC computer. She designed an improved compiler while working for the company and was part of the team which developed Flow-Matic, the first English-language data-processing compiler. In 1951 [12]:... she discovered the first computer "bug." It was a real moth, which she pasted into the UNIVAC I logbook. In 1952 she had an operational compiler. "Nobody believed that," she said. "I had a running compiler and nobody would touch it. They told me computers could only do arithmetic." Hopper's reason for designing a compiler was, she wrote later, because she was lazy and hoped that the introduction of compilers would allow the computer programmer to return to being a mathematician. Indeed it may seem obvious to us today that this would be the route forward for computers but it was an extremely far sighted idea from Hopper. In fact thinking about how computers have developed, particularly with systems such as Mathematica and Maple available today, one sees the rather remarkable vision that Hopper had of how computers would become such an important tool for mathematicians. In 1950 the Remington Rand Corporation had acquired the Eckert-Mauchly Computer Corporation and changed its name to the UNIVAC Division of Remington Rand. Hopper became a Systems Engineer and Director of Automatic Programming Development of the UNIVAC Division. She continued her work on http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopper.html (2 of 5) [2/16/2002 11:15:15 PM]

Hopper

compilers, publishing her first paper on that topic in 1952. She then participated in the work to produce specifications for a common business language. Since Flow-Matic was the only existing business language at that time, it was inevitable that it should provide the foundations for the specification of the language COBOL (COmmon Business-Oriented Language) which eventually came out in 1959. She had another important aim relating to compilers, namely that there should be standardisation. Her aim was that there should be international standardisation of computer languages and she strongly advocated validation procedures. Hopper was never one to hold a single job at any one time. She was involved both with the academic world and with the Navy during the time that she held her positions in the Remington Rand Corporation, then from 1955 in the Sperry Corporation which had merged in that year with Remington Rand. Her connections with the academic world were many, sometimes visiting positions as in 1959 when she was a Visiting Lecturer at the Moore School of Electrical Engineering of the University of Pennsylvania. She was a consultant and lecturer for the United States Naval Reserve up to her retirement in December 1966, by which time she had reached the rank of Commander. The Navy and Hopper were not apart for very long for, in August 1967, she was recalled to active duty in the Navy. At this time she took military leave from the Sperry Corporation and did not return to that job, retiring from it in 1971 when she reached 65 years of age. Her return to the Navy was intended to be for only a six months period [5]:... at the request of Norman Ream, then Special Assistant to the Secretary of the Navy for Automatic Data Processing. After the six months were up, her orders were changed to say her services would be needed indefinitely. She was promoted to Captain in 1973 by Admiral Elmo Zumwalt, Jr., Chief of Naval Operations. And in 1977, she was appointed special advisor to Commander, Naval Data Automation Command, where she stayed until she retired. Active service in the Navy did not prevent Hopper holding academic appointments, and she was a Lecturer in Management Sciences at George Washington University between 1971 and 1978. When Hopper retired from the Navy in August 1986, at 80 years of age, she was the oldest active duty officer in the United States. She had reached the rank of Rear Admiral, being promoted to the rank of Commodore in a White House ceremony in December 1983, then becoming Rear Admiral Hopper in 1985. At a celebration held in Boston on the USS Constitution to celebrate her retirement, Hopper was awarded the Defense Distinguished Service Medal, the highest award possible by the Department of Defense. After a career which involved many jobs in numerous quite different areas, one might have expected her to look forward to a quiet retirement. However, this was not her style and, remarkably, she was appointed a senior consultant to Digital Equipment Corporation after retiring from the Navy, a position she held until 1990. Her job involved representing [3]:... Digital at computer industry forums, making presentations on advanced computing concepts and the value of information and data, and serving as a corporation liaison with educational institutions. In her long career Hopper received so many awards that it would be impossible to note more than a few in this article. She was elected a Fellow of the Institute of Electrical and Electronics Engineers (1962), a Fellow of the American Association for the Advancement of Science (1963), and received Achievement http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopper.html (3 of 5) [2/16/2002 11:15:15 PM]

Hopper

Awards from the Society of Women Engineers (1964) and from the Institute of Electrical and Electronics Engineers (1968). Hopper was named the first computer science Man of the Year by the Data Processing Management Association in 1969. In 1970 she received the Harry M Goode Memorial Award, a medal and $2,000 awarded by the Computer Society:For her pioneering work and leadership in the development of computer software, and for her impact and influence on the computing profession and her fellow colleagues, and for her pioneering work and leadership in the development of important concepts for mathematical and business compilers, and for her contributions to the development and acceptance of English-language, problem-oriented programming, and for her outstanding work and continued efforts in the education and training of men and women for careers in computer science and data processing. She became the first woman to be elected Distinguished Fellow of the British Computer Society in 1973, being the first American elected to this honour. Also in 1973 she was elected to the National Academy of Engineering and the Legion of Merit. Hopper collected a remarkable number of honorary degrees, receiving at least 37 between 1972 and 1987. In 1991 President George Bush awarded Hopper the National Medal of Technology. She was [3]:... the first woman to receive America's highest technology award as an individual. The award recognises her as a computer pioneer, who spent a half century helping keep America on the leading edge of high technology. Article by: J J O'Connor and E F Robertson List of References (12 books/articles) A Poster of Grace Hopper Other Web sites

Mathematicians born in the same country 1. Agnes Scott College 2. San Diego 3. WISE project 4. Virginia Tech 5. Washington DC 6. Arlington 7. Arlington (USN destroyer named (1997) after Hopper) 8. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopper.html (4 of 5) [2/16/2002 11:15:15 PM]

Hopper

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hopper.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hopper.html (5 of 5) [2/16/2002 11:15:15 PM]

Hormander

Lars Hörmander Born: 24 Jan 1931 in Mjällby, Blekinge, Sweden

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lars Hörmander's father was a teacher in the small fishing village on the south coast of Sweden where he was born. Lars attended his first school in the village, then made a daily train journey to attend a higher level school in a larger nearby town. For the next stage in his schooling he had to move to Lund where he attended the gymnasium. There he was able to devote much time to studying university level mathematics, encouraged by a teacher who had been a research student of Marcel Riesz. In 1948 Hörmander graduated from the gymnasium and studied at the University of Lund. There he was taught by Marcel Riesz who lectured to him on classical function theory and harmonic analysis. Hörmander was awarded a master's degree in 1950 and began to undertake research supervised by Marcel Riesz. After Marcel Riesz retired in 1952 Hörmander began working on the theory of partial differential equations. Before completing his doctorate, Hörmander spent the year 1953-54 doing military service but he was able to continue reading mathematics during this time. His doctorate was completed in 1955 and he applied for a professorship at the University of Stockholm. Before a decision was made on the professorship Hörmander left for a visit to the United States where he spent time at the universities of Chicago, Kansas and Minnesota. He also visited the Institute for Mathematical Sciences (now the Courant Institute) which was directed at that time by Courant. Back in Sweden he took up the appointment of professor at Stockholm from the beginning of 1957 which had been offered to him while he was in the United States. He continued to spend time in the United States, particularly at Stanford University and the Institute for Advanced Study. In 1962 the International Congress was held in Stockholm and Hörmander, as well as being heavily involved in the organisation,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hormander.html (1 of 3) [2/16/2002 11:15:17 PM]

Hormander

received a Fields Medal for his work on partial differential equations. Garding, describing Hörmander's contributions that earned him the Fields Medal, writes in [2] of early contributions by Hadamard to linear differential operators which led to the first complete results by Petrovsky. He continues:... in a lecture in 1945 [Petrovsky] explicitly asked for a general theory of linear differential operators including those which do not appear in the mathematical models of physics. ... In his book on distributions Laurent Schwartz ... stated a number of problems about differential operators. Since then a rather comprehensive theory has been worked out. Many people have contributed but the deepest and most significant results are due to Hörmander. In 1963 Hörmander made an arrangement which allowed him to spend the academic teaching year in Stockholm and spring and summer at the University of Stanford in the United States. However, he writes in [4]:I had barely arrived at Stanford when I received an offer to come to the Institute for Advanced Study as a permanent member and professor. Although I had previously been determined not to leave Sweden, the opportunity to do research full time in a mathematically very active environment was hard to resist. Hörmander spent from 1964 to 1968 at Princeton but felt the pressure of a full time research position so returned to Sweden to take up the chair of mathematics at the University of Lund in 1968. However he returned for frequent visits to the United States, in particular to the Institute for Advanced Study and to Stanford. In [5] Hörmander describes the direction of his research after the award of the Fields Medal. In particular he describes how the main areas developed which are covered by his four volume text The analysis of linear partial differential operators the volumes of which appeared between 1983 and 1985. This work updates his original book Linear Partial Differential Operators (1963) which contained the results which led to his award of the Fields Medal. Hörmander describes in [4] the later stages of his career:After five years devoted to writing a four volume monograph on linear partial differential operators, I spent the academic years 1984-86 as director of the Mittag-Leffler Institute in Stockholm. I had only accepted a two year appointment with leave of absence from Lund since I suspected that the many administrative duties there would not agree very well with me. The hunch was right, and since 1986 I have been in Lund where I became professor emeritus in January 1996. Hörmander's text, An Introduction to Complex Analysis in Several Variables, has become a classic dealing with the theory of functions of several complex variables. It developed from lecture notes of a course which he gave in Stanford in 1964 and published in book form two years later. Extra material was added to later expanded editions of the work which appeared in 1973 and in 1990. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hormander.html (2 of 3) [2/16/2002 11:15:17 PM]

Hormander

List of References (5 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1950 to 1960

Honours awarded to Lars Hörmander (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1962

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hormander.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hormander.html (3 of 3) [2/16/2002 11:15:17 PM]

Horner

William George Horner Born: 1786 in Bristol, England Died: 22 Sept 1837 in Bath, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Horner was educated at Kingswood School Bristol. At the almost unbelievable age of 14 he became an assistant master at Kingswood school in 1800 and headmaster 4 years later. He left Bristol and founded his own school in 1809 in Bath. Horner's only significant contribution to mathematics was Horner's method for solving algebraic equations. It was submitted to the Royal Society on 1 July 1819 and was published in the same year in the Philosophical Transactions of the Royal Society. Some years earlier Ruffini had described a similar method which had won him the gold medal offered by the Italian Mathematical Society for Science who had asked for improved methods for numerical solutions to equations. However neither Ruffini nor Horner was the first to discover this method as it was known to Chu Shih-Chieh 500 years earlier. In the 19th and early 20th centuries Horner's method has had a prominent place in English and American textbooks on algebra. It is not unreasonable to ask why that should be. The answer lies simply with De Morgan who gave Horner's name and method wide coverage in many articles which he wrote. After Horner died in 1837 his son, also called William, carried on running the school in Bath. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Other references in MacTutor

Chronology: 1810 to 1820

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Horner.html (1 of 2) [2/16/2002 11:15:18 PM]

Horner

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Horner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Horner.html (2 of 2) [2/16/2002 11:15:18 PM]

Houel

Guillaume Jules Hoüel Born: 7 April 1823 in Thaon, Calvados, France Died: 14 June 1886 in Pétiers (near Caen), France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jules Hoüel attended the Lycée at Caen and then the Collège Rollin. In 1843 he entered the Ecole Normale Supérieure and, after graduating, he taught at Bourges, Bordeaux, Pau, Alençon and Caen. He obtained a doctorate from the Sorbonne in 1855 for research in celestial mechanics. Le Verrier was impressed with Hoüel's work and tried to persuade him to accept a post at the Observatory in Paris. However, Hoüel decided not to accept the offer and spent the next four years undertaking research at his home in Thaon. Hoüel was appointed to the chair of pure mathematics in the Faculty of Science at Bordeaux in 1859 and held this post until his death. As Halsted points out in [4]:Here he found dignity and facilities for work, and considered the position as final. Hoüel published a work on geometry in 1863. At this stage he did not know of the published work on non-euclidean geometry but he clearly was working his way towards the idea. He wrote:Since long, the scientific researches of mathematicians on the fundamental principles of elementary geometry have concentrated themselves almost exclusively on the theory of parallels, and if, hitherto, the efforts of so many eminent minds have produced no satisfactory result, it is perhaps permitted to conclude thence that in pursuing these researches they have followed a false path and attacked an insoluble problem, of which the importance has been exaggerated in consequence of inexact ideas on the nature and origin of the primordial truths of the science of space. Hoüel became interested in non-euclidean geometry once he had been made aware of the work of Bolyai http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Houel.html (1 of 2) [2/16/2002 11:15:19 PM]

Houel

and Lobachevsky. He published translations of many important works by Bolyai, Beltrami, Helmholtz and Riemann. He corresponded with Tilly on non-euclidean geometry. In [4] Halsted explains how Hoüel obtained Bolyai's work:... Hoüel's 'Essai' of 1863 having come by chance into the hands of a young architect of Temesvár in Hungary, this youth Franz Schmidt, desirous of continuing his mathematical studies wrote for counsel to Hoüel. Hoüel had answered helpfully, and later implored the aid of Schmidt to procure Bolyai's work ... Schmidt succeeded in procuring for Hoüel two copies of Bolyai's work. One Hoüel proceeded to translate himself, the other he sent to Battaglini, asking him to make known in Italy this wonderful idea. At Hoüel's suggestion Schmidt collected material which enabled him to write the first biography of Bolyai which he did in 1868. Among his other researches, Hoüel compiled log tables and worked on planetary perturbations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Houel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Houel.html (2 of 2) [2/16/2002 11:15:19 PM]

Householder

Alston Scott Householder Born: 5 May 1904 in Rockford, Illinois, USA Died: 4 July 1993 in Malibu, California, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alston Householder spent his childhood in Alabama where his family had moved shortly after he was born. After attending school, he entered Northwestern University in Evanston, Illinois where he studied philosophy, receiving his BA in 1925. He then went to Cornell University in Ithaca, New York where he continued his study of philosophy, receiving his MA in 1927. Householder then taught mathematics in a number of different places and began to work for his doctorate in mathematics. He was awarded a Ph.D. by the University of Chicago in 1947 for a thesis on the calculus of variations. However his interests were moving towards applications of mathematics, particularly applications of mathematics to biology. From 1947 Householder spent eight years working on mathematical biology as a member of the Committee for Mathematical Biology at the University of Chicago. John Hearon, after retiring from the National Institutes of Health, wrote of Householder's work over this period:Hypothesis, conjecture and tentative theory flew in all directions and there was a period of great ferment. In the midst of this, to every area to which he addressed himself Householder brought organisation and systemisation. He was then, and for some years to come, the only one of the group formally trained as a mathematician. It showed. He brought to every problem he undertook unification, generality of method and, in the end, simplicity. Although this work on mathematical biology occupied Householder for a relatively short period of his career, he wrote 33 papers on the topic and a monograph Mathematical Biophysics of the Central Nervous System written jointly with Herbert D Landahl and published in 1944. The book is:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Householder.html (1 of 4) [2/16/2002 11:15:21 PM]

Householder

... an exposition and an enlargement of work which, to a great extent, was done by the authors themselves towards the establishment of a mathematical neuropsychology. In 1944 Householder left the Committee for Mathematical Biology and became involved in the war effort. In 1946, after the end of the war, Householder joined the Mathematics Division of Oak Ridge National Laboratory. Here he changed topic, leaving behind his research interest of mathematical biology and moving into numerical analysis which was increasing in importance due to the advances in computers. Not surprisingly there is a gap in Householder's prolific publication record while he became a leading expert in this new area. He started publishing on this new topic with Some numerical methods for solving systems of linear equations which appeared in 1950. Even before this first publication in numerical analysis, Householder had been appointed Director of the Oak Ridge National Laboratory in 1948. His role as director certainly did not prevent him from taking a leading role in research in numerical analysis in general and in numerical linear algebra in particular. Wilkinson wrote:In the 1950's our knowledge of this topic was in a rather chaotic state. A large number of algorithms had been developed but no systematic study of their inter-relationships had been undertaken. It is primarily due to the work of Householder that order has emerged from this chaos. In a remarkable series of papers he effectively classified the algorithms for solving linear equations and computing eigensystems, showing that in many cases essentially the same algorithm had been presented in a large variety of superficially quite different algorithms. The resulting classification made it possible to concentrate on the most profitable lines of research and in this way his work was directly responsible for the development of many of the most effective algorithms in use today. Of particular importance is his appreciation of the value of elementary hermitian matrices in numerical analysis. The lasting impact of Householder's work in this area is described in [1]:... Householder transformations are now routinely taught in courses in linear algebra, throughout the world, as is the systematic use of norms in linear algebra, which he pioneered. In 1964 Householder published one of his most important books The theory of matrices in numerical analysis. This book was reviewed by Richard Varga who wrote:Without question, this book represents one of the real highs in scholarly attainment in numerical analysis, and as such, it belongs on the shelves of students and researchers alike in this field. The author has succeeded in bringing all the related contributions of various authors in the field of matrix theory under a single unified point of view. The scholarly depth of this book is indicated by forty-four pages of bibliography, with approximately nine hundred titles; the bibliography alone is one-fifth of the book. Householder will certainly not only be remembered for his research contributions. Equally important as a contribution to mathematics was his organising the Gatlinburg Symposium on Numerical Linear Algebra. The idea for the first symposium was born in 1960 when Householder and several other colleagues discussed the idea at Ann Arbor. The first symposium took place in April 1961 and [1]:... numerical analysts from around the world gathered in the beautiful atmosphere of the Smoky Mountains in Tennessee to share their latest results. There followed more Gatlinburg Symposia. The second was in 1963, the third in 1964 and the fourth in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Householder.html (2 of 4) [2/16/2002 11:15:21 PM]

Householder

1969. Householder left Oak Ridge National Laboratory in 1969, after 25 years service, and became Professor of Mathematics at the University of Tennessee. The year 1969 was important for Householder in other ways too, for in that year he received the Harry M Goode Memorial Award, a medal and $2,000 awarded by the Computer Society:For his impact and influence on computer science in general and particularly for his contributions to the methods and techniques for obtaining numerical solutions to very large problems through the use of digital computers, and for his many publications, including books, which have provided guidance and help to workers in the field of numerical analysis, and for his contributions to professional activities and societies as committee member, paper referee, conference organiser, and society President. After five years as Professor of Mathematics at the University of Tennessee, Householder retired. The Gatlinburg Symposia continued after he left Oak Ridge but they moved to different locations in Europe and North America. The authors of [1] write:... to honour him for his many contributions, these symposia are now called the Householder Symposia. At the 13th in the series of Householder Symposia held in Pontresina, Switzerland in 1996, Friedrich L Bauer spoke on Memories of Alston Householder. He described him in the following way:[Alston Householder] had a full life, with many friends and people who admired him. He was an American in the best sense of the word, liberal and socially conscious. Yet he was a cosmopolitan with a thorough knowledge of foreign languages and cultures. He was a mathematician of distinction. Above all he was a friendly human being. We miss him very much. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Householder.html (3 of 4) [2/16/2002 11:15:21 PM]

Householder

JOC/EFR July 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Householder.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Householder.html (4 of 4) [2/16/2002 11:15:21 PM]

Hsu

Pao-Lu Hsu Born: 1 Sept 1910 in Beijing, China Died: 18 Dec 1970 in Beijing, China

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pao Lu Hsu's name is sometimes transliterated as Xu Bao-lu but the form we have given seems to be the commonest form. His family came from the lake city of Hangzhou in Zhejiang Province but he was born and brought up in Peking. His parents belonged to a mandarin family and in [5] the authors write:Perhaps because of this background he spoke the "common" or "official" dialect with an interesting soft overtone. Hsu's school education was in Peking and he did not choose mathematics as a career at this stage but rather it was chemistry which he decided to study at university. In 1928 he enrolled at Yangjing University to study chemistry. Two years later he decided to change subject, to change universities, and so he went to Tsing Hua University to read for a degree in mathematics. He obtained his Bachelor of Science degree from Tsing Hua University in 1933 and then moved to the Mathematics Department of Peking University where he was employed as an assistant. Hsu passed examinations in 1936 at Peking University and obtained a scholarship to enable him to continue his graduate studies in Britain. He spent four years in Britain mainly at University College, London but he also spent some time studying at Cambridge. Certainly University College, London was an excellent place for Hsu to study as his mathematical interests were in probability and statistics. Egon Pearson, following the retiral of his father Karl Pearson as Galton Professor of Statistics, had been made Reader and became Head of the Department of Applied Statistics three years before Hsu arrived there. Jerzy Neyman had been appointed in 1934 while R A Fisher held Karl Pearson's Galton Chair of Statistics and was Head of the Department of Eugenics at University College. Lehmann writes in [10]:During this period [at University College, London] Hsu wrote a remarkable series of papers on statistical inference which show the strong influence of the Neyman-Pearson point of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hsu.html (1 of 3) [2/16/2002 11:15:23 PM]

Hsu

view. Hsu's first two papers were published in the Statistical Research Memoirs which were edited by Jerzy Neyman and Egon Pearson. One concerned what is now known as the Behrens-Fisher problem, while the second Hsu examined the problem of optimal estimators of the variance in the Gauss-Markov model. In 1938 Hsu, while still undertaking research for his doctorate, too up a position as lecturer in Egon Pearson's Department. He was awarded the degree of Ph.D. and then that of D.Sc. from University College, London, in 1938 and 1940, respectively. Anderson, Chung and Lehmann write in [5]:[Hsu's] British education formed his taste in mathematics; he preferred the hard and concrete to the general and abstract. This is a very fair comment on the style of British statistics during this period, in contrast to the style in Continental Europe. By 1940 China was engaged in World War II fighting against the Japanese invasion and Britain was involved in the war against Germany. Hsu chose to leave Britain to return to his homeland of China where he was appointed as Professor at Peking University. It was a period of great difficulty and hardship for Hsu. He corresponded with Neyman during the years 1943-44, who by this time was at Berkeley in the United States, about statistical matters but he mentions in these letters the great hardship he was suffering, particularly suffering starvation. It is a great tribute to Hsu's determination to devote himself to statistics that he managed to continue his research during these difficult war years. Many of his publications on multivariate analysis from this period show that he had been strongly influenced by R A Fisher while at University College. His role in promoting the use of matrix theory in statistics should also be emphasised. These papers brought him to [3]:... the forefront of the development of the mathematical theory of multivariate analysis. Attempts were made to get Hsu to the United States. In 1945 he arrived in the USA just in time for the First Berkeley Symposium on Probability and Statistics. During the next two years he taught at the University of California, Columbia University, and the University of North Carolina where he was offered an associate professorship. After spending 1946-47 at the University of North Carolina at Chapel Hill, in 1947 Hsu returned to his professorship at Peking University. One of his students at Chapel Hill wrote:In was Hsu's insistence on simplicity combined with depth of understanding, clarity without avoidance of difficulties, and above all a deep and obvious but unspoken commitment to the highest goals and standards of scholarship which attracted us to him. Anderson, Chung and Lehmann describe in [5] (see also [4]) Hsu's personality, particularly during these years in the United States:Setting even higher standards for his own work than for others, he would temper his critical sense with a gentle mocking humour. He could work feverishly on research for spells, but used to lament that life's diverse interests conflicted with a single minded devotion to science. ... A particular hobby was chanting the musical drama of the Yuan dynasty with a small group of connoisseurs accompanied by ancient instruments. partly owing to fragile health he never married but apparently came close to it.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hsu.html (2 of 3) [2/16/2002 11:15:23 PM]

Hsu

Hsu had turned down many offers of position, one particularly attractive one from Wald, from universities in the United States but he felt that there he could be [5]:... part of the emerging new society in his homeland. Hsu had poor health from 1950. He recovered but the extremely hard work which he undertook brought about a recurrence in the summer of 1951 and he spent some time in hospital. His battle against illness is movingly described in Jiang and Duan in [1]:Concerned about his health, the authorities repeatedly suggested that he go abroad to recuperate. all such suggestions met with polite refusal; he insisted on continuing his work ... From 1956 on his health steadily worsened. Having difficulty in moving about, he worked at home. with a blackboard hanging on the wall of his room, he gave lectures to upper-class students, graduate students, and young teachers; he was in charge of seminars and other academic activities. By the early 1960s his health had deteriorated to such an extent that he could stand in front of the blackboard for only a few minutes before he had to sit down and rest. Nevertheless his teaching did not cease at that time, nor did his work cease under the extremely difficult conditions in the later years. It was only a month before his death that his manuscript on the relationship between experimental design and algebraic coding theory was at last completed, being his final legacy. Found beside his bed the day after his death were piles of manuscripts which serve as a testimony to the superhuman fortitude with which he exerted himself over a period of more than 20 years ... Hsu died in his home on the campus of Peking University in 1970. He had published a total of 40 mathematical papers. Article by: J J O'Connor and E F Robertson List of References (11 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hsu.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hsu.html (3 of 3) [2/16/2002 11:15:23 PM]

Hubble

Edwin Powell Hubble Born: 20 Nov 1889 in Marshfield, Missouri, USA Died: 28 Sept 1953 in San Marino, California, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edwin Hubble was a man who changed our view of the Universe. In 1929 he showed that galaxies are moving away from us with a speed proportional to their distance. The explanation is simple, but revolutionary: the Universe is expanding. Hubble was born in Missouri in 1889. His family moved to Chicago in 1898, where at High School he was a promising, though not exceptional, pupil. He was more remarkable for his athletic ability, breaking the Illinois State high jump record. At university too he was an accomplished sportsman playing for the University of Chicago basketball team. He won a Rhodes scholarship to Oxford where he studied law. It was only some time after he returned to the US that he decided his future lay in astronomy. In the early 1920's Hubble played a key role in establishing just what galaxies are. It was known that some spiral nebulae (fuzzy clouds of light on the night sky) contained individual stars, but there was no consensus as to whether these were relatively small collections of stars within our own galaxy, the 'Milky Way' that stretches right across the sky, or whether these could be separate galaxies, or 'island universes', as big as our own galaxy but much further away. In 1924 Hubble measured the distance to the Andromeda nebula, a faint patch of light with about the same apparent diameter as the moon, and showed it was about a hundred thousand times as far away as the nearest stars. It had to be a separate galaxy, comparable in size our own Milky Way but much further away. Hubble was able to measure the distances to only a handful of other galaxies, but he realised that as a rough guide he could take their apparent brightness as an indication of their distance. The speed with which a galaxy was moving toward or away from us was relatively easy to measure due to the Doppler

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hubble.html (1 of 3) [2/16/2002 11:15:25 PM]

Hubble

shift of their light. Just as a sound of a racing car becomes lower as it speeds away from us, so the light from a galaxy becomes redder. Though our ears can hear the change of pitch of the racing car engine our eyes cannot detect the tiny red-shift of the light, but with a sensitive spectrograph Hubble could determine the redshift of light from distant galaxies. The observational data available to Hubble by 1929 was sketchy, but whether guided by inspired instinct or outrageous good fortune, he correctly divined a straight line fit between the data points showing the redshift was proportional to the distance. Since then much improved data has shown the conclusion to be a sound one. Galaxies are receding from us, and one another, as the Universe expands. Within General Relativity, the theory of gravity proposed by Albert Einstein in 1915, the inescapable conclusion was that all the galaxies, and the whole Universe, had originated in a Big Bang, thousands of millions of years in the past. And so the modern science of cosmology was born. Hubble made his great discoveries on the best telescope in the world at that time - the 100-inch telescope on Mount Wilson in southern California. Today his name carried by the best telescope we have, not on Earth, but a satellite observatory orbiting our planet. The Hubble Space Telescope is continuing the work begun by Hubble himself to map our Universe, and producing the most remarkable images of distant galaxies ever seen, many of which are available via the World Wide Web. Article by: David Wands, Portsmouth Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

Some Quotations (2)

Mathematicians born in the same country Cross-references to History Topics

A brief history of cosmology

Honours awarded to Edwin Hubble (Click a link below for the full list of mathematicians honoured in this way) ASP Bruce Medallist

1938

Lunar features

Crater Hubble

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hubble.html (2 of 3) [2/16/2002 11:15:25 PM]

Hubble

JOC/EFR February 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hubble.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hubble.html (3 of 3) [2/16/2002 11:15:25 PM]

Hudde

Johann van Waveren Hudde Born: 23 April 1628 in Amsterdam, Netherlands Died: 15 April 1704 in Amsterdam, Netherlands

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Hudde attended the University of Leiden to study law. However he was introduced to mathematics at Leiden by his teacher van Schooten. From 1654 until 1663 he worked on mathematics as part of van Schooten geometry research group at Leiden. From 1663 he worked in various roles for the Amsterdam City Council. He served for 30 years as burgomaster of Amsterdam being first appointed in 1672. All of Hudde's mathematics was done before he began to work for the city council in 1663. Van Schooten edited and published a second two-volume translation of Descartes's La Géométrie (1659-1661) which contained appendices by de Witt, Hudde and van Heuraet. Hudde worked on maxima and minima and the theory of equations. Hudde gave an ingenious method to find multiple roots of an equation which is essentially the modern method of finding the highest common factor of a polynomial and its derivative. He was the first to treat the coefficients in algebra without considering whether they were positive or negative in De reductione aequationum. In 1656 he gave the power series expansion of ln(1+x). The following year he directed the flooding of parts of Holland to block the advance of the French army. Hudde also worked on optics, producing microscopes and constructing telescope lenses. Hudde corresponded with Huygens on problems of canal maintenance, probability and life expectancy. Leibniz studied Hudde's manuscripts and reported finding many excellent results. The manuscripts must have had an important influence on Leibniz's introduction of the calculus. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hudde.html (1 of 2) [2/16/2002 11:15:27 PM]

Hudde

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Johann Hudde

Mathematicians born in the same country

Cross-references to History Topics

The rise of the calculus

Other references in MacTutor

Chronology: 1650 to 1675

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hudde.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hudde.html (2 of 2) [2/16/2002 11:15:27 PM]

Humbert_Georges

Marie Georges Humbert Born: 7 Jan 1859 in Paris, France Died: 22 Jan 1921 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georges Humbert never knew his parents since he was orphaned when he was very young. He was brought up by his grandparents in Franche-Comté where his grandfather was an industrialist. His grandparents sent Georges to the Oratorian College of Juilly to become a boarder there. He studied classics at the College, then he went on to study at the Collège Stanislas in Paris. He graduated from this college in 1877 and, was placed first for entry to the Ecole Normale Supérieur and seventh for entry to the Ecole Polytechnique in that year. He chose the Ecole Polytechnique and after graduating in second place he studied engineering at the Ecole des Mines. After graduating he spent a few years as a mining engineer. His first position in this capacity took him to Vesoul, but after that he moved to Paris where, in addition to his work as a mining engineer, he was soon employed as a teacher at the Ecole Polytechnique and at the Ecole des Mines. Humbert obtained a doctorate in mathematics in 1885 for his thesis Sur les courbes de genre un. He wrote the important work Application de la théorie des fonctions fuchsiennes à l'étude des courbes algébraiques which was published in the following year. His work was officially recognised when he was awarded the Poncelet Prize from the Académie des Sciences in Paris in 1891 and the prize from the French Mathematical Society in 1893. Humbert married Marie Jagerschmidt in 1890 and in the following year his son Pierre was born. Pierre Humbert grew up to be a mathematician and has a biography in this archive. Georges Humbert's delight at the birth of his son was short-lived, however, for his wife died in 1892. Humbert married again in 1900 and with his second wife Suzanne Lambert-Caillemar had two further children.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Humbert_Georges.html (1 of 3) [2/16/2002 11:15:29 PM]

Humbert_Georges

In 1893 he was elected to the role of president of the French Mathematical Society and, two years later, he was appointed professor of analysis at the Ecole Polytechnique. On the death of Hermite in 1901, Humbert was elected to fill his place in the Académie des Sciences. In fact this was highly appropriate for he continued Hermite's work in number theory and, what is more, did so in a very effective manner producing important contributions. He became Jordan's assistant at the Collège de France in 1904 succeeding to Jordan's chair in 1912. His doctorate extended Clebsch's work on curves. He then studied Abel's work which he developed and put into a geometric setting. It was as a direct consequence of his work on using abelian functions in geometry which won for him the 1892 Académie des Sciences prize for work on Kummer surfaces. As Costabel writes in [1]:He thus enriched analysis and gave the complete solution of the two great questions of the transformation of hyperelliptic functions and of their complex multiplication. He also, as we noted above, extended work of Hermite considering applications to number theory throughout his life. Georges Humbert would be better known today if the area of mathematics in which he worked had remained in favour. Since it has now become merely something of an historical curiosity rather than mainstream mathematics, his contribution is less well known. It does, however, indicate the quality of his mathematics that, despite this, his name and results are known today. To some extent this is a consequence of the fact that although he worked in a specialised area he had a remarkably broad knowledge of mathematics and his results form links between areas. Humbert was a highly respected man who [1]:... was remarkably gifted not only in mathematics but also in clarity of expression and intellectual cultivation. He exerted a great influence and was able, by his discretion and objectivity, to assure respect for his religious convictions during a period of some hostility towards religion in French scientific circles. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Humbert_Georges.html (2 of 3) [2/16/2002 11:15:29 PM]

Humbert_Georges

Glossary index

Quotations index

Mathematicians of the day JOC/EFR May 2000

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Humbert_Georges.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Humbert_Georges.html (3 of 3) [2/16/2002 11:15:29 PM]

Humbert_Pierre

Pierre Humbert Born: 13 June 1891 in Paris, France Died: 17 Nov 1953 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Humbert was a son of Georges Humbert who also has a biography in this archive. Pierre's mother died when he was very young and he therefore never knew her. However, his father remarried when Pierre was about nine years old so he at least had a step-mother from this time. All of Pierre Humbert's school education took place in Paris, and following this he went on to attend the Ecole Polytechnique there entering the famous French university in 1910. After studying in Paris for three years he left for Scotland to undertake research at the University of Edinburgh. At the University of Edinburgh Humbert undertook research under Whittaker whose philosophy of mathematics, and of science more generally, fitted in precisely with those of Humbert. His whole career would be influenced by the one year, 1913-14, which he spent in Edinburgh and in many of his publications Whittaker's influence can be seen. Of course 1914 marks the beginning of World War I and Humbert, despite having rather poor health, joined the army. After receiving a wound Humbert was no longer fit enough to continue to take part in the military action and so he was able to continue his research. He submitted his doctoral thesis Sur les surfaces de Poincaré in 1918 and he received a doctorate in that year. Following this he was appointed to the Faculty of Science at Montpellier as professor of astronomy and, despite travelling widely in France and abroad, he essentially spent his entire teaching career at Montpellier. Humbert's father Georges Humbert had gifts which extended beyond mathematics. This was even more true of his son Pierre Humbert who [1]:... demonstrated a highly refined sensitivity to culture, devoting attention to literature and music as well as to science. Moreover, he was unsatisfied with the simple juxtaposition of knowledge and religious faith. Pierre, like his father, was a Roman Catholic and he mixed his religious beliefs with one of his other great loves, namely teaching, in his work for the Joseph Lotte Association which was an organisation for Catholic public school teachers. Whittaker had instilled in him an aim which went far beyond mathematics or even science, for he aimed at developing the intellect. He had almost an ancient Greek attitude to scholarship and learning and indeed he did have a deep interest in history although it was in general more recent history than that of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Humbert_Pierre.html (1 of 2) [2/16/2002 11:15:30 PM]

Humbert_Pierre

ancient Greece. Humbert married the daughter of the astronomer Henri Andoyer and this, certainly in part, increased his interest in the history of astronomy. He even wrote some articles on the history of mathematics and astronomy with his father-in-law. Specialising in the history of the seventeenth century he wrote particularly on the French astronomers of that period. He also made contributions to mathematics, in particular he wrote on elliptic functions, Lamé functions, and Mathieu functions. His main mathematical work from the mid 1930s onwards was in developing the symbolic calculus. He also wrote on applications of the symbolic calculus to mathematical physics. Humbert had a fine reputation as a lecturer and also was a talented organiser. He showed his organising skills in his involvement with the French Association for the Advancement of Science. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Humbert_Pierre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Humbert_Pierre.html (2 of 2) [2/16/2002 11:15:30 PM]

Hunayn

Abu Zayd Hunayn ibn Ishaq al-Ibadi Born: 808 in al-Hirah (near Baghdad now in Iraq) Died: 873 in Baghdad (now in Iraq) Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hunayn ibn Ishaq is most famous as a translator. He was not a mathematician but trained in medicine and made his original contributions to the subject. However, as the leading translator in the House of Wisdom at one of the most remarkable periods of mathematical revival, his influence on the mathematicians of the time is of sufficient importance to merit his inclusion in this archive. His son Ishaq ibn Hunayn, strongly influenced by his father, is famed for his Arabic translation of Euclid's Elements. Hunayn's father was Ishaq, a pharmacist from Hira. The family were from a group who had belonged to the Syrian Nestorian Christian Church before the rise of Islam, and Hunayn was brought up as a Christian. Hunayn became skilled in languages as a young man, in particular learning Arabic at Basra and also learning Syriac. To continue his education Hunayn went to Baghdad to study medicine under the leading teacher of the time. However, after falling out with this teacher, Hunayn left Baghdad and, probably during a period in Alexandria, became an expert in the Greek language. Hunayn returned to Baghdad and established contact with the teacher with whom he had fallen out. The two became firm friends and were close collaborators on medical topics for many years. Let us go back to a time before Hunayn was born and describe the events which would lead to a remarkeble period of scholarship. Harun al-Rashid became the fifth Caliph of the Abbasid dynasty on 14 September 786. He brought culture to his court and tried to establish the intellectual disciplines which at that time were not flourishing in the Arabic world. It was during al-Rashid's reign that the first Arabic translation of Euclid's Elements was made by al-Hajjaj. The first steps began to be taken which would allow Greek knowledge to spread through the Islamic empire, a process in which Hunayn was to play a major role. Al-Rashid had two sons, the eldest was al-Amin while the younger was al-Ma'mun. Harun al-Rashid died in 809, the year after Hunayn's birth, and there was an armed conflict between his two sons. Al-Ma'mun won the armed struggle, became Caliph and ruled the empire from Baghdad. He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. It should not be thought that the Arabs who were translating these Greek texts simply sat down with a pile of Greek manuscripts and translated them. Most of the difficulty occurred in searching for the manuscripts which were to be translated. In order to find manuscripts of the works of Aristotle and others, al-Ma'mun sent a team of his most learned men to Byzantium. It is thought that Hunayn, being more skilled in the Greek language than any of the other http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hunayn.html (1 of 3) [2/16/2002 11:15:31 PM]

Hunayn

scholars in Baghdad, was on this expedition. As an example of the lengths that Hunayn went in order to find a particular manuscript we quote his description of a search for a medical manuscript (see for example [1]):I sought for [the manuscript] earnestly and travelled in search of it in the lands of Mesopotamia, Syria, Palestine and Egypt, until I reached Alexandria, but I was not able to find anything, except about half of it at Damascus. Al-Ma'mun recruited the most talented men for the House of Wisdom such as al-Khwarizmi, al-Kindi and al-Hajjaj the first translator of Euclid's Elements into Arabic refered to above. There they worked with Hunayn and later also with Thabit ibn Qurra. Hunayn became a close friend of Muhammad Banu Musa although relations between some of the scholars was not good due to rivalry. In 833 al-Ma'mun died and was succeeded by his brother al-Mu'tasim. The house of Wisdom continued to flourish under successive caliphs. Al-Mu'tasim died in 842 and was succeeded by al-Wathiq [1]:Hunayn soon became famous and participated in the scholarly meetings at which physicians and philosophers discussed dificult problems in the presence of Caliph al-Wathiq. Caliph al-Wathiq was succeeded as Caliph in 847 by al-Mutawakkil who appointed Hunayn to the post of chief physician at his court, a position he held for the rest of his life. Under both these Caliphs internal arguments and rivalry arose between the scholars in the House of Wisdom and Hunayn was most certainly involved in this rivalry. The rivalry could certainly become serious and at one point Hunayn had his library confiscated and he was imprisoned. Hunayn is important for the many excellent translations of Greek texts which he made into Arabic. In particular he translated Plato and Aristotle. These translations were spread widely through Mesopotamia, Syria and Egypt. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Arabic mathematics : forgotten brilliance? 2. How do we know about Greek mathematics?

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hunayn.html (2 of 3) [2/16/2002 11:15:31 PM]

Search Form

Hunayn

Glossary index

Quotations index

Mathematicians of the day JOC/EFR November 1999

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hunayn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hunayn.html (3 of 3) [2/16/2002 11:15:31 PM]

Huntington

Edward Vermilye Huntington Born: 26 April 1874 in Clinton, New York, USA Died: 25 Nov 1952 in Cambridge, Massachusetts, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edward Huntington's father was Chester Huntington while his mother was Katherine Hazard Smith. Huntington attended Harvard College, and he was awarded his A.B. in 1895, followed by his A.M. two years later. He had already been appointed as an instructor at Harvard after completing his first degree and he held this post while studying for his Master's degree. After being awarded the Master's degree, Huntington was appointed to Williams College. For two years Huntington worked as a mathematics instructor at Williams College at Williamstown, Massachusetts. The College was a long established one, founded in 1793. In 1899 Huntington left the United States to study for his doctorate in Europe. This was awarded by the University of Strasbourg in 1901. Most American mathematicians of this period who went to Europe to complete their studies chose Germany and, indeed, the city of Strasbourg was German at this time (and in fact called Strassburg). It had been annexed by Germany during the Franco-German War of 1870-71 and would not became French again until after World War I. Huntington's doctorate was awarded for a thesis on algebra. Returning to the United States, Huntington again was appointed as an instructor in Harvard in 1901. He was promoted to an assistant professor in 1905, then in 1909 he married Suzie Edwards Van Volkenburgh - they had no children. Huntington was promoted to associate professor in 1915 and to professor of mechanics at Harvard in 1919. He retired from this chair, and his other duties, in 1941. Huntington was interested in the foundations of mathematics. He devised sets of axioms for many mathematical systems, in particular showing that the sets of axioms were independent by giving examples which satisfied all but one of the axioms. He gave axioms for a group, an abelian group, a boolean algebra, geometry, the real number field, and the complex numbers. In 1904 he gave axioms for a boolean algebra then later, in 1933, he showed that a boolean algebra could be defined in terms of a single binary and a single unary operation. This is now known as Huntington's theorem. Although we have referred above to axioms and axiom systems, we have only done so since this is the modern usage. Huntington himself called these postulates, rather than axioms, and was careful to distinguish between 'postulate' and his own use of the term 'axiom'. For Huntington axiom:... should only be applied to statements of fact [including] obviously true statements about certain definite operations on angles or distances. For Huntington postulates were:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Huntington.html (1 of 3) [2/16/2002 11:15:33 PM]

Huntington

... conditions which a given system may or may not happen to satisfy. Huntington's book The Continuum and other types of serial order (1917) was a standard work on set theory for many years. A second edition of this little book consisting of 82 pages was published (also in 1917) and reprinted as a Dover Publication in 1955. Scanlan writes [3] that the book was:... a widely read introduction to Cantorian set theory and, although now outdated in method, is still a masterful presentation of the mathematical facts. One might wonder why, given these interests, Huntington was appointed as Professor of Mechanics in 1919. In fact this was more as a result of his deep interest in methods of teaching mathematics to engineering students than to a research record in the topic. He was, however, interested in several different aspects of applied mathematics. In 1907 he developed log tables which were especially easy for students to use. He was an advocate of the use of calculating machines and he had a mechanical calculator on his desk in his Harvard office. He was also interested in statistics, which was unusual at this time, and he put his knowledge in this area into use during World War I when he worked on statistical problems for the military. Huntington proposed a method of appointing representatives to the US Congress. This was published as "Edward V Huntington, Methods of Apportionment in Congress. A survey of methods of apportionment in Congress. Senate document no. 304, 76th Congress, Third session. United States Government Printing Office, Washington, 1940". The method was adopted in 1941 and is still used. In 1919 Huntington was president of the Mathematical Association of America, an organisation he helped to found, and his presidential address considered mathematical problems related to statistics. He was also vice-president of the American Mathematical Society in 1924 and vice-president of the Ameriacn Association for the Advancement of Science in 1941. Among the honours he received was election to the American Academy of Arts and Sciences in 1913. He was also elected to the American Philosophical Society in 1933. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Huntington.html (2 of 3) [2/16/2002 11:15:33 PM]

Huntington

JOC/EFR July 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Huntington.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Huntington.html (3 of 3) [2/16/2002 11:15:33 PM]

Hurewicz

Witold Hurewicz Born: 29 June 1904 in Lodz, Russian Empire (now Poland) Died: 6 Sept 1956 in Uxmal, Mexico

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Witold Hurewicz's father was an industrialist. Witold attended school in a Russian controlled Poland but with World War I beginning before he had begun secondary school, major changes occurred in Poland. In August 1915 the Russian forces which had held Poland for many years withdrew. Germany and Austria-Hungary took control of most of the country and the University of Warsaw was refounded and it began operating as a Polish university. Rapidly a strong school of mathematics grew up in the University of Warsaw, with topology being one of the main topics. Although Hurewicz knew intimately the topology that was being studied in Poland he chose to go to Vienna to continue his studies. He studied under Hans Hahn and Karl Menger in Vienna, receiving a Ph.D. in 1926. Hurewicz was awarded a Rockefeller scholarship which allowed him to spend the year 1927-28 in Amsterdam. He was assistant to Brouwer in Amsterdam from 1928 to 1936. He was given study leave for a year which he decided to spend in the United States. He visited the Institute for Advanced Study in Princeton and then decided to remain in the United States and not return to his position in Amsterdam. Given the impending war in Europe this was clearly a wise decision. Hurewicz worked first at the University of North Carolina but during World War II he contributed to the war effort with research on applied mathematics, in particular the work he did on servomechanisms at that time was classified because of its military importance. From 1945 until his death he worked at the Massachusetts Institute of Technology. Hurewicz died falling off a ziggurat (a Mexican pyramid) on a conference outing at the International Symposium on algebraic topology in Mexico. In [1] it is suggested that he was:... a paragon of absentmindedness, a failing that probably led to his death. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hurewicz.html (1 of 3) [2/16/2002 11:15:34 PM]

Hurewicz

Hurewicz's early work was on set theory and topology and [1]:... a remarkable result of this first period [1930] is his topological embedding of separable metric spaces into compact spaces of the same (finite) dimension. In the field of general topology his contributions are centred around dimension theory. He wrote an important text Dimension theory published in 1941. A reviewer writes that the book:... is truly a classic. It presents the theory of dimension for separable metric spaces with what seems to be an impossible mixture of depth, clarity, precision, succinctness, and comprehensiveness. Hurewicz is best remembered for two remarkable contributions to mathematics, his discovery of the higher homotopy groups in 1935-36, and his discovery of exact sequences in 1941. His work led to homological algebra. It was during Hurewicz's time as Brouwer's assistant in Amsterdam that he did the work on the higher homotopy groups; [1]:... the idea was not new, but until Hurewicz nobody had pursued it as it should have been. Investigators did not expect much new information from groups, which were obviously commutative ... Hurewicz had a second textbook published, but this was not until 1958 after his death. Lectures on ordinary differential equations is a beautiful introduction to ordinary differential equations which again reflects the clarity of his thinking and the quality of his writing. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1930 to 1940

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hurewicz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hurewicz.html (2 of 3) [2/16/2002 11:15:34 PM]

Hurewicz

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hurewicz.html (3 of 3) [2/16/2002 11:15:34 PM]

Hurwitz

Adolf Hurwitz Born: 26 March 1859 in Hildesheim, Hanover, (now Germany) Died: 18 Nov 1919 in Zurich, Switzerland

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adolf Hurwitz was first taught by Schubert and then was an undergraduate at the University of Berlin where he attended classes by Kummer, Weierstrass and Kronecker. He then went with Klein to Leipzig where his Ph. D. was supervised by Klein. He received the degree in 1881 with a Ph. D. dissertation on modular functions Grundlagen einer independenten Theorie der elliptischen Modulfunktionen und Theorie der Multiplikatorgleichungen 1. Stufe. In 1884 Hurwitz accepted an invitation from Lindemann to fill a chair at Königsberg and he was to remain there for 8 years. Here he taught Hilbert and Minkowski, becoming a life long friend of Hilbert. In 1892 Frobenius left his chair at Eidgenössische Polytechnikum Zürich to return to Berlin and Hurwitz was appointed to the vacant chair at Zurich. Hurwitz remained at Zurich for the rest of his life, unfortunately continually suffering from ill health. Hurwitz published a paper on a factorisation theory for integer quaternions in 1896 and applied it to the problem of representing an integer as the sum of four squares. A full proof of Hurwitz's ideas appears in a booklet published in the year of his death. This involves studying the ring of integer quaternions in which there are 24 units. He shows that one-sided ideals are principal and introduces prime and primary quaternions. Hurwitz studied the genus of the Riemann surface. He worked on how to derive class number relations from modular equations. He investigated the automorphic groups of algebraic Riemann surfaces of genus http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hurwitz.html (1 of 2) [2/16/2002 11:15:36 PM]

Hurwitz

greater than 1, showing that they were finite. He also studied complex function theory and wrote several papers on Fourier series. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles)

A Quotation

A Poster of Adolf Hurwitz

Mathematicians born in the same country

Cross-references to History Topics

The beginnings of set theory

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hurwitz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hurwitz.html (2 of 2) [2/16/2002 11:15:36 PM]

Hutton

Charles Hutton Born: 14 Aug 1737 in Newcastle-upon-Tyne, England Died: 27 Jan 1823 in London, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Largely self-educated, Charles Hutton taught at the Mathematical School in Newcastle, then became professor of mathematics at the Royal Military Academy in Woolwich in 1773. The following year he became a Fellow of the Royal Society. In 1767 he published A Treatise on Mensuration which he states is adapted particularly to the Uses of Schools, Mathematicians and Mechanics. Many later writers borrowed material from this book. In 1775 Hutton published five volumes of extracts from the Ladies' Diary dealing with entertaining mathematical and poetical parts . He received the Copley Medal of the Royal Society for The Force of Fired Gunpowder and the velocity of Cannon Balls (1778). He also computed the mean density of the Earth based on Maskelyne's data from Schiehallion in An Account of the Calculations made from the Survey and Measures taken at Schiehallion in order to ascertain the mean density of the Earth (1779). In 1781 Hutton published Mathematical Tables for the Board of Longitude. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

Thomas Harriot's manuscripts

Honours awarded to Charles Hutton (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1774

Royal Society Copley Medal

Awarded 1778

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hutton.html (1 of 2) [2/16/2002 11:15:37 PM]

Hutton

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hutton.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hutton.html (2 of 2) [2/16/2002 11:15:37 PM]

Huygens

Christiaan Huygens Born: 14 April 1629 in The Hague, Netherlands Died: 8 July 1695 in The Hague, Netherlands

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Christiaan Huygens came from an important Dutch family. His father Constantin Huygens had studied natural philosophy and was a diplomat. It was through him that Christiaan was to gain access to the top scientific circles of the times. In particular Constantin had many contacts in England and corresponded regularly with Mersenne and was a friend of Descartes. Tutored at home by private teachers until he was 16 years old, Christiaan learned geometry, how to make mechanical models and social skills such as playing the lute. His mathematical education was clearly influenced by Descartes who was an occasional visitor at the Huygens' home and took a great interest in the mathematical progress of the young Christiaan. Christiaan Huygens studied law and mathematics at the University of Leiden from 1645 until 1647. Van Schooten tutored him in mathematics while he was in Leiden. From 1647 until 1649 he continued to study law and mathematics but now at the College of Orange at Breda. Here he was fortunate to have another skilled teacher of mathematics, John Pell. Through his father's contact with Mersenne, a correspondence between Huygens and Mersenne began around this time. Mersenne challenged Huygens to solve a number of problems including the shape of the rope supported from its ends. Although he failed at this problem he did solve the related problem of how to hang weights on the rope so that it hung in a parabolic shape. In 1649 Huygens went to Denmark as part of a diplomatic team and hoped to continue to Stockholm to visit Descartes but the weather did not allow him to make this journey. He followed the visit to Denmark with others around Europe including Rome. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Huygens.html (1 of 7) [2/16/2002 11:15:39 PM]

Huygens

Huygens's first publications in 1651 and 1654 considered mathematical problems. The 1651 publication Cyclometriae showed the fallacy in methods proposed by Gregory of Saint-Vincent, who had claimed to have squared the circle. Huygens' 1654 work De Circuli Magnitudine Inventa was a more major work on similar topics. Huygens soon turned his attention to lens grinding and telescope construction. Around 1654 he devised a new and better way of grinding and polishing lenses. Using one of his own lenses, Huygens detected, in 1655, the first moon of Saturn. In this same year he made his first visit to Paris. He informed the mathematicians in Paris including Boulliau of his discovery and in turn Huygens learnt of the work on probability carried out in a correspondence between Pascal and Fermat. On his return to Holland Huygens wrote a small work De Ratiociniis in Ludo Aleae on the calculus of probabilities, the first printed work on the subject. The following year he discovered the true shape of the rings of Saturn. However others had different theories including Roberval and Boulliau. Boulliau had failed to detect Saturn's moon Titan so Huygens realised that he was using an inferior telescope. By 1656 Huygens was able to confirm his ring theory to Boulliau and the results were reported to the Paris group. In Systema Saturnium (1659), Huygens explained the phases and changes in the shape of the ring. Some, including the Jesuit Fabri, attacked not only Huygens theories but also his observations. However by 1665 even Fabri was persuaded to accept Huygens' ring theory as improving telescopes confirmed his observations. Work in astronomy required accurate timekeeping and this prompted Huygens to tackle this problem. In 1656 he patented the first pendulum clock, which greatly increased the accuracy of time measurement. His work on the pendulum was related to other mathematical work which he had been doing on the cycloid as a result of the challenge by Pascal. Huygens believed that a pendulum swinging in a large are would be more useful at sea and he invented the cycloidal pendulum with this in mind. He built several pendulum clocks to determine longitude at sea and they underwent sea trials in 1662 and again in 1686. In the Horologium Oscillatorium sive de motu pendulorum (1673) he described the theory of pendulum motion. He also derived the law of centrifugal force for uniform circular motion. As a result of this Huygens, Hooke, Halley and Wren formulated the inverse-square law of gravitational attraction. Huygens returned to Paris in 1660 and went to meetings of various scientific societies there. He wrote, in a letter to his brother:... there is a meeting every Tuesday [at Montmor's house] where twenty or thirty illustrious men are found together. I never fail to go ... I have also been occasionally to the house of M Rohault, who expounds the philosophy of M Descartes and does very fine experiments with good reasoning on them. At these societies he met many mathematicians including Roberval, Carcavi, Pascal, Pierre Petit, Desargues and Sorbière. After Pascal visited him in December 1660 Huygens wrote ... we talked of the force of water rarefied in cannons and of flying, I showed him my telescopes... In 1661 Huygens visited London, particularly to find out more about the newly forming Royal Society meeting at that time in Gresham College. He was greatly impressed with Wallis and the other English scientists whom he met and, from this time on, he was to continue his contacts with this group. He http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Huygens.html (2 of 7) [2/16/2002 11:15:39 PM]

Huygens

showed his telescopes to the English scientists and they proved superior to those in use in England. The Duke and Duchess of York came to observe the Moon and Saturn through Huygens' telescope. While in London Huygens saw Boyle's vacuum pump and he was impressed. After his return to the Hague he carried out a number of Boyle's experiments for himself. Huygens was elected to the Royal Society of London in 1663. At this time Huygens patented his design of pendulum clock with the solution of the longitude problem in mind. In 1665 he learnt that the Royal Society was investigating other forms of clock, in particular Hooke was experimenting with a spring regulated clock. Huygens wrote to Hooke doubting this approach which he felt would be unduly affected by temperature changes. Despite this Huygens did begin to experiment with clocks regulated by springs, but their accuracy was poorer than his pendulum clocks. Huygens accepted an invitation from Colbert in 1666 to become part of the Académie Royale des Sciences. He arrived in Paris that year to discover that the Society was not yet organised. After meetings were held with Roberval, Carcavi, Auzout, Frenicle de Bessy, Auzout and Buot in Colbert's library the society moved to the Bibliothèque du Roi where Huygens took up residence. He assumed leadership of the group basing much on his knowledge of the way the Royal Society operated in England. Huygens' work on the collision of elastic bodies showed the error Descartes' laws of impact and his memoir on the topic was sent to the Royal Society in 1668. The Royal Society had posed a question on impact and Huygens' proved by experiment that the momentum in a fixed direction before the collision of two bodies is equal to the momentum in that direction after the collision. Wallis and Wren also answered this question. Circular motion was a topic which Huygens took up at this time but he also continued to think about Descartes' theory of gravity based on vortices. He seems to have shown signs of being unhappy with Descartes' theory around this time but he still addressed the Académie on this topic in 1669 although after his address Roberval and Mariotte argued strongly, and correctly, against Descartes's theory and this may have influenced Huygens. From his youth Huygens' health had never been robust and in 1670 he had a serious illness which resulted in him leaving Paris for Holland. Before he left Paris, believing himself to be close to death he asked that his unpublished papers on mechanics be sent to the Royal Society. The secretary to the English ambassador was called and described Huygens reasons:... he fell into a discourse concerning the Royal Society in England which he said was an assembly of the choicest wits in Christendom .. he said he chose to deposit those little labours ... in their hands sooner than any else. ... he said he did foresee the dissolution of this Académie because it was mixed with tinctures of envy because it was supported upon suppositions of profit because it wholly depended upon the humour of a prince and the favour of a minister... By 1671 Huygens returned to Paris. However in 1672 Louis XIV invaded the Low Countries and Huygens found himself in the extremely difficult position of being in an important position in Paris at a time France was at war with his own country. Scientist of this era felt themselves above political wars and Huygens was able, with much support from his friends, to continue his work.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Huygens.html (3 of 7) [2/16/2002 11:15:39 PM]

Huygens

In 1672 Huygens and Leibniz met in Paris and thereafter Leibniz was a frequent visitor to the Académie. In fact Leibniz owes much to Huygens from whom he learnt much of his mathematics. In this same year Huygens learnt of Newton's work on the telescope and on light. He, quite wrongly, criticised Newton's theory of light, in particular his theory of colour. His own work, Horologium Oscillatorium sive de motu pendulorum appeared in 1673 and showed that Huygens had moved far from Descartes' influence. Horologium Oscillatorium contains work on the pendulum. In it Huygens proves that the cycloid is tautochronous, an important theoretical result but one which had little practical application to the pendulum. He also solves the problem of the compound pendulum. However there is much more than work on pendulums. Huygens describes the descent of bodies in a vacuum, either vertically or along curves. He defines evolutes and involutes of curves and, after giving some elementary properties, finds the evolutes of the cycloid and of the parabola. Huygens attempts for the first time in this work to study the dynamics of bodies rather than particles. Papin worked as an assistant to Huygens around this time and after he left to work with Boyle, Huygens was joined by Tschirnhaus. Another bout of illness in 1676 saw Huygens return to the Hague again. He spent two years there, in particular studying the double refraction Bartholin had discovered in Iceland spar crystal. He also worked on the velocity of light which he believed was finite and was pleased to hear of Römer's experiments which gave an approximate velocity for light determined by observing Jupiter's moons. By 1678 Huygens had returned to Paris. In that year his Traité de la lumiere appeared, in it Huygens argued in favour of a wave theory of light. Huygens stated that an expanding sphere of light behaves as if each point on the wave front were a new source of radiation of the same frequency and phase. However his health became even more unreliable and he became ill in 1679 and then again in 1681 when he returned to the Hague for the last time. La Hire, who had always argued against foreigners in the Académie, sent his best wishes to Huygens but he clearly hoped that he would not return so that he might himself might acquire his position. The longitude problem had remained a constant cause for Huygens to continue work on clocks all his life. Again after his health returned he worked on a new marine clock during 1682 and, with the Dutch East India Company showing interest, he worked hard on the clocks. Colbert died in 1683 and a return to Paris without the support of his patron seemed impossible. His father died in 1687, having reached 91 years of age, and the following year his brother left for England. Huygens missed having people around him with whom he could discuss scientific topics. In 1689 he came to England. In England Huygens met Newton, Boyle and others in the Royal Society. It is not known what discussions went on between Huygens and Newton but we do know that Huygens had a great admiration for Newton but at the same time did not believe the theory of universal gravitation which he said appears to me absurd. In some sense of course Huygens was right, how can one believe that two distant masses attract one another when there is nothing between them, nothing in Newton's theory explains how one mass can possible even know the other mass is there. Writing about Newton and the Principia some time later Huygens wrote:I esteem his understanding and subtlety highly, but I consider that they have been put to ill http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Huygens.html (4 of 7) [2/16/2002 11:15:39 PM]

Huygens

use in the greater part of this work, where the author studies things of little use or when he builds on the improbable principle of attraction. He departed with much sadness at the thoughts of his scientific isolation in Holland. In the final years of his life Huygens composed one of the earliest discussions of extraterrestrial life, published after his death as the Cosmotheoros (1698). He continued to work on improving lenses and on a spring regulated clock and on new pendulum clocks. Huygens described the 31-tone equal temperament in Lettre touchant le cycle harmonique. This has led indirectly to a tradition of 31-tone music in the Netherlands in this century. In a letter to Tschirnhaus written in 1687, Huygens explained his own approach:.. great difficulties are felt at first and these cannot be overcome except by starting from experiments ... and then be conceiving certain hypotheses ... But even so, very much hard work remains to be done and one needs not only great perspicacity but often a degree of good fortune. Huygens scientific achievements are summed up in [4] as follows:... Huygens was the greatest mechanist of the seventeenth century. He combined Galileo's mathematical treatment of phenomena with Descartes' vision of the ultimate design of nature. Beginning as an ardent Cartesian who sought to correct the more glaring errors of the system, he ended up as one of its sharpest critics. ... the ideas of mass, weight, momentum, force, and work were finally clarified in Huygens' treatment of the phenomena of impact, centripetal force and the first dynamical system ever studied - the compound pendulum. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (71 books/articles) A Poster of Christiaan Huygens

Mathematicians born in the same country

Some pages from publications

The title page of Horologium oscillatorium (1673). The first page of De ratiociniis in ludo aleae (1657).

Cross-references to History Topics

1. Longitude and the Académie Royale 2. Squaring the circle 3. Quadratic, cubic and quartic equations 4. The rise of the calculus

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Huygens.html (5 of 7) [2/16/2002 11:15:39 PM]

Huygens

Cross-references to Famous Curves

1. Caustic curves 2. Caustic of a circle 3. Catenary 4. Cissoid of Diocles 5. Cycloid 6. Epicycloid 7. Hypocycloid 8. Involute of a circle 9. Semi-cubical parabola 10. Nephroid 11. Serpentine 12. Tractrix

Other references in MacTutor

1. The theory of the pendulum 2. The theory of collisions 3. Chronology: 1650 to 1675

Honours awarded to Christiaan Huygens (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1663

Lunar features

Mons Huygens

Lunar features

Crater Huygens on Mars

Paris street names

Rue Huyghens (14th Arrondissement)

Other Web sites

1. Fokko J Dijksterhuis 2. Rouse Ball 3. The Galileo Project 4. Science Museum, Florence 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Huygens.html (6 of 7) [2/16/2002 11:15:39 PM]

Huygens

Glossary index

Quotations index

Mathematicians of the day JOC/EFR February 1997

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Huygens.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Huygens.html (7 of 7) [2/16/2002 11:15:39 PM]

Hypatia

Hypatia of Alexandria Born: about 370 in Alexandria, Egypt Died: March 415 in Alexandria, Egypt

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hypatia of Alexandria was the first woman to make a substantial contribution to the development of mathematics. Hypatia was the daughter of the mathematician and philosopher Theon of Alexandria and it is fairly certain that she studied mathematics under the guidance and instruction of her father. It is rather remarkable that Hypatia became head of the Platonist school at Alexandria in about 400 AD. There she lectured on mathematics and philosophy, in particular teaching the philosophy of Neoplatonism. Hypatia based her teachings on those of Plotinus, the founder of Neoplatonism, and Iamblichus who was a developer of Neoplatonism around 300 AD. Plotinus taught that there is an ultimate reality which is beyond the reach of thought or language. The object of life was to aim at this ultimate reality which could never be precisely described. Plotinus stressed that people did not have the mental capacity to fully understand both the ultimate reality itself or the consequences of its existence. Iamblichus distinguished further levels of reality in a hierarchy of levels beneath the ultimate reality. There was a level of reality corresponding to every distinct thought of which the human mind was capable. Hypatia taught these philosophical ideas with a greater scientific emphasis than earlier followers of Neoplatonism. She is described by all commentators as a charismatic teacher. Hypatia came to symbolise learning and science which the early Christians identified with paganism. However, among the pupils who she taught in Alexandria there were many prominent Christians. One of the most famous is Synesius of Cyrene who was later to become the Bishop of Ptolemais. Many of the letters that Synesius wrote to Hypatia have been preserved and we see someone who was filled with admiration and reverence for Hypatia's learning and scientific abilities. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hypatia.html (1 of 3) [2/16/2002 11:15:41 PM]

Hypatia

In 412 Cyril (later St Cyril) became patriarch of Alexandria. However the Roman prefect of Alexandria was Orestes and Cyril and Orestes became bitter political rivals as church and state fought for control. Hypatia was a friend of Orestes and this, together with prejudice against her philosophical views which were seen by Christians to be pagan, led to Hypatia becoming the focal point of riots between Christians and non-Christians. Hypatia, Heath writes, [4]:... by her eloquence and authority ... attained such influence that Christianity considered itself threatened ... A few years later, according to one report, Hypatia was brutally murdered by the Nitrian monks who were a fanatical sect of Christians who were supporters of Cyril. According to another account (by Socrates Scholasticus) she was killed by an Alexandrian mob under the leadership of the reader Peter. What certainly seems indisputable is that she was murdered by Christians who felt threatened by her scholarship, learning, and depth of scientific knowledge. This event seems to be a turning point as described in [2]:Whatever the precise motivation for the murder, the departure soon afterward of many scholars marked the beginning of the decline of Alexandria as a major centre of ancient learning. There is no evidence that Hypatia undertook original mathematical research. However she assisted her father Theon of Alexandria in writing his eleven part commentary on Ptolemy's Almagest. It is also thought that she also assisted her father in producing a new version of Euclid's Elements which has become the basis for all later editions of Euclid. Heath writes of Theon and Hypatia's edition of the Elements [4]:.. while making only inconsiderable additions to the content of the "Elements", he endeavoured to remove difficulties that might be felt by learners in studying the book, as a modern editor might do in editing a classical text-book for use in schools; and there is no doubt that his edition was approved by his pupils at Alexandria for whom it was written, as well as by later Greeks who used it almost exclusively... In addition the the joint work with her father, we are informed by Suidas that Hypatia wrote commentaries on Diophantus's Arithmetica, on Apollonius's Conics and on Ptolemy's astronomical works. The passage in Suidas is far from clear and most historians doubt that Hypatia wrote any commentaries on Ptolemy other than the works which she composed jointly with her father. All Hypatia's work is lost except for its titles and some references to it. However no purely philosophical work is known, only work in mathematics and astronomy. Based on this small amount of evidence Deakin, in [8] and [9], argues that Hypatia was an excellent compiler, editor, and preserver of earlier mathematical works. As mentioned above, some letters of Synesius to Hypatia exist. These ask her advice on the construction of an astrolabe and a hydroscope. Charles Kingsley (best known as the author of The Water Babies) made her the heroine of one of his novels Hypatia, or New Foes with an Old Face. As Kramer writes in [1]:Such works have perpetuated the legend that she was not only intellectual but also beautiful, eloquent, and modest.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hypatia.html (2 of 3) [2/16/2002 11:15:41 PM]

Hypatia

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) A Poster of Hypatia

Mathematicians born in the same country

Cross-references to History Topics

How do we know about Greek mathematics?

Other references in MacTutor

Chronology: 1AD to 500

Honours awarded to Hypatia (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Hypatia and Rimae Hypatia

Other Web sites

1. Alexandria on the Web (A longer biography and some links to other sources, including the account by Socrates Scholasticus mentioned above) 2. H A Landman (Including many further links) 3. Agnes Scott College 4. P Alfeld 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hypatia.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hypatia.html (3 of 3) [2/16/2002 11:15:41 PM]

Hypsicles

Hypsicles of Alexandria Born: about 190 BC in Alexandria, Egypt Died: about 120 BC Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hypsicles of Alexandria wrote a treatise on regular polyhedra. He is the author of what has been called Book XIV of Euclid's Elements, a work which deals with inscribing regular solids in a sphere. What little is known of Hypsicles' life is related by him in the preface to the so-called Book XIV. He writes that Basilides of Tyre came to Alexandria and there he discussed mathematics with Hypsicles' father. Hypsicles relates that his father and Basilides studied a treatise by Apollonius on a dodecahedron and an icosahedron in the same sphere and decided that Apollonius's treatment was not satisfactory. In the so-called Book XIV Hypsicles proves some results due to Apollonius. He had clearly studied Apollonius's tract on inscribing a dodecahedron and an icosahedron in the same sphere and clearly had, as his father and Basilides before him, found it poorly presented and Hypsicles attempts to improve on Apollonius's treatment. Arab writers also claim that Hypsicles was involved with the so-called Book XV of the Elements. Bulmer-Thomas writes in [1] that various aspects are ascribed to him, claiming that either:... he wrote it, edited it, or merely discovered it. But this is clearly a much later and much inferior book, in three separate parts, and this speculation appears to derive from a misunderstanding of the preface to Book XIV. Diophantus quotes a definition of polygonal number due to Hypsicles (see either [1] or [2]):If there are as many numbers as we please beginning from 1 and increasing by the same common difference, then, when the common difference is 1, the sum of all the numbers is a triangular number; when 2 a square; when 3, a pentagonal number [and so on]. And the number of angles is called after the number which exceeds the common difference by 2, and the side after the number of terms including 1. This says that, in modern notation, the nth m-agonal number is n [2 + (n - 1) (m - 2)]/2. We do not know for certain that Hypsicles wrote a text on polygonal numbers, but it is fairly certain that he did write such a text which has been lost. This work on polygonal numbers is related to the ideas on arithmetic progressions that appear in another work by Hypsicles, making it more likely that indeed Hypsicles had indeed done original work on this topic.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hypsicles.html (1 of 3) [2/16/2002 11:15:43 PM]

Hypsicles

The work which involves arithmetic progressions is Hypsicles' On the Ascension of Stars. In this work he was the first to divide the Zodiac into 360 . He says (see [1] or [2]):The circle of the zodiac having been divided into 360 equal arcs, let each of the arcs be called a spatial degree, and likewise, if the time taken by the zodiac circle to return from a point to the same point is divided into 360 equal times, let each of the times be called a temporal degree. Hypsicles considers two problems in this work [2]:-. (i) Given the ratio of the longest to the shortest day at any place, how long does it take any given sign of the zodiac to rise there? (ii) How long does it take any given degree in a sign to rise? Hypsicles makes a false assumption involving arithmetic progressions so that his results are wrong. Heath writes [2]:True, the treatise (if it really be by Hypsicles, and not a clumsy effort by a beginner working from an original by Hypsicles) does no credit to its author; but it is in some respects interesting... The mistake which Hypsicles makes is to assume that the rising times form an arithmetical progression. Having made this assumption his results are correct and Neugebauer [4] certainly values this work much more highly than Heath does. In fact without the aid of the sine function and trigonometry it is hard to see how Hypsicles could have done better. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Other references in MacTutor

Chronology: 500BC to 1AD

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hypsicles.html (2 of 3) [2/16/2002 11:15:43 PM]

Hypsicles

JOC/EFR April 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hypsicles.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Hypsicles.html (3 of 3) [2/16/2002 11:15:43 PM]

Ibrahim

Ibrahim ibn Sinan ibn Thabit ibn Qurra Born: 908 in Baghdad, (now in Iraq) Died: 946 in Baghdad, (now in Iraq) Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ibrahim ibn Sinan was a grandson of Thabit ibn Qurra and studied geometry and in particular tangents to circles. He also studied the apparent motion of the Sun and the geometry of shadows. There is no doubt that had he not died at the young age of thirty-eight, he would have achieved a degree of fame for his mathematical works going even beyond the opinion of Sezgin (see [5] and [6]) that he was:... one of the most important mathematicians in the medieval Islamic world. Perhaps his early death robbed him of the chance to make a contribution even more important than that of his famous grandfather. Ibrahim's most important work was on the quadrature of the parabola where he introduced a method of integration more general than that of Archimedes. His grandfather Thabit ibn Qurra had started to view integration in a different way to Archimedes but Ibrahim realised that al-Mahani had made improvements on what his father had achieved. To Ibrahim it was unacceptable that (see for example [1]):... al-Mahani's study should remain more advanced than my grandfather's unless someone of our family can excel him. Ibrahim is also considered the foremost Arab mathematician to treat mathematical philosophy. He wrote (see for example [1]):I have found that contemporary geometers have neglected the method of Apollonius in analysis and synthesis, as they have in most of the things I have brought forward, and that they have limited themselves to analysis alone in so restrictive a manner that they have led people to believe that this analysis did not correspond to the synthesis effected. We know of Ibrahim's works through his own work Letter on the description of the notions Ibrahim derived in geometry and astronomy in which Ibrahim lists his own works. This is one of seven treatises by Ibrahim given with full Arabic text and English summaries in [2]. Among the works published in [2] are On drawing the three conic sections in which Ibrahim give a pointwise construction for the ellipse, the parabola and the hyperbola. Although based on ideas due to Apollonius there are aspects of this work which illustrate the changed point of view of Arabic mathematicians. For example Ibrahim uses an arithmetical term to denote the product of two geometrical lines. In On the measurement of the parabola Ibrahim ibn Sinan gives a beautiful proof that the area of a segment of the parabola is four-thirds of the area of the inscribed triangle. Another work is On the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ibrahim.html (1 of 3) [2/16/2002 11:15:44 PM]

Ibrahim

method of analysis and synthesis, and the other procedures in geometrical problems which contains a systematic exposition of analysis, synthesis and related subjects, with many easy examples. This is in contrast to The selected problems in which 41 difficult geometrical problems are solved, usually by analysis only, without a discussion of the number of solutions or conditions which make the solutions possible. On the motions of the sun is an astronomical work which discusses of the motion of the solar apogee. It also provides a critical analysis of the observations underlying Ptolemy's solar theory, and Ibrahim ibn Sinan provides his own theory of the sun. The work On the astrolabe includes work on map projections. Ibrahim proves in this work that the stereographic projection maps circles which do not pass through the pole of projection onto circles. In fact geometric transformations figure a great deal in Ibrahim's works and this interesting aspect is discussed in detail in [4]. Examples are given which illustrate how Ibrahim applied an orthogonal compression to transform a circle into an ellipse, and an oblique compression to map a hyperbola into a second hyperbola. In a different work Ibrahim uses a transformation which maps figures keeping invariant the ratio between their areas. Ibrahim's contribution is summed up in [1] as follows:Considering both the problem of infinitesimal determinations and the history of mathematical philosophy, it is obvious that the work of ibn Sinan is important in showing how the Arab mathematicians pursued the mathematics that they had inherited from the Hellenistic period and developed it with independent minds. That is the dominant impression left by his work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ibrahim.html (2 of 3) [2/16/2002 11:15:44 PM]

Ibrahim

JOC/EFR November 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ibrahim.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ibrahim.html (3 of 3) [2/16/2002 11:15:44 PM]

Ingham

Albert Edward Ingham Born: 3 April 1900 in Northampton, England Died: 6 Sept 1967 in Chamonix, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Albert Ingham won a scholarship to Trinity College Cambridge in December 1917, spent a few months in the army, then in January 1919 he began his studies. An outstanding undergraduate career saw him win a Smith's prize and the highest honours. In 1922 he was elected to a fellowship at Trinity for a dissertation on the zeta function and his next four years were occupied only with research, a few months of which were spent at Göttingen. During this time Ingham was greatly influenced by Littlewood who would give him the advice to work at a hard problem: you may not solve it but you'll solve another one. In 1926 Ingham was appointed to Leeds but 4 years later returned to Cambridge, on the death of Ramsey, and remained there for the rest of his life. He was elected a Fellow of the Royal Society in 1945 and became a Reader in Mathematical Analysis in 1953. His book On the distribution of prime numbers published in 1932 was his only book and it is a classic. Many of the ideas here, as in other work of Ingham's, came from the joint work undertaken by Harald Bohr and Littlewood. Ingham's work was on the Riemann zeta function, the theory of numbers, the theory of series and Tauberian theorems. He generalised work on the prime number theorem of Hadamard and Vallée Poussin. The result for which Ingham is best known, however, relates to pn+1 - pn where pn denotes the nth prime. It was proved by Hoheisel in 1930 that there is a constant k such that http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ingham.html (1 of 3) [2/16/2002 11:15:46 PM]

Ingham

pn+1 - pn < pnk for all sufficiently large n. In On the difference between two consecutive primes (1937) Ingham proved that the result holds for k = 5/8. Pólya, in 1919, made the following conjecture. Suppose (n) = 1 if n has an even number of prime factors, -1 if n has an odd number of prime factors (counting multiplicities) then let L(x) be the sum of (n) over all positive integers less than n. The conjecture is that L(x) 0. Ingham, in 1942, was able to find an ingenious method to show how a counterexample could be constructed. It still required computing power to find the counterexample and, using Ingham's method, a counterexample was found by R S Lehman in 1960 when he showed that L(906180359) = 1. Some of Ingham's work on number theory was carried further by Linnik. Ingham also worked on Tauberian theorems. He proved results suggested by Norbert Wiener, and he applied methods first developed by Wiener. Ingham led a life of great simplicity. Burkill describes it in these terms: It did not occur to him to want a car or a radio, let alone a television set. For forty years he used a Sunbeam bicycle that he had won as a school prize. He was an expert photographer: he developed his own colour films and did everything from first principles. He was a good cricketer ... who would have been of minor county class if he had been able to give the time. He died while on a walking holiday in the mountains. He and his wife had taken this type of holiday every summer for many years. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Albert Ingham (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1945

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ingham.html (2 of 3) [2/16/2002 11:15:46 PM]

Search Form

Ingham

Glossary index

Quotations index

Mathematicians of the day JOC/EFR December 1996

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ingham.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ingham.html (3 of 3) [2/16/2002 11:15:46 PM]

Ito

Kiyosi Ito Born: 7 Sept 1915 in Hokusei-cho, Mie Prefecture, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kiyosi Ito studied mathematics in the Faculty of Science of the Imperial University of Tokyo. It was during his student years that he became attracted to probability theory. In [3] he explains how this came about:Ever since I was a student, I have been attracted to the fact that statistical laws reside in seemingly random phenomena. Although I knew that probability theory was a means of describing such phenomena, I was not satisfied with contemporary papers or works on probability theory, since they did not clearly define the random variable, the basic element of probability theory. At that time, few mathematicians regarded probability theory as an authentic mathematical field, in the same strict sense that they regarded differential and integral calculus. With clear definition of real numbers formulated at the end of the19th century, differential and integral calculus had developed into an authentic mathematical system. When I was a student, there were few researchers in probability; among the few were Kolmogorov of Russia, and Paul Levy of France. In 1938 Ito graduated from the University of Tokyo and in the following year he was appointed to the Cabinet Statistics Bureau. He worked there until 1943 and it was during this period that he made his most outstanding contributions:During those five years I had much free time, thanks to the special consideration given me by the then Director Kawashima ... Accordingly, I was able to continue studying probability theory, by reading Kolmogorov's Basic Concept of Probability Theory and Levy's Theory of Sum of Independent Random Variables. At that time, it was commonly believed that Levy's works were extremely difficult, since Levy, a pioneer in the new mathematical field, explained probability theory based on his intuition. I attempted to describe Levy's ideas, using precise logic that Kolmogorov might use. Introducing the concept of regularisation,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ito.html (1 of 4) [2/16/2002 11:15:49 PM]

Ito

developed by Doob of the United States, I finally devised stochastic differential equations, after painstaking solitary endeavours. My first paper was thus developed; today, it is common practice for mathematicians to use my method to describe Levy's theory. In 1940 he published On the probability distribution on a compact group on which he collaborated with Yukiyosi Kawada. The background to Ito's famous 1942 paper On stochastic processes (Infinitely divisible laws of probability) which he published in the Japanese Journal of Mathematics is given in [2]:Brown, a botanist, discovered the motion of pollen particles in water. At the beginning of the twentieth century, Brownian motion was studied by Einstein, Perrin and other physicists. In 1923, against this scientific background, Wiener defined probability measures in path spaces, and used the concept of Lebesgue integrals to lay the mathematical foundations of stochastic analysis. In 1942, Dr. Ito began to reconstruct from scratch the concept of stochastic integrals, and its associated theory of analysis. He created the theory of stochastic differential equations, which describe motion due to random events. Although today we see this paper as a fundamental one, it was not seen as such by mathematicians at the time it was published. Ito, who still did not have a doctorate at this time, would have to wait several years before the importance of his ideas would be fully appreciated and mathematicians would begin to contribute to developing the theory. In 1943 Ito was appointed as Assistant Professor in the Faculty of Science of Nagoya Imperial University. This was a period of high activity for Ito, and when one considers that this occurred during the years of extreme difficulty in Japan caused by World War II, one has to find this all the more remarkable. Volume 20 of the Proceedings of the Imperial Academy of Tokyo contains six papers by Ito: (1) On the ergodicity of a certain stationary process; (2) A kinematic theory of turbulence; (3) On the normal stationary process with no hysteresis; (4) A screw line in Hilbert space and its application to the probability theory; (5) Stochastic integral; and (6) On Student's test. In 1945 Ito was awarded his doctorate. He continued to develop his ideas on stochastic analysis with many important papers on the topic. Among them were On a stochastic integral equation (1946), On the stochastic integral (1948), Stochastic differential equations in a differentiable manifold (1950), Brownian motions in a Lie group (1950), and On stochastic differential equations (1951). In 1952 Ito was appointed to a Professorship at Kyoto University. In the following year he published his famous text Probability theory. In this book, Ito develops the theory on a probability space using terms and tools from measure theory. The years 1954-56 Ito spent at the Institute for Advanced Study at Princeton University. An important publication by Ito in 1957 was Stochastic processes. This book contained five chapters, the first providing an introduction, then the remaining ones studying processes with independent increments, stationary processes, Markov processes, and the theory of diffusion processes. In 1960 Ito visited the Tata Institute in Bombay, India, where he gave a series of lectures surveying his own work and that of other on Markov processes, Levy processes, Brownian motion and linear diffusion. Although Ito remained as a professor at Kyoto University until he retired in 1979, he also held positions as professor at Aarhus University from 1966 to 1969 and professor at Cornell University from 1969 to 1975. During his last three years at Kyoto before he retired, Ito was Director of the Research Institute for Mathematical Sciences there. After retiring from Kyoto University in 1979 he did not retire from mathematics but continued to write research papers. He was also appointed at Professor at Gakushuin University. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ito.html (2 of 4) [2/16/2002 11:15:49 PM]

Ito

Ito gives a wonderful description mathematical beauty in [3] which he then relates to the way in which he and other mathematicians have developed his fundamental ideas:In precisely built mathematical structures, mathematicians find the same sort of beauty others find in enchanting pieces of music, or in magnificent architecture. There is, however, one great difference between the beauty of mathematical structures and that of great art. Music by Mozart, for instance, impresses greatly even those who do not know musical theory; the cathedral in Cologne overwhelms spectators even if they know nothing about Christianity. The beauty in mathematical structures, however, cannot be appreciated without understanding of a group of numerical formulae that express laws of logic. Only mathematicians can read "musical scores" containing many numerical formulae, and play that "music" in their hearts. Accordingly, I once believed that without numerical formulae, I could never communicate the sweet melody played in my heart. Stochastic differential equations, called "Ito Formula," are currently in wide use for describing phenomena of random fluctuations over time. When I first set forth stochastic differential equations, however, my paper did not attract attention. It was over ten years after my paper that other mathematicians began reading my "musical scores" and playing my "music" with their "instruments." By developing my "original musical scores" into more elaborate "music," these researchers have contributed greatly to developing "Ito Formula." Ito received many honours for his outstanding mathematical contributions. He was awarded the Asahi Prize in 1978, and in the same year he received the Imperial Prize and also the Japan Academy Prize. In 1985 he received the Fujiwara Prize and in 1998 the Kyoto Prize in Basic Sciences from the Inamori Foundation. These prizes were all from Japan, and a further Japanese honour was his election to the Japan Academy. However, he also received many honours from other countries. He was elected to the National Academy of Science of the United States and to the Académie des Sciences of France. He received the Wolf Prize from Israel and honorary doctorates from the universities of Warwick, England and ETH, Zurich, Switzerland. In [2] this tribute is paid to Ito:Nowadays, Dr. Ito's theory is used in various fields, in addition to mathematics, for analysing phenomena due to random events. Calculation using the "Ito calculus" is common not only to scientists in physics, population genetics, stochastic control theory, and other natural sciences, but also to mathematical finance in economics. In fact, experts in financial affairs refer to Ito calculus as "Ito's formula." Dr. Ito is the father of the modern stochastic analysis that has been systematically developing during the twentieth century. This ceaseless development has been led by many, including Dr. Ito, whose work in this regard is remarkable for its mathematical depth and strong interaction with a wide range of areas. His work deserves special mention as involving one of the basic theories prominent in mathematical sciences during this century. A recent monograph entitled Ito's Stochastic Calculus and Probability Theory (1996), dedicated to Ito on the occasion of his eightieth birthday, contains papers which deal with recent developments of Ito's ideas:Professor Kiyosi Ito is well known as the creator of the modern theory of stochastic analysis. Although Ito first proposed his theory, now known as Ito's stochastic analysis or Ito's stochastic calculus, about fifty years ago, its value in both pure and applied

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ito.html (3 of 4) [2/16/2002 11:15:49 PM]

Ito

mathematics is becoming greater and greater. For almost all modern theories at the forefront of probability and related fields, Ito's analysis is indispensable as an essential instrument, and it will remain so in the future. For example, a basic formula, called the Ito formula, is well known and widely used in fields as diverse as physics and economics. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ito.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ito.html (4 of 4) [2/16/2002 11:15:49 PM]

Ivory

Sir James Ivory Born: 1765 in Dundee, Scotland Died: 21 Sept 1842 in London, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James Ivory was educated at St Andrews and Edinburgh and taught mathematics in Dundee. He became manager of a flax spinning company and then, in 1804, became professor of mathematics at the Royal Military College, Great Marlow (which became Sandhurst), together with William Wallace. Ill-health caused his resignation in 1816; and, for the rest of his life, he lived simply in London. Ivory was elected a Fellow of the Royal Society in 1815. His difficult personality led him to quarrel with many of the British scientific establishment. Nevertheless, he was awarded the Copley Medal (1814) and the Royal Medal (1838) of the Royal Society of London, he was honoured by many foreign scientific societies, and he received a knighthood in 1831. In 1838 Ivory gave the Bakerian lecture to the Royal Society, entitled On the theory of astronomical refractions. Ivory and Wallace were early supporters of the work of the French analysts, especially Lagrange and Laplace. Ivory's critical commentary of Laplace's Méchanique céleste was praised by Laplace. Ivory wrote several articles for encyclopaedias, including the influential Equations in Encyclopaedia Britannica. But his main research concerned applications of mathematics, most notably: the gravitational attraction of ellipsoids, the shape of self-gravitating rotating fluid bodies, the orbits of comets, and atmospheric refraction. His work on the ellipsoidal equilibrium configuration of self-gravitating fluids was an extension of that of Laplace, and it influenced the achievements of Jacobi and Liouville which followed. Ivory was knighted for his contributions to science. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ivory.html (1 of 2) [2/16/2002 11:15:50 PM]

Ivory

Honours awarded to James Ivory (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1815

Royal Society Copley Medal

Awarded 1814

Royal Society Royal Medal

Awarded 1826 and 1839

Royal Society Bakerian lecturer

1838

Fellow of the Royal Society of Edinburgh Other Web sites

Dundee Library (A biography of Ivory, and a list of the books in his library) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ivory.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ivory.html (2 of 2) [2/16/2002 11:15:50 PM]

Iwasawa

Kenkichi Iwasawa Born: 11 Sept 1917 in Shinshuku-mura (near Kiryu), Gumma Prefecture, Japan Died: 26 Oct 1998 in Tokyo, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kenkichi Iwasawa attended elementary school in the town of his birth but went to Tokyo for his high school studies which were at the Musashi High School. In 1937 he entered Tokyo University where he was taught by Skokichi Iyanaga and Zyoiti Suetuna. At this time Tokyo University had become a centre for the study of algebraic number theory as a result of Teiji Takagi's remarkable contributions. Takagi had retired in 1936, the year before Iwasawa began his studies, but his students Iyanaga and Suetuna were bringing to the university many ideas which they had developed during studies with the leading experts in Europe. Iwasawa graduated in 1940 and remained at Tokyo University to undertake graduate studies. He also was employed as an Assistant in the Mathematics Department. Although the great tradition in number theory at Tokyo inspired him to an interest in that topic, some of his early research contributions were to group theory. The Second World War disrupted life in Japan and essentially ended Suetuna's research career. Iyanaga did not fare much better. He wrote:... towards the end of the War, Tokyo and other Japanese cities were often bombarded and we had to find refuge in the countryside. Everyone was mobilised in one way or another for the War. Clearly Iwasawa found this a most difficult period in which to try to complete work for his doctorate. However, despite the difficulties he succeeded brilliantly, and was awarded the degree of Doctor of Science in 1945. It was not without a high cost, however, for after being awarded his doctorate he became seriously ill with pleurisy and this prevented him returning to the University of Tokyo until April 1947.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Iwasawa.html (1 of 4) [2/16/2002 11:15:52 PM]

Iwasawa

For a glimpse of the research that Iwasawa undertook at this time we look briefly at the paper On some types of topological groups which he published in the Annals of Mathematics in 1949. Iwasawa's results are related to Hibert's fifth problem which asks whether any locally Euclidean topological groups is necessarily a Lie group. In his 1949 paper Iwasawa gives what is now known as the 'Iwasawa decomposition' of a real semisimple Lie group. He gave many results concerning Lie groups, proving in particular that if a locally compact group G has a closed normal subgroup N such that N and G/N are Lie groups then G is a Lie group. In 1950 Iwasawa was invited to give an address at the International Congress of Mathematicians in Cambridge, Massachusetts. He then received an invitation to the Institute for Advanced Study at Princeton and he spent two years there from 1950 until 1952. Artin was at the Institute during Iwasawa's two years there and he was one of the main factors in changing the direction of Iwasawa's research to algebraic number theory. In 1952 Iwasawa published Theory of algebraic functions in Japanese. The book begins with an historical survey of the theory of algebraic functions of one variable, from analytical, algebraic geometrical, and algebro-arithmetical view points. Iwasawa then studies valuations, fields of algebraic functions giving definitions of prime divisors, ideles, valuation vectors and genus. A proof of the Riemann-Roch theorem is given, and the theory of Riemann surfaces and their topology is studied. It was Iwasawa's intention to return to Japan in 1952 after his visit to the Institute for Advanced Study but when he received the offer of a post of assistant professor at the Massachusetts Institute of Technology he decided to accept it. Coates, in [2], describes the fundamental ideas which Iwasawa introduced that have had such a fundamental impact on the development of mathematics in the second half of the 20th century. Iwasawa introduced:... a general method in arithmetical algebraic geometry, known today as Iwasawa theory, whose central goal is to seek analogues for algebraic varieties defined over number field of the techniques which have been so successfully applied to varieties defined over finite fields by H Hasse, A Weil, B Dwork, A Grothendieck, P Deligne, and others. ... The dominant theme of his work in number theory is his revolutionary idea that deep and previously inaccessible information about the arithmetic of a finite extension F of Q can be obtained by studying coarser questions about the arithmetic of certain infinite Galois towers of number fields lying above F. Iwasawa first lectured on his revolutionary ideas at the meeting of the American Mathematical Society in Seattle, Washington in 1956. The ideas were taken up immediately by Serre who saw their great potential and gave lectures to the Seminaire Bourbaki in Paris on Iwasawa theory. Iwasawa himself produced a series of deep papers throughout the 1960s which pushed his ideas much further. R Greenberg, who became a student of Iwasawa's in 1967 wrote:By the time that I became his student, Professor Iwasawa had developed his ideas considerably. The theory had become richer, and at the same time, more mysterious. Even though only a few mathematicians had studied the theory thoroughly at that time, there was a general feeling that the theory was very promising. When I look back at the developments that have taken place in the last three decades, that promise has been fulfilled even beyond expectations. In 1967 Iwasawa left MIT when he was offered the Henry Burchard Fine Chair of Mathematics at http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Iwasawa.html (2 of 4) [2/16/2002 11:15:52 PM]

Iwasawa

Princeton and it was not long after he arrived there that he took on Greenberg as a research student. We learn a lot about Iwasawa if we look at Greenberg's description of how Iwasawa supervised his studies:It was the tradition at Princeton to have tea every afternoon in fine Hall. This provided one of the best opportunities for graduate students to informally discuss mathematics with their professors. Professor Iwasawa usually came to the afternoon teas. It was then that he often suggested problems for me to think about and every few weeks he would ask me if I had made any progress on some of these problems. I recall that these problems seemed quite hard, but sometimes I was able to report some real progress, and then we would go to his office so that he could hear what I had done. He would help me push some of my ideas forward, but it was quite clear that he wanted me to accomplish as much as I could on my own. I often had the feeling that he was purposely not revealing everything that he knew about a specific problem. In the late 1960s Iwasawa made a conjecture for algebraic number fields which, in some sense, was the analogue of the relationship which Weil had found between the zeta function and the divisor class group of an algebraic function field. This conjecture became known as "the main conjecture on cyclotomic fields" and it remained one of the most outstanding conjectures in algebraic number theory until it was solved by Mazur and Wiles in 1984 using modular curves. Iwasawa remained as Henry Burchard Fine Professor of mathematics at Princeton until he retired in 1986. Then he returned to Tokyo where he spent his final years. He published Local class field theory in the year that he retired:This carefully written monograph presents a self-contained and concise account of the modern formal group-theoretic approach to local class field theory. Iwasawa was much honoured for his achievements. He received the Asahi Prize (1959), the Prize of the Academy of Japan (1962), the Cole Prize from the American Mathematical Society (1962), and the Fujiwara Prize (1979). The importance of his work is summed up by Coates [2]:... today it is no exaggeration to say that Iwasawa's ideas have played a pivotal role in many of the finest achievements of modern arithmetical algebraic geometry on such questions as the conjecture of B Birch and H Swinnerton-Dyer on elliptic curve; the conjecture of B Birch, J Tate, and S Lichtenbaum on the orders of the K-groups of the rings of integers of number fields; and the work of A Wiles on the modularity of elliptic curves and Fermat's Last Theorem. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Kenkichi Iwasawa (Click a link below for the full list of mathematicians honoured in this way) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Iwasawa.html (3 of 4) [2/16/2002 11:15:52 PM]

Iwasawa

AMS Cole Prize winner

1962

Other Web sites

1. R Greenberg 2. AMS

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Iwasawa.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Iwasawa.html (4 of 4) [2/16/2002 11:15:52 PM]

Iyanaga

Skokichi Iyanaga Born: 2 April 1906 in Tokyo, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Skokichi Iyanaga entered Tokyo University in 1926. He attended algebra lectures by Takagi in his first year. He wrote (see [1]):It happened to me sometimes that I could not immediately grasp [the lecture's] content, but I could always reconstruct it by taking enough time in reviewing my notes and it was the greatest pleasure to do this in the evening of the day I heard the lecture. In his second year of study Iyanaga took further courses by Takagi which developed group theory, representation theory, Galois theory, and algebraic number theory. Although it was this latter topic which would eventually attract Iyanaga, at this stage of his undergraduate career he was attracted to differential equations. He studied this topic in T Yosiye's course and became interested in conditions under which dy/ = f(x,y) has a unique solution. Yosiye [1]:dx ... encouraged the student to do original work and exempted him from undergoing an examination to obtain credit if he presented a good paper. Iyanaga posed the question of finding some intuitive or geometrical reasons why ... conditions assure the uniqueness of solution. Iyanaga's paper was so good that Yosiye had it published in the Japanese Journal of Mathematics. Surprisingly Iyanaga wrote a second paper while in his second undergraduate year. This came about through talking to the Departmental Assistant T Shimizu who discussed with him questions about power series. They wrote a joint paper which was published in the Proceedings of the Imperial Academy of Tokyo. Both Iyanaga's two papers appeared in print in 1928. In his third undergraduate year Iyanaga was allowed to take part in Takagi's seminar on class field theory. He studied Takagi's famous 1920 paper on extensions of algebraic number fields and also a paper

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Iyanaga.html (1 of 3) [2/16/2002 11:15:54 PM]

Iyanaga

by Hasse written in 1926 to give more details of Takagi's results. A question by Takagi led Iyanaga to prove a result which further led to his first paper on class field theory (and his third published paper while he was an undergraduate). Graduating in 1929, Iyanaga remained at Tokyo University where he worked under Takagi for his doctorate. It was an exciting time with increasing mathematical activity in algebra. For example K Shoda, who was a student of Takagi, had studied with Schur in Berlin and with Emmy Noether in Göttingen, and returned to Japan around the time that Iyanaga began his graduate studies. Most of the graduate students from Tokyo went to study in Germany but Iyanaga decided to go to France as well as Germany. He obtained a scholarship from the French government in 1931 and in that year he went to Hamburg where he studied with Artin. He writes [1]:I was very lucky to follow Artin's course on class field theory together with Chevalley. The International Congress of Mathematicians was held in Zurich in 1932. Takagi came over to Europe for this Congress (he was vice-President) and met up with Iyanaga who had gone from Hamburg to Zurich for the Congress. He writes:I met Takagi in Zurich and accompanied him when he visited Hamburg, Berlin, and Paris. Iyanaga went to Paris in 1932 where he met up with Chevalley whom he had got to know well while in Hamburg. He also met Henri Cartan, Dieudonné and André Weil, and he finally returned to Tokyo in 1934. He was appointed as Assistant Professor in 1934 and began his teaching career. He helped Takagi with his calculus course in 1935-36, the last time Takagi gave it before he retired. Iyanaga writes [1]:I published no research papers in the period 1935-39. I realise now that I was idle in doing research in these years because of the pressure of teaching and other business to which I was not accustomed. In fact Iyanaga did, for various reasons, little original research from this time on. Perhaps his last important idea which he published in his own field was prompted by an invitation to submit a paper to a collection being assembled to honour Furtwängler. Iyanaga managed to solve a question of Artin on generalising the principal ideal theorem and this was published in 1939. He wrote [1]:I received from Professor Furtwängler a letter thanking me for my contribution, which I cherished but lost together with my house which burned in a bombardment in the Spring of 1945. He did publish a number of papers, however, which arose through the various courses such as algebraic topology, functional analysis, and geometry, which he taught. Iyanaga was promoted to Professor at Tokyo University in 1942 but by this time Japan had entered World War II. He wrote:... towards the end of the War, Tokyo and other Japanese cities were often bombarded and we had to fine refuge in the countryside. Everyone was mobilised in one way or another for the War. And just after the War, the whole country was in distress, but on the other hand freed from military oppression, we began to make efforts to rebuild our science and culture in a better form. This kept us busy. I was particularly busy in editing textbooks from primary and secondary schools ... I continued to give courses and organise seminars, in which I used to discuss class field theory with my younger friends. Iyanaga joined the Science Council of Japan in 1947. He was invited by Stone to join the International mathematical union and in 1952 he became a member of the Executive Committee. In this role he helped http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Iyanaga.html (2 of 3) [2/16/2002 11:15:54 PM]

Iyanaga

organise the 1954 International Congress of Mathematicians in Amsterdam, which he attended. As a member of the Science Council of Japan he was the main organiser of an International Symposium on Algebraic Number Theory held in Japan in 1955. Iyanaga spent the year 1961-62 in Chicago [1]:In 1962 I received an invitation from the University of Chicago to join its Department of Mathematics for 1961-62. ... I was most happy to accept this invitation which gave me a chance to escape from administrative work which was making one busier every day. I met there Stone, MacLane, Schilling, Albert, Suzuki, and other old friends. In 1965 Iyanaga became Dean of the faculty of Science at Tokyo University. He held this post until he retired in 1967. However this was not the end of his career for he was visiting Professor at Nancy in France during 1967-68 and then Professor at Gakushuim University for 10 years from 1967 to 1977. One important role which Iyanaga had that we have not yet mentioned was that of President of the International Commission on Mathematical Instruction from 1957 to 1978. Iyanaga received honours such as being awarded the Rising Sun from Japan in 1976, being elected a member of the Japan Academy in 1978, and receiving the Order of Legion d'Honneur from France in 1980. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2001 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Iyanaga.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Iyanaga.html (3 of 3) [2/16/2002 11:15:54 PM]

Jabir_ibn_Aflah

al-Ishbili Abu Muhammad Jabir ibn Aflah Born: about 1100 in possibly Seville Died: about 1160 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jabir ibn Aflah is often known by the Latinised form of his name, namely Geber. Although not he was not in the first rank of Arabic mathematicians, he is important in the development of mathematics since his works were translated into Latin, and so became available to European mathematicians, whereas the work of some of the top rank Arabic mathematicians such as Abu'l-Wafa were not translated into Latin. Very little information is available regarding Jabir ibn Aflah's life. That he came from Seville is known from two sources. Firstly he is described as "al-Ishbili" in manuscripts containing his treatises; this means "from Seville". The other source gives us not only the information that he came from Seville, but also a good estimate for the period in which he lived. The information comes from Maimonides. Moses Maimonides, whose Arabic name was Abu 'Imran Musa ibn Maymun ibn 'Ubayd Allah, was a Jewish philosopher, jurist, and physician who was born in Córdoba in 1135. Among many important works he wrote The Guide of the Perplexed in Arabic in which he writes of:... ibn Aflah of Seville, whose son I have met ... Jabir ibn Aflah invented an observational instrument known as the torquetum, a mechanical device to transform between spherical coordinate systems (see [3] for further details). He also gave his name to a theorem in spherical trigonometry, and his criticisms of Ptolemy's Almagest are well known. These criticisms appears in Jabir ibn Aflah's most famous work Islah al-Majisti (Correction of the Almagest). One sees that ibn Aflah even puts his argument regarding errors made by Ptolemy into the title of the work. In [4] Lorch explains Jabir ibn Aflah's most famous criticism, namely Ptolemy's placement of Venus and Mercury below the Sun. Ptolemy claimed that these planets could never be on a line between an observer on Earth and the sun., but ibn Aflah states that this is an error, and that Venus and Mercury are above the Sun. It is a little difficult to establish the exact degree of originality of Correction of the Almagest. Certainly its general approach bears considerable similarity to the work of Abu'l-Wafa, but it may not be based on Abu'l-Wafa's work. Both may be based on the work of Thabit ibn Qurra, or the work of ibn Aflah, Abu'l-Wafa, and Thabit ibn Qurra may all be based on some still unknown source. The influence of ibn Aflah is quite remarkable. In [4] his influence on astronomers in both the East and West is studied. In particular the author looks at his influence on the Persian astronomer Qutb al-Din http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jabir_ibn_Aflah.html (1 of 2) [2/16/2002 11:15:55 PM]

Jabir_ibn_Aflah

al-Shirazi, who was a pupil of Nasir al-Din al-Tusi; on the Hispano-Arabian philosopher ibn Rushd, who is often known as Averroes, was born in Córdoba in 1126 and integrated Islamic traditions and Greek thought; and on Levi ben Gerson (sometimes known as Gersonides). One of ibn Aflah's more infamous influences was on Regiomontanus who copied large parts of ibn Aflah's work in the fourth book of his publication De triangulis. Regiomontanus did not acknowledge that ibn Aflah was the source of the material and this caused Cardan to strongly criticise Regiomontanus. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1100 to 1300

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Jabir_ibn_Aflah.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jabir_ibn_Aflah.html (2 of 2) [2/16/2002 11:15:55 PM]

Jacobi

Carl Gustav Jacob Jacobi Born: 10 Dec 1804 in Potsdam, Prussia (now Germany) Died: 18 Feb 1851 in Berlin, Germany

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Carl Jacobi came from a Jewish family but he was given the French style name Jacques Simon at birth. His father, Simon Jacobi, was a banker and his family were prosperous. Carl was the second son of the family, the eldest being Moritz Jacobi who eventually became a famous physicist. Moritz Jacobi has an entry in his own right in [1]. There was a sister, Therese Jacobi, and a third brother, Eduard Jacobi, who was younger than Carl. Eduard did not pursue an academic career, but followed instead his father's profession as a banker. Jacobi's early education was given by an uncle on his mother's side, and then, just before his twelfth birthday, Jacobi entered the Gymnasium in Potsdam. He had been well taught by his uncle and he had remarkable talents so in 1817, while still in his first year of schooling, he was put into the final year class. This meant that by the end of the academic year 1816-17 he was still only 12 years old yet he had reached the necessary standard to enter university. The University of Berlin, however, did not accept students below the age of 16, so Jacobi had to remain in the same class at the Gymnasium in Potsdam until the spring of 1821. Of course, Jacobi pressed on with his academic studies despite remaining in the same class at school. He received the highest awards for Latin, Greek and history but it was the study of mathematics which he took furthest. By the time Jacobi left school he had read advanced mathematics texts such as Euler's Introductio in analysin infinitorum and had been undertaking research on his own attempting to solve quintic equations by radicals. Jacobi entered the University of Berlin in 1821 still unsure which topic he would concentrate on. He

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html (1 of 6) [2/16/2002 11:15:57 PM]

Jacobi

attended courses in philosophy, classics and mathematics for two years before realising that he had to make a definite decision between these subjects. He chose mathematics, but this did not mean that he could attend high level courses in mathematics for at this time the standard of university education in mathematics in Germany was rather poor. As he had done at the Gymnasium, Jacobi had to study on his own reading the works of Lagrange and other leading mathematicians. By the end of academic year 1823-24 Jacobi had passed the examinations necessary for him to be able to teach mathematics, Greek, and Latin in secondary schools. Now, of course, one might have expected him to have problems obtaining a teaching position since, as we noted at the beginning of this article, he was Jewish. His brilliance appears to have been sufficient to allow this hurdle to be overcome for, in 1825, he was offered a teaching post at the Joachimsthalsche Gymnasium, one of the leading schools in Berlin. He had submitted his doctoral dissertation to the University of Berlin even before he received the offer of the teaching post, and he was allowed to move quickly to work on his habilitation thesis. Jacobi presented a paper concerning iterated functions to the Academy of Sciences in Berlin in 1825. However, the referees did not consider the results worth publishing and indeed the paper was not published by the Berlin Academy of Sciences. The paper was published eventually, for in 1961 it was published with a commentary in [6]. Biermann, the author of [6], quotes the opinions of the original referees and criticises them strongly. Although this was not the best start for the young Jacobi, it did not hold him back for long and his publication record over the following years would be quite remarkable for both the number and quality of the works. Around 1825 Jacobi changed from the Jewish faith to become a Christian which now made university teaching possible for him. By the academic year 1825-26 he was teaching at the University of Berlin. However prospects in Berlin were not good so, after taking advice from colleagues, Jacobi moved to the University of Königsberg arriving there in May 1826. There he joined Franz Neumann, who had also received his doctorate from Berlin in 1825, and Bessel who was the professor of astronomy at Königsberg. Jacobi had already made major discoveries in number theory before arriving in Königsberg. He now wrote to Gauss to tell him of the results on cubic residues which he had obtained, having been inspired by Gauss's results on quadratic and biquadratic residues. Gauss was impressed, so much so that he wrote to Bessel to obtain more information about the young Jacobi. But Jacobi also had remarkable new ideas about elliptic functions (as Abel did quite independently and at much the same time). On 5 August 1827 Jacobi wrote to Legendre who was the leading expert on the topic and this letter, together with 22 others between Jacobi and Legendre, is given in [4]. Legendre immediately realised that Jacobi had made fundamental advances in his favourite topic. One would have to say that Legendre reacted extremely well to the realisation that his position as the leading expert on elliptic functions had changed overnight with the new theory being developed not only by Jacobi, but also by Abel. Jacobi's promotion to associate professor on 28 December 1827 was mainly due to the praise heaped on him by Legendre. In a letter, sent to Jacobi on 9 February 1828, Legendre wrote:It gives me great satisfaction to see two young mathematicians such as you and [Abel] cultivate with such success a branch of analysis which for such a long time has been my favourite topic of study but which had not been received in my own country as well as it

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html (2 of 6) [2/16/2002 11:15:57 PM]

Jacobi

deserves. By your works you place yourselves in the ranks of the best analysts of our era. In 1829 Jacobi met Legendre and other French mathematicians such as Fourier and Poisson when he made a visit to Paris in the summer vacation. On the journey to Paris he had visited Gauss in Göttingen. Jacobi's fundamental work on the theory of elliptic functions, which had so impressed Legendre, was based on four theta functions. His paper Fundamenta nova theoria functionum ellipticarum published in 1829, together with its later supplements, made fundamental contributions to this theory of elliptic functions. However, despite Jacobi's brilliant contributions to elliptic functions he did not have the field to himself. As we have noted above, Abel was also making fundamental contributions and to some extent a competition had developed between the two. Legendre expressed this clearly in a letter he wrote to Jacobi early in 1829:You proceed so rapidly, gentlemen, in all these wonderful speculations that it is nearly impossible to follow you - particularly for an old man ... I congratulate myself that I have lived long enough to witness these magnanimous contests between two young equally strong athletes, who turn their efforts to the profit of the science whose limits they push back further and further. A few weeks after Legendre wrote this letter Abel died. On 11 September 1831 Jacobi married Marie Schwinck then, a few months later in May 1832, he was promoted to full professor after being subjected to a four hour disputation in Latin. Jacobi's reputation as an excellent teacher attracted many students. He introduced the seminar method to teach students the latest advances in mathematics. Jacobi had a major impact on these students and all others around him [1]:Such were Jacobi's forceful personality and sweeping enthusiasm that none of his gifted students could escape his spell: they were drawn into his sphere of thought, and soon represented a "school". C W Borchardt, E Heine, L O Hesse, F J Richelot, J Rosenhain, and P L von Seidel belonged to this circle; they contributed much to the dissemination not only of Jacobi's mathematical creations but also the new research-oriented attitude in university instruction. The triad of Bessel, Jacobi, and Franz Neumann thus became the nucleus of a revival of mathematics at German universities. In 1833 Jacobi's older brother Moritz joined him in Königsberg where he set himself up as an architect. During the two years Moritz spent there he became more interested in physics and left Königsberg in 1835 when he was appointed to the chair of civil engineering at Dorpat. In 1834 Jacobi received some work from Kummer who was at this time a teacher in a Gymnasium in Liegnitz. The article [20] describes how Jacobi immediately recognised Kummer's mathematical talents. Kummer had made advances beyond what Jacobi had achieved on third-order differential equations and Jacobi wrote to his brother Moritz in 1836 describing how Kummer had managed to solve problems which had defeated him. In 1834 Jacobi proved that if a single-valued function of one variable is doubly periodic then the ratio of the periods is imaginary. This result prompted much further work in this area, in particular by Liouville and Cauchy. Jacobi carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics. He also worked on determinants and studied the functional determinant now called the Jacobian. Jacobi was not the first to study the functional determinant which http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html (3 of 6) [2/16/2002 11:15:57 PM]

Jacobi

now bears his name, it appears first in a 1815 paper of Cauchy. However Jacobi wrote a long memoir De determinantibus functionalibus in 1841 devoted to this determinant. He proved, among many other things, that if a set of n functions in n variables are functionally related then the Jacobian is identically zero, while if the functions are independent the Jacobian cannot be identically zero. In [15] McCleary describes one of Jacobi's most impressive results:One of the prettiest results in the global theory of curves is a theorem of Jacobi (1842): The spherical image of the normal directions along a closed differentiable curve in space divides the unit sphere into regions of equal area. The statement of this theorem is an afterthought to a paper in which Jacobi responds to the published correction by Thomas Clausen (1842) of an earlier paper by Jacobi (1836). In July 1842 Jacobi and Bessel attended the meeting of the British Association for the Advancement of Science in Manchester as representatives of Prussia. Jacobi's wife accompanied the two mathematicians. They returned to Königsberg via Paris where Jacobi lectured at the Académie des Sciences. In the following year Jacobi became unwell and diabetes was diagnosed. He was advised by his doctor to spend time in Italy where the climate would help him recover. However, Jacobi was not a wealthy man and Dirichlet, after visiting Jacobi and discovering his plight, wrote to Alexander von Humboldt asking him to help obtain some financial assistance for Jacobi from Friedrich Wilhelm IV. We should make a small digression to say why Jacobi was not a wealthy man despite having inherited a small fortune from his wealthy father. A severe business depression throughout Prussia (in fact it was a Europe wide depression), had led to a bankruptcy in which Jacobi had lost all his money. Let us now return to Dirichlet and Alexander von Humboldt's attempts to help obtain support for Jacobi's trip to Italy. Jacobi had frequently corresponded with Alexander von Humboldt. The correspondence began in 1828 but only after 1839 did they correspond regularly and the 44 surviving letters between the two men make fascinating reading (see [5] and also [7]). Dirichlet's request to Friedrich Wilhelm IV, supported strongly by Alexander von Humboldt, was successful and Jacobi received a grant to allow him to spend time in Italy. He set off for Italy with Borchardt and Dirichlet and, after stopping in several towns and attending a mathematical meeting in Lucca, they arrived in Rome on 16 November 1843. Schläfli and Steiner were also with them, Schläfli being their interpreter. The climate in Italy did indeed help Jacobi to recover and he began to publish again, his health having prevented him working for some time before this. In fact Jacobi's interests in mathematics were very wide and while in Rome he took the opportunity to satisfy his interest in the history of mathematics working on manuscripts of Diophantus's Arithmetica which were kept in the Vatican. Although his health had improved it was felt that the climate of Königsberg was too extreme for him to return there, so a dispensation was obtained from Friedrich Wilhelm IV to allow him to transfer to Berlin. He was given a supplement to his salary to help offset the higher costs of living in Berlin, and also to help him with his medical expenses. He was in Berlin by June 1844 and although his health prevented him from giving frequent lecture courses, he did lecture at the University of Berlin. In [21] a lecture course which he gave in 1847-48 is discussed by Pulte:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html (4 of 6) [2/16/2002 11:15:57 PM]

Jacobi

Jacobi in his lectures on analytical mechanics (Berlin, 1847 - 1848) ... gave a detailed and critical discussion of Lagrange's mechanics. Lagrange's view that mechanics could be pursued as an axiomatic-deductive science forms the centre of Jacobi's criticism and is rejected on mathematical and philosophical grounds. ... Jacobi's criticism is motivated by a changed evaluation of the role of mathematics in the empirical sciences. In [22] Pulte shows that Jacobi only came to hold these views on analytical mechanics only later in his life, for earlier he had ignored the physical interpretation of mechanics in favour of a purely axiomatic and mathematical approach. By 1848 conditions were bad in the German Confederation. Unemployment and crop failures had led to discontent and disturbances. The news that Louis-Philippe had been overthrown by an uprising in Paris in February 1848 led to revolutions in many states and fighting in Berlin. Republican and socialist feelings meant that the monarchy was in trouble. Jacobi made a political speech in the Constitutional Club in Berlin which managed to upset both the monarchists and the republicans. As a consequence Jacobi's request to be allowed to join the staff of the University of Berlin was refused by the Prussian government. By the summer of 1849 the revolution was completely defeated. The Prussian government, still feeling aggrieved at Jacobi, took away the supplement to his salary which allowed him to live in Berlin. He had to move, and chose the small town of Gotha. He lived there with his family and a few months later accepted a chair at the University of Vienna. The Prussian government suddenly realised what they would lose if they forced Jacobi to leave Prussia, so they made concessions which meant that Jacobi could lecture at the University of Berlin while his family remained in Gotha. It was not a good deal for Jacobi and the fact that he accepted it means that he was strongly attached to his own country. Jacobi planned to spend the university vacations with his family and he spent the summer of 1850 with them in Gotha. In January 1851 he contracted influenza, then he contracted smallpox before he had regained his strength. He died a few days after contracting smallpox. Scriba, in [1], compares Jacobi with Euler:Jacobi and Euler were kindred spirits in the way they created their mathematics. Both were prolific writers and even more prolific calculators; both drew a great deal of insight from immense algorithmical work; both laboured in many fields of mathematics (Euler, in this respect, greatly surpassed Jacobi); and both at any moment could draw from the vast armoury of mathematical methods just those weapons which would promise the best results in the attack of a given problem. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles)

Some Quotations (9)

A Poster of Carl Jacobi

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html (5 of 6) [2/16/2002 11:15:57 PM]

Jacobi

Cross-references to History Topics

1. Orbits and gravitation 2. General relativity 3. Matrices and determinants

Other references in MacTutor

1. Chronology: 1820 to 1830 2. Chronology: 1840 to 1850

Honours awarded to Carl Jacobi (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1833

Lunar features

Crater Jacobi

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Jacobi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jacobi.html (6 of 6) [2/16/2002 11:15:57 PM]

Jacobson

Nathan Jacobson Born: 8 Sept 1910 in Warsaw, Poland Died: 5 Dec 1999

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nathan Jacobson's family moved from Poland to the United States when he was seven years old. The family first settled in Alabama, then moved to Mississippi and finally Georgia. Jacobson entered the University of Alabama and, in 1930, was awarded a B.A. He then went to Princeton where he studied for his doctorate, which was awarded in 1934. Jacobson's career began with relatively short periods in a number of universities throughout the United States. He spent 1934-35 at the Institute for Advanced Studies at Princeton. Then the following session 1935-36 he spent at Bryn Mawr. Jacobson had heard Emmy Noether lecture in Princeton but she died in the spring of 1935, before Jacobson joined the staff at Bryn Mawr where Emmy Noether had been teaching. Session 1936-37 was spent by Jacobson at the University of Chicago where Dickson was working. Then the University of North Carolina was home to him for a number of years. Appointed in 1937 he became an assistant professor there in 1938, then in 1941 an associate professor. In 1943 Jacobson was appointed to Johns Hopkins University where Zariski was on the Faculty. After four years at Johns Hopkins University he moved to Yale in 1947, two years later being promoted to full professor. The professorship at Yale became his permanent position until he retired in 1981. Jacobson is well known for his outstanding contributions to ring theory. He worked on rings satisfying conditions of the type xn = x in 1945. Herstein was to carry on study of rings satisfying this type of condition.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jacobson.html (1 of 3) [2/16/2002 11:15:59 PM]

Jacobson

Jacobson discovered a deep structure theory for rings and has given his name to the Jacobson radical, the intersection of the maximal ideals of a ring. He also made very substantial contributions to nonassociative algebras, in particular Lie algebras and (Pascual) Jordan algebras. The collection of Jacobson's algebra books are beautifully written, presenting deep results which are accessible to advanced undergraduate and postgraduate students. They include The theory of rings (1943) and the three volume work Lectures in abstract algebra (1951-64) covering basic concepts, linear algebra and the theory of fields and Galois theory. He wrote two books which rapidly became classics on Lie algebras, Lie algebras (1962) and Exceptional Lie algebras (1971). On Jordan algebras he wrote Structure and representations of Jordan algebras (1968) and another major work on algebra was PI-algebras : an introduction (1975). Many honours have been bestowed on this outstanding algebraist. Besides the award of a Guggenheim Fellowship, he has been elected to the National Academy of Sciences and the American Academy of Arts and Sciences. He also has served as president of the American Mathematical Society. He was made an honorary member of the London Mathematical Society in 1972. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country Other references in MacTutor

Chronology: 1960 to 1970

Honours awarded to Nathan Jacobson (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1971 - 1972

AMS Colloquium Lecturer

1955

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jacobson.html (2 of 3) [2/16/2002 11:15:59 PM]

Jacobson

JOC/EFR November 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Jacobson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jacobson.html (3 of 3) [2/16/2002 11:15:59 PM]

Jagannatha

Jagannatha Samrat Born: about 1690 in Amber (now Jaipur), India Died: about 1750 in India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jagannatha had Jai Singh Sawai as his patron. Jai Singh Sawai was the ruler of Amber, now Jaipur, in eastern Rajasthan. He began his rule in 1699 and by clever use of tax rights on land that was rented by the state to an individual person he became the most important ruler in the region. His financial success let him finance the scholarly works of people such as Jagannatha. It is worth noting that Jai Singh's importance was recognised by Amber which was then called Jaipur in his honour. Jai Singh ruled Amber throughout the period in which Jagannatha was producing his scientific work. He realised that the health of the country required Indian culture and science to be revitalised and returned to its position of leading importance which it had possessed. So Jai Singh employed Jagannatha to make Sanskrit translations of the important Greek scientific works which at that time were only available in Arabic translations. Jagannatha translated Euclid's Elements from the Arabic translation by Nasir al-Din al-Tusi made nearly 500 years earlier. His Sanskrit version was called Rekhaganita and it was completed by 1727. We know this date since a copy was made by a scribe and he dated the start of his work as 1727. Ptolemy's Almagest had been one of the works which Arabic scientists had studied intently and, in 1247, al-Tusi wrote Tahrir al-Majisti (Commentary on the Almagest) in which he introduced various trigonometrical techniques to calculate tables of sines. Jagannatha translated al-Tusi's Arabic version but he did more than this for he included in the same work, which he called Siddhantasamrat, his own comments on related work of other Arabic mathematical astronomers such as Ulugh Beg and al-Kashi. It is clear from Jagannatha's work that he is working as one of a group of mathematicians and astronomers gathered by Jai Singh in his scheme to bring the best in scientific ideas from outside India to reinvigorate the scientific scene in India. In [3] Gupta looks at the history of the result sin( /10) = ( 5 - 1)/4 in Indian mathematics. The result appears for the first time in the work of Bhaskara II, but there were a number of interesting proofs of the result by later Indian mathematicians. One of the proofs presented by Gupta in [3] was by Jagannatha who gave a proof which was essentially geometric in nature but, interestingly, contained an analytic procedure in terms of trigonometric and algebraic steps. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jagannatha.html (1 of 2) [2/16/2002 11:16:00 PM]

Jagannatha

Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Jagannatha.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jagannatha.html (2 of 2) [2/16/2002 11:16:00 PM]

James

Ioan Mackenzie James Born: 23 May 1928

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ioan James was a Foundation Scholar at St Paul's School. From there he won an Open Scholarship to the University of Oxford where he studied mathematics at Queen's College in the topology school of Henry Whitehead. After completing his education at Oxford, James went to the United States for the year 1954-55. He did this having been awarded a Commonwealth Fund Fellowship which enabled him to spend time at Princeton University, the University of California at Berkeley and the Institute for Advanced Study at Princeton. Returning to England, James was Tapp Research Fellow at Gonville and Caius College in Cambridge in 1956 before returning to the University of Oxford. Back at Oxford in 1957, James was appointed Reader in Pure Mathematics, a post which he held until 1969. From 1959 until 1969 he was a Senior Research Fellow at St John's College. In 1970 James was appointed to the famous chair of Savilian Professor of Geometry. The chair was founded by Savile in 1619 and first occupied by Briggs. James was elected a Fellow of New College Oxford in 1970 when he became Savilian Professor of Geometry and he continues to hold the fellowship. He retired from the Savilian Professor of Geometry in 1995 and became professor emeritus. James has done wide ranging work in topology, particularly in homotopy theory. His first publication was in 1953, followed by four publications in 1954 which were all written jointly with Henry Whitehead. These papers were on fibre spaces and the homotopy theory of sphere bundles over spheres. James then published on suspensions, on multiplication on spheres, on cup-products, and, another publication with Henry Whitehead, Homology with zero coefficients. He then wrote many papers on homotopy groups and Stiefel manifolds.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/James.html (1 of 3) [2/16/2002 11:16:01 PM]

James

The mathematical works of J H C Whitehead appeared in four volumes in 1962 and 1963 edited by James. These volumes covered Henry Whitehead's work in differential geometry, complexes and manifolds, homotopy theory, and algebraic and classical topology. In 1976 James published The topology of Stiefel manifolds which was based on his lecture notes. A reviewer wrote:There are a number of fascinating problems concerning the Stiefel manifolds which are of importance in the geometric applications of homotopy theory, and in homotopy theory itself. These lecture notes present, and solve, some of these problems (many of which the author has worked on), bringing in the necessary tools (e.g. K-theory characteristic classes, J-theory, Samelson products) as needed. general topology and homotopy theory (1984), another book by James which was based on his lectures, is described as follows:In this monograph, based on a set of sixteen lectures to students, the author expounds certain parts of general topology which are particulary relevant to homotopy theory. His book is reasonably self-contained. It is tightly written and elegant. ... This is a beautifully written and much needed book, both as a text and as a reference. It will probably become a classic. In 1987 James published Topological and uniform spaces which was written for advanced undergraduates or new graduate students. Fibrewise topology (1988) is a treatise on general topology, uniform spaces, and homotopy theory from the point of view of fibres. Introduction to uniform spaces (1989) is again based on lectures given by James at the University of Oxford. James published Handbook of algebraic topology in 1995. In the late 1950s Henry Whitehead approached Robert Maxwell, the chairman of Pergamon Press, to start a new journal Topology although Whitehead never lived to see the first part appear. James became an editor of Topology in 1962 and he has continued in that role ever since. On the 25th anniversary of the founding of the journal, James gave an address in which he related much of the early history of the publication with which he has been so closely associated. The address has been published, see [2]. James has received many honours for his major contributions to mathematics. In 1968 he was elected a fellow of the Royal Society of London. Other honours have been the Junior Berwick Prize from the London Mathematical Society in 1959 and the Senior Whitehead Prize from that Society in 1978. He served the London Mathematical Society in a number of ways, in particular as treasurer from 1969 to 1979 and as the 62nd president of the Society from 1984 to 1986. Among other honours he has received have been an honorary fellowship from St John's College Oxford in 1988, an honorary professorship from the University of Wales in 1989 and an honorary doctorate from the University of Aberdeen in 1993. James has been a governor of St Paul's School and St Paul's Girls School since 1970. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/James.html (2 of 3) [2/16/2002 11:16:01 PM]

James

List of References (4 books/articles) Honours awarded to Ioan James (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1968

London Maths Society President

1984 - 1986

LMS Berwick Prize winner

1959

Savilian Professor of Geometry

1969

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/James.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/James.html (3 of 3) [2/16/2002 11:16:01 PM]

Janiszewski

Zygmunt Janiszewski Born: 12 June 1888 in Warsaw, Poland Died: 3 Jan 1920 in Lvov, Poland

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Zygmunt Janiszewski's mother was Julia Szulc-Chojnicka. His father, Czeslaw Janiszewski, was a graduate of the University of Warsaw and was an important person in finance, being the director of the Société du Crédit Municipal in Warsaw. After attending secondary school in Warsaw, Janiszewski decided to go abroad for a university education. This was the typical route for Poles at this time, and we shall give some background to explain why this was so. The first thing to note is that when Janiszewski graduated from secondary school in 1907, Poland did not formally exist. Poland had been partitioned in 1772 and the south was called Galicia and under Austrian control. Russia controlled much of the rest of the country and in the years prior to Janiszewski's birth there had been strong moves by Russia to make "Vistula Land", as it was called, be dominated by Russian culture. In a policy implemented between 1869 and 1874, all secondary schooling was in Russian. Warsaw only had a Russian language university after the University of Warsaw was closed by the Russian regime in 1869. Galicia, although under Austrian control, retained Polish culture and was often where Poles from "Vistula Land" went for their education. Janiszewski, however, decided to go to Zurich for his university education. In Zurich he was part of a group of Polish students which he organised, showing organisational skills which would become evident of a wider scale later in his life. After studying mathematics for a year at Zurich and spending a short time in Munich, Janiszewski went to Göttingen to continue his studies. Having chosen excellent centres of mathematical research at which to study he was taught by many outstanding mathematicians including Burkhardt, Hilbert, Minkowski, and Zermelo

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Janiszewski.html (1 of 4) [2/16/2002 11:16:03 PM]

Janiszewski

Janiszewski next went to one of the other leading centres of mathematics in the world, namely Paris. There his professors included Goursat, Hadamard, Lebesgue, Emile Picard, and Poincaré. Lebesgue supervised Janiszewski's doctoral studies in topology and in 1911 he submitted his thesis Sur les continus irréductibles entre deux points. The examining board for his doctoral thesis was an extremely powerful group of mathematicians, Poincaré, Lebesgue and Fréchet. Janiszewski returned to Congress Poland (the Russian controlled region) and taught for a while in Warsaw; not at the University of Warsaw for this had been closed in 1869 as we noted above. In 1913 he published a paper of fundamental importance On cutting the plane by continua. This paper both won for him the J Mianowski Foundation prize and also qualified him to teach at the University of Lvov. There he taught courses on analytic functions and functional calculus. On 11 July 1913 he delivered his important lecture on the axiom of choice which was published in 1916 as On realism and idealism in mathematics (the realists did mathematics without the axiom of choice, while the idealists accepted the axiom). He continued to teach at Lvov until the outbreak of World War I when he enlisted in the Polish legion. The position of Poland when war broke out was rather complicated. On 16 August 1914, the Austrian government had allowed the organisation of the Polish legion which Janiszewski joined. He joined because he believed that the legion was fighting for Polish independence. However, the Austrians aim was to incorporate Congress Poland into Galicia, the region in which Lvov was situated. Russia tried to win Polish support, particularly in Galicia, by promising the Poles autonomy. By the end of 1914 Russian forces controlled almost all of Galicia. The Central Powers (Germany and Austria- Hungary) recaptured Galicia and large parts of Congress Poland. A German governor general was installed in Warsaw and a new Kingdom of Poland was declared on 5 November 1916. The Polish volunteer troops in the Polish legion, however, were not satisfied with the Austrian-German declaration which left them with a tiny Poland compared to pre-1772 country. When the men in the Polish legion were required to take an oath of loyalty to the Austrian government this became too much for loyal Poles like Janiszewski. He left the legion and hid under the false name of Zygmunt Wicherkiewicz in Boiska near Zwolen. From Boiska he moved on to Ewin, near Wloszczowa, where he directed a refuge for homeless children. The University of Warsaw had become Polish again in November 1915 and before Janiszewski left Ewin he already had links to the university. He now became a professor at the University of Warsaw. Kuratowski attended seminars given by Janiszewski in Warsaw before the end of the war. He writes in [3]:As early as 1917 [Janiszewski and Mazurkiewicz] were conducting a topology seminar, presumably the first in that new, exuberantly developing field. The meeting of that seminar, taken up to a large extent with sometimes quite vehement discussions between Janiszewski and Mazurkiewicz, were a real intellectual treat for the participants. At the end of the war it was Janiszewski who was the main force in the remarkable creation of one of the strongest schools of mathematics in the world. It is all the more remarkable given the position in which Poland found itself at the end of the war. Kuratowski explains in [3] the importance of Janiszewski's vision :In the first volume of [Polish Science: Its Needs, Organisation, and Development], which http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Janiszewski.html (2 of 4) [2/16/2002 11:16:03 PM]

Janiszewski

appeared in 1918, Janiszewski published an article "On the needs of mathematics in Poland", which with amazing clarity and precision presented a blueprint for Polish mathematics. Janiszewski started out with the assumption that Polish mathematicians do not have to be satisfied with the role of followers and customers of foreign mathematical centres but can achieve an independent position for Polish mathematics. One of the best ways of achieving this goal, suggested Janiszewski, was for groups of mathematicians to concentrate on relatively narrow fields in which Polish mathematicians had common interests and more importantly - had already made internationally important contributions. These areas included set theory, and the foundations of mathematics. In fact these were exactly the areas in which Janiszewski himself had already made internationally important contributions. In addition to set theory (which at that time included parts of what we call topology today) Janiszewski produced important results in the foundations of mathematics and other parts of topology. Janiszewski saw that mathematics was one scientific subject where Poland could rapidly reach a leading role, whereas other sciences required a much larger financial investment which Poland was not then in a position to give. He wrote in his article (see for example [2]):It is true that a mathematician does not require laboratories of complex and expensive auxiliary devices. His mathematical contributions, relatively few because of the little time he had to apply himself to research in his short life, are none the less of major importance. His doctoral thesis contains important results on closure properties. He gave a topological characterisation of the plane which simplified considerably the Jordan curve theorem. His paper to the International Mathematical Congress in Cambridge in England in 1912 was of major importance for it sketched the definition of a curve without arcs, so that it had no homeomorphic images of a segment of a straight line. His mathematical contributions are considered in more detail in [1]. Janiszewski played a major role in the setting up of the journal Fundamenta Mathematicae and Kuratowski recalls that it was Janiszewski who proposed the name of the journal in 1919. The first volume appeared in 1920 and, although the intention was for a truly international journal, Janiszewski had quite deliberately decided to make the first volume contain papers by Polish authors only. He wrote (see for example [3]):... it is my intention to present, if possible, all Polish mathematicians working in the field of set theory, to which the journal is devoted. It was an immeasurable loss to mathematics when Janiszewski died during an influenza epidemic. The epidemic spread throughout Europe and many people died. Kuratowski, who knew Janiszewski well, writes in [3]:Janiszewski was an unusual personality, combining great creative talent, organisational talent, faith in the scholar's mission, ardent patriotism, a noble character and kind-heartedness. Receptive to ideas of progress and social justice, he underwent deep ideological changes. In his own words, he "was turning more and more to the left in politics" ... Knaster tells us in [1] that Janiszewski:... donated for public education all the money he received for scientific prizes and an http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Janiszewski.html (3 of 4) [2/16/2002 11:16:03 PM]

Janiszewski

inheritance from his father. Before he died he willed his possessions for social works, his body for medical research, and his cranium for craniological study, desiring to be "useful after his death". Dickstein wrote a commemorative address after Janiszewski's death. Although it repeats much of what we have already written about Janiszewski it is worth ending this biography by quoting part of the address (see for example [3]):Enthusiasm and strong will characterised Janiszewski not only in his scientific work, but in his life generally. His active participation in the Legions, his refusal to take an oath which was not compatible with his patriotic conscience, his work in the field of education, when at a most difficult time he entered that field as an enlightened and wise worker, free of any prejudice and partiality and ardently keen only to propagate light and truth - these facts prove that in the heart of a mathematicians seemingly detached from active life there glowed the purest emotions of affection and self-denial. If we also mention that, having very moderate needs himself, he dispensed all the means at his disposal to educate young talents, and that he bequeathed the property that he had inherited from his parents for educational purposes, and in particular for the education of outstanding individuals, then we may indeed exclaim from the bottom of our hearts that the memory of that life, devoted to the cause and interrupted so early, lives on in its results and deeds and will remain treasured and living for us, the witnesses of his work, and for generations to come. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Zygmunt Janiszewski

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Janiszewski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Janiszewski.html (4 of 4) [2/16/2002 11:16:03 PM]

Janovskaja

Sof'ja Aleksandrovna Janovskaja Born: 31 Jan 1896 in Pruzhany, Poland (now Kobrin, Belarus) Died: 24 Oct 1966 in Moscow, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sof'ja Aleksandrovna Neimark's family moved to Odessa when she was young and she was educated in classics and mathematics in Odessa. She entered the Higher School for Women in Odessa in 1915. When the Russian Revolution arrived Neimark became politically active serving the Red Army and as editor for the Kommunist newspaper in Odessa. In 1923 she returned to her studies attending seminars at Moscow State University. By 1931 she was a professor there, and, four years later, she received her doctorate. Janovskaja worked on the philosophy of mathematics and logic. She argued against the writings of Frege on philosophy. Her work in mathematical logic was important in the development of the subject in Soviet Union. The history of mathematics was another topic which attracted Janovskaja and she published work on Egyptian mathematics, Zeno of Elea's paradoxes, Rolle's criticisms of the calculus, Descartes's geometry and Lobachevsky's work on non-euclidean geometry. Among the honours which were given to Janovskaja was the Order of Lenin in 1951. In 1959 she became the first Head of the new Department of Mathematical Logic at Moscow State University. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Janovskaja.html (1 of 2) [2/16/2002 11:16:05 PM]

Janovskaja

List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Janovskaja.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Janovskaja.html (2 of 2) [2/16/2002 11:16:05 PM]

Jarnik

Vojtech Jarnik Born: 22 Dec 1897 in Prague, Bohemia (now Czech Republic) Died: 22 Sept 1970 in Prague, Czechoslovakia

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vojtech Jarnik studied at the Charles University in Prague. After graduating he was appointed as an assistant at the Charles University. In 1923 he went to the University of Göttingen to work with Landau. Returning to his post in Prague in 1924, he was again to visit Göttingen in session 1927/28 when he worked with Landau. Jarnik was appointed to a chair of mathematics at the Charles University of Prague in 1928. He held this post until he retired in 1968 having taught at the University for a total of 47 years. The main topic of Jarnik's research was number theory. One of the problems which he worked on extensively was related to the Gauss circle problem. Let R(r) denote the number of points (m, n) with m, n Z contained in a circle centre O, radius r. There exists a constant C and a number k with | R(r) - r2 | < Crk. Let d be the minimal value of k. Gauss proved in 1837 that d 1 if c was the smallest number such that the subgroup had an element moving c points. His finiteness theorem showed that for a given c there are only finitely many primitive groups with class c other than the symmetric and alternating groups. Generalising a result of Fuchs on linear differential equations, Jordan was led to study the finite subgroups of the general linear group of n n matrices over the complex numbers. Although there are infinite families of such finite subgroups, Jordan found that they were of a very specific group theoretic structure which he was able to describe. Another generalisation, this time of work by Hermite on quadratic forms with integral coefficients, led n matrices of determinant 1 over the complex numbers Jordan to consider the special linear group of n acting of the vector space of complex polynomials in n indeterminates of degree m. Jordan is best remembered today among analysts and topologists for his proof that a simply closed curve divides a plane into exactly two regions, now called the Jordan curve theorem. It was only his increased understanding of mathematical rigour which made him realise that a proof of such a result was necessary. He also originated the concept of functions of bounded variation and is known especially for his definition of the length of a curve. These concepts appears in his Cours d'analyse de l'Ecole Polytechnique first published in three volumes between 1882 and 1887. The second edition appeared in 1893 while the Jordan curve theorem appeared in the third edition of the text which appeared between 1909 and 1915. Of course by 1882, when the first volume was published, Jordan was lecturing at the Ecole Polytechnique and the book was written as a text for the students there. In some respects this is a little strange since it is a rigorous analysis text built on top of the attempts to put the topic on a firm foundation begun by Cauchy and given considerable impetus by Weierstrass. However, the courses at the Ecole http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jordan.html (3 of 5) [2/16/2002 11:16:32 PM]

Jordan

Polytechnique were supposed to train students to become civil and military engineers and this does not seem to be the approach which one would take trying to teach applications of the calculus to engineers. There had been a tradition of rigorous analysis at the Ecole Polytechnique begun, of course, by Cauchy himself. Jordan was aware that his work was at a level that would be somewhat inappropriate for engineering students for he once said to Lebesgue that he called it "Ecole Polytechnique analysis course" since:... one puts that on the cover to please the publisher... Gispert-Chambaz in [7] contrasts the way that topological concepts are treated by Jordan in the first and second editions of the book. In the first addition most of the topological concepts are dealt with in an supplement to Volume 3. However between the editions Jordan had taught more advanced courses on analysis at the College de France and this may have influenced him to put set topology right up front in the second edition. In this respect one can see the second edition as setting a tone for analysis textbooks which continues today. Among Jordan's many contributions to analysis we should also mention his generalisation of the criteria for the convergence of a Fourier series. The Journal de Mathématiques Pure et Appliquées was a leading mathematical journal and played a very significant part in the development of mathematics throughout the 19th century. It was usually known as the Journal de Liouville since Liouville had founded the journal in 1836. Liouville died in 1882 and in 1885 Jordan became editor of the Journal, a role he kept for over 35 years until his death. In 1912 Jordan retired from his positions. The final years of his life were saddened, however, because of World War I which began in 1914. Between 1914 and 1916 three of his six sons were killed in the war. Of his three remaining sons, Camille was a government minister, Edouard was a professor of history at the Sorbonne, and the third son was an engineer. Among the honours given to Jordan was his election to the Académie des Sciences on 4 April 1881. On 12 July 1890 he became an officer of the Légion d'honneur. He was the Honorary President of the International Congress of Mathematicians at Strasbourg in September 1920. Finally we should note some rather confusing facts. Although given Jordan's work on matrices and the fact that the Jordan normal form is named after him, the Gauss-Jordan pivoting elimination method for solving the matrix equation Ax= b is not. The Jordan of Gauss-Jordan is Wilhelm Jordan (1842 to 1899) who applied the method to finding squared errors to work on surveying. Jordan algebras are called after the German physicist and mathematician Pascual Jordan (1902 to 1980).

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles)

A Quotation

A Poster of Camille Jordan

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jordan.html (4 of 5) [2/16/2002 11:16:32 PM]

Jordan

Cross-references to History Topics

1. Matrices and determinants 2. A history of group theory 3. Topology enters mathematics

Honours awarded to Camille Jordan (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1919

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Jordan.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jordan.html (5 of 5) [2/16/2002 11:16:32 PM]

Jordanus

Jordanus Nemorarius Born: 1225 in Borgentreich (near Warburg), Germany Died: 1260 in At sea Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jordanus was a contemporary of Fibonacci and was the first to correctly formulate the law of the inclined plane. He wrote several books on arithmetic, algebra, geometry and astronomy. Jordanus also used letters to replace numbers and was able to state general algebraic theorems but this early use of algebraic notation was not used by subsequent writers. In astronomy he used letters to denote the magnitudes of stars (nor unrelated to his use of letters for algebraic notation). He wrote a treatise on mathematical astronomy called Planisphaerium as well as Tractatus de Sphaera. Jordanus visited the Holy Land and, on the return journey, he lost his life at sea. Article by: J J O'Connor and E F Robertson List of References (14 books/articles) Mathematicians born in the same country Some pages from publications

A page from De Numeris Datis (1879)

Other references in MacTutor

Chronology: 1100 to 1300

Other Web sites

The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jordanus.html (1 of 2) [2/16/2002 11:16:32 PM]

Jordanus

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Jordanus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jordanus.html (2 of 2) [2/16/2002 11:16:32 PM]

Jourdain

Philip Edward Bertrand Jourdain Born: 16 Oct 1879 in England Died: 1 Oct 1921 in England Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Philip Jourdain suffered from severe disabilities all his life. He was already crippled when he went up to Cambridge in 1898. His undergraduate years were very difficult and he had great successes and great disappointments. He did very poorly in his degree and had to settle for only a pass degree. However he was awarded the Allen studentship for research in 1904. Jourdain worked in mathematical logic. He wrote a number of articles explaining and evaluating Cantor's set theory between 1906 and 1913 under the title Development of the Theory of Transfinite Numbers. In his work he clarified a remark by Russell and formulated precisely the paradox of the largest ordinal. In 1913 Jourdain proposed the card paradox. This was a card on one side of which was printed: The sentence on the other side of this card is TRUE. On the other side of the card the sentence read: The sentence on the other side of this card is FALSE. Jourdain also applied logic to physics in papers such as On some Points in the Foundation of Mathematical Physics (1908). Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jourdain.html (1 of 2) [2/16/2002 11:16:33 PM]

Jourdain

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Jourdain.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jourdain.html (2 of 2) [2/16/2002 11:16:33 PM]

Juel

Sophus Christian Juel Born: 25 Jan 1855 in Randers, Denmark Died: 24 Jan 1935 in Copenhagen, Denmark

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Christian Juel's father was a judge but Christian never knew him for he died before Christian was one year old. He was brought up in the country and attended the Realschule in Svendborg, a coastal town in southern Funen Island, Denmark. When he was fifteen years old he went to Copenhagen and, in the following year, he entered the Technical University of Copenhagen. He studied at the Technical University from 1871 until 1875, then he decided that technical studies were not to his liking and that he would prefer to study pure science in general and pure mathematics in particular. In January 1876 Juel took the entrance examinations for the University of Copenhagen and completed his first degree there in 1879. Continuing with his doctoral studies at the University of Copenhagen he received his doctorate in 1885 for a dissertation on geometry. Juel had spent sixteen years completing his university education and so, despite starting young, was 30 years old before he received his doctorate. Juel taught at the Polytechnic Institute in Copenhagen from 1894, being promoted to a full professorship three years later. He also sometimes lecturer at the University of Copenhagen. Juel married a daughter of the professor of mathematics in Copenhagen, Thorvald N Thiele. He made substantial contributions to projective geometry and wrote an important book on the topic. His approach is similar to that of von Staudt but goes beyond von Staudt's in places. Although he was not alone in making these improvements, since Corrado Segre also proved similar results, but there is no doubt that Juel's results were obtained independently of Corrado Segre.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Juel.html (1 of 2) [2/16/2002 11:16:35 PM]

Juel

Juel also worked on the theory of finite equal polyhedra and on oval surfaces. In 1914 he introduced the concept of an elementary curve [1]:... which is in the projective plane without straight-line segments and has the topological image of a circle and a tangent at every point. Outside these points a convex arc can be described on each side. Thus an elementary curve consists of an infinite number of convex arcs passing smoothly one into another. Although these ideas are clever, Juel did not treat them with care and his writings on this topic, although interesting, are less than precise and leave something to be desired. Juel served mathematics in other important ways. He was an editor of Matematisk Tidsskrift for over 25 years from 1889 until 1915. He also wrote textbooks for school level mathematics as well as for university level mathematics courses. Among the honours which he received was being elected to honorary membership of the Mathematical Association in 1925, then receiving an honorary degree from the University of Oslo four years later. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Juel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Juel.html (2 of 2) [2/16/2002 11:16:35 PM]

Julia

Gaston Maurice Julia Born: 3 Feb 1893 in Sidi Bel Abbès, Algeria Died: 19 March 1978 in Paris, France

Click the picture above to see five larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

When only 25 when Gaston Julia published his 199 page masterpiece Mémoire sur l'iteration des fonctions rationelles which made him famous in the mathematics centres of his days. As a soldier in the First World War, Julia had been severely wounded in an attack on the French front designed to celebrate the Kaiser's birthday. Many on both sides were wounded including Julia who lost his nose and had to wear a leather strap across his face for the rest of his life. Between several painful operations he carried on his mathematical researches in hospital. Later he became a distinguished professor at the Ecole Polytechnique in Paris. In 1918 Julia published a beautiful paper Mémoire sur l'itération des fonctions rationnelles, Journal de Math. Pure et Appl. 8 (1918), 47-245, concerning the iteration of a rational function f. Julia gave a precise description of the set J(f) of those z in C for which the nth iterate fn(z) stays bounded as n tends to infinity. It received the Grand Prix de l'Académie des Sciences. Seminars were organised in Berlin in 1925 to study his work and participants included Brauer, Hopf and Reidemeister. H Cremer produced an essay on his work which included the first visualisation of a Julia set. Although he was famous in the 1920's, his work was essentially forgotten until B Mandelbrot brought it back to prominence in the 1970's through his fundamental computer experiments. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Julia.html (1 of 2) [2/16/2002 11:16:37 PM]

Julia

List of References (5 books/articles) A Poster of Gaston Julia

Mathematicians born in the same country

Other references in MacTutor

An example of a Julia set

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Julia.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Julia.html (2 of 2) [2/16/2002 11:16:37 PM]

Jungius

Joachim Jungius Born: 22 Oct 1587 in Lübeck, Germany Died: 23 Sept 1657 in Hamburg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joachim Jungius attended school in Lübeck until 1605. From 1606 until 1608 he attended the University of Rostock, where he studied metaphysics. On leaving Rostock he entered the University of Giessen where he later received his M.A. In 1609 he was appointed professor of mathematics at Giessen and he held this post until 1614 when he began to become interested in medicine. In 1616 he returned to the University of Rostock to study medicine. Three years later he received a medical degree from the University of Padua. Thereafter Jungius held chairs of mathematics at the University of Rostock from 1624 to 1625 and again 1626 to 1628. For one year in 1625 he held the chair of medicine at the University of Helmstedt. In 1629 he moved to Hamburg where he was professor of natural science until 1640. As well as mathematics, Jungius was interested in natural science and the philosophy of science. In mathematics Jungius proved that the catenary is not a parabola (Galileo assumed it was). He was one of the first to use exponents to represent powers and he used mathematics as a model for the natural sciences. The Logica Hamburgensis (1638) of Jungius presented late medieval theories and techniques of logic. He discussed valid oblique cases of arguments that do not fit into simpler forms of inference. For example The square of an even number is even; 6 is even; therefore, the square of 6 is even. The oblique case of an even number had to be put into the subject position so that standard arguments http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jungius.html (1 of 2) [2/16/2002 11:16:39 PM]

Jungius

could be used. Aristotle had also dealt with this type of logical argument. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Cross-references to Famous Curves

catenary

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Jungius.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jungius.html (2 of 2) [2/16/2002 11:16:39 PM]

Jyesthadeva

Jyesthadeva Born: about 1500 in Kerala, India Died: about 1575 in Kerala, India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jyesthadeva lived on the southwest coast of India in the district of Kerala. He belonged to the Kerala school of mathematics built on the work of Madhava, Nilakantha Somayaji, Paramesvara and others. Jyesthadeva wrote a famous text Yuktibhasa which he wrote in Malayalam, the regional language of Kerala. The work is a survey of Kerala mathematics and, very unusually for an Indian mathematical text, it contains proofs of the theorems and gives derivations of the rules it contains. It is one of the main astronomical and mathematical texts produced by the Kerala school. The work was based mainly on the Tantrasamgraha of Nilakantha. The Yuktibhasa is a major treatise, half on astronomy and half on mathematics, written in 1501. The Tantrasamgraha on which it is based consists of 432 Sanskrit verses divided into 8 chapters, and it covers various aspects of Indian astronomy. It is based on the epicyclic and eccentric models of planetary motion. The first two chapters deal with the motions and longitudes of the planets. The third chapter Treatise on shadow deals with various problems related with the sun's position on the celestial sphere, including the relationships of its expressions in the three systems of coordinates, namely ecliptic, equatorial and horizontal coordinates. The fourth and fifth chapters are Treatise on the lunar eclipse and On the solar eclipse and these two chapters treat various aspects of the eclipses of the sun and the moon. The sixth chapter is On vyatipata and deals with the complete deviation of the longitudes of the sun and the moon. The seventh chapter On visibility computation discusses the rising and setting of the moon and planets. The final chapter On elevation of the lunar cusps examines the size of the part of the moon which is illuminated by the sun and gives a graphical representation of it. The Yuktibhasa is very important in terms of the mathematics Jyesthadeva presents. In particular he presents results discovered by Madhava and the treatise is an important source of the remarkable mathematical theorems which Madhava discovered. Written in about 1550, Jyesthadeva's commentary contained proofs of the earlier results by Madhava and Nilakantha which these earlier authors did not give. In [4] Gupta gives a translation of the text and this is also given in [2] and a number of other sources. Jyesthadeva describes Madhava's series as follows:The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jyesthadeva.html (1 of 2) [2/16/2002 11:16:40 PM]

Jyesthadeva

cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude. This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion. To see how this description of the series fits with Gregory's series for arctan(x) see the biography of Madhava. Other mathematical results presented by Jyesthadeva include topics studied by earlier Indian mathematicians such as integer solutions of systems of first degree equation solved by the kuttaka method, and rules of finding the sines and the cosines of the sum and difference of two angles. Not only does the mathematics anticipate work by European mathematicians a century later, but the planetary theory presented by Jyesthadeva is similar to that adopted by Tycho Brahe. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Jyesthadeva.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Jyesthadeva.html (2 of 2) [2/16/2002 11:16:40 PM]

Kac

Mark Kac Born: 3 Aug 1914 in Krzemieniec, Poland, Russian Empire Died: 26 Oct 1984 in California, USA

Click the picture above to see three larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mark Kac was born into a Jewish family in a Russian part of Poland. His date of birth was given as 3 August on his birth certificate and this was the day on which, throughout his life, he celebrated his birthday. However, this was the date according to the Julian calendar that was used in czarist Russia at the time of his birth, since Poland at that time was part of the Russian Empire. He claimed that his actual date of birth was 16 August. In 1915 Europe was in the midst of World War I and Kac's family was evacuated further east into Russia. Mark Kac's father was an academic with a degree in philosophy from the University of Leipzig and a degree from Moscow in history and philology. During this time he made money tutoring in the family's one-room apartment. One of the topics he tutored was geometry and Mark, although only five years old, became fascinated by what his father was teaching and he persuaded his father to teach him some geometry. Describing this introduction to mathematics in [4], [5] he said:... at that time my father despaired because at the same time I was exceedingly bad learning multiplication tables. That one could know how to prove theorems of elementary geometry without knowing how much seven times nine was seemed more than slightly strange. In 1921 the family returned to Poland and Mark was taught by a French governess. By 1925, when he was eleven, he had learnt Russian, French from the governess and some Hebrew from his father but, despite being Polish, he did not speak Polish. Only in 1925, when he entered the Lycee of Krzemieniec, did he learn Polish. Kac studied Latin and Greek at school as well as mathematics, physics and chemistry. He was equally interested in mathematics and physics but his mother wanted him to study engineering at university. He eventually chose to study mathematics and he recounted how that came about ([4], [5]):... in the summer of 1930 I became obsessed with the problem of solving cubic equations. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kac.html (1 of 4) [2/16/2002 11:16:42 PM]

Kac

Now, I knew the answer, which Cardan had published in 1545, but what I could not find was a derivation that satisfied my need for understanding. When I announced that I was going to write my own derivation, my father offered me a reward of five Polish zlotys (a large sum and no doubt the measure of his scepticism). I spent the days, and some of the nights, of that summer feverishly filling reams of paper with formulas. Never have I worked harder. Well, one morning, there it was -- Cardan's formula on the page. My father paid up without a word, and that fall my mathematics teacher submitted the manuscript to Mlody Matematyk (The Young Mathematician). ... When my gymnasium principal, Mr Rusiecki, heard that I was to study engineering, he said, "No, you must study mathematics; you have clearly a gift for it". Kac entered the Jan Kasimir University of Lvov where he was taught by Steinhaus. After graduating he remained at the University of Lvov and he was awarded his doctorate in June 1937. Kac had decided, before his doctorate was awarded, to try to leave Poland ([4], [5]):... I wanted to get out of Poland very badly. I did not know the disaster was going to be of the magnitude it turned out to be, but it was obvious that Europe, especially eastern Europe, was not the place to stay. His first attempt was to try to obtain an academic post in Britain. He recounted an application he made to Imperial College in London, despite not speaking English at that time ([4], [5]):... in Nature there would be ads of various positions. Most positions required being a British subject, but one of them ...was an ad for a junior lecturer in the Imperial College of Science and Technology at the salary of 150 pounds per annum.... Even then that was not very much money, and I thought that no self-respecting British subject would ever want to apply for a job like this. So I spoke to my teacher, Hugo Steinhaus, and asked whether it would be a good idea to apply, and he, partly in jest, partly seriously, said, "Well, let's estimate your chances of getting the job. I would say it is 1 in 5000. Let's multiply this by the annual salary. If this comes out to be more than the cost of the postage stamp, then you should not apply. If it is less than the cost of the stamp, you should." Well, it turned out to be a little bit less than the cost of the stamp, so I wrote. I got a letter from them later on saying that unfortunately the job was filled, so there had been after all a British subject who wanted the 150 pounds per annum. Kac then tried to go to the United States, helped by Steinhaus who was not only his teacher but also by this time his friend. This was a difficult period, however, since there was a large number of refugees from Germany, particularly people of Jewish descent, who wanted to make a new life in Britain or the United States. While attempting to go to the United States, Kac worked for an insurance company in Lvov. He eventually was given a six month visitor's visa for the United States but was required to buy a return ticket to ensure that he left after six months. After being awarded his doctorate in 1937, he applied for a scholarship to go to Johns Hopkins University but it was not awarded to him. Kac did not give up, however, and the following year he applied again, and this time he was successful. He described how the failure to get the scholarship the first time was an extraordinary piece of good luck ([4], [5]):... that saved my life because if I had gotten it a year earlier, I would have been compelled to go back. This way the war caught me in this country and literally saved my life. I was at Johns Hopkins when the war started, and then I got an offer to Cornell, where I spent twenty-two very happy years.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kac.html (2 of 4) [2/16/2002 11:16:42 PM]

Kac

Certainly Kac was fortunate to have been able to leave Poland at that time. As he writes in [2]:In less than a year the world exploded and much of my part of it was consumed by flames. Millions, including my parents and my brother, were murdered by the Germans, and many dissappeared without a trace in the vastness of the Soviet Union. Kac served at Cornell as an instructor from 1939 to 1943 (the year he became a US citizen), assistant professor from 1943 to 1947 when he was promoted to full professor. When Kac left Cornell in 1961 he went to Rockefeller University, in New York City. His chair in Cornell was filled by Mitchell Feigenbaum, one of the founders of the modern subject of chaos. Kac spent twenty years at Rockefeller University, then decided to spend the rest of his career:.... where there is more sun and less ice ... so he went to the University of Southern California where he spent the rest of his career. Kac has pioneered the modern development of mathematical probability, in particular its applications to statistical physics. The method of quantization now in use involves the Feynman-Kac path integral, named after Richard Feynman and Mark Kac. He published a classic text Statistical Independence in Probability, Analysis and Number Theory in 1959. To many Kac will be remembered best for a paper he wrote for the American Mathematical Monthly in 1966. This is the famous paper Can One Hear the Shape of a Drum? and Kac received the Chauvenet Prize from the Mathematical Association of America in 1968 for the :most outstanding expository article on a mathematical topic by a member of the Association. In addition to the Chauvenet Prize (which in fact he won on two separate occasions), Kac was awarded the George David Birkhoff Prize in Applied Mathematics in 1978. The citation for the fifth award of this prize, which is a joint prize of the American Mathematical Society and the Society for Industrial and Applied Mathematics, was made:To Mark Kac for his important contributions to statistical mechanics and to probability theory and its applications. When asked what pleased him most about the scientific work he had done, Kac replied ([4], [5]):... I was always interested in problems rather than in theories. In retrospect the thing which I am happiest about, and it was done in cooperation with Erdös ... was the introduction of probabilistic methods in number theory. To put it poetically, primes play a game of chance. And also some of the work in mathematical physics. I am amused by things. Can one hear the shape of a drum? I also have a certain component of journalism in me, you see: I like a good headline, and why not? And I am pleased with the sort of thing I did in trying to understand a little bit deeper the theory of phase transitions. I am fascinated, also, with mathematical problems, and particularly ... the role of dimensionality: why certain things happen in 'from three dimensions on' and some others don't. I always feel that that is where the interface, will you pardon the expression, of nature and mathematics is deepest. To know why only certain things observed in nature can happen in the space of a certain dimensionality. Whatever helps understand this riddle is significant, I am pleased that I, in a small way, did something with it. Raimi described Kac, in particular his English accent, in [12]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kac.html (3 of 4) [2/16/2002 11:16:42 PM]

Kac

He spoke his five or six languages beautifully. He never lost his Polish accent - he rolled his Rs mercilessly - but, more important, he never lost that brio, that sparkling juxtaposition of surprising phrases and constructions that made his speech such a delight to anyone with an ounce of poetry in his soul. How much of this was due to his Polish and how much to his own curious outlook on the world one cannot say. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles)

Some Quotations (3)

Mathematicians born in the same country Honours awarded to Mark Kac (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1981

AMS Gibbs Lecturer

1967

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kac.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kac.html (4 of 4) [2/16/2002 11:16:42 PM]

Kaestner

Abraham Gotthelf Kaestner Born: 27 Sept 1719 in Leipzig, Germany Died: 20 June 1800 in Göttingen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Abraham Gotthelf Kästner's father was a university professor of jurisprudence. He hoped that his son would follow in his footsteps and Kästner started out on a university course with the intention of studying the philosophy of law, but he soon found other topics more to his liking and began to concentrate more on philosophy, physics and mathematics. Kästner wrote his habilitation thesis at the University of Leipzig, and was awarded the qualification which allowed him to teach there in 1739. He taught at the University of Leipzig as a Privatdozent until 1746 when he was appointed as an extraordinary professor. Ten years later, in 1756, he was appointed as professor of mathematics and physics at Göttingen where he succeeded to Segner's chair. He was an excellent expositor of mathematics although it is reported that Gauss did not bother to go to his lectures as he found them too elementary. However he did influence Gauss, in particular with his interest in Euclid's parallel postulate. The paper [5] examines the contribution of Kästner who Sinaceur describes as an important German mathematician of the mid- and late 18th century. Kästner is not famed for original research but rather he was involved in compiling encyclopedias and in writing textbooks. He was concerned with philosophical questions in mathematics and other areas such as logic. However Kästner was quite unenthusiastic about logic, but this is not surprising for a mathematician of this period who was interested in geometry. Despite this he was interested in the philosophy of mathematics and he wrote widely, in long volumes, about the applications of mathematics to optics, dynamics and astronomy. Perhaps his two most famous works, both in four volumes, were Mathematische Anfangsgründe and

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaestner.html (1 of 3) [2/16/2002 11:16:44 PM]

Kaestner

Geschichte der Mathematik (1796-1800). This latter work was intended to form the basis for a history of mathematics. For example Volume 2, published in 1787, is considered one of the standard sources on the history of mathematical optics. Perhaps the most important feature of Kästner's contributions was his interest in the parallel postulate which indirectly influenced Bolyai and Lobachevsky too. Kästner taught Bolyai's father and J M C Bartels, one of Kästner's students, taught Lobachevsky. Folta writes in [3] about Kästner's work on geometry:Kästner [is] among the mathematicians of the 18th century whose broad interests compelled [him] to concern [himself] with the principal problems of geometry. [His] results included new features that more precisely formulated the traditional interpretation of elementary geometry. In fact, [he] began the conscious attempt to make a precise axiomatisation of the fundamental concepts. Kästner, in spite of his rather great inclination for Euclid's Elements, based his version of the axiomatics of geometry in his Kompendium on other principles (e.g., on motions) and attempted both to seize on other fundamental properties (continuity, ordering) and to determine the selection of the parallel axiom as a foundation. The article [1] gives us a few more details of Kästner's life, which is described in detail in [2] which is an autobiography. Goe writes in [1]:Kästner is also known in German literature, notably for his epigrams. He was a devout Lutheran. Kästner married twice and had a daughter by his second wife. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Honours awarded to Abraham Gotthelf Kästner (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1789

Lunar features

Crater Kastner

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaestner.html (2 of 3) [2/16/2002 11:16:44 PM]

Kaestner

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Kaestner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaestner.html (3 of 3) [2/16/2002 11:16:44 PM]

Kagan

Benjamin Fedorovich Kagan Born: 10 March 1869 in Shavli, Kovno (now Kaunas, Lithuania) Died: 8 May 1953 in Moscow, USSR Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Benjamin Fedorovich Kagan's father Fedor Kagan was a clerk. Kagan entered Novorossysky University in Odessa in 1887. He was expelled in 1889 for participating in the Democratic Students Movement. He was sent to Ekaterinoslav in south-central Ukraine. The Soviets renamed Ekaterinoslav in 1926 calling it Dnepropetrovsk. In 1892 Kagan received a degree from Kiev University, then in 1895 he was awarded a Master's Degree by St Petersburg University. He taught at Novorossysky from 1897 until 1922, becoming a professor there in 1917. However this was not his only post during this period for he taught higher education classes for women and also taught at the local Jewish school. As if this was not enough, Kagan also edited the Journal of Experimental Physics and Elementary Mathematics from 1902 until 1917 and he was the director of a large publisher of scientific materials Mathesis. In 1922 he went to Moscow when the Department of Differential Geometry was founded at Moscow State University. Kagan was the first Head of Department and he founded an important School of Differential Geometry. He retained his interest in publishing, however, and he was the director of the science department of a state publisher for ten years. He also directed the department of mathematics and natural sciences of the Great Soviet Encyclopaedia. In 1927, Kagan organised a seminar on vector and tensor analysis. He founded a publication associated with this seminar Transactions of the seminar on Vector and Tensor Analysis with its applications to Geometry, Mechanics and Physics in 1933. In 1934 Kagan and other members of his School organised an International conference on differential geometry which took place at Moscow University. Kagan worked on the foundations of geometry and his first work was on Lobachevsky's geometry. In 1902 he proposed axioms and definitions very different from Hilbert. Kagan studied tensor differential geometry after going to Moscow because of an interest in relativity. Kagan wrote a history of non-euclidean geometry and also a detailed biography of Lobachevsky. He edited Lobachevsky's complete works which appeared in five volumes between 1946 and 1951. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kagan.html (1 of 2) [2/16/2002 11:16:45 PM]

Kagan

Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kagan.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kagan.html (2 of 2) [2/16/2002 11:16:45 PM]

Kakutani

Shizuo Kakutani Born: 28 Aug 1911 in Osaka, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Shizuo Kakutani's father, Kakujiro Kakutani, was a lawyer. Shizuo was the youngest of the two sons in the family, the eldest being Seiichi who was eight years older. Seiichi studied physics at Kyoto University and it was through him that Shizuo was first introduced to mathematics. When he was about nine years old his elder brother, on one of his frequent visits back to his home while he studied at university, would explain mathematical ideas to him. Shizuo was fascinated and was enthusiastic to learn more mathematics. However two factors conspired to make this impossible. The first problem was that Kakujiro Kakutani had made the decision that one of his two sons would follow him into law and take over his practice in due course. Clearly Seiichi was training to be a physicist and not studying law so Shizuo would have to be the one to follow his father. The mathematics lessons from Seiichi were tragically cut short when he died of typhoid fever at the age of twenty. After completing his middle school, Shizuo entered Konan High School in Kobe to prepare for his university studies. At this stage he had to choose between arts subjects or sciences and his father gave him no choice since studying law at university required him to graduate from high school which qualifications in literature and arts. By the time Shizuo graduated, his father relented seeing that his son was so keen to study mathematics at university. However, Shizuo was now not qualified to enter a mathematics course at either Tokyo University of Kyoto University since these had absolute rules regarding entry qualifications. There was one possible route for Shizuo which was to enter Tohoko University in Sendai. This did not prevent those without a science qualification entering a mathematics course but it gave preference to those the scientific training. Kakutani applied but there were only fifteen places, and seventeen applicants. Of the seventeen, exactly fifteen had the science qualification from high school so it looked like an easy task to decide to admit those and to turn down Kakutani. However, after due consideration it http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kakutani.html (1 of 3) [2/16/2002 11:16:47 PM]

Kakutani

was decided to admit all seventeen applicants and Kakutani had scrape through. At Tohoko University Kakutani was introduced to the theory of analytic functions. He read various classic texts including those of Stone and Banach and by the time of his graduation at the end of the three year course he had a good foundation in modern analysis. He was appointed as a teaching assistant at Osaka University in 1934 where he collaborated with K Yosida on a paper on Nevanlinna theory. Hajian and Ito write in [2]:During his years at Osaka University Shizuo Kakutani had already established himself as a research mathematician by publishing a number of papers in functional analysis and ergodic theory, and the 1937 paper in the Japanese Journal on Mathematics on Riemann surfaces. It was this work which was later to become the main part of his doctoral dissertation. it was also this paper that caught the attention of Weyl ... Indeed on the strength of this work Weyl invited Kakutani in 1940 to spend two years at the Institute for Advanced Study at Princeton. Kakutani not only took great interest in the work of Weyl's group at Princeton but also the group of mathematicians working with von Neumann on measure theory and ergodic theory. He met many other young mathematicians at Princeton who would influence him such as Ambrose, Halmos, Doob, and Erdös. Kakutani also made many visits during which he met mathematicians such as Garrett Birkhoff, G D Birkhoff, Stone, Wiener, and Hille. In December 1941 with Kakutani still studying at Princeton, war broke out between the United States and Japan with the entry of the U.S.A. into the Second World War. Of course this put Kakutani in a difficult position for he was now a guest in a country at war with his own. He was able to remain at Princeton, however, to complete his visit and he returned to Japan in the summer of 1942. On his return Kakutani accepted the appointment as assistant professor at Osaka University. He also continued his collaboration with Yosida, who by this time was at Nagoya University, and began a new collaboration with Yosida's colleague Kiyosi Ito at Nagoya. The later war years were particularly difficult ones in Japan and many Japanese mathematicians failed to keep their research going through the difficulties of these times. Kakutani, however, managed to continue to produce a stream of papers containing highly original ideas. In 1948 Kakutani was again invited to the Institute for Advanced Study at Princeton. In the summer of that 1949 he worked at the University of Illinois, then later that year accepted the offer of an appointment at Yale University. On one of his frequent visits to New York, Kakutani met Kay Uchida and they were married in 1952; they had one daughter Michiko. Kakutani contributed to several areas of mathematics. Among the areas on which he has written papers we must mention: complex analysis, topological groups, fixed point theorems, Banach spaces and Hilbert spaces, Markov processes, measure theory, flows, Brownian motion, and ergodic theory. Kakutani was to remain at Yale until he retired in 1982. In June of that year he received the Academy Award and the Imperial Award of the Academy of Japan. As to Kakutani's personality, Hajian and Ito write in [2]:Professor Kakutani is a gentleman and a scholar of the old school. His mild manner, gentle graciousness, and total dedication to mathematics leave an indelible impression on all who http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kakutani.html (2 of 3) [2/16/2002 11:16:47 PM]

Kakutani

have gotten to know him. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kakutani.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kakutani.html (3 of 3) [2/16/2002 11:16:47 PM]

Kalmar

László Kalmár Born: 27 March 1905 in Edde (N of Kaposvar), Hungary Died: 2 Aug 1976 in Mátraháza, Hungary

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

László Kalmár's father was Zsigmond Kalmár and his mother was Rósa Krausz. Zsigmond Kalmár was a bailiff on an estate in Transdanubia in northwestern Hungary, the estate being located about 30 km from Lake Balaton. Zsigmond Kalmár died while László was a young boy and Rósa Kalmár then moved to Budapest with László. Kalmár began his secondary schooling in Budapest during World War I. He proved an outstanding pupil and was soon working on his own far in advance of his fellow pupils in mathematics. By the time he was thirteen he had read advanced texts such as ones by Cesàro. However, while still at secondary school Kalmár's mother also died so by the time he entered the University of Budapest in 1922 he was an orphan. The tragedy that had hit his young life did not hold him back in his studies of mathematics and physics, however. Again he was in a different league in mathematics from his fellow students who treated him with the reverence which his brilliance deserved. As a student at the University of Budapest Kalmár had teachers who were leading mathematicians so he received teaching which brought him to the cutting edge of research. Kürschák and Fejér were among these quality teachers. He also had fellow students who also played a role in his development, particularly Rósa Péter. Kalmár would later provide important help to Péter in her career. Kalmár graduated in 1927 and the direction of his work was influenced when he visited Göttingen in 1929. There he became interested in mathematical logic and this was a field in which he was to make major contributions. After the award of his doctorate in Budapest he took up a post in Szeged. After World War I the Hungarian government had been forced to sign the Treaty of Trianon on 4 June, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kalmar.html (1 of 3) [2/16/2002 11:16:49 PM]

Kalmar

1920. Hungary was left with less than one third of the land that had previously been Hungary. Romania, Czechoslovakia and Yugoslavia all took over large areas but Austria, Poland and Italy also gained land from Hungary. Kolozsvár, which was the site of one of Hungary's major universities, was no longer in Hungary after the Treaty of Trianon but rather it was in Romania and was then renamed Cluj, so the Hungarian University there had to move to within the new Hungarian borders. It moved to Szeged in 1920, where there had previously been no university. The appointment of Haar and Riesz resulted in Szeged rapidly becoming a major mathematical centre. At Szeged Kalmár acted at first as a research assistant to both Haar and Riesz. In 1933 Kalmár married Erzsébet Arvay and they had four children. Kalmár was appointed a full professor at Szeged in 1947. He founded at Szeged the first chair for the Foundations of Mathematics and Computer Science anf then became the first to occupy the chair. This was not all he founded at Szeged, for he also set up the Cybernetic Laboratory and the Research Group for Mathematical Logic and Automata Theory. He worked on mathematical logic solving certain cases of the decision problem for the first order predicate calculus, simplified results of Bernays, and worked on ideas of Post, Gödel and Church. He was acknowledged as the leader of Hungarian mathematical logic. Kalmár was also involved in theoretical computer science and promoted the development of computer science and the use of computers in Hungary. His special fields of interest in computer science included programming languages, automatic error correction, non-numerical applications of computers and the connection between computer science and mathematical logic. For his outstanding contributions to mathematics and computer science Kalmár received many honours. He was elected to the Hungarian Academy of Sciences in 1949, becoming a full member in 1961. He received awards such as the Kossuth Prize in 1950 and the Hungarian State Prize in 1975. These were the highest honours which his country could bestow on him. He was also elected to honorary positions such as honorary president of the Janos Bolyai Mathematical Society and the John von Neumann Society for Computer Science. Adam, in [1], writes:Kalmár's scholarly personality was vivid and well rounded. ... He was enthusiastically inclined towards various sorts of personal contacts in his profession: regular teaching of university students, informal discussions with colleagues, lectures to general audiences. His most important lasting contribution to Hungarian science for which he will be remembered most will be [1]:... his ceaseless effort in promoting the development of computer science and the use of computers in his country. Article by: J J O'Connor and E F Robertson List of References (12 books/articles) A Poster of László Kalmár

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kalmar.html (2 of 3) [2/16/2002 11:16:49 PM]

Kalmar

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kalmar.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kalmar.html (3 of 3) [2/16/2002 11:16:49 PM]

Kaluza

Theodor Franz Eduard Kaluza Born: 9 Nov 1885 in Ratibor, Germany (now Raciborz, Poland) Died: 19 Jan 1954 in Göttingen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Theodor Kaluza's father was Max Kaluza. He belonged to a family which had lived in Ratibor for around 300 years. Max Kaluza was a fine scholar and, although German, he was a leading expert on the English language and English literature, with his special field being the study of Chaucer. Theodor, who was an only child, showed great promise at school and he entered the University of Königsberg to study mathematics. Kaluza continued his studies at Königsberg working towards his doctorate under Meyer's supervision. His habilitation thesis was on Tschirnhaus transformations and it was published in the Archiv der Mathematik und Physik in 1910. He had been examined and received the right to lecture in universities in the previous year. Kaluza married in 1909 and, after his habilitation, became a Privatdozent at the University of Königsberg, a position which left him with hardly any income. Most academics are promoted after a few years as a Privatdozent, but Kaluza clearly was not thought to be of sufficient merit for he remained a Privatdozent for 20 years. One person who though very highly of Kaluza, however, was Einstein. He was teaching at Königsberg in April 1919 when he wrote to Einstein and told him about his ideas to unify Einstein's theory of gravity and Maxwell's theory of light. Einstein encouraged him to publish his highly original ideas which he did in 1921 in his paper on the unity problem of physics - Zum Unitätsproblem der Physik published in Sitzungsberichte Preussische Akademie der Wissenschaften 96 (1921), 69. It was Einstein who communicated the paper on 8 December 1921. Kaluza's ideas involved the introduction of a fifth dimension and, although he has been criticised for http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaluza.html (1 of 3) [2/16/2002 11:16:50 PM]

Kaluza

introducing this as a purely mathematical idea, his work is important and was explored by others. He says in this paper that his theory possessed:... virtually unsurpassed formal unity ... which could not amount to the mere alluring play of a capricious accident. The unifying feature of this theory was that it unified Einstein's theory of gravitation and Maxwell's electromagnetic theory. As Kaku writes in [3]:... this unknown scientist was proposing to combine, in one stroke, the two greatest field theories known to science, Maxwell's and Einstein's, by mixing them in the fifth dimension. Kaluza is remembered for this in Kaluza-Klein (named after the mathematician Oskar Klein) field theory, which involved field equations in five-dimensional space. The theory, initially a popular topic of research, quickly lost favour with the introduction of quantum mechanics. In November 1926 Einstein wrote in support of Kaluza to try to impress upon people that he was a mathematician who had produced a remarkably novel idea and was worth far more than the position of Privatdozent which he still occupied. Still Kaluza failed to gain promotion, then in 1929, exactly 20 years after being appointed a Privatdozent, he was appointed as professor in Kiel. In 1935 he was made a full professor at Göttingen where he remained until his death in 1954, two months before he was due to retire. Perhaps Kaluza became too involved with his ideas in theoretical physics to make the contributions to mathematics which he was clearly capable of making. He continued to produce ideas relating to models of the atomic nucleus, and he wrote on relativity. Perhaps his finest mathematical work is the textbook Höhere Mathematik für die Pracktiker which was written jointly with G Joos and published in 1938. Raman writes [1]:in this work he showed himself as a mathematician rather than as a mathematical physicist. His son, also named Theodor Kaluza, contributed personal details of his father which were incorporated into [1]. These give us feeling for Kaluza's character as:... a man of wide-ranging interests. Although mathematical abstraction delighted him tremendously, he was also deeply interested in languages, literature, and philosophy. He studied more than fifteen languages, including Hebrew, Hungarian, Arabic, and Lithuanian. He had a keen sense of humour. A nonswimmer, he once demonstrated the power of theoretical knowledge by reading a book on swimming, then swimming successfully on his first attempt (he was over thirty when he performed this feat). Kaluza loved nature as well as science and was also fond of children. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaluza.html (2 of 3) [2/16/2002 11:16:50 PM]

Kaluza

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Kaluza.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaluza.html (3 of 3) [2/16/2002 11:16:50 PM]

Kaluznin

Lev Arkad'evich Kaluznin Born: 31 Jan 1914 in Moscow, Russia Died: 6 Dec 1990 in Moscow, Russia

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Shortly after Lev Arkad'evich Kaluznin's birth, his parents were divorced and his father, Arkadii Rubin, a well-known businessman, moved to England. Lev was brought up by his mother, Maria Pavlovna Kaluznina. She came from an old noble family many of whose members had become prominent figures in Russian culture, education, and the Arts. She passed her cultural values on to her son, especially her deep love for literature and music. She had a great influence on Lev Arkad'evich throughout her life sometimes markedly so. During the revolution of 1917 and the civil war which followed, mother and son lived in Petrograd (now St Petersburg). In 1923 they moved to Germany. To make ends meet Maria Pavlovna worked as a governess. In 1925 Lev entered a secondary school (Realschule) of high academic standing, from which he graduated in 1933. The school provided a solid background in mathematics, including topics in the foundations of analysis, differential equations and complex variables. In the autumn of 1933 he entered the Humboldt University of Berlin, where he spent the next three years. While at Humboldt he was greatly influenced by Schur, whose lectures on algebra shaped Lev's mathematical interest. In 1936 he moved to Hamburg, where, at the University of Hamburg, he attended lectures of Artin and Hecke and seminars of Zassenhaus and other famous mathematicians. It was here that he obtained his first research result - a generalisation of what is today a well-known theorem of Kurosh on the classification of abelian groups. In the Spring of 1938 Lev Arkad'evich moved with his mother to France, where, about a year later, he started attending lectures at the Sorbonne. World War II and the occupation of Paris by German troops forced Kaluznin to terminate his mathematical studies. To make a living during these difficult times he attended a vocational school and became an electrician. On 22 June 1941 his life changed dramatically - as did the lives of many Soviet citizens, who, like Kaluznin, were interned and sent to a camp in Compiègne near Paris. In the beginning, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaluznin.html (1 of 5) [2/16/2002 11:16:52 PM]

Kaluznin

conditions in the camp were tolerable and some prisoners, being specialists in certain fields, would entertain themselves by lecturing to others on diverse subjects. (In fact, in later years, Lev Arkad'evich could still recall great lectures he had heard there on world history, Roman law, etc.) During this time he did some research in Galois theory. In March 1942 he was transferred, as one of a group of prisoners, to a concentration camp in Wahlsburg. There, the horrors of camp life were felt to their fullest extent. Had it not been for the devotion and efforts of his mother, who found ways of surreptitiously sending him food during this period, Kaluznin may not have survived to see the camp liberated by American soldiers. In the Spring of 1945 Kaluznin returned to Paris. For a while he worked as a translator for the Soviet Embassy, but after a while he returned to his mathematical studies. He worked for the CNRS, published a series of papers on the structure of Sylow p-subgroups of symmetric groups, and in 1948 defended his doctoral thesis on the same topic. The following three years were extremely fruitful: he published several fundamental papers, collaborated with M Krasner, and presented his results at seminars and conferences. At about this time, Lev Arkad'evich and mother had made the decision to return to the USSR. In response to their application, the Soviet authorities him to spend some time in East Germany, where there was an acute shortage of scientists. To meet this condition Kaluznin began working at the Humboldt University in Berlin in 1951 - first as a lecturer, and then later, after habilitation with his thesis Stable automorphism groups, as a full professor. During this period he also held a research position at the Mathematical Institute of the (East-)German Academy of Sciences. In 1955 Lev Arkad'evich returned to the USSR. Through the recruitment efforts of mathematicians Gnedenko and G E Shilov, he was given a professorship at Kiev State University, an appointment he would hold for 31 years. At this time the University, though steeped in tradition as one of the finer mathematical research centres in the USSR, was in a relatively poor state. Memories of Stalin's recent atrocities were fresh in people's minds, and there was a pervasive atmosphere of political denunciation. Many good faculty members were forced to leave. In 1957 Kaluznin defended his postdoctoral thesis (a Soviet version of habilitation) on the topic Sylow p-subgroups of symmetric groups. Complete products of groups. Generalisations of Galois theory. With his successful defence he almost certainly became the only person to have ever received the highest degree in mathematics from three different countries despite having never completed his formal education! (In France permission had been granted by a special committee.) In 1959 Kaluznin became Head of the Department of Algebra and Mathematical Logic, a department created as a result of his own initiative. He became interested in mathematical linguistics and played an important role in the creation of the Department of Mathematical Linguistics at Kiev State University. His activities and achivements during the decade 1960-1970 includes: conducting research, teaching at Kiev State University and the Kiev Pedagogical Institute, consulting for the Department of Mathematical Linguistics, serving as a senior researcher at the Institute of Cybernetics of the Ukrainian Academy of Sciences, organising series of public lectures on mathematics, and serving as a member on editorial boards of several scientific journals. In 1962 he married Zoya Mikhailovna Volotskaya, a well-known linguist. They had two children, but after that lived apart most of the time. In 1968 several friends and students of Kaluznin signed a letter condemning the closed political trials that were then commonplace in Ukraine. In 1970 the political climate deteriorated still further. Having always been perceived as an "alien", Kaluznin was forced to leave his position as Head of the Department of Algebra and Mathematical Logic, though he retained his professorship at Kiev State University until

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaluznin.html (2 of 5) [2/16/2002 11:16:52 PM]

Kaluznin

1985. He was also denied permission to attend numerous conferences abroad to which he had been invited, and his only contact with Western mathematicians was now only by correspondence. During these years he devoted most of his time to his students, to his research activity on permutation groups, and to his newly cultivated interest in computer algebra. In 1984, due to deteriorating health, Lev Arkad'evich relinquished his teaching duties, and in 1985 his position within the faculty was changed to that of a 'senior researcher'. At the same time his son Mikhail returned to religious involvement, left Komsomol (a Russian Communist youth organisation), and became the target of constant attacks. There was no chance that young Mikhail's activities would go unnoticed by the communist authorities since this was the time of his graduation from Kiev State University, so his actions came under close scrutiny. All this contributed heavily to Kaluznin's forced retirement and to his subsequent move to Moscow. As time passed Kaluznin's health deteriorated, and his death came as the result of severe burns caused by an accident. As a researcher, Kaluznin is best known for his work in group theory and in particular permutation groups. He studied the Sylow p-subgroups of symmetric groups and their generalisations. In the case of symmetric groups of degree pn, these subgroups were constructed from cyclic groups of order p by taking their wreath product. His work allowed computations in groups to be replaced by computations in certain polynomial algebras over the field of p elements. Despite the fact that the earliest applications of wreath products of permutation groups was due to C Jordan, W Specht and G Polya, it was Kaluznin who first developed special computational tools for this purpose. Using his techniques, he was able to describe the characteristic subgroups of the Sylow p-subgroups, their derived series, their upper and lower central series, and more. These results have been included in many textbooks on group theory. Kaluznin was also the first to introduce the wreath product of abstract groups and the wreath product of an infinite family of groups. His constructions could be applied to group extensions. A particularly important result is the well-known theorem of Krasner and Kaluznin concerning the embeddings of a group with a subnormal series into the wreath product of the factors of the series. This theorem is widely used in the theory of group varieties, combinatorial group theory, and permutation group theory. Kaluznin made several applications of the wreath product to mathematical logic and mathematical chemistry. Kaluznin's other significant contributions to group theory include his work on stable automorphism groups, the structure of the variety of n-abelian groups, a classification of metabelian groups, work on locally normal groups of higher categories, and characterisations of the maximal subgroups of the symmetric and alternating groups. Another area of algebra which had always attracted Kaluznin's interest was Galois theory. His first papers in this area were devoted to Galois theory of normal extensions of fields. The ideas in this work were later employed by Jacobson in his study of the Galois theory of arbitrary finite extensions of fields. Developing further the methods of abstract Galois theory which had been initiated by Krasner, Kaluznin and his students were able to establish a Galois correspondence between Post algebras and Krasner algebras. His work led to considerable activity in the new area of algebraic combinatorics. Kaluznin also worked in the area of geometrical algebra, particularly on arrangements of subspaces in euclidean and unitary spaces. Though Lev Arkad'evich had never considered himself to be an expert in mathematical linguistics,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaluznin.html (3 of 5) [2/16/2002 11:16:52 PM]

Kaluznin

automata theory, and applications of computers in algebra, his mathematical interests were broad and he did not hesitate to do research in these areas outside of his main strengths. He believed in the fruitfulness of cooperation among people representing different mathematical disciplines, and that there were certain advantages to be had by "amateurs" having solid experience in another field. Often the areas he chose to concentrate on were little known to the scientific communities of the former Soviet Union. Kaluznin was an outstanding teacher. His lectures were inspiring. He was sometimes a bit impatient with the details of proofs, concentrating instead on presenting the broad picture of the mathematical terrain with regard to events that had taken place. At this he was hugely successful. He liked to discuss questions in their proper historical context, to motivate new concepts, and to stress the meaning and importance of a particular result. Lev Arkad'evich had a gift for recognising sparks of talent and potential in his students at a very early stage, often during their first semester at Kiev State University. Despite the absurdities and humiliations of Soviet life, with its constant dependencies on the whims of those who were in power, Lev Arkad'evich was never heard to regret his return to the USSR. Trying to avoid direct political involvement, he nevertheless found ways to make his position on social and political issues known, or to respond to pseudo-scientific political propaganda. He fully realised the extent of his accomplishments as a researcher, teacher, writer, and founder of scientific groups, and he well understood his impact on mathematical education. It is hard to overestimate the influence he has had on the lives of his many students. Finally we make a few remarks on personality. Lev Arkad'evich used to be a heavy smoker, smoking sixty a day even during his lectures. He gave up smoking on 1 January 1970, and never smoked another cigarette. He was well-versed in classical music, classic philosophy and Western prose, all of which he loved passionately. He did not care much for poetry, saying that:... a bear, probably, stepped on my poetic ear. Kaluznin was associated with many members of Kiev's intellectual elite. He organised mathematical gatherings in a cafe (similar to a famous Scottish Café in Lvov during the 30's), and started musical evenings for the students. Lev Arkad'evich liked and valued good red wines and also good beer. He also liked to dress well but at the end of a lecture his clothes would be entirely covered with chalk. At least this happened until the early 70's when, after visiting the GDR, he had returned with a white smock, the purpose of which was to protect his clothing during lectures. This unconventional attire prompted the joke among his students that Kaluznin is the "only real doctor of the Department". On the other hand, Kaluznin sometimes found it difficult to part with old clothing. One day, a well-meaning secretary decided to help him by asking someone to dispose of a dilapidated jacket he liked to wear. The jacket, which normally hung on the wall of his office, was dumped into a waste bin in the men's room. To the secretary's dismay, she later saw a smiling Lev Arkad'evich, jacket in hand, puzzled by how it came to be in the bin. The very next day she disposed of it herself, only this time to a bin in the ladies' room! He was deeply respected and loved by most who knew him. His clever and gentle humour, his aristocratically cultivated manner and free spirit, his extraordinary friendliness and openness to his companions, his clear and strict rejection of all forms of discrimination (be they nationally, religiously or politically based), his everyday deeds and reflections leave memories that will not disappear for a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaluznin.html (4 of 5) [2/16/2002 11:16:52 PM]

Kaluznin

generation. This is a shortened form of an article submitted by M H Klin. Article by: J J O'Connor and E F Robertson List of References (2 books/articles)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kaluznin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaluznin.html (5 of 5) [2/16/2002 11:16:52 PM]

Kamalakara

Kamalakara Born: about 1616 in Benares (now Varanasi), India Died: about 1700 in India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kamalakara was an Indian astronomer and mathematician who came from a family of famous astronomers. Kamalakara's father was Nrsimha who was born in 1586. Two of Kamalakara's three brothers were also famous astronomer/ mathematicians, these being Divakara, who was the eldest of the brothers born in 1606, and Ranganatha who was younger than Kamalakara. As was common throughout the classical period of Indian mathematics, members of the family acted as teachers to other family members. In particular Kamalakara was taught by his elder brother Divakara while Divakara himself had been taught by their uncle Siva. Pingree writes in [1]:[Kamalakara] combined traditional Indian astronomy with Aristotelian physics and Ptolemaic astronomy as presented by Islamic scientists (especially Ulugh Beg). Following his family's tradition he wrote a commentary, Manorama, on Ganesa's Grahalaghava and, like his father, Nrsimha, another commentary on the Suryasiddhanta, called the Vasanabhasya ... Kamalakara's most famous work, the Siddhanta-tattva-viveka, was commented on by Kamalakara himself. The work was completed in 1658. It is a work of fifteen chapters covering standard topics for Indian astronomy texts at this time. It deals with the topics of: units of time measurement; mean motions of the planets; true longitudes of the planets; the three problems of diurnal rotation; diameters and distances of the planets; the earth's shadow; the moon's crescent; risings and settings; syzygies; lunar eclipses, solar eclipses; planetary transits across the sun's disk; the patas of the moon and sun; the "great problems"; and a final chapter which forms a conclusion. The third chapter of the Siddhanta-tattva-viveka contains some of the most interesting mathematical results. In that chapter Kamalakara used the addition and subtraction theorems for the sine and the cosine to give trigonometric formulas for the sines and cosines of double, triple, quadruple and quintuple angles. In particular he gives formulas for sin(A/2) and sin(A/4) in terms of sin(A) and iterative formulas for sin(A/3) and sin(A/5). See for example [7] and [8] for a discussion of the details of Kamalakara's work in this area. The Siddhanta-tattva-viveka is a Sanskrit text and in it Kamalakara makes frequent use of the place-value number system with Sanskrit numerals. This and many other aspects of the work are discussed in [3]. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kamalakara.html (1 of 2) [2/16/2002 11:16:54 PM]

Kamalakara

List of References (10 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kamalakara.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kamalakara.html (2 of 2) [2/16/2002 11:16:54 PM]

Kantorovich

Leonid Vitalyevich Kantorovich Born: 19 Jan 1912 in St Petersburg, Russia Died: 7 April 1986 in USSR

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leonid Vital'evich Kantorovich studied mathematics at Leningrad State University, receiving his doctorate in mathematics in 1930 at the age of eighteen. From 1934 to 1960 he was a professor of mathematics at Leningrad University. He held the chair of mathematics and economics in the Siberian branch of the USSR Academy of Sciences (1961-1971), then directed research at Moscow's Institute of National Economic Planning (1971-76). Kantorovich's background was entirely in mathematics but he showed a considerable feel for the underlying economics to which he applied the mathematical techniques. He was one of the first to use linear programming as a tool in economics and this appeared in a publication Mathematical methods of organising and planning production which he published in 1939. Makarov writes in [24]:This may be considered a historic document, containing the facts about discovery of the linear programming. The mathematical formulation of production problems of optimal planning was presented here for the first time and the effective methods of their solution and economic analysis were proposed. Kantorovich introduced many new concepts into the study of mathematical programming such as giving necessary and sufficient optimality conditions on the base of supporting hyperplanes at the solution point in the production space, the concept of primal-dual methods, the interpretation in economics of multipliers, and the column-generation method used in linear programming. One of his most fundamental works on economics was The best use of economic resources which he wrote in 1942 but was not published until 1959. In this work Kantorovich applies optimisation techniques to a wide range of problems in economics. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kantorovich.html (1 of 3) [2/16/2002 11:16:55 PM]

Kantorovich

He also proposed a theory to handle the economics of technological innovations. This had three components namely the effect on the producer, the effect on the consumer and, the novel part of the theory, the effect derived from the increasing economic potential arising from the innovation. Kantorovich was a joint winner of the 1975 Nobel Prize for economics for his work on the optimal allocation of scarce resources. The article [12] is the autobiography which Kantorovich had to submit to the Nobel Prize committee who were considering him for the award. In [8] Belykh examines the opinions of Western scholars on Kantorovich as a mathematical economist and concludes that:Despite the differences of opinion and attempts to assign Kantorovich to one economic school or another, all the scientists under consideration here emphasise his outstanding contribution to the development of economic sciences. The work of Kantorovich which we have looked at up until now has been on the application of mathematical methods, particularly mathematical programming, to economics. It is this work for which Kantorovich is most famous, but he also worked in many other areas of mathematics. These other areas include functional analysis and numerical analysis and within these topics he published papers on the theory of functions, the theory of complex variables, approximation theory in which he was particularly interested in using Bernstein polynomials, the calculus of variations, methods of finding approximate solutions to partial differential equations, and descriptive set theory. From 1929 he worked on the theory of analytic sets and the Baire classification of functions. This work continued through the early 1930s then in the late 1930s he studied ordered topological vector spaces. However later in his career he also became interested in computer architecture. For example in [9] Fet writes:We describe briefly computers created on the basis of Kantorovich's suggestions. We note the significance of the concept, put forth by Kantorovich, of the large-block organisation of computing processes and the influence of this concept on the development of the architecture of computer systems. Makarov writes in [24] of Kantorovich's:... mathematical genius and the vast range of his interests and knowledge. His remarkable contributions to mathematics, economics and computers was published in over 300 papers and books. Article by: J J O'Connor and E F Robertson List of References (33 books/articles) Mathematicians born in the same country Honours awarded to Leonid Vital'evich Kantorovich (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1975

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kantorovich.html (2 of 3) [2/16/2002 11:16:55 PM]

Kantorovich

Other Web sites

1. Nobel prizes site (An autobiography of Kantorovich and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kantorovich.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kantorovich.html (3 of 3) [2/16/2002 11:16:55 PM]

Kaplansky

Irving Kaplansky Born: 22 March 1917 in Toronto, Ontario, Canada

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Irving Kaplansky's parents were Polish and he was born shortly after they had emigrated to Canada. Irving's early interest was music, an interest which he has kept all his life. Anyone who has heard him play the piano at a conference (as I [EFR] have been fortunate enough to do) will have seen that he exudes the same infectious joy of music as he does for mathematics. However Irving knew from a very young age that mathematics, and not music, was to be his life. He attended the University of Toronto graduating with a B.A. in 1938. He showed his great potential for mathematics at this stage, being on the winning team of the first William Lowell Putnam competition. This is a mathematical contest for students from the USA and Canada. In 1940 Kaplansky received his M.A. from Toronto, continuing his studies at Harvard University after being awarded a Putnam Fellowship (in fact he was the first recipient of this award). He was awarded a doctorate by Harvard in 1941, the year after he had become a citizen of the United States. His thesis supervisor at Harvard was MacLane and Kaplansky's thesis was entitled Maximal Fields with Valuations. Kaplansky was appointed a Benjamin Peirce Instructor in Harvard that year and he continued to hold that post there until 1944. The year 1944-45 Kaplansky spent in the Applied Mathematics Group of the National Defense Council at Columbia University before moving, in 1945, to the University of Chicago. This was to be the main university where he spent most of his career and where he was promoted to professor. During the years 1962-67 Kaplansky was chairman of the department in Chicago. In 1969 he was appointed George Herbert Mead Distinguished Service Professor at Chicago where he remained till his retirement in 1984. Despite holding important positions he remained accessible to colleagues and students alike, and [1]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaplansky.html (1 of 3) [2/16/2002 11:16:57 PM]

Kaplansky

... one could always rely on his availability and on a challenging idea or question as a result of each conversation. After he retired in 1984, Kaplansky went to California where he became director of the Mathematical Sciences Research Institute at the University of California, Berkeley. Kaplansky's work in mathematics is wide ranging although mostly it is in areas of algebra. He has made major contributions to ring theory, group theory and field theory. His book Infinite Abelian Groups was written at a time when this area was causing little interest but it has now blossomed into a major area in its own right. Similarly his many other books are beautiful introductions to various areas of algebra and have been enjoyed for their clarity, style and beauty by large numbers of undergraduate and graduate students. They include Fields and rings (2nd ed, 1972), An introduction to differential algebra (1957), Commutative rings (1970) and Lie algebras and locally compact groups (1971). Kaplansky's books [1]:... at a range of levels, are numerous ... [but] they are certainly not ponderous. He is a man of a few words, writing with polished economy to get the important ideas across. Kaplansky has received numerous awards. He has served for many years on the American Mathematical Society, being on the Council in 1951-53, vice-president in 1975, and he was elected president of the Society shortly after he retired during session 1985-86. There are many other ways in which Kaplansky has served the Society, particularly with respect to the American Mathematical Society publications. From 1945 to 1947 and again from 1979 to 1985 he was on the editorial board of the Bulletin of the American Mathematical Society; from 1947 to 1952 he was on the editorial board of the Transactions of the American Mathematical Society; and from 1957 to 1959 he was on the editorial board of the Proceedings of the American Mathematical Society. Despite this remarkable record of service to the Society, there were still further ways in which Kaplansky used his many talents to its benefit. He served on the Committee on Translations from Russian and other Slavic Languages from 1949 until 1958 and was on the Nominating Committee in 1977-78. Kaplansky was awarded a Guggenheim Fellowship and elected to the National Academy of Sciences and the American Academy of Arts and Sciences. In 1987 he was made an honorary member of the London Mathematical Society. Two years later, in 1989, the American Mathematical Society awarded Kaplansky their Steele Prize. There are three Steele Prizes awarded for different achievements. Kaplansky was awarded one [1]:... in recognition of cumulative influence extending over a career, including the education of doctoral students. The citation for the prize gives an excellent summary of Kaplansky's many achievements. The citation is available from a number of sources, see for example [1]:By his energetic example, his enthusiastic exposition and his overall generosity, he has made striking changes in mathematics and has inspired generations of younger mathematicians. His early works range over number theory, statistics, combinatorics, game theory, as well as his principal interest of commutative algebra. He completed the solution of Kurosh's problem on algebraic algebras of bounded degree, where Jacobson had made a decisive reduction, and considered numerous questions in the area of Banach algebras, always from the algebraist's viewpoint. ... http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaplansky.html (2 of 3) [2/16/2002 11:16:57 PM]

Kaplansky

As commutative algebra took on new life with the infusion of homological methods, he turned his interest once more in this direction, always trying to see past the formalism into "what was really going on". His remarkable success in doing so is witnessed by his publications from the later fifties onwards and the influence they have had on other writers. ... Kaplansky could not be present at the Summer Meeting of the American Mathematical Society in 1989 to reply in person to this citation. However, he did give a written response which was read at the meeting. In this response he showed his modestly by claiming that the "citation ... is too flattering" but he also gave some good advice which he wanted to put into print and it is well worth repeating here [1]:... spend some time every day learning something new that is disjoint from the problem on which you are currently working (remember that the disjointness may be temporary), and read the masters. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article)

A Quotation

A Poster of Irving Kaplansky

Mathematicians born in the same country

Honours awarded to Irving Kaplansky (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1985 - 1986

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kaplansky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kaplansky.html (3 of 3) [2/16/2002 11:16:57 PM]

Karman

Theodore von Kármán Born: 11 May 1881 in Budapest, Hungary Died: 6 May 1963 in Aachen, Germany

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Theodore Von Kármán was a mathematical prodigy and his father, fearing that his son would become a freak, steered him towards engineering. He graduated in 1902 from Budapest and from 1903 to 1906 he worked at the Technical University of Budapest. He left Budapest to study at Göttingen, where he was greatly influenced by Klein, and Paris where he watched some pioneering aviation flights which turned his interest to apply mathematics to aeronautics. In 1911 he made an analysis of the alternating double row of vortices behind a flat body in a fluid flow which is now known as Kármán's Vortex Street. The following year Kármán accepted a post as director of the Aeronautical Institute at Aachen in Germany. He visited the USA in 1926 and four years later he was offered the post of director of the Aeronautical Laboratory at California Institute of Technology. Despite his love for Aachen, the political events in Germany persuaded him to accept. In 1933 he founded the U.S. Institute of Aeronautical Sciences where he continued his research on fluid mechanics, turbulence theory and supersonic flight. He studied applications of mathematics to engineering, aircraft structures and soil erosion. The reference [4] is an autobiography. Article by: J J O'Connor and E F Robertson List of References (13 books/articles)

A Quotation

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Karman.html (1 of 2) [2/16/2002 11:16:59 PM]

Karman

Mathematicians born in the same country Honours awarded to Theodore Von Kármán (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1946

AMS Gibbs Lecturer

1939

Planetary features

Crater Karman on Mars

Other Web sites

1. Budapest, Hungary 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Karman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Karman.html (2 of 2) [2/16/2002 11:16:59 PM]

Karp

Carol Ruth Karp Born: 10 Aug 1926 in Forest Grove, Ottawa County, Michigan, USA Died: 20 Aug 1972 in Maryland, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Carol Karp was born Carol van der Velde. She attended Manchester College in Indiana receiving her BA from there in 1948. Her Master's Degree was obtained two years later from Michigan State University and following this she spent the summer as an instructor at Michigan State University, and then for a time travelled around the United States as a violinist in an all-woman orchestra. She then continued her studies at the University of Southern California, working for a doctorate. Her doctoral thesis was on mathematical logic. The thesis, Languages with expressions of infinite length, was supervised by L Henkin and submitted to the University of Southern California in 1959. However Karp was teaching several years before the award of her Ph.D. having accepted a position as instructor at New Mexico College of Agriculture and Mechanic Arts in 1953. In 1960 the name of the New Mexico College of Agriculture and Mechanic Arts was changed to its present name of New Mexico State University. Karp spent a year at the College in Las Cruces, New Mexico. Her thesis advisor had moved to Berkeley in 1953 and Karp was appointed as a teaching assistant here from 1954 to 1956 while she worked on her doctoral thesis. Karp had married Arthur L Karp in 1952 and, in 1957, she moved to Japan with her husband who was in the US Navy. On her return from Japan, Karp accepted a post as instructor at the University of Maryland. Soon after the award of her doctorate Karp was promoted, in 1960, to assistant professor at the University of Maryland. She was to remain at the University of Maryland until her early death from cancer in 1972 but there she was promoted again, to associate professor in 1963 and then to full professor in 1966.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Karp.html (1 of 3) [2/16/2002 11:17:00 PM]

Karp

Karp was a mathematical logician but, as noted in [1], her work was closely related to algebra:Karp considered herself to be principally an 'algebraic logician'. Her inclination towards algebra was never completely forgotten and she always seemed to draw results concerning Boolean algebras from her results about infinitary languages. In 1964 she published a book on her research Languages with expressions of infinite length but she had hoped to write another work which would take her ideas considerably further. She lectured on this later work as described in [1]:Karp did give lectures at Maryland in the Fall of 1970 on infinitary logic and recursion theory. Basically Karp wanted to return to Gödel's original proof-theoretic definitions of recursive sets but of course using more liberal notions of proof so as to obtain generalisations of recursion theory. It was both as a teacher and researcher that Karp made her reputation. She cared personally for her students and worried greatly for their futures during her illness. Again quoting from [1]:To her, teaching had always been more than a duty, and even during her illness she taught all her classes in addition to carrying out all her administrative tasks. Her research, too, was pushed forward with her usual determination.... Judy Green, who was one of Karp's doctoral students, wrote the article in [2]. In it she writes:Karp's intellectual standards were extremely high, and she was unfailingly honest in applying them. Although she showed almost familial concern for her students and younger colleagues, she was consistently candid in appraising their mathematical contributions and promise. In particular, she advised working towards a doctorate only if one expected to make research the most important part of one's professional career, and she refused to allow her students to graduate until their results met her own high standard for publishability. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Karp.html (2 of 3) [2/16/2002 11:17:00 PM]

Karp

JOC/EFR June 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Karp.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Karp.html (3 of 3) [2/16/2002 11:17:00 PM]

Katyayana

Katyayana Born: about 200 BC in India Died: about 200 BC in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

We cannot attempt to write a biography of Katyayana since essentially nothing is known of him except that he was the author of a Sulbasutra which is much later than the Sulbasutras of Baudhayana and Apastamba. It would also be fair to say that Katyayana's Sulbasutra is the least interesting from a mathematical point of view of the three best known Sulbasutras. It adds very little to that of Apastamba written several hundreds of years earlier. We do not know Katyayana's dates accurately enough to even guess at a life span for him, which is why we have given the same approximate birth year as death year. Katyayana was neither a mathematician in the sense that we would understand it today, nor a scribe who simply copied manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in mathematics for its own sake, merely interested in using it for religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for religious rites and to improve and expand on the rules which had been given by his predecessors. Katyayana would have been a priest instructing the people in the ways of conducting the religious rites he describes. Katyayana lived in a period when the religious rites that the Sulbasutras were written to support were becoming less influential. People were turning to other religions and perhaps this lack of vigour in the religion at this time partly explains why several hundreds of years after Apastamba Katyayana adds little of importance to the Sulbasutra which he wrote. See the article Indian Sulbasutras for more information on the Sulbasutras in general and the mathematical results which they contain. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. An overview of Indian mathematics 2. The Indian Sulbasutras

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Katyayana.html (1 of 2) [2/16/2002 11:17:01 PM]

Katyayana

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Katyayana.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Katyayana.html (2 of 2) [2/16/2002 11:17:01 PM]

Keill

John Keill Born: 1 Dec 1671 in Edinburgh, Scotland Died: 31 Aug 1721 in Oxford, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Keill attended an Edinburgh school, then studied at Edinburgh University under David Gregory obtaining his degree in 1692. Keill went to Oxford with David Gregory in 1691 and studied at Balliol College, obtaining an Oxford degree in 1694. At Oxford Keill lectured on Newton's work and was soon appointed as a lecturer in experimental philosophy. He was appointed deputy to the professor of natural philosophy in 1699, a post he held until 1709. Keill was elected a Fellow of the Royal Society in 1700 and Savilian Professor of Astronomy at Oxford in 1712. Keill acted as a propagator of Newton's philosophy and argued against Whiston and others. He claimed that Leibniz had plagiarised Newton's invention of the calculus and he served as Newton's avowed Champion. Keill wrote in Introductio ad veram (published in Leiden in 1725) The only true Philosophers are those who would account for all Effects and Phenomena by the known established Laws of Motion and Mechanics. His work Euclides elementorum libri priores sex published in 1715 studies trigonometry and logarithms. He also wrote on forces between particles and on theories of the origin of the universe. Two years before he died, Keill was left a fortune on the death of his brother. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Keill.html (1 of 2) [2/16/2002 11:17:02 PM]

Keill

Honours awarded to John Keill (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1700

Savilian Professor of Astronomy

1712

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Keill.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Keill.html (2 of 2) [2/16/2002 11:17:02 PM]

Kelland

Philip Kelland Born: 1808 in Dunster, Somerset, England Died: 7 May 1879 in Bridge of Allen, Stirlingshire, Scotland

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Philip Kelland was the son of the Rev. Phillip Kelland (died 1847), curate of Dunster, Somerset and then rector of Landcross, Devon. He studied at Queens' College, Cambridge and was coached privately by William Hopkins, graduating in 1834 as senior wrangler and first Smith's prizeman. Kelland was ordained in the Church of England and was a fellow of Queens' College during 1834-38. In 1838 he was appointed Professor of Mathematics at Edinburgh University in succession to William Wallace. He was the first English-born and wholly English-educated holder of that chair: the unsuccessful applicants included the Scots Duncan F Gregory, Edward Sang and John Scott Russell. In Edinburgh, Kelland joined with James D Forbes, the Professor of Natural Philosophy, in supporting reforms of the Scottish university system. According to A. Grant, he came to know the Scottish Universities better even than do Scotsmen themselves. He was an effective reformer who won the respect of his colleagues, and a successful and popular teacher who gained the affection of his students. One of his students, the writer Robert Louis Stevenson, wrote kindly that: ... for all his silver hair and worn face, he was not truly old; and he had too much of the unrest and petulant fire of youth, and too much invincible innocence of mind, to play the veteran well ... as I still fancy I behold him frisking actively about the platform, pointer in hand, that which I seem to see most clearly is the way his glasses glittered with affection. For many years, Kelland served as secretary of the Senatus of Edinburgh University. He was elected FRS in 1838 and FRSE in 1839. He was President of the Royal Society of Edinburgh during 1878-79, dying in office. He was also active in the Edinburgh Society of Arts, a member of the Board of Visitors of the Edinburgh Observatory, an examiner for several of Edinburgh's schools and colleges, and a founder and actuarial adviser of the Life Association of Scotland. For the last-mentioned, he toured Canada and the U.S.A., and published an account of his travels entitled Transatlantic Sketches. In addition to his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kelland.html (1 of 3) [2/16/2002 11:17:04 PM]

Kelland

university duties, he gave mathematical lectures to the Edinburgh Ladies' Educational Association, and he took Sunday services in some of Edinburgh's Episcopal churches. (One obituarist records that Preaching, however, was ... one of the few accomplishments in which he did not excel.) He was twice married, first to a Miss Pilkington of Dublin, and then to a Miss Boswell of Wardie. He was survived by his widow, three sons and two daughters. His early research work, undertaken at Cambridge, was influenced by Fourier and Cauchy. This is described in his Theory of Heat (1837, 1842) and in some papers; but this work turned out to be based on unsound principles. In all, 28 papers are listed in the Royal Society Catalogue of Scientific Papers: these are mainly on heat, light and water waves. His theoretical work on water waves (1840, 1844), published in Trans. Roy. Soc. Edin., attempted to explain aspects of the important experiments of John Scott Russell, then being carried out near Edinburgh. This work, though flawed in some respects, anticipated some of the results later obtained more convincingly by George Gabriel Stokes. He wrote analytical papers on General Differentiation (1839), and Differential Equations (1853), and gave a geometrical Theory of Parallels outlining a version of non-Euclidean geometry. Kelland produced a much-revised edition of John Playfair's Elements of Geometry and a successful textbook of Algebra. He wrote on the reform of the Scottish Universities, and edited the collected works of Thomas Young. Late in life, he collaborated with Peter Guthrie Tait on an Introduction to Quaternions. Though not a research mathematician of the highest rank, Kelland had a great influence on the development of education in Scotland, and he attained a high standard of mathematical instruction. Article by: Alex D D Craik, University of St Andrews. List of References (5 books/articles) Mathematicians born in the same country Other Web sites

G Frederickson

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2001 School of Mathematics and Statistics University of St Andrews, Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kelland.html (2 of 3) [2/16/2002 11:17:04 PM]

Kelland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Kelland.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kelland.html (3 of 3) [2/16/2002 11:17:04 PM]

Kellogg

Oliver Dimon Kellogg Born: 10 July 1878 in Linwood, Pennsylvania, USA Died: 26 July 1932 in USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oliver Kellogg studied as an undergraduate at Princeton University. There he attended lectures by Fine who fired his interest in mathematics. After receiving his A.B. in 1899, Kellogg continued his studies for a Master's Degree at Princeton. He was awarded the Master's Degree in 1900 and received a Fellowship to allow him to study in Europe. Kellogg spent a year at the University of Berlin, then moved to Göttingen to work for his doctorate. He attended lectures by Hilbert who suggested he undertake research on the Dirichlet problem for plane regions bounded by a finite number of plane curves which met at points where the boundary was not differentiable. Fredholm had just published a major work on the Dirichlet problem but Fredholm's methods did not apply to the regions which Hilbert suggested Kellogg investigate. In 1902 Kellogg published his first paper giving a direct proof of Fredholm's inversion formula. In January of the following year he received his doctorate for his dissertation Zur Theorie der Integralgleichungen und des Dirichlet'schen Prinzips on the Dirichlet problem but he remained in Germany to the end of the academic year, returning to the United States to take up the post of instructor in mathematics at Princeton. During these years he published two further papers which developed from the work of his thesis. However Kellogg soon became less than happy with these papers. Partly he had failed to answer the questions Hilbert had asked him to solve though this was understandable since they were much harder than Hilbert had realised. Secondly some of Kellogg's results were incomplete and others were incorrect. Again it is hard to criticise him too much over this since very similar errors were later made by both Hilbert and Poincaré. Kellogg was appointed to the University of Missouri in 1905 where, despite a heavy teaching and administrative load he was able to publish impressive papers on potential theory. In 1912 he published the important work Harmonic functions and Green's integral. Although he would return to potential theory, Kellogg next published a number of papers on sets of real orthogonal functions. However his work was disrupted by World War I when he was assigned as scientific advisor to the United States Coast Guard Academy at New London, Connecticut. At the end of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kellogg.html (1 of 2) [2/16/2002 11:17:05 PM]

Kellogg

the war he was appointed to Harvard University. In addition to his work on potential theory and orthogonal set of functions he published a short paper on the problem of the maximum value an of a positive integer belonging to a set of n positive integers whose reciprocals add to 1. In fact the answer is given by an+1 = an(an + 1) where a1 = 1. Kellogg also wrote a number of papers on the existence of certain sets of functions in analysis as well as generalisations of polynomials due to Sergi Bernstein. Kellogg continued to work at Harvard until his death which resulted from a heart attack which he suffered while climbing. Birkhoff writes in [1]:His quick, generous nature and unusual charm of personality were united with a versatile and original mind. The full story of Kellogg's many successful efforts to help others would be an extraordinary one ... in order to judge his mathematical achievements, it is necessary not only to consider his published work but to take into account his modesty and his readiness to share his nascent ideas with others. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kellogg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kellogg.html (2 of 2) [2/16/2002 11:17:05 PM]

Kemeny

John Kemeny Born: 31 May 1926 in Budapest, Hungary Died: 26 Dec 1992 in Hanover, New Hampshire, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Kemeny attended primary school in Budapest. He came from a Jewish family and, in 1940, his father had the good sense to take his family to the United States. Understandably, not all the family wanted to leave their home but Kemeny's grandfather, who refused to leave, died in the Holocaust. An aunt and uncle of Kemeny's also failed to survive the Nazis. Kemeny's family settled in New York and John attended high school in New York City. Kemeny entered Princeton where he studied mathematics and philosophy, but he took a year off during his undergraduate course to work on the Manhattan Project in Los Alamos. His boss in Los Alamos was Richard Feynman and he also worked there with von Neumann. Returning to Princeton, Kemeny graduated with his B.A. in 1947, then worked for his doctorate under Alonzo Church's supervision. Kemeny was awarded his doctorate in 1949 for a dissertation entitled Type-Theory vs. Set-Theory. While a doctoral student be was appointed as Albert Einstein's mathematical assistant. Kemeny later wrote:People would ask - did you know enough physics to help Einstein? My standard line was: Einstein did not need help in physics. But contrary to popular belief, Einstein did need help in mathematics. By which I do not mean that he wasn't good at mathematics. He was very good at it, but he was not an up-to-date research level mathematician. His assistants were mathematicians for two reasons. First of all, in just ordinary calculations, anybody makes mistakes. There were many long calculations, deriving one formula from another to solve a differential equation. They go on forever. Any number of times we got the wrong answer.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kemeny.html (1 of 3) [2/16/2002 11:17:07 PM]

Kemeny

Sometimes one of us got the wrong answer, sometimes the other. The calculations were long enough that if you got the same answer at the end, you were confident. So he needed an assistant for that, and, frankly, I was more up-to-date in mathematics than he was. Kemeny had continued to study both mathematics and philosophy and his first full-time teaching position, in 1951, was a philosophy appointment at Princeton. But Snell writes in [2]:He was on his way to a traditional distinguished scholarly career as a promising young assistant professor at Princeton with a joint appointment in mathematics and philosophy when he decided instead to accept the challenge of developing a new mathematics department at a college which he had barely heard of and which was certainly not known as a centre of excellence in mathematics. So Kemeny was appointed to the Mathematics Department at Dartmouth in 1953 and, two years later, he became chairman of the Department. He held this post until 1967. He was president of Dartmouth between 1970 and 1981 and, in 1982, he returned to full-time teaching. In fact he never gave up teaching while he was president, he had made it a condition of taking the post. Kemeny will be remembered by most people as the co-inventor of the BASIC (Beginners All-purpose Symbolic Instruction Code) computer language. It was in 1963 that Kemeny with Thomas Kurtz decided that they wanted to give students easy access to computing. Snell in [2] writes:Just as von Neumann realised that a computer that did only ordinary arithmetic operations could have extraordinary power, Kemeny realised that to make this power available to everyone, a programming language could and should be exceedingly simple. This led him to develop with Tom Kurtz the computer language BASIC... Kemeny and Kurtz designed the first "time sharing" system so that many students could a single computer at the same time. BASIC was designed to allow students to write programs easily. The first BASIC program was run at Dartmouth at 2 am on 4 May 1964. A teaching innovation which Kemeny introduced was in developing a Finite Mathematics course including topics that are no surprise to us today; logic, probability and matrix algebra. It was designed because he was unhappy that mathematics (entirely calculus in first year courses at that time) was [2]:...the only subject you can study for 14 years and not learn a single thing that has been done since 1800. Kemeny was well known outside mathematical circles however for, in 1979, President Jimmy Carter asked him to chair the commission investigating the Three Mile Island nuclear accident [1]:The Kemeny Commission, as it came to be called, was very critical of the nuclear power industry and its federal regulators. Kemeny's character was painted by his wife:He liked science fiction, football games, shrimp, all kinds of puzzles, Agatha Christie, and solitude (for two). He did not enjoy socialising. Before he retired, John recognised only two flowers, the tulip and the rose, and two pieces of music, the 1812 Overture and Poor Little Buttercup. These last years he had time to enjoy Mozart, wildflowers, pileated woodpeckers, eclipses. Sometimes he liked just to sit still and think. Many awards and honours were bestowed on Kemeny. He was given the New York Academy of Sciences Award in 1984, the Institute of Electrical Engineers Computer Medal in 1986 and the Louis

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kemeny.html (2 of 3) [2/16/2002 11:17:07 PM]

Kemeny

Robinson Award on 1990. He received twenty honorary degrees. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other Web sites

Bellevue College, USA

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kemeny.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kemeny.html (3 of 3) [2/16/2002 11:17:07 PM]

Kempe

Alfred Bray Kempe Born: 6 July 1849 in Kensington, London, England Died: 21 April 1922

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alfred Kempe was educated at Cambridge and graduated in 1872. He was a barrister by profession, and an authority on ecclesiastical law, while his favourite recreations were mathematics and music. He published a false "proof" of the four colour theorem in 1879 which stood until Heawood showed the mistake 11 years later. The 'proof' is however still the basis for the computer aided proof discovered 100 years later. His other work (which was correct!) on straight line linkages was inspired by Sylvester. Kempe published How to draw a straight line: A lecture on linkages in 1877, which was a classic on the topic. He was elected to the Royal Society on 1881. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Alfred Kempe

Mathematicians born in the same country

Cross-references to History Topics

The four colour theorem

Cross-references to Famous Curves

Watt's Curve

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kempe.html (1 of 2) [2/16/2002 11:17:09 PM]

Kempe

Other references in MacTutor

1. Chronology: 1870 to 1880 2. Chronology: 1890 to 1900

Honours awarded to Alfred Kempe (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1881

London Maths Society President

1892 - 1894

Other Web sites

Amsterdam, Netherlands (Some details of linkages)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kempe.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kempe.html (2 of 2) [2/16/2002 11:17:09 PM]

Kendall

David George Kendall Born: 15 Jan 1918 in Ripon, Yorkshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

David Kendall attended Ripon Grammar School and the entered Queen's College, Oxford. He was awarded his M.A. in 1943 but he had aleardy been involved in war work. During the years of World War II and Kendall worked as an Experimental Officer with the Ministry of Supply from 1940 until the end of the war in 1945. Other mathematicians such as Rogers also held similar posts with the Ministry of Supply. In 1946 Kendall was elected a fellow of Magdalen College, Oxford and appointed a lecturer in mathematics. He spent the academic year 1952-53 in the United States as a visiting lecturer at Princeton University. Then in 1962 Kendall was appointed as Professor of Mathematical Statistics at the University of Cambridge. At the same time he was elected a fellow of Churchill College Cambridge. Kendall held this chair of mathematical statistics until he retired in 1985 and which time he became professor emeritus. He also became an Emeritus Fellow of Magdalen College, Oxford in 1989. Kendall is a leading world authority on applied probability and data analysis. He has written on stochastic geometry and its applications, and the statistical theory of shape. His recent work includes two articles How to look at objects in a five-dimensional shape space (1994-95) and The Riemannian structure of Euclidean shape spaces: a novel environment for statistics (1993). He has received many honours and awards for his outstanding work in these areas of mathematical statistics including the Guy Medal in Silver of the Royal Statistical Society in 1955 and the Gold Medal of the Royal Statistical Society in 1981. He was also awarded the Weldon Memerial Prize and Medal for Biometric Science from Oxford University in 1974 and Princeton University awarded him their Wilks Prize in 1980.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kendall.html (1 of 3) [2/16/2002 11:17:11 PM]

Kendall

An exceptional lecturer, Kendall has been the Larmor Lecturer at the Cambridge Philosophical Society in 1980, the Milne Lecturer at Wadham College, Oxford in 1983, the Hoteling Lecturer at the University of North Carolina in 1985, the Rietz Lecturer at the Institute of Mathematical Statistics in 1989 and the Kolmogorov Lecturer at the Bernoulli Society for Mathematical Statistics and Probability in 1990. In 1964, Kendall was elected a fellow of the Royal Society and served on the Council of that Society during 1967-69 and then for a second spell during 1982-83. The Royal Society awarded Kendall their Sylvester Medal in 1976. He was the 56th president of the London Mathematical Society during 1972-74 and the London Mathematical Society awarded him their Whitehead Prize in 1980 and their De Morgan Medal in 1989. However it is not only the London Mathematical Society that has elected Kendall to be their president. He has also been president of the Bernoulli Society for Mathematical Statistics and Probability in 1975 and for the Mathematics and Physics Sections of the British Association in 1982. He has also received a number of honorary degrees to mark his outstanding contributions to statistics. For example he was elected an honorary member of the Romanian Academy in 1992 and has received honorary degrees from the University of Paris René Descartes (1976) and the University of Bath (1986). Kendall has been joint editor of a number of important works, including Mathematics in the Archaeological and Historical Sciences (1971), Stochastic Analysis (1973), Stochastic Geometry (1974), Analytic and Geometric Stochastics (1986). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to David Kendall (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1964

Royal Society Sylvester Medal

Awarded 1976

London Maths Society President

1972 - 1974

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kendall.html (2 of 3) [2/16/2002 11:17:11 PM]

Search Form

Kendall

Glossary index

Quotations index

Mathematicians of the day JOC/EFR June 1998

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kendall.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kendall.html (3 of 3) [2/16/2002 11:17:11 PM]

Kendall_Maurice

Maurice George Kendall Born: 6 Sept 1907 in Kettering, Northamptonshire, England Died: 29 March 1983 in Redhill, Surrey, Emgland

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Maurice Kendall's father, John Roughton Kendall, was an engineering worker who was brought up in Kettering, while his mother was Georgina Brewer who came from Hertfordshire. As a young boy Maurice sometimes helped his grandfather who owned a public house, The Woolsack, in Kettering. It is rather a miracle that Maurice ever survived to adulthood, for he contracted cerebral meningitis, a disease which at that time was almost always fatal. In 1914 World War I started and Maurice's father moved to Derby to work for Rolls Royce. It was in Derby that Maurice received his early education but he showed little signs of the great academic achievements which were to come. He sat the entrance examinations for Grammar School but did not gain admission so he went to Derby Central School. His early interests were in languages but near the end of his secondary schooling he began to show an aptitude for mathematics. The Headmaster of the Central School must get much credit for making the highly unusual move of arranging for Maurice to attend the Grammar School in his final year. It is rather remarkable that Maurice's improvement was such that he was awarded a scholarship to study at St John's College, Cambridge. His father, however, did not want his son to go to Cambridge, but rather he wanted him to train as an engineer. Maurice was always extremely grateful to his mother who supported him in wishing to take up the scholarship. Stuart writes in [3]:Life at St John's was in striking contrast to that in Derby, and Maurice's naturally gregarious nature brought him several circles of friends apart from the group reading

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kendall_Maurice.html (1 of 4) [2/16/2002 11:17:13 PM]

Kendall_Maurice

mathematics at St John's ... Two interests outside mathematics were cricket and chess. One of his close friends who shared his interest in chess was Bronowski. Of the staff he got to know MacMahon, who was a Fellow of St John's but very elderly, but failed to ever come in contact with Yule at this time despite Yule being a lecturer and Fellow of St John's. Kendall graduated a mathematics Wrangler in 1929 and, in the following year, he passed the Civil Service Examinations and joined the Ministry of Agriculture, It was at the Ministry of Agriculture that Kendall became involved in statistical work. The quality of this work was such that he was elected a Fellow of the Royal Statistical Society in 1934. One of his first papers came through his work at the Ministry of Agriculture and was on factor analysis applied to crop productivity. In 1935 Kendall met Yule for the first time. He spent part of his holidays reading statistics books at the St John's College library and he had to seek out Yule who had the key to the library. Their brief conversation would prove significant for when Yule discussed a revision of his text An Introduction to the Theory of Statistics (first published in 1911) with his publisher the suggestion was made that a second author might be brought in to help Yule. Yule remembered his meeting with Kendall and after discussions joint authorship of a new edition was agreed, the work being undertaken in 1937. It is worth noting that by 1950 the 14th edition of this book had appeared. This work prompted an even greater enthusiasm for statistics from Kendall who attended lecturers in advanced statistical topics at University College London, and began publishing a stream of quality papers on statistical topics. An advanced treatise on mathematical statistics was proposed in 1939 and Maurice Kendall, Egon Pearson, John Wishart, and others held preliminary discussions. However, the project came to nothing with the outbreak of war since the various authors were dispersed around the country. Kendall remained in London and in 1940 left the Civil Service to take up the post of statistician to the British Chamber of Shipping. As Ord writes in [2]:Despite his heavy workload by day and air-raid warden duties by night, he somehow contrived to find time to work on the project [the advanced treatise on mathematical statistics] single-handedly Volume One of the Advanced Theory of Statistics was published in 1943, and Volume Two appeared in 1946. In 1994, more than ten years after Kendall death, the sixth edition which is now called Kendall's advanced theory of statistics. Vol. 1 appeared authored by Alan Stuart and Keith Ord. The preface states:It is fifty years since the first edition of Maurice Kendall's Volume 1 appeared, so it is fitting that a new edition sees a major restructuring of The advanced theory that, we hope, remains true to his two goals of presenting a 'systematic treatment of (statistical) theory as it exists at the present time' and keeping the volumes first and foremost as a treatment of 'statistics, not statistical mathematics'. Kendall continued a remarkable stream of research papers on topics such as the theory of k-statistics, time series, and rank correlation methods and a monograph Rank Correlation in 1948. In 1949 he accepted the second chair of statistics at the London School of Economics. He held this post until 1961 when he resigned to take on the position of Managing Director of a computer consultancy which became SciCon. He later became Chairman of the company. In 1972, having reached 65 years of age, he retired from SciCon. He most certainly did not retire from

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kendall_Maurice.html (2 of 4) [2/16/2002 11:17:13 PM]

Kendall_Maurice

work, however, for in that year he took on the role of Director of the World Fertility Survey. The survey was carried out for the United Nations working with the International Statistical Institute. In 1974 he was knighted for his services to statistics, and in 1980 the United Nations awarded him their Peace Medal for his work on the World Fertility Survey. We have mentioned above some of Kendal's work and some of the textbooks which he wrote. Among other publications which he edited (some jointly) were the two volume Statistical Sources in the United Kingdom (1952, 1957), Dictionary of Statistical Terms (1957) which followed his aim of making new ideas in statistics more widely available, and Bibliography of Statistical Literature which appeared in three volumes (1962, 1965, 1968). There are many more major texts by Kendall and their importance and popularity is seen by the number of editions which have appeared over the years and continue to appear many years after his death. For example the fifth edition of Rank Correlation appeared in 1990 while we mentioned above that the first edition appeared in 1948. Other monographs are A course in the geometry of n dimensions (1961) which aims to present that part of the theory of n-dimensional geometry which has statistical applications, and to sketch very briefly what those applications are. Indeed the brief aim is achieved for the book only contains 63 pages. In 1963 he published (jointly with P A P Moran) Geometrical probability followed by Time series (1973) in which Kendall states his objectives to bridge the gap between "sophisticated theory and practical applications" in the field of time series and to "treat the subject in its entirety for the benefit of the practising statistician". He also published A course in multivariate analysis and Cluster analysis as well a whole series of articles Studies in the history of probability and statistics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kendall_Maurice.html (3 of 4) [2/16/2002 11:17:13 PM]

Kendall_Maurice

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kendall_Maurice.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kendall_Maurice.html (4 of 4) [2/16/2002 11:17:13 PM]

Kepler

Johannes Kepler Born: 27 Dec 1571 in Weil der Stadt, Württemberg, Holy Roman Empire (now Germany) Died: 15 Nov 1630 in Regensburg (now in Germany)

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johannes Kepler is now chiefly remembered for discovering the three laws of planetary motion that bear his name published in 1609 and 1619). He also did important work in optics (1604, 1611), discovered two new regular polyhedra (1619), gave the first mathematical treatment of close packing of equal spheres (leading to an explanation of the shape of the cells of a honeycomb, 1611), gave the first proof of how logarithms worked (1624), and devised a method of finding the volumes of solids of revolution that (with hindsight!) can be seen as contributing to the development of calculus (1615, 1616). Moreover, he calculated the most exact astronomical tables hitherto known, whose continued accuracy did much to establish the truth of heliocentric astronomy (Rudolphine Tables, Ulm, 1627). A large quantity of Kepler's correspondence survives. Many of his letters are almost the equivalent of a scientific paper (there were as yet no scientific journals), and correspondents seem to have kept them because they were interesting. In consequence, we know rather a lot about Kepler's life, and indeed about his character. It is partly because of this that Kepler has had something of a career as a more or less fictional character (see historiographic note). Childhood Kepler was born in the small town of Weil der Stadt in Swabia and moved to nearby Leonberg with his parents in 1576. His father was a mercenary soldier and his mother the daughter of an innkeeper. Johannes was their first child. His father left home for the last time when Johannes was five, and is believed to have died in the war in the Netherlands. As a child, Kepler lived

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html (1 of 9) [2/16/2002 11:17:16 PM]

Kepler

with his mother in his grandfather's inn. He tells us that he used to help by serving in the inn. One imagines customers were sometimes bemused by the child's unusual competence at arithmetic. Kepler's early education was in a local school and then at a nearby seminary, from which, intending to be ordained, he went on to enrol at the University of Tübingen, then (as now) a bastion of Lutheran orthodoxy. Kepler's opinions Throughout his life, Kepler was a profoundly religious man. All his writings contain numerous references to God, and he saw his work as a fulfilment of his Christian duty to understand the works of God. Man being, as Kepler believed, made in the image of God, was clearly capable of understanding the Universe that He had created. Moreover, Kepler was convinced that God had made the Universe according to a mathematical plan (a belief found in the works of Plato and associated with Pythagoras). Since it was generally accepted at the time that mathematics provided a secure method of arriving at truths about the world (Euclid's common notions and postulates being regarded as actually true), we have here a strategy for understanding the Universe. Since some authors have given Kepler a name for irrationality, it is worth noting that this rather hopeful epistemology is very far indeed from the mystic's conviction that things can only be understood in an imprecise way that relies upon insights that are not subject to reason. Kepler does indeed repeatedly thank God for granting him insights, but the insights are presented as rational. University education At this time, it was usual for all students at a university to attend courses on "mathematics". In principle this included the four mathematical sciences: arithmetic, geometry, astronomy and music. It seems, however, that what was taught depended on the particular university. At Tübingen Kepler was taught astronomy by one of the leading astronomers of the day, Michael Maestlin (1550 - 1631). The astronomy of the curriculum was, of course, geocentric astronomy, that is the current version of the Ptolemaic system, in which all seven planets - Moon, Mercury, Venus, Sun, Mars, Jupiter and Saturn - moved round the Earth, their positions against the fixed stars being calculated by combining circular motions. This system was more or less in accord with current (Aristotelian) notions of physics, though there were certain difficulties, such as whether one might consider as 'uniform' (and therefore acceptable as obviously eternal) a circular motion that was not uniform about its own centre but about another point (called an 'equant'). However, it seems that on the whole astronomers (who saw themselves as 'mathematicians') were content to carry on calculating positions of planets and leave it to natural philosophers to worry about whether the mathematical models corresponded to physical mechanisms. Kepler did not take this attitude. His earliest published work (1596) proposes to consider the actual paths of the planets, not the circles used to construct them. At Tübingen, Kepler studied not only mathematics but also Greek and Hebrew (both necessary for reading the scriptures in their original languages). Teaching was in Latin. At the end of his first year Kepler got 'A's for everything except mathematics. Probably Maestlin was trying to tell him he could do better, because Kepler was in fact one of the select pupils to whom he chose to teach more advanced astronomy by introducing them to the new, heliocentric cosmological system of Copernicus. It was from Maestlin that Kepler learned that the preface to On the revolutions, explaining that this was 'only mathematics', was not by Copernicus. Kepler seems to have accepted almost instantly that the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html (2 of 9) [2/16/2002 11:17:16 PM]

Kepler

Copernican system was physically true; his reasons for accepting it will be discussed in connection with his first cosmological model (see below). It seems that even in Kepler's student days there were indications that his religious beliefs were not entirely in accord with the orthodox Lutheranism current in Tübingen and formulated in the 'Augsburg Confession' (Confessio Augustana). Kepler's problems with this Protestant orthodoxy concerned the supposed relation between matter and 'spirit' (a non-material entity) in the doctrine of the Eucharist. This ties up with Kepler's astronomy to the extent that he apparently found somewhat similar intellectual difficulties in explaining how 'force' from the Sun could affect the planets. In his writings, Kepler is given to laying his opinions on the line - which is very convenient for historians. In real life, it seems likely that a similar tendency to openness led the authorities at Tübingen to entertain well-founded doubts about his religious orthodoxy. These may explain why Maestlin persuaded Kepler to abandon plans for ordination and instead take up a post teaching mathematics in Graz. Religious intolerance sharpened in the following years. Kepler was excommunicated in 1612. This caused him much pain, but despite his (by then) relatively high social standing, as Imperial Mathematician, he never succeeded in getting the ban lifted. Kepler's first cosmological model (1596) Instead of the seven planets in standard geocentric astronomy the Copernican system had only six, the Moon having become a body of kind previously unknown to astronomy, which Kepler was later to call a 'satellite' (a name he coined in 1610 to describe the moons that Galileo had discovered were orbiting Jupiter, literally meaning 'attendant'). Why six planets? Moreover, in geocentric astronomy there was no way of using observations to find the relative sizes of the planetary orbs; they were simply assumed to be in contact. This seemed to require no explanation, since it fitted nicely with natural philosophers' belief that the whole system was turned from the movement of the outermost sphere, one (or maybe two) beyond the sphere of the 'fixed' stars (the ones whose pattern made the constellations), beyond the sphere of Saturn. In the Copernican system, the fact that the annual component of each planetary motion was a reflection of the annual motion of the Earth allowed one to use observations to calculate the size of each planet's path, and it turned out that there were huge spaces between the planets. Why these particular spaces?

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html (3 of 9) [2/16/2002 11:17:16 PM]

Kepler

Kepler's answer to these questions, described in his Mystery of the Cosmos (Mysterium cosmographicum, Tübingen, 1596), looks bizarre to twentieth-century readers (see the figure on the right). He suggested that if a sphere were drawn to touch the inside of the path of Saturn, and a cube were inscribed in the sphere, then the sphere inscribed in that cube would be the sphere circumscribing the path of Jupiter. Then if a regular tetrahedron were drawn in the sphere inscribing the path of Jupiter, the insphere of the tetrahedron would be the sphere circumscribing the path of Mars, and so inwards, putting the regular dodecahedron between Mars and Earth, the regular icosahedron between Earth and Venus, and the regular octahedron between Venus and Mercury. This explains the number of planets perfectly: there are only five convex regular solids (as is proved in Euclid's Elements , Book 13). It also gives a convincing fit with the sizes of the paths as deduced by Copernicus, the greatest error being less than 10% (which is spectacularly good for a cosmological model even now). Kepler did not express himself in terms of percentage errors, and his is in fact the first mathematical cosmological model, but it is easy to see why he believed that the observational evidence supported his theory. Kepler saw his cosmological theory as providing evidence for the Copernican theory. Before presenting his own theory he gave arguments to establish the plausibility of the Copernican theory itself. Kepler asserts that its advantages over the geocentric theory are in its greater explanatory power. For instance, the Copernican theory can explain why Venus and Mercury are never seen very far from the Sun (they lie between Earth and the Sun) whereas in the geocentric theory there is no explanation of this fact. Kepler lists nine such questions in the first chapter of the Mysterium cosmographicum. Kepler carried out this work while he was teaching in Graz, but the book was seen through the press in Tübingen by Maestlin. The agreement with values deduced from observation was not exact, and Kepler hoped that better observations would improve the agreement, so he sent a copy of the Mysterium cosmographicum to one of the foremost observational astronomers of the time, Tycho Brahe (1546 1601). Tycho, then working in Prague, had in fact already written to Maestlin in search of a mathematical assistant. Kepler got the job. The 'War with Mars' Naturally enough, Tycho's priorities were not the same as Kepler's, and Kepler soon found himself working on the intractable problem of the orbit of Mars [[(See Appendix below)]]. He continued to work on this after Tycho died (in 1601) and Kepler succeeded him as Imperial Mathematician. Conventionally, orbits were compounded of circles, and rather few observational values were required to fix the relative radii and positions of the circles. Tycho had made a huge number of observations and Kepler determined http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html (4 of 9) [2/16/2002 11:17:16 PM]

Kepler

to make the best possible use of them. Essentially, he had so many observations available that once he had constructed a possible orbit he was able to check it against further observations until satisfactory agreement was reached. Kepler concluded that the orbit of Mars was an ellipse with the Sun in one of its foci (a result which when extended to all the planets is now called "Kepler's First Law"), and that a line joining the planet to the Sun swept out equal areas in equal times as the planet described its orbit ("Kepler's Second Law"), that is the area is used as a measure of time. After this work was published in New Astronomy ... (Astronomia nova, ..., Heidelberg, 1609), Kepler found orbits for the other planets, thus establishing that the two laws held for them too. Both laws relate the motion of the planet to the Sun; Kepler's Copernicanism was crucial to his reasoning and to his deductions. The actual process of calculation for Mars was immensely laborious - there are nearly a thousand surviving folio sheets of arithmetic - and Kepler himself refers to this work as 'my war with Mars', but the result was an orbit which agrees with modern results so exactly that the comparison has to make allowance for secular changes in the orbit since Kepler's time. Observational error It was crucial to Kepler's method of checking possible orbits against observations that he have an idea of what should be accepted as adequate agreement. From this arises the first explicit use of the concept of observational error. Kepler may have owed this notion at least partly to Tycho, who made detailed checks on the performance of his instruments (see the biography of Brahe). Optics, and the New Star of 1604 The work on Mars was essentially completed by 1605, but there were delays in getting the book published. Meanwhile, in response to concerns about the different apparent diameter of the Moon when observed directly and when observed using a camera obscura, Kepler did some work on optics, and came up with the first correct mathematical theory of the camera obscura and the first correct explanation of the working of the human eye, with an upside-down picture formed on the retina. These results were published in Supplements to Witelo, on the optical part of astronomy (Ad Vitellionem paralipomena, quibus astronomiae pars optica traditur, Frankfurt, 1604). He also wrote about the New Star of 1604, now usually called 'Kepler's supernova', rejecting numerous explanations, and remarking at one point that of course this star could just be a special creation 'but before we come to [that] I think we should try everything else' (On the New Star, De stella nova, Prague, 1606, Chapter 22, KGW 1, p. 257, line 23). Following Galileo's use of the telescope in discovering the moons of Jupiter, published in his Sidereal Messenger (Venice, 1610), to which Kepler had written an enthusiastic reply (1610), Kepler wrote a study of the properties of lenses (the first such work on optics) in which he presented a new design of telescope, using two convex lenses (Dioptrice, Prague, 1611). This design, in which the final image is inverted, was so successful that it is now usually known not as a Keplerian telescope but simply as the astronomical telescope. Leaving Prague for Linz Kepler's years in Prague were relatively peaceful, and scientifically extremely productive. In fact, even when things went badly, he seems never to have allowed external circumstances to prevent him from getting on with his work. Things began to go very badly in late 1611. First, his seven year old son died. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html (5 of 9) [2/16/2002 11:17:16 PM]

Kepler

Kepler wrote to a friend that this death was particularly hard to bear because the child reminded him so much of himself at that age. Then Kepler's wife died. Then the Emperor Rudolf, whose health was failing, was forced to abdicate in favour of his brother Matthias, who, like Rudolf, was a Catholic but (unlike Rudolf) did not believe in tolerance of Protestants. Kepler had to leave Prague. Before he departed he had his wife's body moved into the son's grave, and wrote a Latin epitaph for them. He and his remaining children moved to Linz (now in Austria). Marriage and wine barrels Kepler seems to have married his first wife, Barbara, for love (though the marriage was arranged through a broker). The second marriage, in 1613, was a matter of practical necessity; he needed someone to look after the children. Kepler's new wife, Susanna, had a crash course in Kepler's character: the dedicatory letter to the resultant book explains that at the wedding celebrations he noticed that the volumes of wine barrels were estimated by means of a rod slipped in diagonally through the bung-hole, and he began to wonder how that could work. The result was a study of the volumes of solids of revolution (New Stereometry of wine barrels ..., Nova stereometria doliorum ..., Linz, 1615) in which Kepler, basing himself on the work of Archimedes, used a resolution into 'indivisibles'. This method was later developed by Bonaventura Cavalieri (c. 1598 - 1547) and is part of the ancestry of the infinitesimal calculus. The Harmony of the World Kepler's main task as Imperial Mathematician was to write astronomical tables, based on Tycho's observations, but what he really wanted to do was write The Harmony of the World, planned since 1599 as a development of his Mystery of the Cosmos. This second work on cosmology (Harmonices mundi libri V, Linz, 1619) presents a more elaborate mathematical model than the earlier one, though the polyhedra are still there. The mathematics in this work includes the first systematic treatment of tessellations, a proof that there are only thirteen convex uniform polyhedra (the Archimedean solids) and the first account of two non-convex regular polyhedra (all in Book 2). The Harmony of the World also contains what is now known as 'Kepler's Third Law', that for any two planets the ratio of the squares of their periods will be the same as the ratio of the cubes of the mean radii of their orbits. From the first, Kepler had sought a rule relating the sizes of the orbits to the periods, but there was no slow series of steps towards this law as there had been towards the other two. In fact, although the Third Law plays an important part in some of the final sections of the printed version of the Harmony of the World, it was not actually discovered until the work was in press. Kepler made last-minute revisions. He himself tells the story of the eventual success: ...and if you want the exact moment in time, it was conceived mentally on 8th March in this year one thousand six hundred and eighteen, but submitted to calculation in an unlucky way, and therefore rejected as false, and finally returning on the 15th of May and adopting a new line of attack, stormed the darkness of my mind. So strong was the support from the combination of my labour of seventeen years on the observations of Brahe and the present study, which conspired together, that at first I believed I was dreaming, and assuming my conclusion among my basic premises. But it is absolutely certain and exact that "the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances ..." (Harmonice mundi Book 5, Chapter 3, trans. Aiton, Duncan and Field, p. 411). Witchcraft trial http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html (6 of 9) [2/16/2002 11:17:16 PM]

Kepler

While Kepler was working on his Harmony of the World, his mother was charged with witchcraft. He enlisted the help of the legal faculty at Tübingen. Katharina Kepler was eventually released, at least partly as a result of technical objections arising from the authorities' failure to follow the correct legal procedures in the use of torture. The surviving documents are chilling. However, Kepler continued to work. In the coach, on his journey to Württemberg to defend his mother, he read a work on music theory by Vincenzo Galilei (c.1520 - 1591, Galileo's father), to which there are numerous references in The Harmony of the World. Astronomical Tables Calculating tables, the normal business for an astronomer, always involved heavy arithmetic. Kepler was accordingly delighted when in 1616 he came across Napier's work on logarithms (published in 1614). However, Maestlin promptly told him first that it was unseemly for a serious mathematician to rejoice over a mere aid to calculation and second that it was unwise to trust logarithms because no-one understood how they worked. (Similar comments were made about computers in the early 1960s.) Kepler's answer to the second objection was to publish a proof of how logarithms worked, based on an impeccably respectable source: Euclid's Elements Book 5. Kepler calculated tables of eight-figure logarithms, which were published with the Rudolphine Tables (Ulm, 1628). The astronomical tables used not only Tycho's observations, but also Kepler's first two laws. All astronomical tables that made use of new observations were accurate for the first few years after publication. What was remarkable about the Rudolphine Tables was that they proved to be accurate over decades. And as the years mounted up, the continued accuracy of the tables was, naturally, seen as an argument for the correctness of Kepler's laws, and thus for the correctness of the heliocentric astronomy. Kepler's fulfilment of his dull official task as Imperial Mathematician led to the fulfilment of his dearest wish, to help establish Copernicanism. Wallenstein By the time the Rudolphine Tables were published Kepler was, in fact, no longer working for the Emperor (he had left Linz in 1626), but for Albrecht von Wallenstein (1583 - 1632), one of the few successful military leaders in the Thirty Years' War (1618 - 1648). Wallenstein, like the emperor Rudolf, expected Kepler to give him advice based on astrology. Kepler naturally had to obey, but repeatedly points out that he does not believe precise predictions can be made. Like most people of the time, Kepler accepted the principle of astrology, that heavenly bodies could influence what happened on Earth (the clearest examples being the Sun causing the seasons and the Moon the tides) but as a Copernican he did not believe in the physical reality of the constellations. His astrology was based only on the angles between the positions of heavenly bodies ('astrological aspects'). He expresses utter contempt for the complicated systems of conventional astrology. Death Kepler died in Regensburg, after a short illness. He was staying in the city on his way to collect some money owing to him in connection with the Rudolphine Tables. He was buried in the local church, but this was destroyed in the course of the Thirty Years' War and nothing remains of the tomb. Historiographic note Much has sometimes been made of supposedly non-rational elements in Kepler's scientific activity. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html (7 of 9) [2/16/2002 11:17:16 PM]

Kepler

Believing astrologers frequently claim his work provides a scientifically respectable antecedent to their own. In his influential Sleepwalkers the late Arthur Koestler made Kepler's battle with Mars into an argument for the inherent irrationality of modern science. There have been many tacit followers of these two persuasions. Both are, however, based on very partial reading of Kepler's work. In particular, Koestler seems not to have had the mathematical expertise to understand Kepler's procedures. Closer study shows Koestler was simply mistaken in his assessment. The truly important non-rational element in Kepler's work is his Christianity. Kepler's extensive and successful use of mathematics makes his work look 'modern', but we are in fact dealing with a Christian Natural Philosopher, for whom understanding the nature of the Universe included understanding the nature of its Creator. Article by: J. V. Field, London Click on this link to see a list of the Glossary entries for this page List of References (121 books/articles)

Some Quotations (16)

A Poster of Johannes Kepler

Mathematicians born in the same country

Cross-references to History Topics

1. Orbits and gravitation 2. Thomas Harriot's manuscripts 3. A brief history of cosmology 4. General relativity 5. The rise of the calculus

Other references in MacTutor

1. Kepler's elliptical orbit for Mars 2. Chronology: 1600 to 1625

Honours awarded to Johannes Kepler (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Kepler

Planetary features

Crater Kepler on Mars

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html (8 of 9) [2/16/2002 11:17:16 PM]

Kepler

Other Web sites

1. Pass Magazine (Kepler's proofs) 2. NASA 3. The Galileo Project 4. High Altitude Observatory 5. Science Museum, Florence 6. Kevin Brown (Kepler and the third law) 7. Thomas C Hales (The Kepler sphere packing problem) 8. The Daily Telegraph (Its solution) 9. AMS (More sphere packing) 10. George W Hart (Kepler's polyhedra) 11. Linda Hall Library (Star Atlas) 12. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kepler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kepler.html (9 of 9) [2/16/2002 11:17:16 PM]

Kerekjarto

Béla Kerékjártó Born: 1 Oct 1898 in Budapest, Hungary Died: 26 June 1946 in Gyöngyos, Hungary

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Béla Kerékjártó received his Ph.D. from Budapest, then taught at Szeged from 1922 and Budapest from 1938. The first three monographs on topology were by Veblen (1922), Kerékjártó (1923), and Lefschetz (1924). Kerékjártó's book sold best but it is a very bad book with large parts of it being wrong. Similarly most of Kerékjártó's papers are full of gaps and mistakes, although around 1940 he wrote some papers which appear to be correct. He never knew the notation for set inclusion, union and intersection. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kerekjarto.html (1 of 2) [2/16/2002 11:17:17 PM]

Kerekjarto

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kerekjarto.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kerekjarto.html (2 of 2) [2/16/2002 11:17:17 PM]

Keynes

John Maynard Keynes Born: 5 June 1883 in Cambridge, Cambridgeshire, England Died: 21 April 1946 in Firle, Sussex, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Maynard Keynes (pronounces "Canes") was born into an academic family. His father, John Nevile Keynes, was a lecturer at the University of Cambridge teaching logic and political economy. John Nevile published Formal Logic four months after John Maynard was born. John Maynard's mother, Florence Ada Brown, was a remarkable woman who was a highly successful author, and also a great pioneer in social reform. It is worth commenting at this stage that, although John Maynard Keynes lived to the age of 63, his parents both outlived him. At the age of seven, Keynes entered Perse School Kindergarten but he learnt more from lessons given at home. Two years later he entered St Faith's preparatory school but there was little sign at this stage that he was an exceptional pupil. As time went by he did begin to show more promise, however, and in 1894 he topped the class for the first time and received a prize for mathematics. By 1896 he was described by the headmaster as (see for example [5]):... head and shoulders above all the other boys in the school. The following year Keynes sat the entrance examination for Eton and came tenth out of the twenty boys who were accepted into the school in that year. He did, however, come first equal in mathematics. Keynes enjoyed his school days in Eton. Harrod writes in [5]:... Eton greatly helped his development. He found there associates who were congenial to him, youths of intellectual distinction with whom he could quickly get on to terms of intimacy on the basis of common interests. Keynes did well at Eton winning the Senior Mathematics Prize in 1899, and again in 1900. But it was not

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Keynes.html (1 of 5) [2/16/2002 11:17:19 PM]

Keynes

only in mathematics that he did well: for example in 1901 he was first in mathematics, first in history, and first in the English essay. In 1902 he won a scholarship to King's College Cambridge in mathematics and classics. Before we leave our description of Keynes' time at Eton, we should note that it was here that he continued with one of his passions (begun at the age of twelve), namely collecting old books. He had purchased 329 old books before he entered the University of Cambridge in October 1902. At Cambridge Keynes was tutored mathematics by E W Hobson whom he called "Hobbema". Although he studied mathematics he was no mathematical genius. His [5]:... logical faculty, his accuracy and his lightning speed of thought made him a thoroughly competent mathematician. He had no specific genius for mathematics; he had to take pains with his work; ... he did not seek out those abstruse regions which are a joy to the heart of the professional mathematician. He had many interests at Cambridge beyond his academic work, spending much time with literary friends, reading, and involving himself in political activity. He put in some effort as the examinations approached to achieve a reasonable degree and he was placed twelfth Wrangler in the Mathematical Tripos of 1905, that is twelfth in the ranked list of those receiving a First Class degree. Immediately following taking the Tripos examinations, Keynes began a serious study of economics, reading major texts on the subject. He did briefly consider taking a second Tripos examination in Economics but decided against it. After a holiday in Switzerland, he returned to Cambridge in October 1905 and attended lectures there by Alfred Marshall on economics. In August 1906 he took the Civil Service examinations and was placed second of the ten who were accepted that year. The top person had first choice of which department to enter, and chose to enter the treasury (which Keynes would have done had he come top). Keynes, with the next choice, entered the India Office. Keynes was very unhappy when he received detailed results of the examination. He came top in logic, psychology, and the essay while his worst subjects were mathematics and economics. He expressed disbelief at both the mathematics and economics results, and commented, probably accurately, that he knew more about economics than his examiners. The India Office did not provide a career to Keynes' liking. He worked mostly on his own work, devoting all his spare time to the study of the theory of probability. He then submitted a dissertation on probability for a Fellowship at King's College. Johnson and Whitehead were appointed as assessors of the dissertation but, after a close contest in March 1908, Keynes was not elected. On 5 June 1908 he resigned from the India Office and, with some financial assistance from his father, went to King's with the hope that he would be successful in the Fellowship competition the following year. Using the detailed comments on his probability dissertation by both Johnson and Whitehead, Keynes worked had to improve it. He also discussed his work with Russell. After submitting a new version of his dissertation on probability, Keynes was elected to a Fellowship in March 1909. The reports were impressive: Whitehead wrote:... his axioms are good; they are simple and few and by the aid of the symbolism he deduces the whole subject from them by rigid reasoning. The very certainty and ease by which he is enabled to solve difficult questions and to detect ambiguities and errors in the work of his predecessors exemplifies and at the same time almost conceals the advance which he has made. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Keynes.html (2 of 5) [2/16/2002 11:17:19 PM]

Keynes

Russell, writing about the book which Keynes eventually published on probability, praised the work highly:The mathematical calculus is astonishingly powerful, considering the very restricted premises which form its foundation... The book as a whole is one which it is impossible to praise too highly and it is hoped that it will stimulate further work on a most important subject which philosophers and logicians have unduly neglected. Keynes now taught economics at Cambridge. He published papers in statistics, in particular he attacked strongly work by Karl Pearson in letters published in 1910 and 1911 in the Journal of the Royal Statistical Society. Whether Pearson or Keynes had the better scientific case is open to question, but there is no doubt that Keynes was by far the more skilful in his style of letter writing, making Pearson (probably unfairly) look rather silly. Keynes also wrote on economics related to India and he published a major book Indian Currency and Finance in 1913. The book is consider a classic and contains a description of the "gold exchange standard". Keynes was appointed secretary of a Commission to examine Indian Finance and Currency in 1913 and he began to seek a publisher for his major treatise on probability based on his fellowship dissertation. His life, however, changed markedly with the beginning of World War I in August 1914. At first he continued much as before, publishing War and the Financial System, August 1914 in the Economic Journal. During the first term of the academic year1914-15 he carried out his duties as normal at Cambridge but already Cambridge was a different place. In November 1914 he published a paper on The City of London and the Bank of England but a letter he wrote at this time shows the effect that the war was having (see for example [5]):For myself I am absolutely and completely desolated. It is utterly unbearable to see day by day the youths going away, first to boredom and discomfort, and then to slaughter. By 1915 Keynes was working at the Treasury where [2]:...he was daily concerned with the economic management of the war. His special responsibility covered relations with allies and the conservation of England's scant supply of foreign currencies. His position at the Treasury meant that he could no longer publish. In particular his treatise on probability had to be put to one side until the war was over. In many ways these war years saw Keynes at the height of his powers and at his most influential. Certainly he had the confidence to believe his opinions were right while others in the highest positions of power could be seriously wrong. Not long after he began to work at the Treasury he was present when Lloyd George, then Chancellor of the Exchequer, made a statement regarding the position in France. He then asked for comments to which Keynes replied:With the utmost respect, I must, if asked for my opinion, tell you that I regard your account as rubbish. After the war ended, Keynes represented the Treasury at the Versailles Peace Conference, but, in June 1919, he resigned on the grounds that the proposals being put forward for German reparations were both unfair and impractical. He was then in a position to publish again and he attacked the conclusions of the Versailles Peace Conference in December 1919 with The Economic Consequences of the Peace. In this work he attacked the leading political figures in no uncertain terms and as a consequence he was never fully trusted by the government again.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Keynes.html (3 of 5) [2/16/2002 11:17:19 PM]

Keynes

In 1920 Keynes began to prepare his Treatise on Probability for publication. This he found a little difficult, for he had not looked at the work for six years. Its publication in 1921 is the most important of his works as far as this mathematical archive is concerned. In this work he argues that probability is a logical relation and so it is objective. A statement involving probability relations has a truth-value independent of people's opinions. In 1926 Ramsey published a paper Truth and probability arguing against these arguments of Keynes. The paper [16] examines the two points of view of Keynes and Ramsey on probability. Other important ideas discussed by Keynes in Treatise on Probability is that probability relations forms only a partially ordered set in the sense that two probabilities cannot necessarily always be compared. Keynes also argues that probability is a basic concept which cannot be reduced to other concepts. Another important period of Keynes' career was during the 1930s. This was a period of unemployment and the depression. Conventional economics could not cope with the extraordinary events which took place leaving traditional economic theory with no answer. Keynes first major work which indicates the direction his ideas were taking away from the conventional approach was A Treatise on Money published in 1930. His most important work giving the culmination of his ideas was The General Theory of Employment, Interest and Money published in 1935-36. The two main messages of this work are [2]:... the existing theory of unemployment nonsense. In a depression ... there was no wage so low that it could eliminate unemployment. Accordingly, it was wicked to blame the unemployed for their plight. The second proposition proposed an alternative explanation about the origins of unemployment and depression. This centred upon aggregate demand i.e. the total spending of consumers, business investors, and public agencies. When aggregate demand was low, sales and jobs suffered. When it was high, all was well. By 1937 Keynes' health began to deteriorate. He would never be fully fit again. However, his expertise was such that he was given an honorary role in the Treasury during World War II. One of the most important projects he was involved in during his last years was the setting up of the International Monetary Fund. There are a couple of other aspects of Keynes' interests which we should comment on. We have already mentioned his interest in old books, which he had from a very young age. He had a similar interest in modern paintings. One of his main interests was in works of the seventeenth and eighteenth centuries and in particular he was fascinated by Newton's manuscripts. In 1936 Newton's papers were sold at Sotheby's and they were dispersed by this sale. Keynes made strenuous attempts to acquire the manuscripts after the sale and these attempts are described in [17]. The year 1942 marked three hundred years from Newton's birth [Newton was born on Christmas day 1642 although this became 4 January 1643 in the new calendar]. Keynes wrote an article Newton, the Man for the celebrations. Unlike most accounts of Newton's life and work which concentrate on Newton's achievements in mathematics and physics, Keynes gave equal weight to Newton's writings on alchemy and religion. The reason for this was that he based his account on the manuscripts of Newton's which he owned and these clearly showed him that, to Newton, his work on these other topics was as important as his work on mathematical physics. In 1942 Keynes was elevated to the peerage and took his seat in the House of Lords, where he sat on the Liberal benches. Around the same time he became chairman of the newly formed Committee for the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Keynes.html (4 of 5) [2/16/2002 11:17:19 PM]

Keynes

Encouragement of Music and the Arts which, before the end of the war, was renamed the British Arts Council. Keynes described the purpose of the Arts Council in a radio broadcast:The purpose of the Arts Council of Great Britain is to create an environment, to breed a spirit, to cultivate an opinion, to offer a stimulus to such purpose that the artist and the public can each sustain and live on the other in that union which has occasionally existed in the past at the great ages of a communal civilised life. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (17 books/articles)

Some Quotations (5)

Mathematicians born in the same country Other references in MacTutor

Chronology: 1920 to 1930

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Keynes.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Keynes.html (5 of 5) [2/16/2002 11:17:19 PM]

Khayyam

Omar Khayyam Born: 18 May 1048 in Nishapur, Persia (now Iran) Died: 4 Dec 1131 in Nishapur, Persia (now Iran)

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Omar Khayyam's full name was Ghiyath al-Din Abu'l-Fath Umar ibn Ibrahim Al-Nisaburi al-Khayyami. A literal translation of the name al-Khayyami (or al-Khayyam) means 'tent maker' and this may have been the trade of Ibrahim his father. Khayyam played on the meaning of his own name when he wrote:Khayyam, who stitched the tents of science, Has fallen in grief's furnace and been suddenly burned, The shears of Fate have cut the tent ropes of his life, And the broker of Hope has sold him for nothing! The political events of the 11th century played a major role in the course of Khayyam's life. The Seljuq Turks were tribes that invaded southwestern Asia in the 11th century and eventually founded an empire that included Mesopotamia, Syria, Palestine, and most of Iran. The Seljuq occupied the grazing grounds of Khorasan and then, between 1038 and 1040, they conquered all of north-eastern Iran. The Seljuq ruler Toghrïl Beg proclaimed himself sultan at Nishapur in 1038 and entered Baghdad in 1055. It was in this difficult unstable military empire, which also had religious problems as it attempted to establish an orthodox Muslim state, that Khayyam grew up. Khayyam studied philosophy at Naishapur and one of his fellow students wrote that he was:... endowed with sharpness of wit and the highest natural powers ... However, this was not an empire in which those of learning, even those as learned as Khayyam, found life easy unless they had the support of a ruler at one of the many courts. Even such patronage would not provide too much stability since local politics and the fortunes of the local military regime decided who at any one time held power. Khayyam himself described the difficulties for men of learning during this period in the introduction to his Treatise on Demonstration of Problems of Algebra (see for example http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Khayyam.html (1 of 5) [2/16/2002 11:17:21 PM]

Khayyam

[1]):I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him. However Khayyam was an outstanding mathematician and astronomer and, despite the difficulties which he described in this quote, he did write several works including Problems of Arithmetic, a book on music and one on algebra before he was 25 years old. In 1070 he moved to Samarkand in Uzbekistan which is one of the oldest cities of Central Asia. There Khayyam was supported by Abu Tahir, a prominent jurist of Samarkand, and this allowed him to write his most famous algebra work, Treatise on Demonstration of Problems of Algebra from which we gave the quote above. We shall describe the mathematical contents of this work later in this biography. Toghril Beg, the founder of the Seljuq dynasty, had made Esfahan the capital of his domains and his grandson Malik-Shah was the ruler of that city from 1073. An invitation was sent to Khayyam from Malik-Shah and from his vizier Nizam al-Mulk asking Khayyam to go to Esfahan to set up an Observatory there. Other leading astronomers were also brought to the Observatory in Esfahan and for 18 years Khayyam led the scientists and produced work of outstanding quality. It was a period of peace during which the political situation allowed Khayyam the opportunity to devote himself entirely to his scholarly work. During this time Khayyam led work on compiling astronomical tables and he also contributed to calendar reform in 1079. Cowell quotes The Calcutta Review No 59:When the Malik Shah determined to reform the calendar, Omar was one of the eight learned men employed to do it, the result was the Jalali era (so called from Jalal-ud-din, one of the king's names) - 'a computation of time,' says Gibbon, 'which surpasses the Julian, and approaches the accuracy of the Gregorian style.' Khayyam measured the length of the year as 365.24219858156 days. Two comments on this result. Firstly it shows an incredible confidence to attempt to give the result to this degree of accuracy. We know now that the length of the year is changing in the sixth decimal place over a person's lifetime. Secondly it is outstandingly accurate. For comparison the length of the year at the end of the 19th century was 365.242196 days, while today it is 365.242190 days. In 1092 political events ended Khayyam's period of peaceful existence. Malik-Shah died in November of that year, a month after his vizier Nizam al-Mulk had been murdered on the road from Esfahan to Baghdad by the terrorist movement called the Assassins. Malik-Shah's second wife took over as ruler for two years but she had argued with Nizam al-Mulk so now those whom he had supported found that support withdrawn. Funding to run the Observatory ceased and Khayyam's calendar reform was put on hold. Khayyam also came under attack from the orthodox Muslims who felt that Khayyam's questioning mind did not conform to the faith. He wrote in his poem the Rubaiyat :http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Khayyam.html (2 of 5) [2/16/2002 11:17:21 PM]

Khayyam

Indeed, the Idols I have loved so long Have done my Credit in Men's Eye much Wrong: Have drowned my Honour in a shallow cup, And sold my reputation for a Song. Despite being out of favour on all sides, Khayyam remained at the Court and tried to regain favour. He wrote a work in which he described former rulers in Iran as men of great honour who had supported public works, science and scholarship. Malik-Shah's third son Sanjar, who was governor of Khorasan, became the overall ruler of the Seljuq empire in 1118. Sometime after this Khayyam left Esfahan and travelled to Merv (now Mary, Turkmenistan) which Sanjar had made the capital of the Seljuq empire. Sanjar created a great centre of Islamic learning in Merv where Khayyam wrote further works on mathematics. The paper [18] by Khayyam is an early work on algebra written before his famous algebra text. In it he considers the problem:Find a point on a quadrant of a circle in such manner that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal. Khayyam shows that this problem is equivalent to solving a second problem:Find a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse. This problem in turn led Khayyam to solve the cubic equation x3 + 200x = 20x2 + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by ruler and compass methods, a result which would not be proved for another 750 years. Khayyam also wrote that he hoped to give a full description of the solution of cubic equations in a later work [18]:If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared. Indeed Khayyam did produce such a work, the Treatise on Demonstration of Problems of Algebra which contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations. Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of al-Khwarizmi). However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations. Khayyam wrote (see for example [9] or [10]):In the science of algebra one encounters problems dependent on certain types of extremely difficult preliminary theorems, whose solution was unsuccessful for most of those who attempted it. As for the Ancients, no work from them dealing with the subject has come down to us; perhaps after having looked for solutions and having examined them, they were

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Khayyam.html (3 of 5) [2/16/2002 11:17:21 PM]

Khayyam

unable to fathom their difficulties; or perhaps their investigations did not require such an examination; or finally, their works on this subject, if they existed, have not been translated into our language. Another achievement in the algebra text is Khayyam's realisation that a cubic equation can have more than one solution. He demonstrated the existence of equations having two solutions, but unfortunately he does not appear to have found that a cubic can have three solutions. He did hope that "arithmetic solutions" might be found one day when he wrote (see for example [1]):Perhaps someone else who comes after us may find it out in the case, when there are not only the first three classes of known powers, namely the number, the thing and the square. The "someone else who comes after us" were in fact del Ferro, Tartaglia and Ferrari in the 16th century. Also in his algebra book, Khayyam refers to another work of his which is now lost. In the lost work Khayyam discusses the Pascal triangle but he was not the first to do so since al-Karaji discussed the Pascal triangle before this date. In fact we can be fairly sure that Khayyam used a method of finding nth roots based on the binomial expansion, and therefore on the binomial coefficients. This follows from the following passage in his algebra book (see for example [1], [9] or [10]):The Indians possess methods for finding the sides of squares and cubes based on such knowledge of the squares of nine figures, that is the square of 1, 2, 3, etc. and also the products formed by multiplying them by each other, i.e. the products of 2, 3 etc. I have composed a work to demonstrate the accuracy of these methods, and have proved that they do lead to the sought aim. I have moreover increased the species, that is I have shown how to find the sides of the square-square, quatro-cube, cubo-cube, etc. to any length, which has not been made before now. the proofs I gave on this occasion are only arithmetic proofs based on the arithmetical parts of Euclid's "Elements". In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to non-euclidean geometry, although this was not his intention. In trying to prove the parallels postulate he accidentally proved properties of figures in non-euclidean geometries. Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios. The importance of Khayyam's contribution is that he examined both Euclid's definition of equality of ratios (which was that first proposed by Eudoxus) and the definition of equality of ratios as proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions. Khayyam proved that the two definitions are equivalent. He also posed the question of whether a ratio can be regarded as a number but leaves the question unanswered. Outside the world of mathematics, Khayyam is best known as a result of Edward Fitzgerald's popular translation in 1859 of nearly 600 short four line poems the Rubaiyat. Khayyam's fame as a poet has caused some to forget his scientific achievements which were much more substantial. Versions of the forms and verses used in the Rubaiyat existed in Persian literature before Khayyam, and only about 120 of the verses can be attributed to him with certainty. Of all the verses, the best known is the following:The Moving Finger writes, and, having writ, Moves on: nor all thy Piety nor Wit Shall lure it back to cancel half a Line, Nor all thy Tears wash out a Word of it.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Khayyam.html (4 of 5) [2/16/2002 11:17:21 PM]

Khayyam

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (27 books/articles)

Some Quotations (8)

A Poster of Omar Khayyam

Mathematicians born in the same country

Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Other references in MacTutor

1. A picture of a mausoleum to Omar Khayyam 2. Khayyam's construction for solving a cubic equation 3. Chronology: 900 to 1100

Honours awarded to Omar Khayyam (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Omar Khayyam

Other Web sites

1. Yi V Takhteyev 2. Muslim scientists 3. MIT (The Rubaiyat) 4. Kevin Brown (On the cubic) 5. Georgia (Geometric solution of the cubic) 6. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Khayyam.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Khayyam.html (5 of 5) [2/16/2002 11:17:21 PM]

Khinchin

Aleksandr Yakovlevich Khinchin Born: 19 July 1894 in Kondrovo, Kaluzhskaya guberniya, Russia Died: 18 Nov 1959 in Moscow, USSR

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aleksandr Yakovlevich Khinchin's father was an engineer. Khinchin attended the technical high school in Moscow where he became fascinated by mathematics. However mathematics was certainly not his only interest when he was at secondary school for he also had a passionate love of poetry and of the theatre. He completed his secondary education in 1911 and entered the Faculty of Physics and Mathematics of Moscow University in that year. At university in Moscow Khinchin worked with Luzin and others. He was an outstanding student being particularly interested in the metric theory of functions and before he graduated in 1916 he had already written his first paper on a generalisation of the Denjoy integral. This first paper began a series of publications by Khinchin on properties of functions which are retained after deleting a set of density zero at a given point. He summarised his contributions to this area with the paper Recherches sur la structure des fonctions measurables in Fundamanta mathematica in 1927. After graduating in 1916, Khinchin remained at Moscow University undertaking research for his dissertation which would allow him to become a university teacher. After a couple of years he began teaching in a number of different colleges both in Moscow and Ivanovo. The town of Ivanovo, east of Moscow, was a centre for the textile industry and it plays a surprisingly important part in the development of Russian mathematics with several of the major figures teaching in the town. Around 1922 Khinchin took up new mathematical interests when he began to study the theory of numbers and probability theory. In the following year he strengthened results of Hardy and Littlewood with his introduction of the iterated logarithm published in Mathematische Zeitschrift. With these ideas

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Khinchin.html (1 of 3) [2/16/2002 11:17:23 PM]

Khinchin

he also strengthened the law of large numbers due to Borel. In 1927 Khinchin was appointed as a professor at Moscow University and, in the same year, he published Basic laws of probability theory. Between 1932 and 1934 he laid the foundations for the theory of stationary random processes culminating in a major paper in Mathematische Annalen in 1934. Khinchin left Moscow in 1935 to spend two years at Saratov University but returned to Moscow University in 1937 to continue his role of building the school of probability theory there in partnership with Kolmogorov and others, including in particular their student Gnedenko. From the 1940s his work changed direction again and this time he became interested in the theory of statistical mechanics. In the last few years of his life his interests turned to developing Shannon's ideas on information theory. We shall look at some of Khinchin's major publications and in this way get a feel for the large number of important contributions he made in a remarkably large range of topics. Some of these publications we have already mentioned in the brief description of his career which we gave above. Khinchin first published the book Continued Fractions in 1936 with a second edition being published in 1949. The book consists of three chapters, the first two of which present the classical theory of continued fractions. The third chapter, the longest and most important, contains an account of Khinchin's own contributions to the topic of the metrical theory of Diophantine approximations. Another contribution by Khinchin to number theory is the short book Three pearls of number theory which appeared in an English translation in 1952. The book Eight lectures on mathematical analysis by Khinchin ran to several editions. It was first published in 1943 and the eight lectures it contains are: Continuum; Limits; Functions; Series; Derivative; Integral; Series expansions of functions; and Differential equations. The book was designed to be used to supplement a standard course on the calculus and gives a careful treatment of some of the basic notions of mathematical analysis. Ivanov, reviewing the fourth edition, wrote:The presentation is smooth, elegant and interesting and makes very enjoyable reading ... Khinchin published Mathematical Principles of Statistical Mechanics in 1943. It showed how to make classical statistical mechanics into a mathematically rigorous subject, developing a consistent presentation of the topic. In 1951 he extended the work of this 1943 book when he published Mathematical foundations of quantum statistics. This new publication on the topic appeared in a German translation in 1956 and then in an English translation in 1960. The book was written in such a way as to be useful both to mathematicians who wanted to become better acquainted with some applications of analysis to physics, and also to physicists who wanted to understand more about the mathematical foundations for their subject. Topics covered included: local limit theorems for sums of identically distributed random variables; the foundations of quantum mechanics; general principles of quantum statistics; the foundations of the statistics of photons; entropy; and the second law of thermodynamics. The book has been rated as being equal in quality to von Neumann's masterpiece Mathematical foundations of quantum mechanics. Khinchin's book Mathematical Foundations of Information Theory , translated into English from the original Russian in 1957, is important. It consists of English translations of two articles: The entropy concept in probability theory and On the basic theorems of information theory which were both published earlier in Russian. The second of these articles provides a refinement of Shannon's concepts of the capacity of a noisy channel and the entropy of a source. Khinchin generalised some of Shannon's

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Khinchin.html (2 of 3) [2/16/2002 11:17:23 PM]

Khinchin

results in this book which was written in an elementary style yet gave a comprehensive account with full details of all the results. In [6] Gnedenko, who was a student of Khinchin, lists 151 publications by Khinchin on the mathematical theory of probability (the list is given again in [4]). Among the many honours which Khinchin received for his work was election to the Soviet Academy of Sciences in 1939 and the award of a State Prize for scientific achievements in the following year. Vere-Jones writes [9]:Khinchin was a fascinating figure ..., not least because of his early enthusiasms for poetry and acting, and his links with such figures of the revolution as the poet Mayakovsky and members of the Moscow Arts Theatre. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Khinchin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Khinchin.html (3 of 3) [2/16/2002 11:17:23 PM]

Killing

Wilhelm Karl Joseph Killing Born: 10 May 1847 in Burbach (near Siegen), Westphalia, Germany Died: 11 Feb 1923 in Münster, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Killing began his university studies at Münster in 1865 but soon moved to Berlin were he was influenced by Kummer and Weierstrass. His doctorate, supervised by Weierstrass, was presented in 1872 and applied the theory of elementary divisors of a matrix to surfaces. From 1868 to 1882 Killing taught at schools in Berlin and Brilon. At Weierstrass's recommendation he was appointed to a chair of mathematics at the Lyzeum Hosianum in Braunsberg. Killing spent 10 years in Braunsberg isolated mathematically but during this period he produced some of the most original mathematics ever produced. He published work on (i) non-euclidean geometry in n-dimensions (1883), (ii)The extension of the concept of space (1886) which contains Killing's original classification of the simple Lie algebras, (iii) Lie's transformation groups. In 1892 he returned to Münster as professor of mathematics and he spent the rest of his life there submerged in teaching, administration and charitable work. At the age of 39 he entered the Order of Franciscans. Killing was a great patriot and collapse of social cohesion in Germany after 1918 caused him much pain in his last years. Lie algebras were introduced by Lie in about 1870 in his work on differential equations. Killing introduced them independently with quite a different purpose since his interest was in non-euclidean

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Killing.html (1 of 2) [2/16/2002 11:17:25 PM]

Killing

geometry. The main tools in the classification of the simple Lie algebras are Cartan subalgebras and the Cartan matrix both first introduced by Killing. He also introduced the idea of a root system which now appears throughout much of the algebra of today. Finally it is worth noting that Killing introduced the term 'characteristic equation' of a matrix. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Wilhelm Killing

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Killing.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Killing.html (2 of 2) [2/16/2002 11:17:25 PM]

Kingman

John Frank Charles Kingman Born: 28 Aug 1939 in Beckenham, Kent, England

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Kingman was educated at Christ's College, Finchley in London. He then entered Pembroke College Cambridge from where he was awarded an M.A. In 1961 he became a Fellow of Pembroke College, being a Smith's Prizeman in 1962. Also in 1962 Kingman was appointed as an assistant lecturer in mathematics. He was a visiting professor at the University of Western Australia in 1963, then he returned to Cambridge where he was promoted to lecturer in mathematics in 1964. In 1965 Kingman moved from Cambridge when he was appointed as reader in mathematics and statistics at the University of Sussex. Sussex was a very new university when Kingman was appointed there since it had only been founded a few years earlier. After only a year at Sussex, Kingman was promoted to a chair of mathematics and statistics in 1966. Two important books by Kingman were published in that year, namely Introduction to Measure and Probability (written jointly with S J Taylor) and The Algebra of Queues. In 1967 he became a member of the International Statistical Institute. Kingman was appointed as professor of mathematics at the University of Oxford in 1969 and he held this post until 1985. Two years after his appointment, in 1971, his tremendous contributions to mathematical statistics were recognised when he was elected a Fellow of the Royal Society. While at Oxford he became a fellow of St Anne's College, holding this position from 1978 until 1985. During his time at Oxford, Kingman held several visiting appointments, in particular at the University of Western Australia in 1974 and the Australian National University in 1978. He held a number of important national positions during this time, such as chairman of the Science Board of the Science Research Council from 1979 to 1981 and then chairman of the Science and Engineering Research Council from 1981 until 1985. In 1985 Kingman was appointed as Vice-Chancellor of the University of Bristol. Although he was at this stage in a position which no longer required him to retain his interests in mathematical statistics, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kingman.html (1 of 3) [2/16/2002 11:17:27 PM]

Kingman

Kingman certainly did not give up his mathematical interests. He was president of the Royal Statistical Society from 1987 to 1989 and president of the London Mathematical Society from 1990 to 1992. He continued his position as vice-president of the Institute of Statisticians which he had held from 1976 and he continued in this role until 1992. In addition to these mathematical activities, he undertook some less mathematical one. He was a member of the board of the British Council between 1986 and 1991 and chairman of the Committee of Inquiry into the Teaching of the English Language in 1987-88. He also held a number of directorships such as IBM UK (1985-95) and Beecham Group (1986-89). He was associated with the British Technology Group, serving on the Council from 1984 until 1992 when he became a director. Kingman's research was mostly in the area of mathematical statistics, more precisely stochastic analysis, random processes, regenerative phenomena and mathematical genetics. He wrote on the theory of Markov processes and published a series of articles on Markov transition probabilities. Among a large number of other topics to which he made major contributions were ergodic theorems, random walks and the theory of queues, in particular applying queuing theory to problems of traffic flow. He also wrote on a number of fascinating topics such as what happens to a piece of string when it is thrown. In 1979 Kingman gave a series of lectures at Iowa State University on the contributions of mathematics to the study of genetic evolution. This was an area in which Kingman had made many important contributions himself and these are detailed in the lecture notes published as Mathematics of Genetic Diversity in the following year. Kingman discusses deterministic models and stochastic models relating to genetic evolution. In 1993 Kingman published Poisson processes which provides a systematic treatment of the subject. A reviewer writes that the book:... fulfill the expectations one might have when a famous elder author writes a book on a classic topic. It gives the basic facts in a clear and lucid way. It is shown how the theory can be applied to interesting problems of astronomy, queuing and traffic etc., and these examples are studied very thoroughly and deeply, giving even the specialist new insights. Kingman has received many honours for his work in mathematics and statistics in addition to those, such as election as a fellow of the Royal Society, mentioned above. He was awarded the Junior Berwick Prize of the London Mathematical Society in 1967, the Guy medal in Silver from the Royal Statistical Society in 1981 and the Royal Medal of the Royal Society in 1983. He has been awarded honours by the universities of Sussex (1983), Southampton (1985), Bristol (1989), West of England and Hannover (1991). Kingman was knighted in 1985. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kingman.html (2 of 3) [2/16/2002 11:17:27 PM]

Kingman

Honours awarded to John Kingman (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1971

Royal Society Royal Medal

Awarded 1983

London Maths Society President

1990 - 1992

LMS Berwick Prize winner

1967

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 1998 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Kingman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kingman.html (3 of 3) [2/16/2002 11:17:27 PM]

Kirchhoff

Gustav Robert Kirchhoff Born: 12 March 1824 in Königsberg, Prussia (now Kaliningrad, Russia) Died: 17 Oct 1887 in Berlin, Germany

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gustav Kirchhoff was a student of Gauss. He taught at Berlin (an unpaid post) from 1847, then at Breslau. In 1854 he was appointed professor of physics at Heidelberg where he collaborated with Bunsen. He was a physicist who made important contributions to the theory of circuits using topology and to elasticity. Kirchhoff's laws, announced in 1854, allow calculation of currents, voltages and resistances of electrical circuits extending the work of Ohm. His work on black body radiation was fundamental in the development of quantum theory. His work on spectrum analysis led on to a study of the composition of light from the Sun. Kirchhoff was the first to explain the dark lines in the Sun's spectrum as caused by absorption of particular wavelengths as the light passes through a gas. This started a new era in astronomy. In 1875 he was appointed to the chair of mathematical physics at Berlin. Disability meant he had to spend much of his life on crutches or in a wheelchair. His best known work is the four volume masterpiece Vorlesungen über mathematische Physik (1876-94). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kirchhoff.html (1 of 2) [2/16/2002 11:17:28 PM]

Kirchhoff

List of References (8 books/articles) Mathematicians born in the same country Cross-references to History Topics

The quantum age begins

Honours awarded to Gustav R Kirchhoff (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1875

Lunar features

Crater Kirchhoff

Other Web sites

1. High Altitude Observatory 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kirchhoff.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kirchhoff.html (2 of 2) [2/16/2002 11:17:28 PM]

Kirkman

Thomas Penyngton Kirkman Born: 31 March 1806 in Bolton (near Manchester), England Died: 4 Feb 1895 in Bowdon (near Manchester), England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Kirkman published over 60 substantial mathematical papers and many more minor ones. He solved the problem of Steiner triples in 1846 in On a Problem in Combinatorics, 6 years before Steiner proposed it. He also constructed finite projective planes. Thomas attended the grammar school in Bolton where he was taught Greek and Latin but no mathematics. He did well at school but although his schoolmaster and the vicar saw that he was a potential Cambridge fellow, Thomas's father could not be persuaded and Thomas was forced to leave school at the age of 14. He worked in his father's office, continuing his study of Greek and Latin in his own time and extending his knowledge of languages by also learning French and German. After 9 years working in the office, Thomas went against his father's wishes and he entered Trinity College Dublin to study mathematics, philosophy, classics and science for his B.A. On returning to England in 1835 he entered the Church of England. He spent five years as a curate, first in Bury, then in Lymm. By 1839 he became vicar in the Parish of Southworth in Lancashire, a position he held for 52 years. As a graduate of Dublin University, Kirkman was naturally interested when Hamilton published his work on quaternions. Kirkman's interest in mathematics was rapidly increasing and his first paper was presented in 1846 when he was 40 years old. It answered a problem which appeared in the Lady's and Gentleman's Diary of 1845 and shows the existence of Steiner systems seven years before Steiner's article which asked whether such systems existed. This work of Kirkman appeared in the Cambridge and Dublin Mathematical Journal. After Steiner asked his question, a solution was given by M Reiss in 1859.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kirkman.html (1 of 3) [2/16/2002 11:17:30 PM]

Kirkman

Kirkman sarcastically wrote ..... how did the Cambridge and Dublin Mathematical Journal Vol II p. 191, contrive to steal so much from a later paper in Crelle's Journal Vol LVI p. 326 on exactly the same problem in combinatorics? Despite Kirkman's clear priority, we call such systems today Steiner systems and not Kirkman systems. In 1848 Kirkman published a work, described by De Morgan as the most curious crochet I ever saw in which Kirkman attempted to make mathematical formulas more memorable by asking the student ...to teach them to your ear and to your tongue, each of which has a memory of its own, by saying them again and again with a sing-song repetition... The book was not popular but it is fair to say that school teaching of mathematics today sometimes resorts to similar memory aids. Kirkman then investigated generalisations of the quaternions. For example the Cayley numbers and generalisations are discussed. He also at this time examined certain questions in geometry. He examined points of congruence of Pascal lines and his work on this area came to be part of standard texts such as Salmon's Conics. Kirkman is best known for the Fifteen Schoolgirls Problem. He published this in the Lady's and Gentleman's Diary of 1850. Fifteen young ladies of a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk abreast more than once. The solution to the Fifteen Schoolgirls Problem is not particularly hard. Cayley published a solution first, then Kirkman published his own solution, which of course he knew before asking the question. Sylvester also studied aspects of this problem and later disputed with Kirkman on who had thought of it first. There is a more general problem of when n schoolgirls can be arranged into n/3 triples on each of (n 1)/2 days so that no two are in the same triple more than once. Clearly n must be congruent to 3 modulo 6 if such a set with n elements exists, but it was not until a paper in 1971 that it was proved that such an arrangement is possible for every such n. As Biggs comments in [2] regarding the Fifteen Schoolgirls Problem, It is unfortunate that such a trifle should overshadow the many more significant contributions which its author was to make to mathematics. Nevertheless it is his most lasting memorial. From 1853 Kirkman began a large piece of work on the enumeration of polyhedra, publishing many major papers in the Royal Society. Kirkman became a Fellow of the Royal Society in 1857, mainly for this work on polyhedra which had been communicated to the Royal Society by Cayley. On seeing that the Académie des Sciences in Paris were awarding a prize for the study of 'group theory' in 1860, Kirkman decided to enter. This meant that he had just two years to become an expert in group theory. Indeed he achieved this and submitted a memoir of high quality. Three memoirs were submitted, the other two by Emile Mathieu and Jordan. The three submissions were praised but no prize awarded. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kirkman.html (2 of 3) [2/16/2002 11:17:30 PM]

Kirkman

Kirkman continued to work on group theory, his last paper on the subject being The complete theory of groups (1863). The paper, which is an abstract of his Grand Prize Memoir, gives a recursive method for compiling lists of transitive groups and a complete list of transitive groups of degree 10 is given. Kirkman also planned to enter for the Grand Prix of the Académie des Sciences of 1861 on the topic of polyhedra. However although much of this work had been completed, he changed his mind after his disappointment in the 1860 competition. He submitted a long work of 21 sections on polyhedra to the Royal Society in 1862. They decided to publish only the first 2 sections which themselves take up over 40 pages of the Proceedings. Again disappointed, Kirkman blamed Cayley and wrote to John Herschel suggesting Cayley wanted to prevent publication because he had a paper of his own on polyhedra. Kirkman continued to work on combinatorial questions. Then in 1884, at the age of 78, he published his first paper on knots. This was followed by a series of papers. In joint work with Tait they produced tables of knots with 8, 9 and 10 crossings. Kirkman continued to study mathematics until his 89th year sending questions and solutions to the Educational Times up to a few months before his death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Thomas Kirkman

Mathematicians born in the same country

Cross-references to History Topics

Mathematical games and recreations

Honours awarded to Thomas Kirkman (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1857

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kirkman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kirkman.html (3 of 3) [2/16/2002 11:17:30 PM]

Kleene

Stephen Cole Kleene Born: 5 Jan 1909 in Hartford, Connecticut, USA Died: 25 Jan 1994 in Madison, Wisconsin, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stephen C Kleene studied for his first degree at Amherst College. He went on to receive a doctorate from Princeton University in 1934, supervised by Church, for a thesis entitled A Theory of Positive Integers in Formal Logic. Then Kleene taught at Princeton until he joined the University of Wisconsin at Madison in 1935. He became a full professor at the University of Wisconsin at Madison in 1948 and remained on the staff there until he retired in 1979. Kleene's research was on the theory of algorithms and recursive functions. He developed the field of recursion theory with Church, Gödel, Turing and others. He contributed to mathematical Intuitionism which had been founded by Brouwer. His work on recursion theory helped to provide the foundations of theoretical computer science. By providing methods of determining which problems are soluble, Kleene's work led to the study of which functions can be computed. At a lecture in the University of Chicago in 1995, Robert Soare described his work in these terms:Kleene's formulation of computable function via six schemata is one of the most succinct and useful, and his previous work on lambda functions played a major role in supporting Church's Thesis that these classes coincide with the intuitively calculable functions. From 1930's on Kleene more than any other mathematician developed the notions of computability and effective process in all their forms both abstract and concrete, both mathematical and philosophical. He tended to lay the foundations for an area and then

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kleene.html (1 of 2) [2/16/2002 11:17:32 PM]

Kleene

move on to the next, as each successive one blossomed into a major research area in his wake. Kleene developed a diverse array of topics in computability: the arithmetical hierarchy, degrees of computability, computable ordinals and hyperarithmetic theory, finite automata and regular sets with enormous consequences for computer science, computability on higher types, recursive realizability for intuitionistic arithmetic with consequences for philosphy and for program correctness in computer science. Kleene's best known books are Introduction to Metamathematics (1952) and Mathematical Logic (1967). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kleene.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kleene.html (2 of 2) [2/16/2002 11:17:32 PM]

Klein

Felix Christian Klein Born: 25 April 1849 in Düsseldorf, Prussia (now Germany) Died: 22 June 1925 in Göttingen, Germany

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Felix Klein is best known for his work in non- euclidean geometry, for his work on the connections between geometry and group theory, and for results in function theory. Klein attended the Gymnasium in Düsseldorf. After graduating, he entered the University of Bonn and studied mathematics and physics there during 1865-1866. He started out on his career with the intention of becoming a physicist. While still studying at University of Bonn, he was appointed to the post of laboratory assistant to Plücker in 1866. Plücker held a chair of mathematics and experimental physics at Bonn but, by the time Klein became his assistant, Plücker's interests had become very firmly rooted in geometry. Klein received his doctorate, which was supervised by Plücker, from the University of Bonn in 1868, with a dissertation Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien- Koordinaten auf eine kanonische Form on line geometry and its applications to mechanics. In his dissertation Klein classified second degree line complexes using Weierstrass's theory of elementary divisors. However in the year Klein received his doctorate Plücker died leaving his major work on the foundations of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes and this work led him to become acquainted with Clebsch. Clebsch moved to Göttingen in 1868 and, during 1869, Klein made visits to Berlin and Paris and Göttingen. In July 1870 Klein was in Paris when Bismarck, the Prussian chancellor, published a message which infuriated the French government. France declared war on Prussia on the 19th of July and Klein felt he could no longer remain in Paris and returned. Then, for a short period, he did military service as a medical orderly before

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein.html (1 of 5) [2/16/2002 11:17:34 PM]

Klein

being appointed as a lecturer at Göttingen in early 1871. Klein was appointed professor at Erlangen, in Bavaria in southern Germany, in 1872. He was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day, and so Klein held a chair from the remarkably early age of 23. However Klein did not build a school at Erlangen where there were only a few students, so he was pleased to be offered a chair at the Technische Hochschule at Munich in 1875. There he, and his colleague Brill, taught advanced courses to large numbers of excellent students and Klein's great talent at teaching was fully expressed. Among the students that Klein taught while at Munich were Hurwitz, von Dyck, Rohn, Runge, Planck, Bianchi and Ricci-Curbastro. Also in 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel. After five years at the Technische Hochschule at Munich, Klein was appointed to a chair of geometry at Leipzig. There he had as colleagues a number of talented young lecturers, including von Dyck, Rohn, Study and Engel. The years 1880 to 1886 that Klein spent at Leipzig were in many ways to fundamentally change his life. As D E Rowe writes in [12]:Leipzig seemed to be a superb outpost for building the kind of school he now had in mind: one that would draw heavily on the abundant riches offered by Riemann's geometric approach to function theory. But unforeseen events and his always delicate health conspired against this plan. .. [In him were] two souls ... one longing for the tranquil scholar's life, the other for the active life of an editor, teacher, and scientific organiser. ... It was during the autumn of 1882 that the first of these two worlds came crashing down upon him ... his health collapsed completely, and throughout the years 1883-1884 he was plagued by depression. His career as a research mathematician essentially over, Klein accepted a chair at the University of Göttingen in 1886. He taught at Göttingen until he retired in 1913 but he now sought to re-establish Göttingen as the foremost mathematics research centre in the world. His own role as the leader of a geometrical school at Leipzig was never transferred to Göttingen. At Göttingen he taught a wide variety of courses, mainly on the interface between mathematics and physics, such as mechanics and potential theory. Klein established a research centre at Göttingen which was to serve as a model for the best mathematical research centres throughout the world. He introduced weekly discussion meetings, a mathematical reading room with a mathematical library. Klein brought Hilbert from Königsberg to join his research team at Göttingen in 1895. The fame of the journal Mathematische Annalen is based on Klein's mathematical and management abilities. The journal was originally founded by Clebsch but only under Klein's management did it first rival, and then surpass in importance, Crelle's Journal. In a sense these journals represented the rival teams of the Berlin school of mathematics who ran Crelle's Journal and the followers of Clebsch who supported the Mathematische Annalen. Klein set up a small team of editors who met regularly and made democratic decisions. The journal specialised in complex analysis, algebraic geometry and invariant theory. It also provided an important outlet for real analysis and the new area of group theory. Klein retired due to ill health in 1913. However he continued to teach mathematics at his home during the years of World War I. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein.html (2 of 5) [2/16/2002 11:17:34 PM]

Klein

It is a little hard to understand the significance of Klein's contributions to geometry. This is not because it is strange to us today, quite the reverse, it has become so much a part of our present mathematical thinking that it is hard for us to realise the novelty of his results and also the fact that they were not universally accepted by all his contemporaries. Klein's first important mathematical discoveries were made in 1870 in collaboration with Lie. They discovered the fundamental properties of the asymptotic lines on the Kummer surface. Further collaboration with Lie followed and they worked on an investigation of W-curves, curves invariant under a group of projective transformations. In fact Lie played an important role in Klein's development, introducing him to the group concept which played a major role in his later work. It is fair to add that Camille Jordan also played a part in teaching Klein about groups. During his time at Göttingen in 1871 Klein made major discoveries regarding geometry. He published two papers On the So-called Non-Euclidean Geometry in which he showed that it was possible to consider euclidean geometry and non-euclidean geometry as special cases of a projective surface with a specific conic section adjoined. This had the remarkable corollary that non-euclidean geometry was consistent if and only if euclidean geometry was consistent. The fact that non-euclidean geometry was at the time still a controversial topic now vanished. Its status was put on an identical footing to euclidean geometry. Cayley never accepted Klein's ideas believing his arguments to be circular. Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm (1872), profoundly influenced mathematical development. This was written for the occasion of Klein's inaugural address when he was appointed professor at Erlangen in 1872 although it was not actually the speech he gave on that occasion. The Erlanger Programm gave a unified approach to geometry which is now the standard accepted view. Transformations play a major role in modern mathematics and Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. In this way the Erlanger Programm defined geometry so that it included both Euclidean geometry and non-Euclidean geometry. However Klein himself saw his work on function theory as his major contribution to mathematics. As W Burau and B Schoenberg write in [1]:Klein considered his work in function theory to be the summit of his work in mathematics. He owed some of his greatest successes to his development of Riemann's ideas and to the intimate alliance he forged between the later and the conception of invariant theory, of number theory and algebra, of group theory, and of multidimensional geometry and the theory of differential equations, especially in his own fields, elliptic modular functions and automorphic functions. By considering the action of the modular group on the complex plane, Klein showed that the fundamental region is moved around to tessellate the plane. In 1879 he looked at the action of PSL(2,7), thought of as an image of the modular group, and obtained an explicit representation of a Riemann surface. He showed it had equation x3y + y3z + z3x = 0 given in homogeneous coordinates as a curve in the projective plane and its group of symmetries was PSL(2,7) of order 168. He wrote Riemanns Theorie der algebraischen Funktionen und ihre Integrale in 1882 which treats function theory in a geometric way

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein.html (3 of 5) [2/16/2002 11:17:34 PM]

Klein

connecting potential theory and conformal mappings. He also used physical ideas in this work, especially those of fluid dynamics. Klein considered equations of degree greater than 4 and was particularly interested in using transcendental methods to solve the general equation of the fifth degree. After building on methods due to Hermite and Kronecker, producing similar results to Brioschi, he went on to completely solve the problem using the group of the icosahedron. This work led him to consider elliptic modular functions which he studied in a series of papers. He developed a theory of automorphic functions, connecting algebraic and geometric results in his important 1884 book on the icosahedron. However Poincaré began publishing an outline of his theory of automorphic functions in 1881 and, as explained in [12], this led to a competition between the two:Klein initiated a correspondence with Poincaré, and soon a friendly rivalry ensued as both sought to formulate and prove a grand uniformization theorem that would serve as a capstone to this theory. Working under great stress, Klein succeeded in formulating such a theorem and in sketching a strategy for proving it. However it was during this work that Klein's health collapsed as mentioned above. With Robert Fricke who came to Leipzig in 1884, Klein wrote a major four volume classic on automorphic and elliptic modular functions produced over the following 20 years. We should also mention the Klein bottle, a one-sided closed surface named after Klein. A Klein bottle cannot be constructed in Euclidean space. It is best pictured as a cylinder looped back through itself to join with its other end. However this is not a continuous surface in 3-space as the surface cannot go through itself without a discontinuity. It is possible to construct a Klein bottle in non-Euclidean space. In the 1890s Klein became interested in mathematical physics, although throughout his career he showed he was never far from this area in attitude. Following from this interest, he wrote an important work on the gyroscope with A Sommerfeld. Later in his career Klein became interested in teaching at school level. W Burau and B Schoenberg write in [1]:Starting in 1900 he began to take a lively interest in mathematical instruction below university level while continuing to pursue his academic functions. An advocate of modernizing mathematics instruction in Germany, in 1905 he played a decisive role in formulating the "Meraner Lehrplanentwürfe". The essential change recommended was the introduction in secondary schools of the rudiments of differential and integral calculus and the function concept. Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Mathematical Congress of 1908. Under his guidance the German branch of the Commission published many volumes on the teaching of mathematics at all levels in Germany. Another project he worked on around the turn of the century was the Enzyklopädie der Mathematischen Wissenschaften. He took an active part in this project, editing with K Müller the four volume section on mechanics. Klein was elected a member of the Royal Society in 1885 and received the Copley medal of the Society

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein.html (4 of 5) [2/16/2002 11:17:34 PM]

Klein

in 1912. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles)

Some Quotations (7)

A Poster of Felix Klein

Mathematicians born in the same country

Cross-references to History Topics

1. The development of group theory 2. Non-Euclidean geometry 3. General relativity

Other references in MacTutor

Chronology: 1870 to 1880

Honours awarded to Felix Klein (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1885

Royal Society Copley Medal

Awarded 1912

LMS De Morgan Medal

Awarded 1893

Other Web sites

1. Munich, Germany (A biography in German) 2. Minnesota (A Klein bottle rotated in 3 and 4 dimensional space) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Klein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein.html (5 of 5) [2/16/2002 11:17:34 PM]

Klein_Oskar

Oskar Klein Born: 15 Sept 1894 in Mörby, Sweden Died: 5 Feb 1977 in Stockholm, Sweden

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oskar Klein was was the youngest son of Sweden' s first rabbi, Gottlieb Klein, who was originally from the Southern Carpathian. Gottlieb Klein received his doctorate from Heidelberg and moved to Sweden in 1883. He evidently instilled an interest in learning in his young son, as Oskar became quite fond of biology at an early age. This interest changed to chemistry around the age of 15 and soon after, in 1910, Svante Arrhenius, at what seems to be the behest of Gottlieb, invited Oskar to work in his laboratory at the Nobel Institute. Here he took up an interest in solubility and he published his first paper in 1912 on the solubility of zinc hydroxide in alkalis. This was the very same year that he finished his secondary education. He waited, however, until 1914 to take the University exam. Arrhenius wanted to send Klein to work with Jean-Baptiste Perrin in his laboratory at the University of Paris but the plan was foiled by the outbreak of World War I. Klein found himself caught up in the tempest and saw military service in 1915 and 1916. After his service concluded, but with the war still raging, he returned to work with Arrhenius. Their work now centred around studying dielectric constants of alcohols in various solvents. During this particular stay in Stockholm, he met Hendrik A Kramers, who, at the time (1917), was a student of Niels Bohr in Copenhagen. Kramers and Klein met several times during the next few years both in Stockholm and in Copenhagen, which was to be Klein's next destination. In 1917 Klein received a fellowship to study abroad and, subsequently, arrived in Copenhagen in 1918. Over the course of the next two years he would travel between Stockholm and Copenhagen performing work for both Bohr and Arrhenius, spending the summer of 1919 with Kramers in Copenhagen, and finally returning to Stockholm in 1920. But that was not to be the end of his Copenhagen experience. In fact, it was merely the beginning. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein_Oskar.html (1 of 6) [2/16/2002 11:17:37 PM]

Klein_Oskar

Bohr traveled to Stockholm in 1920 to visit Klein and convinced him to return to Copenhagen once more to work at Bohr's Institute. Klein agreed and began what would prove to be quite a fruitful relationship that eventually would lead him to his first teaching position. Around this time, Bohr was working with Svein Rosseland on the statistical equilibrium of a mixture of atomic and free electrons. At the time, it was believed that electrons colliding with atoms always lost energy. However, Klein, in conjunction with Rosseland, introduced "collisions of the second kind" where the electrons actually gained energy! Klein continued his work on the other side of the 'molecular aisle' by turning his attention to ions. In fact, this led him to his thesis research in which he studied the forces between ions in strong electrolytes using Gibbs' statistical mechanics. The result was a generalized formulation of Brownian motion. He defended his doctorate in 1921 at Stockholm Högskola and was opposed by Erik Ivar Fredholm the mathematical physicist best known for his work on integral equations and spectral theory. After his successful defence, Klein returned to Copenhagen, later assisting Bohr on a trip to Göttingen. Around this time Klein turned to publishing semi-popular writings on physics. His first work in this new arena was a philosophical paper that was a refutation of an objection to relativity theory by Swedish philosophers. Not surprisingly, it was around this time that he began to look for a job. In 1923, Oskar Klein married Gerda Agnete Koch and moved to Ann Arbor, Michigan to take up a post at the University of Michigan, a post he won with no small thanks to his venerable friend Niels Bohr. His first work in Ann Arbor dealt with the anomalous Zeeman effect which was a problem that arose out of the fact that no one at the time understood the behavior of atoms in a magnetic field. The classical Zeeman effect was explained, in a nutshell, as the splitting of spectral lines by the magnetic field. The problem was that the classical theory only effectively described atoms with a total electron spin of zero. The difference can be seen in the Hamiltonians of the two. For the normal Zeeman effect, the Hamiltonian reads: H1 = e/2mc L . B For the anomalous Zeeman effect, the Hamiltonian becomes: H1 = e/2mc (L + 2S) . B The extra term arises from the intrinsic dipole moment of an object with spin, where S is the spin angular momentum. For the time (1923), this was a fairly large problem to tackle, but Klein did not stop there. He went on to work on the interaction of diatomic molecules with precessing electrons, studying the angular momentum within the molecule itself. The following year, in 1924, he taught a course on electromagnetism and lectured on an electric particle in a combined gravitational and electromagnetic field. This was the beginning of his landmark work on a unified field theory. Klein chose to solve the problem by essentially extending his work to a fifth dimension, though his early unification ideas centred around quantum physics as the catalyst. He did this by setting p52 = m2. Brink [5] has said that Klein was driven by:... the wish to have a formalism which includes the wave aspect and the particle aspect as a limit. After a time Klein argued less and less that quantum physics could lead to a unified picture, in fact he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein_Oskar.html (2 of 6) [2/16/2002 11:17:37 PM]

Klein_Oskar

later abandoned the idea entirely. However, he did see the possibility of unification in five dimensions, which seems to have been present in his initial attempt. At this time, Klein apparently was unaware of the work of Theodor Kaluza. Kaluza, in 1919, sent a paper to Albert Einstein proposing a unification of gravity with Maxwell's theory of light. Einstein initially was uninterested in the paper, but later realized the highly original ideas contained within it and encouraged Kaluza to publish his ideas. In fact the paper was communicated by Einstein himself on 8 December 1921. In 1925, Klein returned to Copenhagen and contracted hepatitis. He was ill for half a year, though he was visited by Heisenberg in July of 1925 and Schrödinger in January of 1926. This was around the time he was finally able to return to work. It was at this time that he finally became aware of Kaluza's work. Wolfgang Pauli communicated this work to him and Klein:... tried to rescue what I could from the shipwreck. Klein's adaptation of Kaluza's work had a major difference from the original in that the extra or fifth dimension was curled up into a ball that was on the order of the Planck length, 10-33 cm. It is important to note, however, that the extra dimension, though curled up, was still Euclidean in nature. Basically, the fifth coordinate was not observable but was a physical quantity that was conjugate to the electrical charge. As Kragh [4] explains, Klein attempted to explain the atomicity of electricity as a quantum law. He also attempted to account for the electron and the proton. Klein assumed the fifth dimension to be periodic with a period l = c(2k)1/2/e where e was the charge of the electron and k was Einstein's constant of gravitation. The dimension was on the order of the Planck length. Klein's results were published in Nature in the autumn of 1926 and generated interest from such eminent theorists as Vladimir Fock, Leon Rosenfeld, Louis de Broglie, and Dirk Struik. Unfortunately, despite a lot of initial interest in unification, most physicists eventually went on to more promising and experimentally testable research leaving Kaluza-Klein theory to be explored by another generation of physicists nearly half a century later. In Klein's own words:Dirac may well say that my main trouble came from trying to solve too many problems at a time. It was also in 1926 that Klein was appointed as docent at Lund University and became, for the next five years, Bohr's closest collaborator both on correspondence and complimentarity, and apparently contributed to the development of the uncertainty principle, as Heisenberg recalled:After several weeks of discussion, which were not devoid of stress, we soon concluded, not least thanks to Oskar Klein's participation, that we really meant the same, and that the uncertainty relations were just a special case of the more general complementarity principle. In fact, 1926 was a banner year for Klein. In addition to finally recovering from the hepatitis and becoming docent at Lund, it was in this same year that he made his next great theoretical breakthrough. In a paper in which he determined the atomic transition probabilities (prior to Dirac), he introduced the initial form of what would become known as the Klein-Gordon equation. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein_Oskar.html (3 of 6) [2/16/2002 11:17:37 PM]

Klein_Oskar

The Klein-Gordon equation was the first relativistic wave equation. The equation can be written:

It is interesting to note that this equation appeared exactly as it has been written in David Bohm's 1951 book Quantum Theory but was not called the Klein-Gordon equation. However, Bethe and Jackiw's Intermediate Quantum Mechanics, originally written in 1964, does refer to the same equation as the Klein-Gordon equation. Klein and Walter Gordon were thus eventually honoured with having the equation named after them, though it seems to have taken over a quarter of a century to receive the honour. Oddly enough, Schrödinger himself privately developed a relativistic wave equation from his original wave equation, which, in reality, was not that difficult to do, and did so prior to Klein and Gordon, though he never published his results. The trouble came when the equation did not result in the correct fine structure of the hydrogen atom and when Pauli introduced the concept of spin a year later (1927). The equation turned out to be incompatible with spin and, as a result, is only useful for calculations involving spinless particles. But, nonetheless, it was an important point in quantum theory and, along with his unification theory, was to ensure a lasting legacy for Klein and cemented 1926 as a pivotal year in his life. In the years following 1926, Klein turned to teaching and continued his research, though possibly at a reduced pace. Brink [5] quotes a friend and mentor to Klein as having said:You will now fulfill the words: go and teach the people. Your great pedagogical talents always were one of your strongest qualities. I am not of the opinion that finding new laws of nature and indicating new directions is one of your great strengths, although you always have developed a certain ambition in this direction. In 1927, Klein was appointed Lektor in Copenhagen but nonetheless continued his research working with Pascual Jordan on the second quantization in quantum mechanics. In his work with Jordan, he demonstrated the close connection between quantum fields and quantum statistics. It was known that second quantization guarantees that photons obey Bose-Einstein statistics, but Klein showed that second quantization is not confined to free particles only. He and Jordan showed that one can quantize the non-relativistic Schrödinger equation and, in honour of this work, he was the recipient of yet another named mathematical tool, the Jordan-Klein matrices. In subsequent years he collaborated with the Japanese physicist Yoshio Nishina who was in Copenhagen on an extended research visit and worked on the problem of Compton scattering of a Dirac electron. Despite the so-called Klein paradox, that being that the positron was not completely understood by physicists, he was able to convince physicists of the soundness of Dirac's relativistic wave equation. His continued work included the quantum mechanics of the second law of thermodynamics and Klein's lemma. In 1930, he was offered Fredholm's position at Stockholm Högskala and he finally returned to his native city to take up a post that he held until his retirement in 1962. During the 1930's, Klein helped many refugee physicists who were expelled from Germany and other nations largely due to their Jewish heritage. Of the many he helped, one included Walter Gordon who

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein_Oskar.html (4 of 6) [2/16/2002 11:17:37 PM]

Klein_Oskar

would later join Klein in being the beneficiaries of the named equation we have just discussed. In 1943, Klein also aided in Bohr's escape from Copenhagen. During the 1930's Klein also found time to attend conferences, not the least of which included the 1938 Warsaw Conference where he spoke on (almost) non-Abelian gauge theories. This conference included some of the leading theorists of the day including Sir Arthur Eddington, Eugene Wigner, and others. It was at this conference that Klein suggested that a spin-1 particle mediated beta decay and played a role in weak interactions in a similar manner to the photon in electromagnetism. Klein's hypothesis was yet another crack at a unified field theory, this time in attempt to unify the strong, weak, and electromagnetic forces. The work was not noticed until nearly twenty years later when it was resurrected by Julian Schwinger in 1957. In the 1940's Klein worked on a wide variety of subjects including superconductivity (with Jens Lindhard in 1945), biochemistry, universal p-decay, general relativity, and stellar evolution. Sometime after 1947 he, and independently Giovanni Puppi, realized that both the electron and the particles.

-meson were "weak"

In the 1950's and 1960's Klein remained active, addressing the 11th Solvay Conference in 1958, developing a new model for cosmology in conjunction with Hannes Alfven in 1963, and tackling Einstein's General Relativity in a paper published in Astrophisica Norvegica in 1964. During his later years, he also became very interested in philosophy and especially in analogies between science and religion. In addition, he took to writing a few popular books, most of which are out of print. Oskar Klein died in Stockholm, one of the finest theoretical physicists of the twentieth century. Article by: Ian Durham, University of St Andrews. Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other Web sites

1. Niels Bohr Archive (Oskar Klein Papers) 2. J P van der Schaar (Kaluza-Klein Theory) 3. University of Michigan, USA (Oskar Klein Meeting) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein_Oskar.html (5 of 6) [2/16/2002 11:17:37 PM]

Klein_Oskar

Glossary index

Quotations index

Mathematicians of the day JOC/EFR January 2001

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Klein_Oskar.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klein_Oskar.html (6 of 6) [2/16/2002 11:17:37 PM]

Klingenberg

Wilhelm P A Klingenberg Born: 28 Jan 1924 in Rostock, Mecklenburg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Klingenberg's family moved to Berlin in 1934 and there Klingenberg attended schools where he learnt Latin, Greek and French but he had to study mathematics on his own. He entered the Joachimsthalsces Gymnasium in 1937 and received his school diploma in 1941. He applied to enter the University of Berlin but this was not permitted and he had to serve in the army. He writes in [1]:When the end of the war finally gave me my freedom, I changed my handwriting and started looking for a place to study. The devastated and Soviet occupied city of Berlin was out of the question, Göttingen and Hamburg were filled up, so I went to Kiel University. Klingenberg obtained a doctorate in 1950 with a thesis on affine differential geometry. from 1950 to 1952 he was a research assistant at Kiel where F Bachmann interested him in the foundations of geometry. At this time he solved a problem on equivalences of configurations in an affine plane which Ruth Moufang had worked on. Blaschke advised him regarding trips to Italy and he spent time in 1952/53 at the University of Rome where he was strongly influenced by F Severi, E Bompiani and Beniamino Segre. After returning to Germany he completed his habilitation thesis at Hamburg and then obtained a permanent position at Göttingen working with Reidemeister. He wrote in [1]:I have fond memories of our years there - Reidemeister had a brilliant mind and a wide range of interests, his wife Elisabeth was a renowned photographer. Klingenberg spent 1954/55 at Bloomington in the United States visiting Morse at Princeton during his time at Bloomington. His interests had turned away from affine and projective differential geometry and turned towards Riemannian geometry. Although he remained on the staff at Göttingen until 1963,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klingenberg.html (1 of 2) [2/16/2002 11:17:39 PM]

Klingenberg

Klingenberg spent 1956/57 and 1957/58 at the Institute for Advanced Study at Princeton. He also spent 1962 at the University of California at Berkeley at the invitation of S S Chern. Klingenberg wrote in [1]:I had met him in Hamburg in 1953 when he visited his former teacher Blaschke, and since that time he had actively helped my career whenever he had a chance to do so. While at Berkeley, Klingenberg received offers of chairs at Würzburg and Mainz - he chose Mainz. Three years later, in 1966, he was offered chairs at Zurich and at Bonn. He chose, not without some difficulty, to accept the offer from Bonn. However Bonn grew rapidly with additional staff and students:... and some of the intimate charm of a close-knit group thereby went down the drain. Not without some pain and struggle, I finally accepted the change and concentrated my activities on my own differential geometry group. Klingenberg worked during his years at Bonn on closed geodesics. He retired in 1989. His major books include A course in differential geometry (1978), Lectures on closed geodesics (1978) and Riemannian geometry (1982). Of this last work Klingenberg comments:It was the first book on this subject since the monograph of L P Eisenhart in 1926. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Klingenberg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klingenberg.html (2 of 2) [2/16/2002 11:17:39 PM]

Kloosterman

Hendrik Douwe Kloosterman Born: 9 April 1900 in Rottevalle, The Netherlands Died: 1968 in Leiden, The Netherlands

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hendrik Kloosterman was brought up as a young boy in the village of Rottevalle which is a small farming village. Although he began his schooling in Rottevalle, he completed his studies at a high school in The Hague. After graduating from the high school, Kloosterman entered the University of Leiden. Here he studied mathematics and graduated with a Master's degree in 1922. The education he received in the mathematics department was a good one, but the department did not have stars of the quality that worked in Leiden on mathematical physics. Lorentz, although officially retired from 1912, continued to lecture at Leiden and his successor Ehrenfest was an outstanding theoretical physicist whose friends Niels Bohr and Einstein were frequent visitors to Leiden. Kloosterman continued his studies at Leiden with J C Kluyver as the supervisor of his number theory thesis. It was Ehrenfest, however, who looked after Kloosterman as he would his own students, making sure that he had the opportunity to study with the experts on the subject of his thesis. Ehrenfest arranged for Kloosterman to spend some time in Copenhagen working with Harald Bohr, Niels Bohr's brother. The topic which Kloosterman was working on for his doctorate was concerned with Waring type problems. Kloosterman was examining the number of solutions in integers xn, to the equation m = a1x12 + ax2 + ... + asxs2 (*) The method which Kloosterman was using was based on one due to Littlewood and Hardy so the most natural person to advise Kloosterman was G H Hardy. Again it was due to Ehrenfest that Kloosterman was able to visit Hardy in Oxford.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kloosterman.html (1 of 3) [2/16/2002 11:17:41 PM]

Kloosterman

Kloosterman presented his doctoral thesis to the University of Leiden in 1924. He had managed to find, provided s 5 and the an satisfy suitable congruence conditions, an asymptotic formula for the number of solution to the equation (*). Under these conditions (1) always has a solution for large values of m. However for s = 4 his application of the Littlewood-Hardy method failed and Kloosterman noted in his thesis that it is rather strange that this powerful technique fails to show Lagrange's result that every positive integer is the sum of four squares. The case when s = 4 which Kloosterman had failed to solve in his thesis gave him the challenge which he attacked once his PhD had been awarded. His solution of this case appeared in his paper On the representation of numbers in the form ax2 + by2 + cz2 + dt2 which was published in Acta Mathematica in 1926. In this paper Kloosterman introduced what are today called 'Kloosterman sums'. These have proved important in many areas of number theory. The award of a Rockefeller Scholarship allowed Kloosterman to spend 1926-27 at the University of Göttingen and 1927-28 at the University of Hamburg. During his visit to Hamburg Kloosterman applied his idea of 'Kloosterman sums' to obtain estimates for the Fourier coefficients of modular forms. After this two years of travel, Kloosterman was offered a post at the University of Münster. He accepted this position and remained there for two years before returning to the University of Leiden in 1930 to a post equivalent to that of associate professor. Springer writes in [4]:This was mainly a teaching position. Kloosterman turned out to be an exceptional teacher. He was able to expose with great clarity and great economy the essentials of a piece of mathematics, be it elementary or advanced. In 1941 the University of Leiden closed during the German occupation of The Netherlands. This in fact presented an opportunity to Kloosterman to undertake research since he had no teaching duties. The university remained closed unto 1945 and the outcome of this period was two major publications on the irreducible representations of finite groups. The group he studied was the special linear group of 2 by 2 matrices over the ring of integers modulo pn. Schur had solved the problem for the case n = 1, where the matrices are over a field, and the case of n = 2 had been solved in the 1930s. Kloosterman solved the general case in two papers The behaviour of general theta functions under the modular group and the characters of binary modular congruence groups which occupy 130 pages of the Annals of Mathematics in 1946. Kloosterman was promoted to professor at the University of Leiden in 1947, a post he retained until his death. In [4], as well as looking at Kloosterman's contributions, Springer looks at further developments of his techniques. He writes:Although he was not a prolific writer, his work had a significant impact and is still of considerable interest. Article by: J J O'Connor and E F Robertson List of References (6 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kloosterman.html (2 of 3) [2/16/2002 11:17:41 PM]

Kloosterman

Mathematicians born in the same country Other Web sites

1. J P Murre 2. AMS

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kloosterman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kloosterman.html (3 of 3) [2/16/2002 11:17:41 PM]

Klugel

Georg Simon Klügel Born: 19 Aug 1739 in Hamburg, Germany Died: 4 Aug 1812 in Halle, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Klügel's father was a businessman. Klügel attended the Johanneum Grammar School, the renowned humanistic school in Hamburg. From that school he progressed to the Hamburg Gymnasium Academicum where he received a solid mathematical education. He did not decide to follow a mathematical career at this stage, however, and he entered Göttingen University in 1760 with the intention of reading for a degree in theology. At Göttingen University Klügel took a mathematics course as part of his degree and so he met Kaestner who quickly saw his talents in mathematics. Soon Klügel was fascinated by the topic was ready to follow Kaestner's advice and change his course to read for a degree in mathematics. Again following Kaestner's advice he wrote a thesis on the parallel postulate. In this work he listed nearly 30 attempts to prove the fifth axiom and correctly concluded that the 'proofs' were all false. His work is cited by almost all later contributors to non-euclidean geometry. He defended his thesis on 20 August 1763 and after this he continued to undertake mathematical research in Göttingen. Klügel remained in Göttingen until 1765 when he moved to Hannover to take up the appointment as editor of the Intelligenzblatt. In 1767 he was appointed professor of mathematics at Helmstedt then he moved to the chair of mathematics and physics at the University of Halle. He remained in this post for the rest of his career. It was in Helmstedt and Halle that Klügel made his most important contributions to mathematics. These were somewhat of a mixture between encyclopaedic style accumulation of facts together with some real innovative ideas. Klügel made an exceptional contribution to trigonometry, unifying formulas and introducing the concept of trigonometric function, in his Analytische Trigonometrie. Euler, who studied similar problems 9 years later, in some respects achieved less than Klügel in this area. Folta writes in [1]:Klügel's trigonometry was very modern for its time and was exceptional among the contemporary textbooks. It was his mathematical dictionary, however, which led to his fame. This was a three volume work which appeared between 1803 and 1808. In 1808 Klügel became seriously ill and could do no further work on the project. Another three volumes were added between 1823 and 1836 by Mollweide and Grunert and

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klugel.html (1 of 2) [2/16/2002 11:17:42 PM]

Klugel

the dictionary was widely used for several generations making Klügel's name widely known. Among the honours which Klügel received for his contributions to mathematics was election to the Berlin Academy which took place on 27 January 1803. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Klugel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Klugel.html (2 of 2) [2/16/2002 11:17:42 PM]

Kneser

Adolf Kneser Born: 19 March 1862 in Grüssow, Germany Died: 24 Jan 1930 in Breslau, Germany (now Wroclaw, Poland)

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adolf Kneser was taught by Kronecker and also influenced by Weierstrass. After writing a thesis on algebraic functions and equations, he worked on space curves. Kneser was appointed to the chair in Dorpat, and it was while he was holding the chair there that his son Hellmuth Kneser was born. Later Adolf Kneser was appointed to the chair of mathematics at Breslau where he spent the rest of his career. Adolf Kneser's main work was mainly in two areas. One area was that of linear differential equations; in particular he worked on the Sturm-Liouville problem and integral equations in general. He wrote an important text on integral equations. The second main area of his work was the calculus of variations. Wielandt, writing in his obituary of Hellmuth Kneser, describes Adolf Kneser as:... the first to introduce Hilbert's new methods into analysis in his textbook on integral equations. He devoted himself to the task of putting general results into concrete form by applying them to the functions of mathematical physics. In a sense, he made the boundary between the old and new mathematics his field of work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kneser.html (1 of 2) [2/16/2002 11:17:44 PM]

Kneser

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kneser.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kneser.html (2 of 2) [2/16/2002 11:17:44 PM]

Kneser_Hellmuth

Hellmuth Kneser Born: 16 April 1898 in Dorpat, Russia (now Tartu, Estonia) Died: 23 August 1973 in Tübingen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hellmuth Kneser was the son of Adolf Kneser. Hellmuth entered the University of Breslau in 1916 where his father was the Professor of Mathematics. Schmidt's lectures at Breslau were to prove an important influence on Hellmuth Kneser's mathematical development. From Breslau Kneser went to Göttingen to undertake research. His doctoral studies there were directed by Hilbert and he submitted a dissertation on the mathematics of quantum mechanics in 1921 Untersuchungen zur Quantentheorie. After the award of his doctorate Kneser remained at Göttingen. There, after one year, he was appointed to a teaching post on the strength of impressive work determining all the regular families of curves on closed surfaces. His first student was Baer and, at Göttingen, Kneser supervised Baer's doctoral thesis on the classification of curves on surfaces. Kneser did not remain long at Göttingen for, in 1925, he succeeded Radon to a chair in Greifswald. Kneser spent twelve years at Greifswald before he accepted the chair at Tübingen in 1937. He played an important role in assisting Wilhelm Süss to found the Mathematical Research Institute at Oberwolfach in 1944. When the survival of the Institute became increasingly difficult in the years following World War II, it was Kneser's support which proved significant in the battle to retain this wonderful asset for mathematical research. Large numbers of mathematicians like myself [EFR] who have benefited from visits to this unique conference centre must have said a quiet thank you to Süss, Kneser and their colleagues. When Süss died in 1958 it was Kneser who took over the scientific leadership of the Oberwolfach Institute. Describing the areas of mathematics on which Kneser worked is difficult since his work was so wide http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kneser_Hellmuth.html (1 of 3) [2/16/2002 11:17:45 PM]

Kneser_Hellmuth

ranging throughout mathematics. In fact he made a very definite decision after completing his doctoral dissertation that he would refuse to specialise. As Wielandt writes in [2]:He wanted to gain an overview over and an opinion on all parts of his science and be able to do research in each area. He would get close to the realisation of this, in his words, bold desire to an extent that filled his colleagues with amazement, but at times threatened to discourage his students. After his doctoral work on quantum theory he turned toward topology and the theory of analytic functions in several indeterminates. While at Greifswald he really achieved his aim of working in all areas of mathematics. He published 30 papers in the time he held the chair there, publishing important contributions in every area of current interest. Kneser published on sums of squares in fields, on groups, on non-Euclidean geometry, on Harald Bohr's almost periodic functions, on iteration of analytic functions, on the differential geometry of manifolds, on local uniformisation and boundary values. He succeeded in pushing forward Weierstrass and Hadamard's ideas to open up the area of the value distribution of meromorphic functions. Kneser, writing of his work on this last topic said:I hope that this theory will also prove fruitful for the special functions used in analysis, this has to be required of a new theory, in particular, if one considers that the general theory of rational functions of one indeterminate came from the treatment of special functions, namely the gamma and sigma functions by Weierstrass and of the Riemann zeta function by Hadamard. After Kneser moved to Tübingen the emphasis in his work changed. Although he still produced papers of great significance, he now became interested in a variety of other topics related to the teaching and the relation of mathematics to other sciences. It was not just the relation between mathematics and the physical sciences that fascinated him. He now became interested in the mathematical theory of economics and of sociology. As a mathematical basis of these topics he studied applications to them of game theory. Still, despite his ever widening range of activities such as organising teaching seminars and courses for school teacher of mathematics, his research continued to answer fundamental questions. For example he produced a beautiful solution to the functional equation f( f(x) ) = ex which he published in 1950, and the deep understanding he achieved of the strange properties of manifolds without a countable basis of neighbourhoods between 1958 and 1964. Wielandt comments in [2] on Kneser's influence and personality:The high reputation, which Kneser enjoyed as a mathematician with an unusually broad horizon, predetermined him to take over tasks with a wide area of influence; despite his rather shy nature he never avoided these responsibilities. For many years he made his well-informed judgement available to mathematical publications as editor: to the Mathematische Zeitschrift, Archiv der Mathematik and the Aequationes Matheamticae. He received many honours. He was elected President of the Deutsche Mathematiker Vereinigung and served on the executive committee of the International Mathematical Union. His work is summed up in [2]:Kneser served his science for years with his vision, care and tactfulness... http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kneser_Hellmuth.html (2 of 3) [2/16/2002 11:17:45 PM]

Kneser_Hellmuth

Hellmuth Kneser was the son of Adolf Kneser. Martin Kneser, another mathematician, is Hellmuth Kneser's son. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Cross-references to History Topics

The fundamental theorem of algebra

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kneser_Hellmuth.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kneser_Hellmuth.html (3 of 3) [2/16/2002 11:17:45 PM]

Knopp

Konrad Hermann Theodor Knopp Born: 22 July 1882 in Berlin, Germany Died: 20 April 1957 in Annecy, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Konrad Knopp spent one semester at the University of Lausanne in 1901. He then went to the University of Berlin receiving a qualification to teach in 1906 and a doctorate in 1907. Knopp taught in Nagasaki in western Kyushu, Japan and travelled in India and China. In 1910 he returned to Germany and married the painter Gertrud Kressner. They moved to the German- Chinese academy in Tsingtao, eastern Shantung province, China. Germany had occupied Tsingtao in 1897, modern port facilities were installed and a modern European-style city was created. In 1911 Knopp and his wife returned from Tsingtao to Germany and Knopp became an officer in the army, being wounded in World War I. After 1914 he taught at Berlin University, at Königsberg where he was a professor in 1915 and at Tübingen from 1926 until he retired in 1950. Knopp worked on generalised limits and wrote excellent books on complex functions. He was the co-founder of Mathematische Zeitschrift in 1918. He was editor from 1934 to 1952. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) A Poster of Konrad Knopp

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Knopp.html (1 of 2) [2/16/2002 11:17:47 PM]

Knopp

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Knopp.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Knopp.html (2 of 2) [2/16/2002 11:17:47 PM]

Kober

Hermann Kober Born: 1888 in Beuthen (now Bytom), Upper Silesia (now Poland) Died: 4 Oct 1973 in Birmingham, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hermann Kober studied at Breslau and Göttingen where he was one of Landau's first students. His doctorate in 1911 was directed by Kneser and his thesis studied the calculus of variations. He became a teacher at Breslau until 1934 when, as a Jew, he was forced out of his post. He then taught at the Jewish school in Breslau but spent much time at Cambridge doing research. Hardy helped him obtain a research grant at Birmingham University and he emigrated to England in 1939. Kober was a highly productive mathematician working on special functions, functional analysis (in this area Kober's Theorem is named after him), approximation theory and the theory of functions of a real variable. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kober.html (1 of 2) [2/16/2002 11:17:49 PM]

Kober

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kober.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kober.html (2 of 2) [2/16/2002 11:17:49 PM]

Koch

Niels Fabian Helge von Koch Born: 25 Jan 1870 in Stockholm, Sweden Died: 11 March 1924 in Danderyd, Stockholm, Sweden

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Helge von Koch's father was Richert Vogt von Koch, who had a military career, and his mother was Agathe Henriette Wrede. Von Koch attended a good school in Stockholm, completing his studies there in 1887. He then entered Stockholm University. Stockholm University was the third university in Sweden and it was planned from 1865, opening in 1880 with Mittag-Leffler as its first professor of mathematics. We should note that although we shall refer to it here by its present name of Stockholm University, it was known in Sweden as Stockholms Högskola until 1950 which literally means "Stockholm High School". Von Koch spent some time at Uppsala University from 1888. He was a student of Mittag-Leffler at Stockholm University. Von Koch's first results were on infinitely many linear equations in infinitely many unknowns. In 1891 he wrote the first of two papers on applications of infinite determinants to solving systems of differential equations with analytic coefficients. The methods he used were based on those published by Poincaré about six years earlier. The second of von Koch's papers was published in 1892, the year in which von Koch was awarded a doctorate for his thesis which contained the results of the two papers. Von Koch was awarded a doctorate in mathematics by Stockholm University on 26 May 1892. Garding writes in [2] that his doctoral thesis was:... a fantastically mature work ... Bernkoff writes in [1], however, that this work by von Koch:... cannot be called pioneering. His results were all fairly accessible, although many of the calculations are lengthy. He was aware, through a knowledge of Poincaré's work, of the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Koch.html (1 of 3) [2/16/2002 11:17:50 PM]

Koch

possibility of obtaining pathological results but did little to explore them. Yet this work can be said to be the first step on the long road which eventually led to functional analysis, since it provided Fredholm with the key for the solution of his integral equation. Garding writes in [2]:After the thesis von Koch wrote many papers, among others some on infinite determinants, for instance in 1901, but the subject did not have many possibilities for extension and growth and present interest is nil. Between the years 1893 and 1905 von Koch had several appointements as an assistant professor of mathematics. He failed in his application for the chair of algebra and number theory at Uppsala University. In 1905 Bendixson, who had also been a student of Mittag-Leffler, resigned his professorship at KTH, (in Swedish Kungliga Teknologiska Högskolan; in English the Royal Technological Institute in Stockholm), when he accepted a chair at Stockholm University. Von Koch was then appointed to the chair of pure mathematics at the Royal Technological Institute in Stockholm. In July 1911 von Koch succeeded Mittag-Leffler as professor of mathematics at Stockholm University. Von Koch is famous for the Koch curve which appears in his paper Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes plane published in 1906. This is constructed by dividing a line into three equal parts and replacing the middle segment by the other two sides of an equilateral triangle constructed on the middle segment. Repeat on each of the (now 4) segments. Repeat indefinitely. It gives a continuous curve which is of infinite length and nowhere differentiable. If one starts with an equilateral triangle and applies the construction, one gets the von Koch snowflake (sometimes called the von Koch star) as the limit of the construction. Here is a Von Koch's snowflake. The von Koch snowflake is a continuous curve which does not have a tangent at any point. Von Koch's 1906 paper mainly consists of a proof of this fact. He also shows in the paper that there are two functions f and g which are both nowhere differentiable such that the snowflake curve is x = f(t), y = g(t) where -1 t 1. The first person to give an example of an analytic construction of a function which is continuous but nowhere differentiable was Weierstrass. At the end of his paper, von Koch gives a geometric construction, based on the von Koch curve, of such a function which he also expresses analytically. Von Koch also wrote papers on number theory, in particular he wrote several papers on the prime number theorem such as Sur la distribution des nombres premiers in 1901 and Contribution à la théorie des nombres premiers in 1910.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Koch.html (2 of 3) [2/16/2002 11:17:50 PM]

Koch

A Poster of Helge von Koch

Mathematicians born in the same country

Other references in MacTutor

1. Von Koch's curve 2. Chronology: 1900 to 1910

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Koch.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Koch.html (3 of 3) [2/16/2002 11:17:50 PM]

Kochin

Nikolai Yevgrafovich Kochin Born: 1901 in St Petersburg, Russia Died: 31 Dec 1944 in Moscow, USSR Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kochin graduated from Petrograd University (St Petersburg changed its name to Petrograd in 1914 and then to Leningrad in 1924) in 1923. He taught mathematics and mechanics there (called Leningrad University by this time) from 1924 to 1934. Kochin moved Moscow University in 1934 and worked there for the rest of his life. In addition he was head of the mechanics section of the Mechanics Institute of the Academy from 1939 to 1944. Kochin's research was on meteorology, gas dynamics and shock waves in compressible fluids. He gave the solution to the problem of small amplitude waves on the surface of an uncompressed liquid in Towards a Theory of Cauchy-Poisson Waves in 1935. He also worked on the pitch and roll of ships. In aerodynamics he introduced formulas for aerodynamic force and for the distribution of pressure. He wrote textbooks on hydromechanics and vector analysis. Article by: J J O'Connor and E F Robertson List of References (9 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kochin.html (1 of 2) [2/16/2002 11:17:52 PM]

Kochin

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kochin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kochin.html (2 of 2) [2/16/2002 11:17:52 PM]

Kodaira

Kunihiko Kodaira Born: 16 March 1915 in Tokyo, Japan Died: 26 July 1997 in Kofu, Japan

Click the picture above to see three larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kunihiko Kodaira graduated from the University of Tokyo in 1938 with a degree in mathematics. Not content with one degree, he graduated from the physics department at the University of Tokyo in 1941. At this time Kodaira was interested in topology, Hilbert spaces, Haar measure, Lie groups and almost periodic functions. Of course, World War II was at this time having a severe affect on Japan, in particular it effectively isolated Japanese scientists from contacts with other scientists around the world. Despite this Kodaira was able to obtain papers to read of mathematical developments and he was most influenced by reading the works of Weyl, Stone, von Neumann, Hodge, Weil and Zariski. From 1944 until 1949, Kodaira was an associate professor at the University of Tokyo but by this time his work was well known to mathematicians world-wide and he received an invitation from Weyl to come to Princeton. Kodaira accepted Weyl's invitation and, from 1949 until 1961, he was a member of the Institute for Advanced Study at Princeton. Following this he spent a year at Harvard, then in 1962 he was appointed to the chair of mathematics at Johns Hopkins University. In 1965 Kodaira left Johns Hopkins to take up the chair of mathematics at Stanford University. However, after two years at Stanford, he returned to Japan and held the chair of mathematics at the University of Tokyo from 1967. Kodaira's work covers many topics. These include applications of Hilbert space methods to differential equations which was an important topic in his early work and was largely the result of influence by Weyl. This time through the influence of Hodge, he worked on harmonic integrals and later he applied

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kodaira.html (1 of 2) [2/16/2002 11:17:53 PM]

Kodaira

this work to problem in algebraic geometry. Another important area of Kodaira's work was to apply sheaves to algebraic geometry. In around 1960 he became involved in the classification of compact, complex analytic spaces. One of the themes running through much of his work is the Riemann-Roch theorem and this plays an important role in much of his research. Kodaira received many honours for his outstanding research. Perhaps the most noteworthy was the award of a Fields Medal in 1954. He was made an honorary member of many learned societies throughout the world, including the London Mathematical Society in 1979. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Kunihiko Kodaira (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1954

Other Web sites

1. AMS 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kodaira.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kodaira.html (2 of 2) [2/16/2002 11:17:53 PM]

Koebe

Paul Koebe Born: 15 Feb 1882 in Luckenwalde, Germany Died: 6 Aug 1945 in Leipzig, Germany

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Koebe's father was Hermann Koebe and his mother was Emma Kramer. Hermann Koebe owned a factory and was able to give his son a good education. Koebe attended a realgymnasium in Berlin. He entered this school in 1891 and there he studied religion, Latin and modern languages, history and geography, and mathematics and science. The course, which was based more on practical applications than that of the more academic gymnasium, still qualified Koebe to enter university. He studied first at Kiel University which he entered in 1900 but after one semester he moved to Berlin University where he was to study for five years. At Berlin his thesis was directed by Herman Schwarz and his additional examiner for the oral on his thesis was Friedrich Schottky who had been appointed to Berlin in 1902 while Koebe was in the middle of his studies. Between 1904 and 1905 Koebe studied at the Charlottenburg Technische Hochschule, then he undertook research at Göttingen for his habilitation presenting his thesis in 1907. Koebe was appointed to Leipzig University in 1910 as an extraordinary professor of mathematics. He became an ordinary professor in 1914 when he accepted a position at Jena university. He returned to Leipzig, this time as an ordinary professor, in 1926. Koebe's work was all on complex functions, his most important results being on the uniformisation of Riemann surfaces. Shortly after 1900 Koebe established the general principle of uniformisation which had been originally conceived by Klein and Poincaré. Koebe's proof of the uniformisation theorem has been described as:... arguably one of the great theorems of the century. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Koebe.html (1 of 2) [2/16/2002 11:17:55 PM]

Koebe

The article [5] describes his contributions in some detail and gives a list of 68 publications by Koebe. These are not, however, a collection of great works on a par with his proof of the uniformisation theorem. Koebe's style was pompous and chaotic and Koebe anecdotes were famous in Germany between the two wars. He did make other important contributions, however, and his circle domain conjecture is still being attacked. A special case was proved in 1993 by Z-X He and O Schramm. Freudenthal writes in [1]:He tended to deal broadly with special cases of a general theory by a variety of methods ... Freudenthal also tells us that Koebe's life-style was, as his mathematics, chaotic. It is unclear from what Freudenthal writes whether he is implying that Koebe required a wife to help organise his life but certainly he had no wife, remaining a bachelor all his life. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Koebe.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Koebe.html (2 of 2) [2/16/2002 11:17:55 PM]

Koenigs

Gabriel Koenigs Born: 17 Jan 1858 in Toulouse, France Died: 29 Oct 1931 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gabriel Koenigs was appointed lecturer in mechanics at Bresancon, then in analysis at Toulouse. He returned to Ecole Normale Supérieure in Paris in 1886 where he had obtained a doctorate 4 years earlier. He became professor of mechanics at the Sorbonne in 1897. Koenigs was greatly influenced by Darboux and his first work was on geometry following work of Plücker and Klein. Then he looked at iteration theory and analytic mechanics where he applied Poincaré's theory. In [3] his work on roulettes is discussed and put into context relating it to the work of several other mathematicians:The construction of centers of curvature of plane roulettes by Bobillier (1831), Gilbert (1858) and Koenigs (1897) was based on the theory of centroids by M Chasles (1830). This problem has kinematical origins (a roulette is the plane curve described by points of a plane figure moving in its plane) but each plane curve may be considered as a roulette. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Koenigs.html (1 of 2) [2/16/2002 11:17:56 PM]

Koenigs

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Koenigs.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Koenigs.html (2 of 2) [2/16/2002 11:17:56 PM]

Kolmogorov

Andrey Nikolaevich Kolmogorov Born: 25 April 1903 in Tambov, Tambov province, Russia Died: 20 Oct 1987 in Moscow, Russia

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Andrei Nikolaevich Kolmogorov's parents were not married and his father took no part in his upbringing. His father Nikolai Kataev, the son of a priest, was an agriculturist who was exiled. He returned after the Revolution to head a Department in the Agricultural Ministry but died in fighting in 1919. Kolmogorov's mother also, tragically, took no part in his upbringing since she died in childbirth at Kolmogorov's birth. His mother's sister, Vera Yakovlena, brought Kolmogorov up and he always had the deepest affection for her. In fact it was chance that had Kolmogorov born in Tambov since the family had no connections with that place. Kolmogorov's mother had been on a journey from the Crimea back to her home in Tunoshna near Yaroslavl and it was in the home of his maternal grandfather in Tunoshna that Kolmogorov spent his youth. Kolmogorov's name came from his grandfather, Yakov Stepanovich Kolmogorov, and not from his own father. Yakov Stepanovich was from the nobility, a difficult status to have in Russia at this time, and there is certainly stories told that an illegal printing press was operated from his house. After Kolmogorov left school he worked for a while as a conductor on the railway. In his spare time he wrote a treatise on Newton's laws of mechanics. Then, in 1920, Kolmogorov entered Moscow State University but at this stage he was far from committed to mathematics. He studied a number of subjects, for example in addition to mathematics he studied metallurgy and Russian history. Nor should it be thought that Russian history was merely a topic to fill out his course, indeed he wrote a serious scientific thesis on the owning of property in Novgorod in the 15th and 16th centuries. There is an anecdote told by D G Kendall in [10] regarding this thesis, his teacher saying:You have supplied one proof of your thesis, and in the mathematics that you study this would http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kolmogorov.html (1 of 6) [2/16/2002 11:17:58 PM]

Kolmogorov

perhaps suffice, but we historians prefer to have at least ten proofs. Kolmogorov may have told this story as a joke but never-the-less jokes are only funny if there is some truth in them and undoubtedly this is the case here. In mathematics Kolmogorov was influenced at an early stage by a number of outstanding mathematicians. P S Aleksandrov was beginning his research (for the second time) at Moscow around the time Kolmogorov began his undergraduate career. Luzin and Egorov were running their impressive research group at this time which the students called 'Luzitania'. It included M Ya Suslin and P S Urysohn, in addition to Aleksandrov. However the person who made the deepest impression on Kolmogorov at this time was Stepanov who lectured to him on trigonometric series. It is remarkable that Kolmogorov, although only an undergraduate, began research and produced results of international importance at this stage. He had finished writing a paper on operations on sets by the spring of 1922 which was a major generalisation of results obtained by Suslin. By June of 1922 he had constructed a summable function which diverged almost everywhere. This was wholly unexpected by the experts and Kolmogorov's name began to be known around the world. The authors of [7] and [8] note that:Almost simultaneously [Kolmogorov] exhibited his interest in a number of other areas of classical analysis: in problems of differentiation and integration, in measures of sets etc. In every one of his papers, dealing with such a variety of topics, he introduced an element of originality, a breadth of approach, and a depth of thought. Kolmogorov graduated from Moscow State University in 1925 and began research under Luzin's supervision in that year. It is remarkable that Kolmogorov published eight papers in 1925, all written while he was still an undergraduate. Another milestone occurred in 1925, namely Kolmogorov's first paper on probability appeared. This was published jointly with Khinchin and contains the 'three series' theorem as well as results on inequalities of partial sums of random variables which would become the basis for martingale inequalities and the stochastic calculus. In 1929 Kolmogorov completed his doctorate. By this time he had 18 publications and Kendall writes in [10]:These included his versions of the strong law of large numbers and the law of the iterated logarithm, some generalisations of the operations of differentiation and integration, and a contribution to intuitional logic. His papers ... on this last topic are regarded with awe by specialists in the field. The Russian language edition of Kolmogorov's collected works contains a retrospective commentary on these papers which [Kolmogorov] evidently regarded as marking an important development in his philosophical outlook. An important event for Kolmogorov was his friendship with Aleksandrov which began in the summer of 1929 when they spent three weeks together. On a trip starting from Yaroslavl, they went by boat down the Volga then across the Caucasus mountains to Lake Sevan in Armenia. There Aleksandrov worked on the topology book which he co-authored with Hopf, while Kolmogorov worked on Markov processes with continuous states and continuous time. Kolmogorov's results from his work by the Lake were published in 1931 and mark the beginning of diffusion theory. In the summer of 1931 Kolmogorov and Aleksandrov made another long trip. They visited Berlin, Göttingen, Munich, and Paris where Kolmogorov spent many hours in deep discussions with Paul Lévy. After this they spent a month at the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kolmogorov.html (2 of 6) [2/16/2002 11:17:58 PM]

Kolmogorov

seaside with Fréchet Kolmogorov was appointed a professor at Moscow University in 1931. His monograph on probability theory Grundbegriffe der Wahrscheinlichkeitsrechnung published in 1933 built up probability theory in a rigorous way from fundamental axioms in a way comparable with Euclid's treatment of geometry. One success of this approach is that it provides a rigorous definition of conditional expectation. As noted in [10]:The year 1931 can be regarded as the beginning of the second creative stage in Kolmogorov's life. Broad general concepts advanced by him in various branched of mathematics are characteristic of this stage. After mentioning the highly significant paper Analytic methods in probability theory which Kolmogorov published in 1938 laying the foundations of the theory of Markov random processes, they continue to describe:... his ideas in set-theoretic topology, approximation theory, the theory of turbulent flow, functional analysis, the foundations of geometry, and the history and methodology of mathematics. [His contributions to] each of these branches ... [is] a single whole, where a serious advance in one field leads to a substantial enrichment of the others. Aleksandrov and Kolmogorov bought a house in Komarovka, a small village outside Moscow, in 1935. Many famous mathematicians visited Komarovka: Hadamard, Fréchet, Banach, Hopf, Kuratowski, and others. Gnedenko and other graduate students went on ([7] and [8]):... mathematical outings [which] ended in Komarovka, where Kolmogorov and Aleksandrov treated the whole company to dinner. Tired and full of mathematical ideas, happy from the consciousness that we had found out something which one cannot find in books, we would return in the evening to Moscow. Around this time Malcev and Gelfand and others were graduate students of Kolmogorov along with Gnedenko who describes what it was like being supervised by Kolmogorov ([7] and [8]):The time of their graduate studies remains for all of Kolmogorov's students an unforgettable period in their lives, full of high scientific and cultural strivings, outbursts of scientific progress and a dedication of all one's powers to the solutions of the problems of science. It is impossible to forget the wonderful walks on Sundays to which [Kolmogorov] invited all his own students (graduates and undergraduates), as well as the students of other supervisors. These outings in the environs of Bolshevo, Klyazma, and other places about 30-35 kilometres away, were full of discussions about the current problems of mathematics (and its applications), as well as discussions about the questions of the progress of culture, especially painting, architecture and literature. In 1938-1939 a number of leading mathematicians from the Moscow University joined the Steklov Mathematical Institute of the Academy of Sciences while retaining their positions at the University. Among them were Aleksandrov, Gelfand, Kolmogorov, Petrovsky, and Khinchin. The Department of Probability and Statistics was set up at the Institute and Kolmogorov was appointed as Head of Department. Kolmogorov later extended his work to study the motion of the planets and the turbulent flow of air from a jet engine. In 1941 he published two papers on turbulence which are of fundamental importance. In http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kolmogorov.html (3 of 6) [2/16/2002 11:17:58 PM]

Kolmogorov

1954 he developed his work on dynamical systems in relation to planetary motion. He thus demonstrated the vital role of probability theory in physics. We must mention just a few of the numerous other major contributions which Kolmogorov made in a whole range of different areas of mathematics. In topology Kolmogorov introduced the notion of cohomology groups at much the same time, and independently of, Alexander. In 1934 Kolmogorov investigated chains, cochains, homology and cohomology of a finite cell complex. In further papers, published in 1936, Kolmogorov defined cohomology groups for an arbitrary locally compact topological space. Another contribution of the highest significance in this area was his definition of the cohomology ring which he announced at the International Topology Conference in Moscow in 1935. At this conference both Kolmogorov and Alexander lectured on their independent work on cohomology. In 1953 and 1954 two papers by Kolmogorov, each of four pages in length, appeared. These are on the theory of dynamical systems with applications to Hamiltonian dynamics. These papers mark the beginning of KAM-theory, which is named after Kolmogorov, Arnold and Moser. Kolmogorov addressed the International Congress of Mathematicians in Amsterdam in 1954 on this topic with his important talk General theory of dynamical systems and classical mechanics. N H Bingham [10] notes Kolmogorov's major part in setting up the theory to answer the probability part of Hilbert's Sixth Problem "to treat ... by means of axioms those physical sciences in which mathematics plays an important part; in the first rank are the theory of probability and mechanics" in his 1933 monograph Grundbegriffe der Wahrscheinlichkeitsrechnung. Bingham also notes:... Paul Lévy writes poignantly of his realisation, immediately on seeing the Grundbegriffe, of the opportunity which he himself had neglected to take. A rather different perspective is supplied by the eloquent writings of Mark Kac on the struggles that Polish mathematicians of the calibre Steinhaus and himself had in the 1930s, even armed with the Grundbegriffe, to understand the (apparently perspicuous) notion of stochastic independence. If Kolmogorov made a major contribution to Hilbert's sixth problem, he completely solved Hilbert's Thirteenth Problem in 1957 when he showed that Hilbert was wrong in asking for a proof that there exist continuous functions of three variables which could not be represented by continuous functions of two variables. Kolmogorov took a special interest in a project to provide special education for gifted children [10]:To this school he devoted a major proportion of his time over many years, planning syllabuses, writing textbooks, spending a large number of teaching hours with the children themselves, introducing then to literature and music, joining in their recreations and taking them on hikes, excursions, and expeditions. ... [Kolmogorov] sought to ensure for these children a broad and natural development of the personality, and it did not worry him if the children in his school did not become mathematicians. Whatever profession they ultimately followed, he would be content if their outlook remained broad and their curiosity unstifled. Indeed it must have been wonderful to belong to this extended family of [Kolmogorov]. Such an outstanding scientist as Kolmogorov naturally received a whole host of honours from many different countries. In 1939 he was elected to the Academy of Sciences of the USSR. He received one of the first State Prizes to be awarded in 1941, the Lenin Prize in 1965, the Order of Lenin on six separate occasions, and the Lobachevsky Prize in 1987. He was also elected to the many other academies and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kolmogorov.html (4 of 6) [2/16/2002 11:17:58 PM]

Kolmogorov

societies including the Romanian Academy of Sciences (1956), the Royal Statistical Society of London (1956), the Loopoldina Academy of Germany (1959), the American Academy of Arts and Sciences (1959), the London Mathematical Society (1959), the American Philosophical Society (1961), The Indian Statistical Institute (1962), the Netherlands Academy of Sciences (1963), the Royal Society of London (1964), the National Academy of the United States (1967), the French Academy of Sciences (1968). In addition to the prizes mentioned above, Kolmogorov was awarded the Balzan International Prize in 1962. Many universities awarded him an honorary degree including Paris, Stockholm, and Warsaw. Kolmogorov had many interests outside mathematics, in particular he was interested in the form and structure of the poetry of the Russian author Pushkin. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (20 books/articles)

A Quotation

A Poster of Andrey Kolmogorov

Mathematicians born in the same country

Other references in MacTutor

1. 2. Chronology: 1920 to 1930 3. Chronology: 1930 to 1940 4. Chronology: 1950 to 1960

Honours awarded to Andrey Kolmogorov (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1964

Other Web sites

1. CWI, Netherlands 2. Landau-Kolmogorov Constants 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kolmogorov.html (5 of 6) [2/16/2002 11:17:58 PM]

Kolmogorov

JOC/EFR January 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kolmogorov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kolmogorov.html (6 of 6) [2/16/2002 11:17:58 PM]

Kolosov

Gury Vasilievich Kolosov Born: 25 Aug 1867 in Ust, Novgorod guberniya, Russia Died: 7 Nov 1936 in Leningrad (now St Petersburg), Russia Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gury Kolosov was educated at St Petersburg and after working at Yurev University from 1902 to 1913, he returned to St Petersburg where he spent the rest of his career. He studied the mechanics of solid bodies and the theory of elasticity. In 1907 Kolosov derived the solution for stresses around an elliptical hole. It showed that the concentration of stress could become far greater, as the radius of curvature at an end of the hole becomes small compared with the overall length of the hole. Engineers have to understand Kolosov results so that stresses can be kept to safe levels. From 1908 onwards Kolosov devoted himself exclusively to the study of elasticity where the formulas he discovered are still used. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kolosov.html (1 of 2) [2/16/2002 11:17:59 PM]

Kolosov

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kolosov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kolosov.html (2 of 2) [2/16/2002 11:17:59 PM]

Konig_Denes

Denes König Born: 21 Sept 1884 in Budapest, Hungary Died: 19 Oct 1944 in Budapest, Hungary Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Denes König was the son of Gyula (Julius) König. He studied at Budapest and Göttingen, obtaining his doctorate in 1907. That year he joined the staff of the Technische Hochschule in Budapest, where his father was professor. He remained there until his death, becoming a full professor in 1935. At Göttingen, König had been influenced by Minkowski's lectures on the four colour problem. These lectures contributed to his growing interest in graph theory, on which he lectured in Budapest from 1911. His book, Theorie der endlichen und unendlichen Graphen, was published in 1936, and was a major factor in the growth of interest in graph theory worldwide. It was eventually translated into English under the title Theory of finite and infinite graphs (translated by R McCoart), Birkhauser, 1990; this also contains a biographical sketch by Tibor Gallai. König's work on the factorisation of bipartite graphs relates closely to the marriage problem of Philip Hall. König's use of graphs to give a simpler proof of a determinant result of Frobenius seems to have lead to some hostility between the two men. After the Nazi occupation of Hungary, König worked to help persecuted mathematicians. This lead to his death a few days after the Hungarian National Socialist Party took over the country. Article by: Ian Anderson, Glasgow A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Konig_Denes.html (1 of 2) [2/16/2002 11:18:00 PM]

Konig_Denes

Glossary index

Quotations index

Mathematicians of the day JOC/EFR September 2000

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Konig_Denes.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Konig_Denes.html (2 of 2) [2/16/2002 11:18:00 PM]

Konig_Julius

Julius König Born: 1849 in Györ, Hungary Died: 1913 in Budapest, Hungary

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Julius König studied at Vienna and Heidelberg. His doctorate was from Heidelberg in 1870. He became a lecturer in Budapest in 1872, becoming professor at the Technical University of Budapest in 1873. In Heidelberg König was influenced by Helmholtz who suggested research topics. However he ended up working with Leo Königsberger on elliptic functions. His work in Hungary was on algebra and analysis. His most important work in 1903 is based on a fundamental study by Kronecker published in 1892. König developed Kronecker's polynomial ideals and presented many results on discriminants of forms, elimination theory and Diophantine problems. König's work on polynomial ideals influenced Hilbert, Lasker, Macaulay, Emmy Noether, van der Waerden and Gröbner but they simplified his ideas so König's work is now only of historical interest. In the last eight years of his life König's interests turned towards set theory and he contributed to the continuum hypothesis. In 1904 he announced that the continuum hypothesis was false but Zermelo found the error in the proof. He spent the last part of his life working on his own approach to set theory, which was published in 1914, the year after his death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Konig_Julius.html (1 of 2) [2/16/2002 11:18:01 PM]

Konig_Julius

List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Konig_Julius.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Konig_Julius.html (2 of 2) [2/16/2002 11:18:01 PM]

Konig_Samuel

Johann Samuel König Born: 31 July 1712 in Büdingen, Germany Died: 21 Aug 1757 in Zuilenstein (near Amerongen), Netherlands

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Samuel König's father was a mathematician and theologian who spent the last 20 years of his life as professor of Oriental Studies at the University of Bern. König's early education was at home where he studied with his father's instruction. His father was a great enthusiast and he passed this on to his son with exciting lessons in science and mathematics. After studying at Bern, then in 1729 at Lausanne, König went to Basel in 1730 to study under Johann Bernoulli. After three years of study with Johann Bernoulli he also began to be taught by Daniel Bernoulli. Along with Clairaut and Maupertuis he studied Newton's publications, in particular he did thorough and detailed work on the Principia. In 1731 Hermann left the chair of mathematics at St Petersburg to return to Basel where he was appointed to the the chair of ethics and natural law. König studied under Hermann from 1731, in particular he learn of Leibniz's philosophy. In 1735 König went to Marburg to continue his study of Leibniz's philosophy under Christian von Wolff's supervision. In 1737 König returned to Bern, having begun to publish mathematical articles two years earlier. In Bern he wrote on dynamics, and two articles on this topic appeared in 1738, but he practiced law (which earned him more money than mathematics). Near the end of 1738 König went to Paris where he met Maupertuis. Through Maupertuis, König was introduced to Voltaire and to the Marquise du Châtelet. He began to give the Marquise du Châtelet lessons in mathematics and on the philosophy of Leibniz. Together with Voltaire and du Châtelet, König visited René-Antoine Ferchault de Réaumur, the leading entomologist of the day who was conducting research in widely varied fields. Through his discussions

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Konig_Samuel.html (1 of 3) [2/16/2002 11:18:03 PM]

Konig_Samuel

with Réaumur, König was led to publish a work on the structure of honeycombs. This work was so highly thought of that König was elected to the Paris Academy of Sciences. Soon after this König fell out with the Marquess du Châtelet and, although the reason is not certain, it has been suggested that they argued over payments for the lessons that König was giving to the Marquess. Certainly König did not leave Paris after his disagreement but continued to live there for about 18 months. After returning to Bern, he continued to earn his living from his law practice but also continued to study mathematics, in particular studying the works of Clairaut and Maupertuis. König wrote a work on the shape of the Earth, which was published in 1747, based on what he had learnt from studying Clairaut's Théorie de la figure de la Terre and other works on this topic. In 1744 König was exiled from Bern for 10 years because he signed a liberal petition. König obtained a chair of philosophy and mathematics at Franeker in the Netherlands. Then, in 1749, he went to The Hague where he was librarian to Prince William IV of Orange. Maupertuis proposed König for the Berlin Academy and he was honoured by election to the Academy in 1749. However, he was to cause a split in the Academy through an essay, written before 1749 but published in 1751, on the principle of least action. This work attacked Maupertuis accusing him of having plagiarized Leibniz's work on this principle. König had touched on a sore point with the Berlin Academy for a number of reasons. Firstly the Academy had just been involved in a dispute on the prize question for 1746, which Euler had made it clear required the entrants to criticise the philosophy of Leibniz and Wolff. Euler was a strong opponent of the Leibniz and Wolff philosophy. Also for König, as a newly elected member of the Academy, to be seen to be attacking its President was not going to be well received. König had studied the philosophy of Leibniz under Wolff so he was a strong follower of their philosophy. He claimed in his 1751 article that Leibniz had stated the Principle of Least Action in a letter of 1707 to Jacob Hermann. A heated dispute followed the publication of König's article and he was asked to produce the letter. However König failed to do so and, after an investigation, Euler gave a report on the affair accusing König of fraud. As a result of the dispute Maupertuis was greatly upset and soon left the Berlin Academy. König's article was to have other repercussions as Voltaire, d'Alembert and Euler all ended at the centre of the dispute. König lived for six years after his article causing [1]:... the ugliest of all scientific disputes. These last six years of his life were completely dominated by the dispute. An important kinetic law of energy, published in a work of 1751 (the same year as his controversal article appeared), is named after König and this law is a major contribution which he made to mathematics. König's character and role in science is summed up in [1] as follows:A candid and amiable man, he was distinguished by erudition of unususl breadth even for his time. ... The opinion is occasonally voiced that were it not for the controversy over the principle of least action, König would be completely forgotten in the history of science. His formulation of the law (named after him) of the kinetic energy of the motion of a mass point system relative to its centre of gravity is sufficient in itself to refute this view.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Konig_Samuel.html (2 of 3) [2/16/2002 11:18:03 PM]

Konig_Samuel

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Konig_Samuel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Konig_Samuel.html (3 of 3) [2/16/2002 11:18:03 PM]

Konigsberger

Leo Königsberger Born: 15 Oct 1837 in Posen, Germany (now Poznan, Poland) Died: 15 Dec 1921 in Heidelberg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leo Königsberger came from a rich family, his father being a wealthy merchant. He was educated in Posen, Germany (now Poznan, Poland) and as a young man he became friendly with Lazarus Fuchs who was educated in the same town. Königsberger, like his friend Fuchs, studied at the University of Berlin. In his first semester at the university, which was in the year 1857, he was taught by Weierstrass. In fact he attended Weierstrass's lectures on the theory of elliptic functions (which was Weierstrass's main research topic) and many years later published his account of these lectures. This was the first course that Weierstrass gave on elliptic functions and Königsberger's publication in 1917 was of considerable historical importance. Königsberger graduated from Berlin in 1860 and then spent the three years from 1861 to 1864 teaching mathematics and physics to the Berlin cadet corps. In 1864 he obtained his first academic appointment as an extraordinary professor at the University of Greifswald. Five years later he was appointed to a chair of mathematics at Heidelberg then, in 1875 he moved to the Technische Hochschule in Dresden. After two years in Dresden Königsberger moved again, this time to the University of Vienna. He returned to Heidelberg in 1884, teaching there until he retired in 1914. So from a academic career spanning 50 years, despite the many moves, Königsberger spent 36 of these years at the University of Heidelberg. As Burau writes in [1]:Königsberger was one of the most famous mathematicians of his time, member of many academies, and universally accepted. He contributed to many fields of mathematics, most http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Konigsberger.html (1 of 2) [2/16/2002 11:18:05 PM]

Konigsberger

notably to analysis and analytic mechanics. His early work must have been in influenced by Weierstrass's lectures on elliptic functions, for this was the topic of much of his early research. He wrote an important text on elliptic functions in 1874 and another important textbook on hyperelliptic integrals four years later. Much of Königsberger's work was on differential equations, influenced by Fuchs's function theory (as we noted above he was a friend of Fuchs from his youth). His work on differential equations was influenced by Bunsen, Kirchhoff and Helmholtz, with whom he was close friends in Heidelberg, and he considered the differential equations of analytic mechanics. Königsberger is also famed for his biography of Helmholtz (1902) and his biographical Festschrift for Jacobi (1904). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Konigsberger.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Konigsberger.html (2 of 2) [2/16/2002 11:18:05 PM]

Korteweg

Diederik Johannes Korteweg Born: 31 March 1848 in 'sHertogenbosch, Netherlands Died: 10 May 1941 in Amsterdam, Netherlands

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Diederik Korteweg studied at the Polytechnical School at Delft. He originally intended to become an engineer but he turned to mathematics. He became a secondary school teacher, teaching in schools at Tilberg and Breda. He then entered the University of Amsterdam, receiving a doctorate in 1878. He remained at Amsterdam becoming professor there in 1881. He held this post until he retired in 1918. Korteweg's main work was in applied mathematics. His doctoral dissertation was on the velocity of wave propogation in elastic tubes. His subsequent work included work on electricity, statistical mechanics, thermodynamics and further work on wave propagation. He established a criterion for the stability of orbits of particles moving under a central force. He also studied a stationary wave in a rectangular canal. He is remembered in particular for the Korteweg-de Vries equation on solitary waves. In 1894 de Vries wrote a dissertation Bijdrage tot de kennis der lange golven supervised by Korteweg. The results of this thesis were written up for publication in a joint paper published in 1905. His books studying Huygens contributed greatly to the history of mathematics. He was also the editor of Huygens complete works. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Korteweg.html (1 of 2) [2/16/2002 11:18:07 PM]

Korteweg

List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Korteweg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Korteweg.html (2 of 2) [2/16/2002 11:18:07 PM]

Kotelnikov

Aleksandr Petrovich Kotelnikov Born: 20 Oct 1865 in Kazan, Russia Died: 6 March 1944 in Moscow, USSR Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aleksandr Kotelnikov's father P I Kotelnikov was a colleague of Lobachevsky and, in fact, the only one of his colleagues to publicly praise his great geometrical achievements during his lifetime. This connection between Kotelnikov's father and Lobachevsky is important since the Lobachevsky connection was to play a large role in Kotelnikov's work throughout his life. Kotelnikov was educated at the University of Kazan, graduating in 1884. After this he taught at a gymnasium in Kazan before entering the Department of Mechanics at Kazan to work for his university teachers qualification. He began teaching at Kazan University in 1893, then in 1896 he received his Master's Degree. The thesis he presented for the Master's Degree was The Cross-Product Calculus and Certain of its Applications in Geometry and Mechanics. This thesis applied vector methods in theoretical mechanics, and he was to teach this vector approach to mechanics throughout his life. Kotelnikov obtained his doctorate in 1899 for a thesis The Projective Theory of Vectors which generalised the vector calculus to the non-euclidean spaces of Lobachevsky and Riemann. He also applied this to mechanics in non-euclidean spaces. Much of his career is spent working on physics and non-euclidean geometry. In 1899, the year he received his doctorate, he was appointed professor and Head of the Department of Pure Mathematics at the University of Kiev. Perhaps it is worth pointing out that in many countries today a doctorate is a lower degree, essentially a degree of a level to start university teaching. This was equivalent to the Master's Degree referred to here, where the doctorate was really the qualification necessary to become a professor. In 1904 Kotelnikov left Kiev and returned to Kazan to become professor and Head of the Department of Pure Mathematics at the University. After 10 years he went back to Kiev again, this time o become Head of Theoretical Mechanics at the Polytechnical Institute. After another 10 years he moved, in 1924, to Moscow, where he worked at the Bauman Technical College until his death in 1944. In 1927 he published one of his most important works, The Principle of Relativity and Lobachevsky's Geometry. He also worked on quaternions and applied them to mechanics and geometry.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kotelnikov.html (1 of 2) [2/16/2002 11:18:08 PM]

Kotelnikov

Among his other major pieces of work was to edit the Complete Works of two mathematicians, Lobachevsky and Zhukovsky. He received many honours for his work, being named Honoured Scientist in 1934, then one year before he died he was awarded the State Prize of the USSR. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kotelnikov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kotelnikov.html (2 of 2) [2/16/2002 11:18:08 PM]

Kovalevskaya

Sofia Vasilyevna Kovalevskaya Born: 15 Jan 1850 in Moscow, Russia Died: 10 Feb 1891 in Stockholm, Sweden

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sofia Kovalevskaya was the middle child of Vasily Korvin-Krukovsky, an artillery general, and Velizaveta Shubert, both well-educated members of the Russian nobility. Sofia was educated by tutors and governess's, lived first at Palabino, the Krukovsky country estate, then in St. Petersburg, and joined her family's social circle which included the author Dostoevsky. Sofia was attracted to mathematics at a very young age. Her uncle Pyotr Vasilievich Krukovsky, who had a great respect for mathematics, spoke about the subject. Sofia wrote in her autobiography:The meaning of these concepts I naturally could not yet grasp, but they acted on my imagination, instilling in me a reverence for mathematics as an exalted and mysterious science which opens up to its initiates a new world of wonders, inaccessible to ordinary mortals. When Sofia was 11 years old, the walls of her nursery were papered with pages of Ostrogradski's lecture notes on differential and integral analysis. She noticed that certain things on the sheets she had heard mentioned by her uncle. Studying the wallpaper was Sofia's introduction to calculus. It was under the family's tutor, Y I Malevich, that Sofia undertook her first proper study of mathematics, and she says that it was as his pupil that I began to feel an attraction for my mathematics so intense that I started to neglect my other studies. Sofia 's father decided to put a stop to her mathematics lessons but she borrowed a copy of Bourdeu's Algebra which she read at night when the rest of the household was asleep.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kovalevskaya.html (1 of 4) [2/16/2002 11:18:10 PM]

Kovalevskaya

A year later a neighbour, Professor Tyrtov, presented her family with a physics textbook which he had written, and Sofia attempted to read it. She did not understand the trigonometric formulas and attempted to explain them herself. Tyrtov realised that in her working with the concept of sine, she had used the same method by which it had developed historically. Tyrtov argued with Sofia's father that she should be encouraged to study mathematics further but it was several years later that he permitted Sofia to take private lessons. Sofia was forced to marry so that she could go abroad to enter higher education. Her father would not allow her to leave home to study at a university, and women in Russia could not live apart from their families without the written permission of their father or husband. At the age of eighteen, she entered a nominal marriage with Vladimir Kovalevski, a young palaeontologist. This marriage caused problems for Sofia and, throughout its fifteen years, it was a source of intermittent sorrow, exasperation and tension and her concentration was broken by her frequent quarrels and misunderstandings with her husband . In 1869 Sofia travelled to Heidelberg to study mathematics and the natural sciences, only to discover that women could not matriculate at the university. Eventually she persuaded the university authorities to allow her to attend lectures unofficially, provided that she obtain the permission of each of her lecturers. Sofia studied there successfully for three semesters and, according to the memoirs of a fellow student, she immediately attracted the attention of her teachers with her uncommon mathematical ability. Professor Königsberger, the eminent chemist Kirchhoff, .... and all of the other professors were ecstatic over their gifted student and spoke about her as an extraordinary phenomenon . In 1871 Kovalevskaya moved to Berlin to study with Weierstrass, Königsberger's teacher. Despite the efforts of Weierstrass and his colleagues the senate refused to permit her to attend courses at the university. Ironically this actually helped her since over the next four years Weierstrass tutored her privately. By the spring of 1874, Kovalevskaya had completed three papers. Weierstrass deemed each of these worthy of a doctorate. The three papers were on Partial differential equations, Abelian integrals and Saturn's Rings. The first of these is a remarkable contribution which was published in Crelle's Journal in 1875. The paper on the reduction of abelian integrals to simpler elliptic integrals is of less importance but it consisted of a skilled series of manipulations which showed her complete command of Weierstrass's theory. In 1874 Kovalevskaya was granted her doctorate, summa cum laude, from Göttingen University. Despite this doctorate and letters of strong recommendation from Weierstrass, Kovalevskaya was unable to obtain an academic position. This was for a combination of reasons, but her sex was a major handicap. Her rejections resulted in a six year period during which time she neither undertook research nor replied to Weierstrass's letters. She was bitter to discover that the best job she was offered was teaching arithmetic to elementary classes of school girls, and remarked I was unfortunately weak in the multiplication table . In 1878, Kovalevskaya gave birth to a daughter, but from 1880 increasingly returned to her study of mathematics. In 1882 she began work on the refraction of light, and wrote three articles on the topic. In

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kovalevskaya.html (2 of 4) [2/16/2002 11:18:10 PM]

Kovalevskaya

1916, Volterra discovered that Kovalevskaya had made the same mistake as Lamé, on whose work these papers were based, though she had pointed out several others which he had made in his presentation of the problem. The first of these three articles was still a valuable paper however, because it contained an exposition of Weierstrass's theory for integrating certain partial differential equations. In the spring of 1883, Vladimir, from whom Sofia had been separated for two years, committed suicide. After the initial shock, Kovalevskaya immersed herself in mathematical work in an attempt to rid herself of feelings of guilt. Mittag-Leffler managed to overcome opposition to Kovalevskaya in Stockholm, and obtained for her a position as privat docent. She began to lecture there in early 1884, was appointed to a five year extraordinary professorship in June of that year, and in June 1889 became the first woman since the physicist Laura Bassi and Maria Gaetana Agnesi to hold a chair at a European university. During Kovalevskaya's years at Stockholm, she carried out what many consider her most important research She taught courses on the latest topics in analysis and became an editor of the new journal Acta Mathematica. She took over the task of liaison with the mathematicians of Paris and Berlin and took part in the organisation of international conferences. Her status brought her attention from society, and she began again to write reminiscences and dramas that she had enjoyed doing when young. The topic of the Prix Bordin of the French Academy of Sciences was announced in 1886. Entries were to be significant contributions to the problem of the study of a rigid body. Kovalevskaya entered and, in 1886, was awarded the Prix Bordin for her paper Mémoire sur un cas particulier du problème de le rotation d'un corps pesant autour d'un point fixe, ou l'intégration s'effectue à l'aide des fonctions ultraelliptiques du temps. In recognition of the brilliance of this work the prize money was raised from 3,000 to 5,000 francs. Kovalevskaya's further research on this subject won a prize from the Swedish Academy of Sciences in 1889, and in the same year, on the initiative of Chebyshev, Kovalevskaya was elected a corresponding member of the Imperial Academy of Sciences. Although the Tsarist government had repeatedly refused her a university position in her own country, the rules at the Imperial Academy were changed to allow the election of a woman. Kovalevskaya's last published work was a short article Sur un théorème de M. Bruns in which she gave a new, simpler proof of Bruns' theorem on a property of the potential function of a homogeneous body. In early 1891, at the height of her mathematical powers and reputation, Kovalevskaya died of influenza complicated by pneumonia. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles)

A Quotation

A Poster of Sofia Kovalevskaya

Mathematicians born in the same country

Cross-references to History Topics

A comment from Thomas Hirst's diary

Other references in MacTutor

A biography of Kovalevskaya based on a project by Tom Burslem

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kovalevskaya.html (3 of 4) [2/16/2002 11:18:10 PM]

Kovalevskaya

Honours awarded to Sofia Kovalevskaya (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Kovalevskaya

Other Web sites

1. Tom Burslem 2. R Cooke 3. Agnes Scott College 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kovalevskaya.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kovalevskaya.html (4 of 4) [2/16/2002 11:18:10 PM]

Kramer

Edna Ernestine Kramer Lassar Born: 11 May 1902 in Manhattan, New York, USA Died: 9 July 1984 in Manhattan, New York, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edna Ernestine Kramer Lassar was born on 11 May 1902, in New York City in the borough of Manhattan. Named after her uncle, Edward Elowitch who had died shortly before her birth and who had shown a gift for mathematics, Kramer's childhood ambition was to do well in his honour. She was the eldest child of Joseph and Sabine Kramer who were both Jewish immigrants from Europe. They had come to Manhattan as children, having to go out to work at an early age but nevertheless attending Night School to obtain their high school education. Both grew up to become intellectual, ambitious adults; a trait they passed to their children. They were both excellent linguists and while speaking and writing English well they also retained their native German. Kramer's parents believed strongly in the importance of education and her father was proud to serve on the New York City Board of Education, despite having no academic affiliation. Kramer and her two siblings, Martha, who was two years younger and Herbert, born in 1911 were held to high standards by their parents with the result that they were all prize-winning students, who were all elected to the Phi Beta Kappa and became teachers. Kramer was a precocious child who intrigued her relatives. Their interest in her was to have a huge effect. She was especially influenced by her young Aunt Therese Elowitch, who lived nearby and was a lawyer by profession and by a cousin, Josephine Schwartz, who lived with the Kramers as a child. They challenged Kramer to card games, such as Hearts and feats of memory, including reciting poetry. By the time Kramer entered first grade, in May 1908, she had already studied many of her cousin's higher-level elementary school assignments. Kramer was also impressed by her mother and aunt's involvement within the Suffragette movement. Kramer's initial career plans; to become a German teacher were dampened by the First World War and consequently she had to review her future. However she did not have to look far for inspiration. As a freshman at Wadleigh High School, Manhattan she had a teacher whom led her to see her future differently. John A. Swenson was the chairman of the mathematics department and he inspired Kramer with a love of mathematics that was to last throughout her life. Her friendship with Swenson remained constant as he guided and inspired her to a career as a mathematician. Kramer majored in mathematics at Hunter College, receiving her B.A degree summa cum laude in 1922. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kramer.html (1 of 5) [2/16/2002 11:18:12 PM]

Kramer

There she was elected to Pi Mu Epsilon Honorary Mathematics Society and Phi Beta Kappa. With help from Svenson, who arranged her program to fit her university classes she continued her studies while teaching high school mathematics (DeWitt Clinton High School, Bronx, New York, 1922-23 and Wadleigh High School 1923-29). She obtained an M.A in mathematics from Columbia University in 1925 and her PhD in mathematics (with a minor in physics) in 1930. Her PhD dissertation discussed the geometric properties of polygenic functions, extending the work of Edward Kasner, her thesis supervisor, George Scheffers and Edmond Laguerre. Kasner had published a number of articles on polygenic functions (his coinage) of the ordinary complex variable Kramer used the first part of her thesis to develop an analogous theory of polygenic functions of the dual variable Although similarities between the two theories were found no general principle of transference from one theory to the other appeared to exist. Other antecedents of Kramer's work were in the papers of Scheffers on monogenic functions of the dual variable In the second part of her thesis Kramer studied the Laguerre group, a set of linear fractional polygenic transformations of the dual variable and gave a more analytic treatment than had previously been done. While Kasner influenced her dissertation, Kramer's earliest pedagogical publication reflects the influence of her mentor, John A. Swenson and her job. In this publication she showed how prospective teachers could learn both content and method simultaneously, to the enrichment of both. She recommended bringing appropriate college-level mathematics to the high school level, emphasising concepts over mechanics to avoid the common occurrence of [6]:... not being able to see the basic ideas through the haze of technique with which they are surrounded. The developments of Kramer's ideas, which form the basis of future books, are apparent in this article. Years later she returned to formal studies as a post-graduate student at the Courant Institute of New York University (1939-1940, 1965-1969) and the University of Chicago in 1941. In 1929 she rejected an offer for a position in the Education Department of Hunter College because she preferred to teach mathematics and she was still hoping to do some mathematical research. However that same year, strongly endorsed by Swenson, Kramer became the first female instructor of mathematics at the New Jersey State Teachers College in Montclair, where she was promoted to assistant professor in 1932. It was there that she was invited to become a co-author of high school texts, however she declined the offer because of loyalty to Swenson's new teaching and curricular proposals: integration of mathematical topics and incorporation of advanced concepts, and her compunction against writing books that might compete with his. She did however help the other authors, John Stone and Virgil Mallory with ideas, corrections, exercises and applications and gave them credit for influencing her to write a statistics textbook in 1935. Her only text, A First Course in Educational Statistics contained an exposition of modern mathematical statistics that to the credit of the author's writing was also accessible to non mathematicians. Illustrative tables and exercises used actual data from journals of special interest to teachers. The Depression of the 1930s saw a scarcity of college positions, low wages and there was much

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kramer.html (2 of 5) [2/16/2002 11:18:12 PM]

Kramer

anti-Semitism and discrimination against women, particularly married ones. Planning to marry and fearing the hostility of the new chairman at Montclair College, Kramer decided to return to the New York City School system and she obtained a teaching position at Thomas Jefferson High school in Brooklyn, where her salary doubled. Before long she was acting chairman of the department and as soon as appointments were made she was promoted to chairman. On 2 July 1935, Kramer married Benedict Taxier Lassar, a French teacher and guidance counsellor, who later became a clinical psychologist. Kramer continued with her school teaching after marriage, while also writing, consulting and teaching college courses, including teaching methods courses in the graduate division of Brooklyn College between 1935 and 1938. From 1943-45 while Kramer was still teaching at Jefferson High School she worked at Columbia University as a statistical consultant to the university's Division of War Research, under the Office of Scientific Research and Development in Washington D.C. Her work was concerned with probabilistic strategic tactics of the war in Japan, and with anti-aircraft fire control. In 1948 Kramer began work at the New York Polytechnic Institute as an adjunct instructor and rose to an adjunct professor in 1953. She retired from school teaching in 1956 and from the New York Polytechnic Institute in 1965. As well as her published article on strategies in mathematics teaching she wrote other pedagogical publications concerned with the applications of mathematics. Mathematics Takes Wings (1942) related aeronautics to many different topics in the high school mathematics curriculum. The Integration of Trigonometry with Physical Science (1948) showed how trigonometry could be taught with applications to electricity, sound and light. As co-author of Experiences in Mathematical Discovery (1966), she developed special materials for the student of general mathematics. Kramer was not only interested in the applications of mathematics; she was also keen to collect all sorts of historical, cultural and recreational materials to accompany each mathematical concept. She helped Edward Kasner to prepare Mathematics and the Imagination and she served as an advisor to Richard Courant in the writing of What is Mathematics? By 1951 her extensive collection of applications and other material had grown into a book, The Main Stream of Mathematics. A combination of mathematical concepts and history up to the early part of the 20th century, together with applications to the fields of science, art and music it has received many favourable reviews and has been translated into many languages [7]:The book develops the theme that mathematics and mathematicians can be interesting. Kramer's style of writing makes it an enjoyable read for both mathematician and layman alike. Her examination of Omar Khayyam and algebra, Newton and calculus, Fermat and probability, Lewis Carroll and logic and Einstein and relativity provides an intriguing book for non-mathematicians and a valuable reference source for teachers and students. In 1970 Kramer published the voluminous expansion and sequel, The Nature and Growth of Modern Mathematics. It covers in a popular, but unusually comprehensive fashion 20th century mathematics and the people responsible for creating it. Carl Boyer who reviewed the book praised the historical allusions and thorough mathematical content [8]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kramer.html (3 of 5) [2/16/2002 11:18:12 PM]

Kramer

One cannot easily think of a topic within layman's comprehension which is not presented in considerable detail, including analysis, algebra, logic and foundations. Both books use a spiral approach and emphasise concepts over chronology. Kramer achieved her purpose in giving the reader access to an understanding of the importance of mathematics and it's relationship to other areas of scientific thought. She managed to give an all-round picture with balance among computational, historical, recreational and cultural points of view and to [9]:... promote interest and diminish awe while also providing much valuable background for the specialist as well. As a scientist with a keen interest in history, Kramer books also included details about the lives of the mathematicians whose ideas and accomplishments she discussed. Women mathematicians, past and present became a special interest and she wrote biographies of these women and their achievements for the journal Scripta Mathematica and the Dictionary of Scientific Biography. She also travelled to Europe to interview eminent female mathematicians of the twentieth century, including Hanna Neumann. During the course of her career Kramer was a member of the American Mathematical Society, the Mathematical Association of America, the Societe Mathematique de France, the Association Of Women in Mathematics, the American Association for the Advancement of Science, the History of Science Society and the New York Academy of Sciences. Although retired she remained active for many years, continuing studying, publishing and travelling. She attended classes at the Courant Institute from 1965 to 1969, and in 1973 she travelled to Singapore where she gave an invited lecture entitled: The Contributions of Women Past and Present to the Development of Mathematics, at Nanyang University. For the last ten years of her life Kramer suffered from Parkinson's disease and she died of pneumonia at her home in Manhattan on 9 July 1984. Edna Ernestine Kramer Lassar was a mathematician gifted with respect to both mathematics and language. On completion of her PhD the Depression began and hence she was unable to find a job conductive to the mathematical research she expected to do. However she utilised her other talents, for writing and for clear explanation of very complicated ideas. She explored mathematics to a great breath and depth, writing about its most significant and profound aspects from a variety of perspectives. She was well received by one of the biggest audiences a mathematical writer can hope to reach. Article by: J J O'Connor and E F Robertson based on a project by Rose-Marie Monahan.

Mathematicians born in the same country Other Web sites

Agnes Scott College

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kramer.html (4 of 5) [2/16/2002 11:18:12 PM]

Kramer

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Famous curves

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Kramer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kramer.html (5 of 5) [2/16/2002 11:18:12 PM]

Kramp

Christian Kramp Born: 8 July 1760 in Strasbourg, France Died: 13 May 1826 in Strasbourg, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Christian Kramp's father was a teacher in the Gymnasium at Strasbourg. Kramp studied medicine and, after graduating, practised medicine in the region around where he lived travelling to patients in a fairly wide area. However his interests certainly ranged outside medicine for, in addition to a number of medical publications, he published a work on crystallography in 1793. In 1795 France annexed the Rhineland region in which Kramp was carrying out his medical work and after this he became a teacher at Cologne, teaching mathematics, chemistry and physics. Kramp was appointed professor of mathematics at Strasbourg, the town of his birth, in 1809. He was elected to the geometry section of the Académie des Sciences in 1817. As Bessel, Legendre and Gauss did, Kramp worked on the generalised factorial function which applied to non-integers. His work on factorials is independent of that of Stirling and Vandermonde. He did achieve one "first" in that he was the first to use the notation n! although he seems not to be remembered today for this widely used mathematical notation. Kramp sent his work on factorials of non-integers to Bessel who was influenced by it. Among Kramp's other work was a paper on crystallography and some quite important work on double refraction. He also wrote a number of elementary treatises on pure mathematics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kramp.html (1 of 2) [2/16/2002 11:18:13 PM]

Kramp

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kramp.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kramp.html (2 of 2) [2/16/2002 11:18:13 PM]

Krawtchouk

Mykhailo Pilipovich Krawtchouk Born: 27 Sept 1892 in Chovnitsy, Ukraine Died: 9 March 1942 in Kolyma, Siberia, USSR

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

There are many different transliterations of Mikhail Krawtchouk's name, the forms Kravchuk and Krawtschuk also being used. The form used here "Krawtchouk" is the spelling used in the papers which he wrote in French. Krawtchouk studied at St Vladimir University in Kiev and obtained his first degree in 1914. The First World War broke out shortly after Krawtchouk graduated and because of problems at Kiev University he had to move to Moscow. However, it was a time of severe political problems with one disruption following another for Krawtchouk. In 1917 the Bolsheviks seized power in St Petersburg and fighting broke out in Moscow. Bolshevik power was soon firmly established and Krawtchouk returned to Kiev. For two years Krawtchouk taught in a number of different institutions until the outbreak of the civil war. This was not an easy period for, in January 1918, an independent Ukrainian state was proclaimed with Kiev as its capital. Red Army troops entered Kiev in the following month. Later in 1918 an independent Ukraine was again declared in Kiev but there followed a series of struggles between Ukrainian nationalist, White, and Red forces. In November 1919 Kiev was briefly taken by the White armies before being occupied by the Red Army. There was still no peace in Kiev for, in May 1920 the Poles captured Kiev but were driven out in a counterattack. During this stormy period Krawtchouk was headmaster at a country school not far from Kiev. After working on his doctoral thesis, he was awarded a doctorate for a dissertation On Quadratic Forms and Linear Transforms in 1924. In the same year he attended the International Mathematical Congress in Toronto and made contacts with many mathematicians. Four years later he attended the International Mathematical Congress in Bologna. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krawtchouk.html (1 of 3) [2/16/2002 11:18:14 PM]

Krawtchouk

Krawtchouk's contacts with other mathematicians were extremely valuable, particularly those with Hadamard, Hilbert, Courant and Tricomi. In 1929 Krawtchouk published his most famous work, Sur une généralisation des polynômes d'Hermite. In this paper he introduced a new system of orthogonal polynomials now known as the Krawtchouk polynomials, which are polynomials associated with the binomial distribution. However his mathematical work was very wide and, despite his early death, he was the author of around 180 articles on mathematics. He wrote papers on differential and integral equations, studying both their theory and applications. Other areas he wrote on included algebra (where among other topics he studied the theory of permutation matrices), geometry, mathematical and numerical analysis, probability theory and mathematical statistics. He was also interested in the philosophy of mathematics, the history of mathematics and mathematical education. Krawtchouk edited the first three-volume dictionary of Ukrainian mathematical terminology. In 1929 Krawtchouk was elected a full member of the Ukrainian Academy of Sciences. He taught at the Kiev Polytechnic Institute (now the National Technical University of Ukraine) where he became professor and was head of the Mathematical Chair. However, this was a time of new political problems. At this time political life in the USSR revolved round the exposure and suppression of alleged plotters against the regime. The country was subjected to an vigorous campaign against so-called enemies of the people. There was a series of public trials and in a massive terror campaign against the population as a whole. The worst phase of the terror took place during the period 1937-38 when at least five million people were sent to camps mostly in the Arctic. In 1937 Krawtchouk was accused of being a Polish spy and also a bourgeois nationalist. He was arrested, tried and sentenced to twenty years in prison and five years in exile. As part of the punishment imposed on him, Krawtchouk was stripped of his membership of the Ukrainian Academy of Sciences and his exile was to be spent at the Kolyma camp in northeastern Siberia, one of the two most notorious of the Corrective Labour Camps. He died at the age of 49 in Kolyma, one of the Labour Camps set up by the Gulag. In March 1992, fifty years after his death, he was restored as a member of the Ukrainian Academy of Sciences. In September 1992 the First International M Krawtchouk Conference was held in the Ukraine. Since then these conferences have been a yearly event. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krawtchouk.html (2 of 3) [2/16/2002 11:18:14 PM]

Krawtchouk

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Krawtchouk.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krawtchouk.html (3 of 3) [2/16/2002 11:18:14 PM]

Krein

Mark Grigorievich Krein Born: 3 April 1907 in Kiev, Ukraine Died: 17 Oct 1989 in Odessa, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mark G Krein's father was involved in the wood trade which meant that the family had little money. The fact that the family were Jewish meant that Krein grew up in an atmosphere of persecution. This was also highly significant for Krein's subsequent career, for discrimination against Jews in the Ukraine was bad and, by misfortune for Krein, was particularly bad in Odessa where he lived from the age of 17. Krein showed a remarkable talent for mathematics at a young age. By the time he reached 14 years of age he was already attending research seminars in mathematics in Kiev. However, Krein never completed his undergraduate degree for he left his home in Kiev when he was 17 years old and ran away to Odessa. Despite the lack of an undergraduate degree, his talents were clearly visible to the mathematicians at Odessa University and, in 1926, Krein was accepted for doctoral studies under Chebotaryov. Chebotaryov himself was not in the happiest of positions for, after working at Odessa University, he had accepted a permanent post in Moscow in 1924. The political situation surrounding this appointment induced him to leave Moscow and return to Odessa after only a few months. Krein therefore began his doctoral studies well aware of the difficulties of the time. In 1928 Chebotaryov left Odessa and became professor at Kazan University. Krein completed his doctorate at Odessa in the following year and remained on the staff at the university building up one of the most important centres for functional analysis research in the world. During this time he worked on topics such as Banach spaces, the moment problem, integral equations and matrices, and on spectral theory for linear operators. As Osilenker writes in [28] and [29]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krein.html (1 of 4) [2/16/2002 11:18:16 PM]

Krein

In the creative legacy of M G Krein a large place is occupied by the moment problem and the study of associated Jacobian matrices.... Krein's work on the moment problem, which played such an important part in his mathematical development, is discussed in detail in [25] and [26]. Kolmogorov had laid the foundations for the study of extremal problems in 1935 and Krein began to work on extremal problems for the class of differentiable periodic functions. Together with N I Akhiezer, Krein made a major contribution to this field in 1937. In 1941 Krein had to leave Odessa when the university was evacuated as the German armies advanced. He was appointed as professor of theoretical mechanics at Kuibyshev Industrial Institute but he returned to Odessa in 1944. However, soon after taking up his post again he was dismissed. Gohberg writes in [13]:[Krein] was accused of Jewish nationalism, presumably for having had too many Jewish students before the War. This accusation was certainly included in his classified file and was presumably held against him all his life. ... He was not allowed to have Jewish students and was deprived of a university base. Not only was Krein dismissed but the whole of the functional analysis school at Odessa was closed down. Potapov, who had been one of Krein's non-Jewish students, tried hard to influence the university authorities to reverse their decisions. He wrote:The leaders of the university have to correct their mistakes and to revive immediately the famous traditions of the Odessa School of Mathematics. The Faculty of Physics and Mathematics has to become active again, as a really creative centre of scientific and mathematical thought in our city .... Krein was not reinstated, however, but held the chair of theoretical mechanics at Odessa Marine Engineering Institute from 1944. Also from 1944, he held a part-time post as head of the functional analysis and algebra department at the Mathematical Institute of the Ukranian Academy of Science in Kiev. This latter position came to a sudden end in 1952 when he was dismissed for a second time. Officially the reason given was that he did not live in Kiev, but in reality it seems more likely that the accusation of Jewish nationalism in his classified file was again the reason. Although the 1940s must have been difficult times for Krein, his mathematical research did not suffer. Among other important work, Krein wrote eight papers on harmonic analysis and representation theory in the 1940s. In these papers he studied the use of algebras of operators applied to obtain results on positive definite kernels and functions. He also studied analysis on homogeneous spaces and duality theorems. From 1954 until his retirement Krein occupied the chair of theoretical mechanics at the Odessa Civil Engineering Institute. During the 1940s and 1950s there were many unsuccessful attempts to have Krein and his students reinstated at Odessa University but all attempts failed. During the many years of very active research Krein had around him a group of very active mathematicians although this group was based more on an informal arrangement than that of a proper research group. This group provided him with strong support, frequently meeting in Krein's own house. For example one of the Jewish student he ahd supervised before World War II was Livsic and he, like

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krein.html (2 of 4) [2/16/2002 11:18:16 PM]

Krein

Krein, was not welcome back at Odessa University after the War. Livsic, again like Krein, did return to Odessa, teaching at the Hydrometerological Institute until 1957 and forming part of Krein's unofficial research group in Odessa. As well as Krein's house, the group often met at the Scientists' Club in Odessa and, in [16], Thomas Kailath describes a visit he made to Odessa where he talked at the Scientists' Club at the end of 1984. Despite the support of those around him, Krein did have to work at a mathematical disadvantage in one sense, however, for he was not allowed to travel abroad and so was unable to attend international conferences that many mathematicians feel to be almost essential for someone so preeminent at an international level. International recognition came to Krein despite his inability to visit centres of mathematical research around the world. In 1968 the American Academy of Arts and Sciences elected him an honorary member, and he was elected a Foreign Member of the National Academy of Sciences in 1979. Krein received many other honours, but perhaps the most prestigious award made to him was the Wolf Prize in Mathematics in 1982. After the years of discrimination on account of being Jewish it must have been particularly pleasing to receive this Prize from Israel. The citation for the prize summarises well the contribution that Krein made to mathematics:His work is the culmination of the noble line of research begun by Chebyshev, Stieltjes, Sergi Bernstein and Markov and continued by F Riesz, Banach and Szego. Krein brought the full force of mathematical analysis to bear on problems of function theory, operator theory, probability and mathematical physics. His contributions led to important developments in the applications of mathematics to different fields ranging from theoretical mechanics to electrical engineering. His style in mathematics and his personal leadership and integrity have set standards of excellence. Gohberg writes of the quality and style of Krein's work in [13]:He is the author of more than 270 papers and monographs of unsurpassed breadth and quality. ... A profound intrinsic unity and close interlacing of general abstract and geometrical ideas with concrete and analytical results and applications and characteristic of Krein's work. Despite a life of persecution, in which he often feared arrest, Krein remained enthusiastic, friendly and kind, showing great mathematical generosity towards his students and colleagues. Towards the end of his life, however, the struggle which he continually had to make took its toll and he suffered from depression which became worse after the death of his wife. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (31 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krein.html (3 of 4) [2/16/2002 11:18:16 PM]

Krein

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Krein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krein.html (4 of 4) [2/16/2002 11:18:16 PM]

Kreisel

Georg Kreisel Born: 15 Sept 1923 in Graz, Austria

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Kreisel came from a Jewish background so growing up in Graz in the 1930s was very difficult. Kreisel's parents saw the approaching political problems and, before Hitler took over Austria, they sent Georg and his brother to England. Kreisel studied mathematics at Trinity College, Cambridge graduating with a B.A. in 1944. During his undergraduate years Kreisel was influenced by Wittgenstein who was also at Trinity. Wittgenstein said that Kreisel was:... the most able philosopher he had ever met who was also a mathematician. Kreisel was sent to do War Service with the Admiralty immediately his university courses were over and he began work at West Leigh near Havant and close to the naval base at Portsmouth. The head of West Leigh at that time was Collingwood. After a while, Kreisel was moved to Fanum House in central London where he studied the effects of waves on the harbours which were being designed for the Normandy landings. In 1946 Kreisel returned to Cambridge to undertake research, studying mathematical logic. After the award of his doctorate Kreisel hoped for a Fellowship at Trinity but this was not forthcoming. He applied for academic positions and was appointed to Reading in 1949. Freeman Dyson was an undergraduate at Cambridge in the same year as Kreisel and by the 1950s was at the Institute for Advanced Study. He persuaded Gödel to invite Kreisel to the Institute for Advanced Study and Kreisel arrived there in the summer of 1955. Verena Huber-Dyson writes in [1]:Although he did not come with the explicit intention of staying, and the possibility of return to Europe was open to him, Kreisel did not have strong ties to his position at Reading, a rather puzzling domicile for a person of his peculiar qualities. Kreisel returned to Reading in 1957 but kept up a mathematically important correspondence with Gödel. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kreisel.html (1 of 3) [2/16/2002 11:18:18 PM]

Kreisel

In 1958-59 Kreisel was back in the United States, this time at Stanford. After returning to Reading for his last year on the staff there in 1959-60 he spent the two years 1960-62 in Paris. In 1962 Kreisel returned to the United States and was appointed to Stanford where he remained on the staff until he retired in 1985. S Feferman in [1] writes about Kreisel's contributions:Through his own contributions (individual and collaborative) and his extraordinary personal influence, Georg Kreisel did perhaps more than anyone else to promote the development of proof theory and the metamathematics of constructivity in the last forty years. One important aspect which Kreisel worked on over a period of 30 years was his "unwinding". In 1958 in a paper Mathematical significance of consistency proof in 1958 Kreisel wrote:There is a different general program which does not seem to suffer the defects of [Hilbert's] consistency program: To determine the constructive (recursive) content or the constructive equivalent of the non-constructive concepts and theorems used in mathematics, particularly in arithmetic and analysis. This "different general program" he later gave the more colourful name "unwinding program". As Feferman writes [1]:It aimed to substitute clear mathematical results for what were said to be vague, misplaced, crude foundational goals. But, as with his work on constructivity, Kreisel also sought to replace those by a more sophisticated stance about foundations. Kreisel was elected a fellow of the Royal Society in 1966. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kreisel.html (2 of 3) [2/16/2002 11:18:18 PM]

Kreisel

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kreisel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kreisel.html (3 of 3) [2/16/2002 11:18:18 PM]

Kronecker

Leopold Kronecker Born: 7 Dec 1823 in Liegnitz, Prussia (now Legnica, Poland) Died: 29 Dec 1891 in Berlin, Germany

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leopold Kronecker's parents were well off, his father, Isidor Kronecker, being a successful business man while his mother was Johanna Prausnitzer who also came from a wealthy family. The family were Jewish, the religion that Kronecker kept until a year before his death when he became a convert to Christianity. Kronecker's parents employed private tutors to teach him up to the stage when he entered the Gymnasium at Liegnitz, and this tutoring gave him a very sound foundation to his education. Kronecker was taught mathematics at Liegnitz Gymnasium by Kummer, and it was due to Kummer that Kronecker became interested in mathematics. Kummer immediately recognised Kronecker's talent for mathematics and he took him well beyond what would be expected at school, encouraging him to undertake research. Despite his Jewish upbringing, Kronecker was given Evangelical religious instruction at the Gymnasium which certainly shows that his parents were openminded on religious matters. Kronecker became a student at Berlin University in 1841 and there he studied under Dirichlet and Steiner. He did not restrict himself to studying mathematics, however, for he studied other topics such as astronomy, meteorology and chemistry. He was especially interested in philosophy studying the philosophical works of Descartes, Leibniz, Kant, Spinoza and Hegel. After spending the summer of 1843 at the University of Bonn, which he went to because of his interest in astronomy rather than mathematics, he then went to the University of Breslau for the winter semester of 1843-44. The reason that he went to Breslau was certainly because of his interest in mathematics because he wanted to study again with his old school teacher Kummer who had been appointed to a chair at Breslau in 1842.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kronecker.html (1 of 5) [2/16/2002 11:18:20 PM]

Kronecker

Kronecker spent a year at Breslau before returning to Berlin for the winter semester of 1844-45. Back in Berlin he worked on his doctoral thesis on algebraic number theory under Dirichlet's supervision. The thesis, On complex units was submitted on 30 July 1845 and he took the necessary oral examination on 14 August. Dirichlet commented on the thesis saying that in it Kronecker showed:... unusual penetration, great assiduity, and an exact knowledge of the present state of higher mathematics. It may come as a surprise to many Ph.D. students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy. Jacobi had health problems which caused him to leave Königsberg, where he held a chair, and return to Berlin. Eisenstein, whose health was also poor, lectured in Berlin around this time and Kronecker came to know both men well. The direction that Kronecker's mathematical interests went later had much to do with the influence of Jacobi and Eisenstein around this time. However, just as it looked as if he would embark on an academic career, Kronecker left Berlin to deal with family affairs. He helped to manage the banking business of his mother's brother and, in 1848, he married the daughter of this uncle, Fanny Prausnitzer. He also managed a family estate but still found the time to continue working on mathematics, although he did this entirely for his own enjoyment. Certainly Kronecker did not need to take on paid employment since he was by now a wealthy man. His enjoyment of mathematics meant, however, that when circumstances changed in 1855 and he no longer needed to live on the estate outside Liegnitz, he returned to Berlin. He did not wish a university post, rather he wanted to take part in the mathematical life of the university and undertake research interacting with the other mathematicians. In 1855 Kummer came to Berlin to fill the vacancy which occurred when Dirichlet left for Göttingen. Borchardt had lectured at Berlin since 1848 and, in late 1855, he took over the editorship of Crelle's Journal on Crelle's death. In 1856 Weierstrass came to Berlin, so within a year of Kronecker returning to Berlin, the remarkable team of Kummer, Borchardt, Weierstrass and Kronecker was in place in Berlin. Of course since Kronecker did not hold a university appointment, he did not lecture at this time but was remarkably active in research publishing a large number of works in quick succession. These were on number theory, elliptic functions and algebra, but, more importantly, he explored the interconnections between these topics. Kummer proposed Kronecker for election to the Berlin Academy in 1860, and the proposal was seconded by Borchardt and Weierstrass. On 23 January 1861 Kronecker was elected to the Academy and this had a surprising benefit. Members of the Berlin Academy had a right to lecture at Berlin University. Although Kronecker was not employed by the University, or any other organisation for that matter, Kummer suggested that Kronecker exercise his right to lecture at the University and this he did beginning in October 1862. The topics on which he lectured were very much related to his research: number theory, the theory of equations, the theory of determinants, and the theory of integrals. In his lectures [1]:He attempted to simplify and refine existing theories and to present them from new perspectives. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kronecker.html (2 of 5) [2/16/2002 11:18:20 PM]

Kronecker

For the best students his lectures were demanding but stimulating. However, he was not a popular teacher with the average students [1]:Kronecker did not attract great numbers of students. Only a few of his auditors were able to follow the flights of his thought, and only a few persevered until the end of the semester. Berlin was attractive to Kronecker, so much so that when he was offered the chair of mathematics in Göttingen in 1868, he declined. He did accept honours such as election to the Paris Academy in that year and for many years he enjoyed good relations with his colleagues in Berlin and elsewhere. In order to understand why relations began to deteriorate in the 1870s we need to examine Kronecker's mathematical contributions more closely. We have already indicated that Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers. However the topics he studied were restricted by the fact that he believed in the reduction of all mathematics to arguments involving only the integers and a finite number of steps. Kronecker is well known for his remark:God created the integers, all else is the work of man. Kronecker believed that mathematics should deal only with finite numbers and with a finite number of operations. He was the first to doubt the significance of non-constructive existence proofs. It appears that, from the early 1870s, Kronecker was opposed to the use of irrational numbers, upper and lower limits, and the Bolzano-Weierstrass theorem, because of their non-constructive nature. Another consequence of his philosophy of mathematics was that to Kronecker transcendental numbers could not exist. In 1870 Heine published a paper On trigonometric series in Crelle's Journal, but Kronecker had tried to persuade Heine to withdraw the paper. Again in 1877 Kronecker tried to prevent publication of Cantor's work in Crelle's Journal, not because of any personal feelings against Cantor (which has been suggested by some biographers of Cantor) but rather because Kronecker believed that Cantor's paper was meaningless, since it proved results about mathematical objects which Kronecker believed did not exist. Kronecker was on the editorial staff of Crelle's Journal which is why he had a particularly strong influence on what was published in that journal. After Borchardt died in 1880, Kronecker took over control of Crelle's Journal as the editor and his influence on which papers which would be published increased. The mathematical seminar in Berlin had been jointly founded in 1861 by Kummer and Weierstrass and, when Kummer retired in 1883, Kronecker became a codirector of the seminar. This increased Kronecker's influence in Berlin. Kronecker's international fame also spread, and he was honoured by being elected a foreign member of the Royal Society of London on 31 January 1884. He was also a very influential figure within German mathematics [1]:He established other contacts with foreign scientists in numerous travels abroad and in extending to them the hospitality of his Berlin home. For this reason his advice was often solicited in regard to filling mathematical professorships both in Germany and elsewhere; his recommendations were probably as significant as those of his erstwhile friend Weierstrass. Although Kronecker's view of mathematics was well known to his colleagues throughout the 1870s and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kronecker.html (3 of 5) [2/16/2002 11:18:20 PM]

Kronecker

1880s, it was not until 1886 that he made these views public. In that year he argued against the theory of irrational numbers used by Dedekind, Cantor and Heine giving the arguments by which he opposed:... the introduction of various concepts by the help of which it has frequently been attempted in recent times (but first by Heine) to conceive and establish the "irrationals" in general. Even the concept of an infinite series, for example one which increases according to definite powers of variables, is in my opinion only permissible with the reservation that in every special case, on the basis of the arithmetic laws of constructing terms (or coefficients), ... certain assumptions must be shown to hold which are applicable to the series like finite expressions, and which thus make the extension beyond the concept of a finite series really unnecessary. Lindemann had proved that is transcendental in 1882, and in a lecture given in 1886 Kronecker complimented Lindemann on a beautiful proof but, he claimed, one that proved nothing since transcendental numbers did not exist. So Kronecker was consistent in his arguments and his beliefs, but many mathematicians, proud of their hard earned results, felt that Kronecker was attempting to change the course of mathematics and write their line of research out of future developments. Kronecker explained his programme based on studying only mathematical objects which could be constructed with a finite number of operation from the integers in Über den Zahlbergriff in 1887. Another feature of Kronecker's personality was that he tended to fall out personally with those who he disagreed with mathematically. Of course, given his belief that only finitely constructible mathematical objects existed, he was completely opposed to Cantor's developing ideas in set theory. Not only Dedekind, Heine and Cantor's mathematics was unacceptable to this way of thinking, and Weierstrass also came to feel that Kronecker was trying to convince the next generation of mathematicians that Weierstrass's work on analysis was of no value. Kronecker had no official position at Berlin until Kummer retired in 1883 when he was appointed to the chair. But by 1888 Weierstrass felt that he could no longer work with Kronecker in Berlin and decided to go to Switzerland, but then, realising that Kronecker would be in a strong position to influence the choice of his successor, he decided to remain in Berlin. Kronecker was of very small stature and extremely self-conscious about his height. An example of how Kronecker reacted occurred in 1885 when Schwarz sent him a greeting which included the sentence:He who does not honour the Smaller, is not worthy of the Greater. Here Schwarz was joking about the small man Kronecker and the large man Weierstrass. Kronecker did not see the funny side of the comment, however, and never had any further dealings with Schwarz (who was Weierstrass's student and Kummer's son-in-law). Others however displayed more tact and, for example, Helmholtz who was a professor in Berlin from 1871, managed to stay on good terms with Kronecker. The Deutsche Mathematiker-Vereinigung was set up in 1890 and the first meeting of the Association was organised in Halle in September 1891. Despite the bitter antagonism between Cantor and Kronecker, Cantor invited Kronecker to address this first meeting as a sign of respect for one of the senior and most eminent figures in German mathematics. However, Kronecker never addressed the meeting, since his wife was seriously injured in a climbing accident in the summer and died on 23 August 1891. Kronecker http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kronecker.html (4 of 5) [2/16/2002 11:18:20 PM]

Kronecker

only outlived his wife by a few months, and died in December 1891. We should not think that Kronecker's views of mathematics were totally eccentric. Although it was true that most mathematicians of his day would not agree with those views, and indeed most mathematicians today would not agree with them, they were not put aside. Kronecker's ideas were further developed by Poincaré and Brouwer, who placed particular emphasis upon intuition. Intuitionism stresses that mathematics has priority over logic, the objects of mathematics are constructed and operated upon in the mind by the mathematician, and it is impossible to define the properties of mathematical objects simply by establishing a number of axioms.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles)

Some Quotations (3)

A Poster of Leopold Kronecker

Mathematicians born in the same country

Cross-references to History Topics

1. The beginnings of set theory 2. Matrices and determinants 3. A history of group theory

Honours awarded to Leopold Kronecker (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1884

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kronecker.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kronecker.html (5 of 5) [2/16/2002 11:18:20 PM]

Krull

Wolfgang Krull Born: 26 Aug 1899 in Baden-Baden, Germany Died: 12 April 1971 in Bonn, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wolfgang Krull's father was Helmuth Krull and his mother was Adele Siefert Krull. Helmuth Krull had a dentist's practice in Baden-Baden and it was in that town that Krull attended school. After graduating from secondary school in 1919 he entered the University of Freiberg. It was the custom in those days for students in Germany to move around various universities during their period of study and Krull was no exception. He spent time at the University of Rostock before moving to Göttingen in 1920. From 1920 to 1921 he studied at Göttingen with Klein but was most influenced by Emmy Noether. He attended Klein's seminar in the session 1920-21 and he then returned to Freiberg and presented his doctoral thesis on the theory of elementary divisors in 1922. Ring theory results from this thesis have recently been found important in the area of coding theory. Appointed as an instructor at Freiberg on 1 October 1922 he was promoted to extraordinary professor in 1926. He remained there until 1928 when he moved to Erlangen. His inaugural address on becoming a full professor at Erlangen was one which says much of how Krull saw mathematics. He saw the role of a mathematician as [1] (see also [2] and [4]):... not merely ... finding theorems and proving them. He wants to arrange and group these theorems together in such a way that they appear not only as correct but also as imperative and self-evident. To my mind such an aspiration is an aesthetic one and not one based on theoretical cognition If Emmy Noether had the greatest influence on the topics which Krull would spend his life researching, it http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krull.html (1 of 3) [2/16/2002 11:18:22 PM]

Krull

can be seen from this inaugural address that it was Klein who had the greatest influence on Krull's large scale view of mathematics. In 1929 he married Gret Meyer and they would have two daughters. The ten years Krull spent in Erlangen were the most productive period of his career. Schoeneborn writes in [1]:The years Krull spent as a full professor in Erlangen were the high point of his creative life. About thirty-five publications of fundamental importance for the development of commutative algebra and algebraic geometry date from this period. At Erlangen he was involved in university life as well as concentrating on his research, being elected Head of the Faculty of Science. In 1939 Krull left Erlangen to take up a chair at Bonn. However, his career was disrupted by the Second World War which began shortly after Krull was appointed to the University of Bonn. During the war he undertook war duties, working in the naval meteorological service. When his war service had ended in 1946, Krull took up again his post at the University of Bonn and he would remain there for the rest of his life. In this final period of his career Krull continued his high level of productivity (he wrote 50 papers in his post-war years in Bonn) and also broadened his mathematical interests. He continued [1]:... his earlier studies, but also dealt with other fields of mathematics: group theory, calculus of variations, differential equations, Hilbert spaces. Krull's first publications were on rings and algebraic extension fields. In 1925 he proved the Krull-Schmidt theorem for decomposing abelian groups of operators. He then studied Galois theory and extended the classical results on Galois theory of finite field extensions to infinite field extensions. In passing from the finite to the infinite case Krull introduced topological ideas. In 1928 he defined the Krull dimension of a commutative Noetherian ring and brought ring theory into in new setting in which he was able to show that the principal ideal theorem held. Perhaps the reason that the idea of the Krull dimension is such a natural concept is that it encapsulates in an abstract setting the analogues of geometric dimensions. The principal ideal theorem [2]:... was quickly recognised as a decisive advance in Noether's programme of emancipating abstract ring theory from the theory of polynomial rings. Krull carried his work forward, defining further concepts which are today central to modern research in ring theory. In 1932 he defined valuations which are today known as Krull valuations. He then wrote the remarkable treatise Ideal Theory which remains a beautiful introduction to ring theory but is simply a theory built from the results that Krull had himself proved. One could say that Krull had achieved the goal he had in some sense set himself in his Erlangen address and arranged his theory to be self-evident. Another major topic in ring theory is the study of local rings, that is rings having a unique maximal ideal, and they are used in the study of local properties of algebraic varieties. The concept was introduced by Krull in 1938 and his fundamental results were developed into a major theory by mathematicians such as Chevalley and Zariski. He supervised 35 doctoral students, and rather remarkably, 32 of these were students which he supervised after the end of World War II. Gray, in [2], writes that Krull's papers are:... marked by the profundity of his ideas, the rigour of his proofs, and also by a strong aesthetic sense. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krull.html (2 of 3) [2/16/2002 11:18:22 PM]

Krull

Indeed much of modern ring theory is still following the path which Krull took, building on the foundations which Emmy Noether had laid. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Wolfgang Krull

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1920 to 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Krull.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krull.html (3 of 3) [2/16/2002 11:18:22 PM]

Krylov_Aleksei

Aleksei Nikolaevich Krylov Born: 15 Aug 1863 in Visyaga, Simbirskoy (now Ulyanovskaya), Russia Died: 26 Oct 1945 in Leningrad, USSR (now St Petersburg, Russia)

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aleksei Krylov entered the Maritime High School in St Petersburg in 1878. He graduated in 1884 and was appointed to the compass unit of the Main Hydrographic Administration. There he began work on compass deviation, a topic he would return to many times. In 1880 Krylov joined the department of ship construction of St Petersburg Maritime Academy. There he was taught mathematics by A N Korkin, a student of Chebyshev. He graduated in 1890 and stayed there to teach for almost 50 years. Krylov worked on what Euler called 'naval science', namely the theories of buoyancy, stability, rolling and pitching, vibrations, compass theories etc. In 1914 Krylov was awarded a doctorate in applied mathematics from Moscow University. From 1927 until 1932 he was director of the Physics - Mathematics Institute of the Soviet Academy of Sciences. Krylov improved Fourier's method for solving boundary value problems in a 1905 paper and gave many applications. His work led him to study the acceleration of convergence of Fourier series and the approximate solutions to differential equations. In 1931 he found a new method of solving the secular equation which determines the frequency of vibrations in mechanical systems which is better than methods due to Lagrange, Laplace, Jacobi and Le Verrier.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krylov_Aleksei.html (1 of 2) [2/16/2002 11:18:23 PM]

Krylov_Aleksei

In 1943 Krylov was awarded a state prize for his compass theory work and made a hero of socialist labour. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (17 books/articles) Mathematicians born in the same country Honours awarded to Aleksei N Krylov (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Krylov

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Krylov_Aleksei.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krylov_Aleksei.html (2 of 2) [2/16/2002 11:18:23 PM]

Krylov_Nikolai

Nikolai Mitrofanovich Krylov Born: 29 Nov 1879 in St Petersburg, Russia Died: 11 May 1955 in Moscow, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nikolai Krylov graduated from the St Petersburg Institute of Mines in 1902. He was professor there from 1912 until 1917 when he went to the Crimea to become professor at the Crimea University. He held this post until 1922 when he was appointed chairman of the mathematical physics department of the Ukranian Academy of Sciences. In 1939 Krylov became an honoured scientist of the Ukranian Soviet Socialist Republic. He worked mainly on interpolation and numerical solutions to differential equations, where he obtained very effective formulas for the errors. He also applied his methods to non-linear oscillatory problems in 1932 and, in so doing, laid the foundations for non-linear mechanics. Krylov published over 200 papers on analysis and mathematical physics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krylov_Nikolai.html (1 of 2) [2/16/2002 11:18:25 PM]

Krylov_Nikolai

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Krylov_Nikolai.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Krylov_Nikolai.html (2 of 2) [2/16/2002 11:18:25 PM]

Kulik

Yakov Kulik Born: 1 April 1783 in Lvov, Poland (now Ukraine) Died: 28 Feb 1863 in Prague, Czech Republic Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yakov Kulik attended the law and philosophy faculties of Lvov University. He graduated in 1814 and then taught at a High School while he continued to work for his doctorate. He received his doctorate in 1822, then, from 1826, he was Professor of Mathematics at the Charles University of Prague until his death in 1863. In 1848 the library of the University of Lvov was destroyed by fire and Kulik donated over 1000 books to help rebuild the collection at the University in the town of his birth where he had studied. Kulik wrote texts on mathematics and mechanics, for example publishing Lehrbuch der höheren mechanik in 1846. He also published Der transcendjährige Kalender. Kulik produced numerous mathematical tables including an unpublished table of divisors of integers consisting of 4212 pages. Kulik did publish a description of the unpublished tables in 1860 and, in 1866, Petzval also described Kulik's tables. In [3] Howard Eves states that Kulik's greatest achievement was the construction of these factor tables:His as yet unpublished manuscript is the result of a twenty-year hobby, and covers all numbers up to 100,000,000. Similar statements are made in many other books, see for example [2]. Kulik's manuscripts are kept in the archives of the Viennese Academy of Sciences, and Novy has studied them and has written [5] to correct false statements about them such as the one by Eves above:... the manuscript of Kulik's tables of divisors is essentially useless beginning with the third volume; the second volume, which has been lost, could perhaps tell us more about the real value of the manuscript. Kulik's methods of calculating his tables and other manuscripts left by Kulik are, however, very interesting and are discussed in [5]. Article by: J J O'Connor and E F Robertson List of References (5 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kulik.html (1 of 2) [2/16/2002 11:18:26 PM]

Kulik

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kulik.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kulik.html (2 of 2) [2/16/2002 11:18:26 PM]

Kumano-Go

Hitoshi Kumano-Go Born: 4 Oct 1935 in Arita, Wakayama Prefecture, Japan Died: 24 Aug 1982 in Osaka, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hitoshi Kumano-Go was educated in the standard way for gifted students in Japan at this time. Before entering university he studied at Taikyu Senior High School, graduating in 1958. Entering Osaka University after graduating from high school, Kumano-Go studied mathematics and graduated in 1958. He continued to study at Osaka University for his doctorate. His supervisor was M Nagumo who supervised his work on the singular perturbation of second order partial differential equations. As is frequently the case, the problem which Kumano-Go studied was to extend work which had been completed earlier by his supervisor. In 1962 Kumano-Go was appointed as an assistant at Osaka University. He submitted his doctoral dissertation in 1963 and received his Ph.D. in September of that year. He continued on the staff at Osaka, being promoted to assistant professor in 1964 and to associate professor in 1967. During these years Kumano-Go published a series of papers which studied the local and global uniqueness of the solutions of the Cauchy problem for partial differential equations. This work used ideas from earlier contributions to the topic by Calderon and Zygmund. In two papers Kumano-Go also studied non-uniqueness of solutions of the Cauchy problem. Kumano-Go spent the two academic years 1967-69 visiting the Courant Institute of Mathematical Sciences at New York University. These were years of great benefit to Kumano-Go who was able to develop many ideas in conversations with Kurt Friedrichs, Peter Lax, Louis Nirenberg and others. He became involved in founding the theory of pseudo-differential operators and after his return to Osaka he continued to publish important contributions to this topic.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kumano-Go.html (1 of 3) [2/16/2002 11:18:28 PM]

Kumano-Go

In 1971 Kumano-Go was promoted to full professor. His major treatise on Pseudo Differential Operators was published in Japanese in 1974. It is [2]:... an outstanding contribution to the field of pseudo-differential operators in view of its uniqueness and the wealth of material. The book has been translated into English and appeared, published by MIT Press, in 1982. One other important monograph written by Kumano-Go should be mentioned. This is Partial differential equations which was again written in Japanese and was published in 1978. This is a textbook which in addition to studying partial differential equations provides an introduction to pseudo-differential operators. In addition to his work on pseudo-differential operators, Kumano-Go published a series of papers on the product of Fourier integral operators. This collaborative work with his colleagues led to results which were applied to [2]:... the construction of the fundamental solution of a first order hyperbolic system and the study of the wave front sets of solutions. H Tanabe writes in [2] about Kumano-Go mathematical contributions other than those contained in his research:Kumano-Go's contributions were not limited to his own personal mathematical achievements. He also trained many young mathematicians of high ability. He was well known for the kindly interest which he took in the careers of young mathematicians. He was always ready to help, encourage and advise them. He looked through all their manuscripts, helping to improve and develop their ideas. Kumano-Go suffered ill health and was admitted to Osaka Hospital in May 1981. The doctors discovered that he was suffering from a brain tumour from which no cure was possible. At the height of his mathematical contributions at the age of 46, Kumano-Go sadly died. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kumano-Go.html (2 of 3) [2/16/2002 11:18:28 PM]

Kumano-Go

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kumano-Go.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kumano-Go.html (3 of 3) [2/16/2002 11:18:28 PM]

Kummer

Ernst Eduard Kummer Born: 29 Jan 1810 in Sorau, Brandenburg, Prussia (now Germany) Died: 14 May 1893 in Berlin, Germany

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eduard Kummer's father, Carl Gotthelf Kummer was a physician. However he died when Eduard was only three years old and Eduard and his elder brother were brought up by their mother. Eduard received private coaching before entering the Gymnasium in Sorau when he was nine years old. In 1828 Kummer entered the University of Halle with the intention of studying Protestant theology. Fortunately for the good of mathematics, Kummer received mathematics teaching as part of his degree, designed to provide a proper foundation to the study of philosophy. Kummer's mathematics lecturer H F Scherk inspired his interest in mathematics and Kummer soon was studying mathematics as his main subject, although at this stage he still saw it as leading to a later study of philosophy. In 1831 Kummer was awarded a prize for a mathematical essay he wrote on a topic set by Scherk. In the same year he was awarded his certificate to enable him to teach in schools and, on the strength of his prize winning essay, he was awarded a doctorate. During session 1831-32 Kummer taught a probationary year at the Gymnasium in Sorau where he had been educated. Then he was appointed to a teaching post at the Gymnasium in Liegnitz, now Legnica in Poland. Kummer held this teaching post in Liegnitz for 10 years. There he taught mathematics and physics and sometimes other topics. Some of his pupils had great ability, and conversely they were extremely fortunate to find a school teacher of Kummer's quality and ability to inspire. His two most famous pupils were Kronecker and Joachimsthal and, under Kummer's guidance, they were undertaking mathematical research while at school. Kummer himself was undertaking mathematics research while teaching at Liegnitz. He published a paper

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kummer.html (1 of 4) [2/16/2002 11:18:30 PM]

Kummer

on hypergeometric series in Crelle's Journal in 1836 and he sent a copy of the paper to Jacobi. This led to Jacobi, and later Dirichlet, corresponding with Kummer on mathematical topics and they soon realised the great potential for the highest level of mathematics that Kummer possessed. In 1839, although still a school teacher, Kummer was elected to the Berlin Academy of Sciences on Dirichlet's recommendation. Jacobi now realised that he had to find Kummer a university professorship. In 1840 Kummer married a cousin of Dirichlet's wife. This marriage would only last eight years as his wife sadly died in 1848. However these where years of great achievement for Kummer. In 1842, with strong support from Jacobi and Dirichlet, he was appointed a full professor at the University of Breslau, now Wroclaw in Poland. There he quickly established himself as an outstanding university teacher of mathematics and, starting with his move to Breslau, he began to undertake research in number theory. After the death of his wife in 1848, Kummer remarried fairly soon. During the 22 to 24 February 1848 insurrection in Paris, king Louis-Philippe was overthrown. This resulted in a series of sympathetic revolutions against the governments of the German Confederation. Most of them were tame affairs but in the case of the fighting in Berlin it was bitter and bloody. Nobody could fail to have strong political views at this time and indeed Kummer had strong right wing political views. He supported the constitutional monarchy in the 1848 revolution and was against a republic. When on 13 March 1848 Metternich, a symbol of the establishment, was forced to resign his position in the Austrian Cabinet, the princes quickly made peace with the opposition to prevent republican and socialist experiments like those in France. In 1855 Dirichlet left Berlin to succeed Gauss at Göttingen. He recommended to Berlin that they offer the vacant chair to Kummer, which indeed they did. Now Kummer played a clever political trick. He wanted Weierstrass as a colleague at Berlin, yet he realised that Weierstrass was a strong candidate for the chair he was leaving vacant in Breslau. Hence he recommended to Breslau that they appoint his former student Joachimsthal. All went according to plan for Kummer and, in 1856, Weierstrass was appointed to Berlin. Kronecker had also been appointed to Berlin in 1855 so Berlin became one of the leading mathematical centres in the world. In 1861 Kummer and Weierstrass established Germany's first pure mathematics seminar in Berlin. K-R Biermann writes of Kummer's teaching in Berlin in [1]:Kummer's Berlin lectures, always carefully prepared, covered analytic geometry, mechanics, the theory of surfaces, and number theory. The clarity and vividness of his presentations brought him great numbers of students - as many as 250 were counted at his lectures. While Weierstrass and Kronecker offered the most recent results of their research in their lectures, Kummer in his restricted himself, after instituting the seminar, to laying firm foundations. In the seminar, on the other hand, he discussed his own research in order to encourage the participants to undertake independent investigations. Explaining Kummer's popularity Biermann writes:Kummer's popularity as a professor was based mot only on the clarity of his lectures but on his charm and sense of humour as well. Moreover, he was concerned for the well-being of his students and willingly aided them when material difficulties arose... Surprisingly, given Kummer's outstanding qualities as a teacher of mathematics, he never wrote any

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kummer.html (2 of 4) [2/16/2002 11:18:30 PM]

Kummer

textbooks. He did publish mathematical lectures and, of course, many extremely influential mathematical papers. Having first been elected to the Berlin Academy while still a school teacher, Kummer ended up with high office in the Academy. He was secretary of the Mathematics/ Physics Section of the Academy from 1863 to 1878. He also held high office in the University of Berlin, being Dean in 1857-58 and again in 1865-66. He was rector of the university in 1868-69. While at Berlin, Kummer supervised a large number of doctoral students including many who went on to hold mathematics chairs at universities, including Bachmann, Cantor, du Bois-Reymond, Gordan, Schönflies and Schwarz. In fact Schwarz became related to Kummer when he married one of his daughters. Kummer also appointed many talented young lecturers including Clebsch, Christoffel and Fuchs. It was Fuchs who succeeded Kummer when he decided to retire in 1883 on the grounds that his memory was failing, although nobody other than Kummer himself ever detected this. During Kummer's first period of mathematics he worked on Function theory. He extended Gauss's work on hypergeometric series, giving developments that are useful in the theory of differential equations. He was the first to compute the monodromy groups of these series. In 1843 Kummer, realising that attempts to prove Fermat's Last Theorem broke down because the unique factorisation of integers did not extend to other rings of complex numbers, attempted to restore the uniqueness of factorisation by introducing 'ideal' numbers. Not only has his work been most fundamental in work relating to Fermat's Last Theorem, since all later work was based on it for many years, but the concept of an ideal allowed ring theory, and much of abstract algebra, to develop. The Paris Academy of Sciences awarded Kummer the Grand Prize in 1857 for this work. In fact the prize of 3000 francs was offered for a solution to Fermat's Last Theorem but when no solution was forthcoming, even after extending the date, the Prize was given to Kummer even though he had not submitted an entry for the Prize. Soon after Kummer was awarded the Grand Prix he was elected to membership of the Paris Academy of Sciences and then, in 1863, he was elected a Fellow of the Royal Society of London. He received numerous other honours in his long career. Kummer's geometric period was one when he devoted himself to the study of the ray systems that Hamilton had examined, but Kummer treated these problems algebraically. He also discovered the fourth order surface, now named after him, based on the singular surface of the quadratic line complex. The Kummer surface has 16 isolated conical double points and 16 singular tangent planes and was published in 1864. The three great mathematician of Berlin, Kummer, Weierstrass and Kronecker were close friends for twenty years as they worked closely and effectively together, However, around 1875 Weierstrass and Kronecker fell out. Kummer continued his friendship with Kronecker but this put a strain on his relation with Weierstrass. Perhaps it is not too surprising that this should happen to these three great mathematical talents, particularly given that Kronecker vigorously attacked personally anyone with whom he had a mathematical difference.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kummer.html (3 of 4) [2/16/2002 11:18:30 PM]

Kummer

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles)

A Quotation

A Poster of Eduard Kummer

Mathematicians born in the same country

Cross-references to History Topics

Fermat's last theorem

Other references in MacTutor

Chronology: 1840 to 1850

Honours awarded to Eduard Kummer (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1863

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kummer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kummer.html (4 of 4) [2/16/2002 11:18:30 PM]

Kuratowski

Kazimierz Kuratowski Born: 2 Feb 1896 in Warsaw, Poland Died: 18 June 1980 in Warsaw, Poland

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kazimierz Kuratowski's father, Marek Kuratowski was a leading lawyer in Warsaw. To understand what Kuratowski's school years were like it is necessary to look a little at the history of Poland around the time he was born. The first thing to note is that really Poland did not formally exist at this time. Poland had been partitioned in 1772 and the south was called Galicia and under Austrian control. Russia controlled much of the rest of the country and in the years prior to Kuratowski's birth there had been strong moves by Russia to make "Vistula Land", as it was called, be dominated by Russian culture. In a policy implemented between 1869 and 1874, all secondary schooling was in Russian. Warsaw only had a Russian language university after the University of Warsaw became a Russian university in 1869. From 1906, however, the Underground Warsaw University was set up to provide a Polish university education for those prepared to risk teaching and learning in this illegal institution. Galicia, although under Austrian control, retained Polish culture and was often where Poles from "Vistula Land" went for their education. When Kuratowski was nine years old the policy of Russian schooling was softened, but although Polish language schools were allowed, a student could not proceed from such a secondary school to university without taking the Russian examinations as an external candidate. As a consequence most Poles in "Vistula Land" at this time went abroad for their university education. Some went to Galicia where, although under Austrian control, Polish education still flourished. Kuratowski, however, when he left secondary school decided that he wanted to become an engineer. The University of Glasgow, in Scotland, had an engineering school with a long established history, the chair of engineering being established in 1840. It rightly appeared to Kuratowski as an outstanding place to study engineering.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kuratowski.html (1 of 5) [2/16/2002 11:18:32 PM]

Kuratowski

After Kuratowski made the decision to study in Glasgow, he matriculated there as a student there in October 1813. Interestingly, Sneddon relates in [15]:He must have feared that his name would present difficulty to his fellow students for it appears in the registry of the Ordinary Class in Mathematics as Casimir Kuratov. At the end of his first year Kuratowski was awarded the Class Prize in Mathematics. He then studied chemistry at the Technical College during the summer and returned to Poland for a holiday before starting his second year of study. However, back in Poland in August 1914 at the outbreak of World War I, returning to Scotland became impossible for Kuratowski. Although his education was disrupted, one benefit to mathematics was that Kuratowski could no longer study engineering and mathematics would gain enormously. In August 1915 the Russian forces which had held Poland for many years withdrew from Warsaw. Germany and Austria-Hungary took control of most of the country and a German governor general was installed in Warsaw. One of the first moves after the Russian withdrawal was the refounding of the University of Warsaw and it began operating as a Polish university in November 1915. Kuratowski was one of the first students to study mathematics when the university reopened. He attended seminars given by Janiszewski and Mazurkiewicz in Warsaw before the end of the war. He writes in [2]:As early as 1917 [Janiszewski and Mazurkiewicz] were conducting a topology seminar, presumably the first in that new, exuberantly developing field. The meeting of that seminar, taken up to a large extent with sometimes quite vehement discussions between Janiszewski and Mazurkiewicz, were a real intellectual treat for the participants. There were two others on the staff at the University of Warsaw who were also to have a major influence on Kuratowski. One was Lukasiewicz, a professor of philosophy who worked on mathematical logic. The second person, who arrived in 1918, was Sierpinski. In fact the first paper which Kuratowski wrote was On the definitions in mathematics, written in 1917, which was a consequence of discussions which he had while attending Lukasiewicz's seminar. After graduating in 1919, Kuratowski undertook his doctoral studies working under Janiszewski and Mazurkiewicz. In 1921 Kuratowski was awarded his doctorate, but sadly one of his supervisors Janiszewski had died in 1920. Janiszewski had been the leader in a move to set up the new journal Fundamenta Mathematicae and the first volume, which appeared in 1920, contained a joint paper Sur les continus indécomposable by Janiszewski and Kuratowski. Kuratowski was appointed as a professor at the Technical University of Lvov in 1927. In [4] Ulam, who began his university undergraduate career the year Kuratowski began lecturing in Lvov, wrote:He was a freshman professor, so to speak, and I was a freshman student. From the very first lecture I was enchanted by the clarity, logic, and polish of his exposition and the material he presented. ... Soon I could answer some of the more difficult questions in the set theory course, and I began to pose other problems. Right from the start I appreciated Kuratowski's patience and generosity in spending so much time with a novice. The mathematicians of Lvov did a great deal of mathematical research in the cafés of the city. The Scottish Café was the most popular with the mathematicians in general but not with Kuratowski who, together with Steinhaus (according to Ulam [4]):-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kuratowski.html (2 of 5) [2/16/2002 11:18:32 PM]

Kuratowski

... usually frequented a more genteel tea shop that boasted the best pastry in Poland. This café was Ludwik Zalewski's Confectionery at 22 Akademicka Street. It was in the Scottish Café, however, that the famous Scottish Book consisting of open questions posed by the mathematicians working there came into being. Kuratowski (and Steinhaus) sometimes joined their colleagues in the Scottish Café but he had left Lvov before the mathematicians began writing down the problems in the Scottish Book. You can see a picture of the Scottish Café. At Lvov, however, Kuratowski worked with Banach and they answered some fundamental problems on measure theory. Ulam, who had become Banach's research student also worked with them. As Arboleda writes in [1]:This was a beautiful example of scientific collaboration and understanding, and of the ability to organise and encourage creative activity at its height. Kuratowski retained his links with Warsaw while in Lvov, returning each summer to his house outside the capital. In 1934 he left Lvov and became professor of mathematics at the University of Warsaw. He was to spend the rest of his career at the University of Warsaw although he became involved in mathematical activities which saw hime travelling world-wide. It was now that Kuratowski began to devote his energies to the cause of Polish mathematics rather than to give all his efforts to his research. He was still extremely active in research, however, and while spending a month at Princeton in 1936 he wrote a joint paper with von Neumann. During his time in the United States he also made contact with Robert Moore's topology group, meeting mathematicians who he would keep in contact with for many years. Janiszewski had made the case for Polish mathematics concentrating on its areas of strength when he wrote his report at the end of World War I. In 1936 a committee was set up by the Polish Academy of Learning to look at the way forward for Polish science. Kuratowski became secretary to the mathematics committee and his report was made in 1937. He recommended that the time had come to go beyond the era of concentrating on strengths, proposed by Janiszewski, and to develop across the whole of the mathematical spectrum. In particular there was a need:... to raise applied mathematics to such a standard that it can fulfil its tasks as required by other branches of science, as well as those tasks connected with the problems of the country. The recommendations of the report to set up two research institutes, one for pure mathematics and one for applied mathematics, may have been implemented had it not been for the political situation. After the German invasion of Poland in 1939 life there became extremely difficult. There was a strategy by the invaders to put an end to the intellectual life of Poland and to achieve this they sent many academics to concentration camps and murdered others. The Poles had experience of surviving such attacks, however, and they employed the same tactics as they had during the period of Russian domination and organised an underground university in Warsaw. Kuratowski risked his life to teach in this illegal educational establishment through the war. He writes in [2]:Almost all our professors of mathematics lectured at these clandestine universities, and quite a few of the students then are now professors or docents themselves. Due to that underground organisation, and in spite of extremely difficult conditions, scientific work and teaching continued, though on a considerably smaller scale of course. The importance of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kuratowski.html (3 of 5) [2/16/2002 11:18:32 PM]

Kuratowski

clandestine education consisted among others in keeping up the spirit of resistance, as well as optimism and confidence in the future, which was so necessary in the conditions of occupation. The conditions of a scientist's life at that time were truly tragic. Most painful were the human losses. Between the two world wars Poland had made a remarkable leap forward in mathematical teaching and research. At the end of World War II the whole educational system was destroyed and had to be completely rebuilt. It was Kuratowski who now took on the role of leader in this rebuilding process and, through the Polish Mathematical Society of which he was president for eight years immediately following the war, he set about arguing for the implementation of the recommendations of his 1937 report. The two research institutes, one for pure mathematics and one for applied mathematics, were merged into a plan for a single mathematics institute and accepted in 1948. Kuratowski was appointed the Director of the Mathematical Institute of the Polish Academy of Sciences in 1949. Despite being 53 years of age when appointed, Kuratowski held this position of director for 19 years. He held other positions of importance in the Polish scientific scene. For example, he was served as a vice president of the Polish Academy of Sciences. Kuratowski also played a major role in the publishing of mathematics in general and Polish mathematics in particular. He served on the editorial board of Fundamenta Mathematicae from 1928, replacing Sierpinski as editor-in-chief in 1952 and continuing in this role for the rest of his life. He was also one of the founders and an editor of the important Mathematical Monographs series. He contributed the third volume in this series with his monograph on topology which we will mention again below. As an ambassador for Polish mathematics, Kuratowski did a remarkable job with many foreign visits and lecture tours. He lectured in London (1946), Geneva (1948), many universities in the United States during 1948-49, Prague, Berlin, Budapest, Amsterdam, Rome, Peking (1955), Canton (1955), Shanghai (1955), Agra (1956), Lucknow (1956), and Bombay (1956). All this was during the Stalinist era when travel was restricted, and after travel became easier Kuratowski did indeed take full advantage with many visits to western Europe, Britain, USA, and Canada. Kuratowski's main work was in the area of topology and set theory. He used the notion of a limit point to give closure axioms to define a topological space. In 1922 [1]:... he used Boolean algebra to characterise the topology of an abstract space independently of the notion of points. Subsequent research showed that, together with Felix Hausdorff's definition of topological space in terms of neighbourhoods, the closure operator yielded more fertile results than the axiomatic theories based on Maurice Fréchet's convergence (1906) and Frigyes Riesz's point of accumulation (1907). Other major contributions by Kuratowski were to compactness and metric spaces. He was the author of Topologie, referred to above, which was the crowning achievement of the Warsaw School in point set topology. The first volume of this work was the major source on metric spaces for several decades. His 1930 work on non-planar graphs is of fundamental importance in graph theory, he showed that a necessary and sufficient condition for a graph G to be planar is that it does not contain a subgraph homeomorphic to either K5 or K3,3. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kuratowski.html (4 of 5) [2/16/2002 11:18:32 PM]

Kuratowski

His work in set theory considered a function as a set of ordered pairs and this made the function notion as proposed by Frege, Charles Peirce and Schröder redundant. He also considered the topology of the continuum, the theory of connectivity, dimension theory, and answered measure theory questions. Kuratowski was honoured with prizes and election to academies. The Academy of Sciences of the USSR, the Hungarian Academy, the Austrian Academy, the Academy of the German Democratic Republic, the Academy of Sciences of Argentina, the Accademia Nazionale dei Lincei, the Academy of Arts and Letters of Palermo, and the Royal Society of Edinburgh all elected him to membership. He received honorary degrees from many universities including Glasgow, the Sorbonne, Prague and Wroclaw. Ulam, in the preface which he wrote to [2], sums up Kuratowski's contribution in the following words:Professor Kuratowski stands out not only as a great figure in mathematical research, but in his ability, so rare among original scientists, to organise and direct schools of mathematical research and education. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles) A Poster of Kazimierz Kuratowski

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1930 to 1940

Honours awarded to Kazimierz Kuratowski (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kuratowski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kuratowski.html (5 of 5) [2/16/2002 11:18:32 PM]

Kurosh

Aleksandr Gennadievich Kurosh Born: 19 Jan 1908 in Yartsevo (near Smolensk), Russia Died: 18 May 1971 in Moscow, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aleksandr Gennadievich Kurosh's father was a clerk in a cotton factory. His life became harder at age six with the outbreak of World War I and his family wrote that, despite his young age, Shura (as he was known) followed the events in the newspapers. In the middle of the war, in 1916, Shura entered school, going straight into the third class. After the war Shura's life soon became even harder when, in 1920, his father died of tuberculosis. The family were poor and, in order to survive, Kurosh had to work as well attend school. Kurosh left school at the age of 15 and went to Moscow to take the examinations for the Textile Institute. He achieved very high grade passes in the entrance examinations but was judged to be too young to enter the Institute. He was sent back to his home in Yartsevo where he obtained a job as an accountant. Kurosh never intended to spend his life as an accountant and he studied at evening classes after a full working day. Up to this time, unlike many great mathematicians, he had no particular passion for the subject. His special topic at the evening classes was the stream engine. In 1924 Kurosh became a student at Smolensk University. It was at this time that his interest moved towards mathematics and he later wrote how he was influenced towards algebra:In 1926 Pavel Sergeevich Aleksandrov began to give lectures at the university. ... I attended his lectures on the theory of sets, the theory of functions, and topology. In 1928 I was kept on at the university by Aleksandrov as a postgraduate student. At that time Emmy Noether was in Moscow and gave a course at Moscow State University on abstract algebra, which Aleksandrov attended. Under the fresh impression of these lectures he gave a course on modern algebra at Smolensk. It was then that my scientific interests were formed. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kurosh.html (1 of 4) [2/16/2002 11:18:34 PM]

Kurosh

In 1929 Kurosh was assigned to the Institute of Mathematics and Mechanics at Moscow State University. This was done at Aleksandrov's request since he was by this time in Moscow and wished to continue to supervise Kurosh's work. However, although Kurosh's first results were in topology, solving problems posed by Aleksandrov, he was already interested in the theory of groups. He had read O Yu Schmidt's group theory papers while still in Smolensk so when he found himself able to attend Schmidt's seminar at Moscow State University, his interest in groups increased further. However, after attending Schmidt's group theory course in 1930, he found himself taking over some of Schmidt's duties when he left the university in the autumn of that year. Kurosh was appointed an assistant at Moscow State University in 1930 becoming a lecturer in 1932 and a professor there in 1937. The professorial appointment came after Kurosh was awarded his doctorate for a thesis entitled Research on infinite groups which he defended on 22 April 1936. It should be mentioned that the doctorate was a much higher degree than the present British/American PhD which compares in level with the Master's Degree in Russia. In 1949 he became a director at Moscow State University where he continued to work throughout his career. The Chair of Higher Algebra at Moscow State University was created in 1929 and O Yu Schmidt was the first occupant of the Chair. Kurosh was the second holder of the Chair, holding the Chair from 1949 until his death in 1971. Like most Russian academics of that period, Kurosh had a variety of other attachments teaching courses and lectured at a number of other institutions in Moscow. As mentioned above, Kurosh's first significant results were in topology, solving a problem set by Aleksandrov. Quickly he moved to research in group theory and his first paper on this topic appeared in 1932 on direct decompositions of groups. Shortly after completing this work, Kurosh came across Schreier's papers on free products of groups. Soon he was producing important results on the topic and two papers came out of his work on free products. The second of these papers appeared in Mathematische Annalen and contains a proof of the celebrated Kurosh subgroup theorem, which describes subgroups of a free product of groups. This paper brought Kurosh international fame. Kurosh is best known for his book The Theory of Groups which was written in two volumes. He completed writing the book in 1940 but the events of World War II prevented publication of the book until 1944. In 1952 Kurosh brought out a second edition of the book which was almost a new book given that it attempted to cover the large amount of progress during the years 1940-52. The book, in the words of Kurt Hirsch who translated the book into English in 1955:has been widely acclaimed as the first modern text on the general theory of groups, with major emphasis on infinite groups. The book includes many of Kurosh's own results on groups, in particular the Kurosh Subgroup Theorem mentioned above. However, Kurosh did not spend all his research efforts on group theory. In [1] the progression of his work is described in these terms:Gradually, along with papers on group theory, Kurosh began to publish papers on ring theory, linear algebra and lattices; later, also papers on category theory and the theory of multi-operator groups, rings and linear algebras. Some of this work was described in his other famous textbook Lectures on General Algebra published in 1960 which became an internationally famous text. As with The Theory of Groups this text was also translated into English by Hirsch.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kurosh.html (2 of 4) [2/16/2002 11:18:34 PM]

Kurosh

During the 1950s he concentrated on Universal algebra and category theory, organising a major seminar on category theory in 1958. Many mathematicians participated in this seminar and it led to the birth of the Moscow School of Category Theory. In his last years Kurosh worked in an area best described in his own words taken from a lecture that he gave in 1970:... between the theory of universal algebras and the classical branches of general algebra there exists a big uncultivated space. Research has begun only in a few places, isolated, sometimes random ... One has to expect that it is precisely in this no-man's land where the basic branches of general algebra will move in the next decades. In accordance with the general tendencies of contemporary science ... new objects of study, new theories will appear ... more and more often. To stop this process is impossible, to try to stop it is unreasonable. One can only direct this process. Kurosh's whole life was involved in teaching at all levels from directing the research of young people, to lecturing, to designing new courses. As the authors of [1] write:Kurosh belonged to that category of scientists who could not conceive of their creative work without attracting wide circles of youths to science. Not only did Kurosh work with university students but he gave many popular lectures to school children. Twice he organised the Moscow University Olympiads for school mathematics. There were other ways too in which he served mathematics. One way was his long involvement with the Moscow Mathematical Society. In 1931, while he was still an assistant, Kurosh joined the Moscow Mathematical Society and he was elected to its governing body in 1933, only a year after his appointment as a lecturer. In that year he gave his first lecture to the Society on Fundamental trends in finite group theory. During the years 1943-48 Kurosh served the Society as its Librarian and then for 22 years from 1948 he served continuously on the governing body of the Society. During six of these years he was Vice-President. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Aleksandr G Kurosh

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kurosh.html (3 of 4) [2/16/2002 11:18:34 PM]

Kurosh

JOC/EFR February 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kurosh.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kurosh.html (4 of 4) [2/16/2002 11:18:34 PM]

Kurschak

József Kürschák Born: 14 March 1864 in Buda (now part of Budapest), Hungary Died: 26 March 1933 in Budapest, Hungary Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

József Kürschák received his doctorate from Budapest in 1890 and then taught in Budapest, at the Technical University, for the whole of his career. In [3] the authors describe a paper by Kürschák written in 1898 in which a regular dodecagon inscribed in a unit circle is investigated. A trigonometric argument can be used to show that its area of the dodecagon is 3 but Kürschák gives a purely geometric proof. He proves that the dodecagon can be dissected into a set of triangles which can be rearranged so as to fill three squares with sides having length 1. Kürschák's most important work, however, was in 1912 when he founded the theory of valuations. His work was inspired by earlier work of Julius König, Steinitz and Hensel. The importance of Kürschák's valuations is that they allow notions of convergence and limits be used in the theory of abstract fields and greatly enrich the topic. Von Neumann was one of Kürschák's students. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kurschak.html (1 of 2) [2/16/2002 11:18:35 PM]

Kurschak

Glossary index

Quotations index

Mathematicians of the day JOC/EFR February 1997

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kurschak.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kurschak.html (2 of 2) [2/16/2002 11:18:35 PM]

Kutta

Martin Wilhelm Kutta Born: 3 Nov 1867 in Pitschen, Upper Silesia (now Byczyna, Poland) Died: 25 Dec 1944 in Fürstenfeldbruck, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Martin Kutta studied at Breslau from 1885 to 1890. Then he went to Munich where he studied from 1891 to 1894, later becoming an assistant to von Dyck at Munich. During this period he spent the year 1898-99 in England at the University of Cambridge. Kutta held posts at Munich, Jena and Aachen. He became professor at Stuttgart in 1911 and remained there until he retired in 1935. He is best known for the Runge-Kutta method (1901) for solving ordinary differential equations and for the Zhukovsky- Kutta aerofoil. Runge presented Kutta's methods. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Other references in MacTutor

Chronology: 1900 to 1910

Other Web sites

Munich, Germany (A biography in German)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kutta.html (1 of 2) [2/16/2002 11:18:37 PM]

Kutta

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kutta.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kutta.html (2 of 2) [2/16/2002 11:18:37 PM]

Kuttner

Brian Kuttner Born: 11 April 1908 in London, England Died: 2 Jan 1992 in Birmingham, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Brian Kuttner attended University College School in London and from there he won a scholarship to study at Christ's College Cambridge. He graduated in 1929 and then continued to undertake research at Cambridge. Kuttner spent a while in Göttingen studying with Landau and received his doctorate in 1934. However, before the award of his doctorate Kuttner had been appointed in 1932 as an assistant lecturer at the University of Birmingham. At Birmingham Kuttner joined the Mathematics Department headed by G N Watson. He was to spend the rest of his career at Birmingham being promoted to lecturer in 1936, Senior Lecturer in 1952, Reader in 1955 and then to the chair of Mathematical Analysis in 1969. Kuttner's work was on [1]:... Fourier series, strong summability, Riesz means, Nörland methods, and Tauberian theory. Most of Kuttner's early work is on Fourier series and summability. Hardy quotes some of these early results of Kuttner's in his treatise Divergent series (1949). Maddox, in [1], writes:... at the age of only 26, Kuttner proved a basic theorem in the general theory of trigonometric series, a result delightful for both the deceptive simplicity of its statement and the elegance of its proof. ... Zygmund greatly admired this theorem of Kuttner, which now occupies an honoured place in Zygmund's monumental work on trigonometric series. Throughout his career he continued to publish a steady stream of high quality research papers right up to the time of his death. There was no signs that his output was decreasing when he retired, rather the reverse since the publication of 8 papers in 1978 indicates that his research activity increased after he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kuttner.html (1 of 2) [2/16/2002 11:18:38 PM]

Kuttner

retired from the Birmingham chair in 1975. Mathematical Reviews lists over 120 of his papers, and the continuation of joint papers appearing after his death show clearly that even into his 80s his love for his favourite topics of analysis remained as strong as ever. Kuttner's interests outside mathematics included travelling and walking. He regularly attended the annual British Mathematical Colloquium and I [EFR] remember him as someone held in great respect by my colleagues who were working in analysis when I began attending these Colloquia in the second half of the 1960s. Maddox sums up Kuttner's many fine qualities [1]:Brian Kuttner was a kind, helpful and gentle man, revered by his research students, admired by the numerous analysts who collaborated with him in research, and greatly respected by all who were fortunate to have known him. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Kuttner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Kuttner.html (2 of 2) [2/16/2002 11:18:38 PM]

La_Condamine

Charles Marie de La Condamine Born: 28 Jan 1701 in Paris, France Died: 4 Feb 1774 in Paris, France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles-Marie La Condamine studied at the Jesuit College of Louis-le-Grand in Paris. There he was taught mathematics by Père Louis Castel. On leaving the College he decided to take up a military career and, when war broke out with Spain he joined the army. He distinguished himself with his bravery at the siege of Rosas in 1719 but decided that army life did not suit him. At this point La Condamine made contact with scientists in Paris and became a member of the Académie Royale des Sciences in 1730. The quite life in Paris did not suit him either and he sailed on a voyage to Algiers, Alexandria, Palestine, Cyprus and Constantinople (now Istanbul) where he spent five months. On his return to Paris he published mathematical and physical observations of his voyage. The Académie Royale was impressed and sent him on an expedition to Peru. In April 1735 La Condamine set out on the expedition to Peru to measure the length of a degree of meridian at the equator. Bouguer was a member of the same expedition and its third scientific member was the leader of the expedition Louis Godin. The three finished their journey by different routes, La Condamine going overland from Manta, the other two sailing to Quito where they joined up. The three were soon involved in disagreements. Godin began to work on his own while La Condamine worked with Bouguer. In 1741 Bouguer discovered a small error in their joint measurements and these two fell out when Bouguer refused to allow La Condamine to recheck the results. All three made independent measurements, the work being completed in 1743. The three returned by different routes. In 1743 La Condamine began his return journey which included a four month raft journey down the Amazon river. His was the first scientific account of the Amazon which he published as Journal du http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/La_Condamine.html (1 of 2) [2/16/2002 11:18:40 PM]

La_Condamine

voyage fait par ordre du roi a l'équateur (1751). La Condamine spent five months in Cayenne on his journey home and here he repeated Richer's experiments on the variation of weights at different latitudes. By February 1745 La Condamine was back in Paris after his ten year journey. He returned with many notes, 200 natural history specimens and works of art which he gave to Buffon. Y Laissus writing in the Dictionary of Scientific Biography says:The last survivor of the expedition, La Condamine, who was a less gifted astronomer than Godin and a less reliable mathematician than Bouguer often received the major part of the credit, probably because of his amiable nature and his talent as a writer. La Condamine was a close friend of Maupertuis for many years. He spent much effort in the last part of his life campaigning for inoculation against small-pox. His passion on this topic was partly due to the fact that he had suffered from small-pox as a child. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Honours awarded to Charles-Marie La Condamine (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1748

Lunar features

Crater La Condamine

Paris street names

Rue La Condamine (17th Arrondissement)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/La_Condamine.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/La_Condamine.html (2 of 2) [2/16/2002 11:18:40 PM]

La_Faille

Jean Charles de La Faille Born: 1 March 1597 in Antwerp, Belgium Died: 4 Nov 1652 in Barcelona, Spain

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jan-Karel della Faille or Jean Charles de La Faille attended the Jesuit school in Antwerp, joining the Jesuit Order in 1613. He spent two years at a Jesuit College in Malines before returning to Antwerp. De la Faille became a disciple of Saint-Vincent, who he met in Antwerp. In 1620 he went to France taking a theology degree at Dole and teaching mathematics there. De la Faille taught at the Jesuit College of Louvain from 1626 until 1628 when he went to Spain and was appointed to the Imperial College in Madrid. Philip IV had become king of Spain and Portugal in 1621. He led Spain and Portugal to victories up to 1635 but then France declared war and he suffered defeats. De la Faille advised Philip IV on questions of defence and of military engineering during this period. La Faille also taught mathematics and military engineering in Madrid. In 1640 Philip IV suffered the loss of Portugal when it declared its independence. De la Faille helped Philip IV by serving as adviser on fortifications along the Spanish - Portuguese border from 1641 until 1644. In 1644 De la Faille, still working for Philip IV, made military expeditions to Naples, Sicily and Catalonia. De la Faille wrote Theses mechanicae in 1625. He is famed, however, for a work Theoremata de centro gravitatis partium circuli et ellipsis in 1632 in which he was the first to determine the centre of gravity of the sector of a circle. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/La_Faille.html (1 of 2) [2/16/2002 11:18:42 PM]

La_Faille

List of References (3 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/La_Faille.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/La_Faille.html (2 of 2) [2/16/2002 11:18:42 PM]

La_Hire

Philippe de La Hire Born: 18 March 1640 in Paris, France Died: 21 April 1718 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Philippe de la Hire was educated as an artist and became skilled in drawing and painting. He visited Italy where he spent four years developing his artistic skills and learning geometry. The interest in geometry arose from his study of perspective in art. La Hire was elected to the Académie Royal des Sciences in 1678. In 1683 he was appointed to the chair of mathematics at the Collège Royale. Four years later he was appointed, in addition, to the chair of architecture at the Académie Royale. Much influenced by the work of Desargues, La Hire worked on conic sections which he treated projectively. He published his first work on conic sections Nouvelle methode en géometrie pour les sections des superficies coniques et cylindriques in 1673. In 1675 he published a more comprehensive work on conic sections Sectiones conicae which contained a description of Desargues' projective geometry. In 1708 he calculated the length of the cardioid. Other topics to which he made important contributions included astronomy, physics and geodesy. In astronomy he installed the first transit instrument in the Paris Observatory. He also produced tables giving the movements of the Sun, Moon and the planets. He did much work on surveying, in particular taking measurements of the French coastline. He designed an instrument to find the level at a site. He was also a major contributor to a project to map France. La Hire's maps of the Earth were made with the centre of projection, not at the pole, but at r/ 2 along a radius produced through the pole (where r is the radius of the Earth). http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/La_Hire.html (1 of 3) [2/16/2002 11:18:44 PM]

La_Hire

La Hire also wrote on magic squares. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

Longitude and the Académie Royale

Cross-references to Famous Curves

1. Cardioid 2. Epicycloid 3. Epitrochoid 4. Hypocycloid 5. Hypotrochoid

Other references in MacTutor

1. Chronology: 1675 to 1700 2. Chronology: 1700 to 1720

Honours awarded to Philippe de la Hire (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Mons La Hire

Paris street names

Rue LaHire (13th Arrondissement)

Other Web sites

1. Linda Hall Library (Star Atlas) 2. The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/La_Hire.html (2 of 3) [2/16/2002 11:18:44 PM]

La_Hire

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/La_Hire.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/La_Hire.html (3 of 3) [2/16/2002 11:18:44 PM]

La_Roche

Estienne de La Roche Born: 1470 in Lyon, France Died: 1530 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Estienne de la Roche's family lived in Lyon and also owned property near the town of Villefranche. Because of where he lived when young, La Roche was sometimes known as Villefranche. He learned mathematics from Chuquet and had in his possession many manuscripts in Chuquet's hand so it appears that he was on good terms with Chuquet. La Roche taught commercial arithmetic in Lyon for 25 years. Clearly he was well thought of as a teacher of arithmetic since he was often called master of ciphers. La Roche published Larismetique in 1520 which was considered an excellent arithmetic book with good notation for powers and roots. However in 1880 Aristide Marre published Chuquet's Triparty. It was immediately discovered that the first part of La Roche's Larismetique is essentially a copy of Chuquet's Algebra. It was thought at this time that La Roche was simply a plagiarist. However more recent work has come to a less harsh conclusion about La Roche. It now appears that La Roche was trying to teach important mathematics which was not available to the French public he tried to teach. The rest of La Roche's work is a commercial arithmetic. La Roche says of his own work that it is the flower of several masters, experts in the art such as Pacioli. This statement itself indicates that La Roche was probably in way trying to hide his dependence on others. Indeed La Roche took parts of Chuquet, parts of Pacioli and parts of Philippe Frescobaldi, a French banker who was lesser known as a mathematics writer, and without any real skill on his part formed them into a teaching book. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/La_Roche.html (1 of 2) [2/16/2002 11:18:45 PM]

La_Roche

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/La_Roche.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/La_Roche.html (2 of 2) [2/16/2002 11:18:45 PM]

Lacroix

Sylvestre François Lacroix Born: 28 April 1765 in Mâcon, France Died: 24 May 1843 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sylvestre Lacroix's parents were poor and he was fortunate to be a student of Monge who was influential in his career. Through Monge's influence he was appointed Professor of Mathematics at the Ecole Gardes de Marine at Rochefort in 1782. From Rochefort Lacroix went to a chair at the Ecole Normale, then he moved to the Ecole Centrale des Quatre Nations. He was appointed professor of mathematics at the Ecole Polytechnique from 1799 where he held the chair of analysis. In the same year he became a member of the Institute. He also became professor at Faculté des Sciences in 1810. In 1815 he left the Ecole Polytechnique to take up a chair at the Sorbonne and a chair at the Collège de France. Lacroix was the writer of important textbooks in mathematics and through these he made a major contribution to the teaching of mathematics throughout France. He published a three volume text Traité de Calcul differéntiel et intégral (1797-1800). During the same period he was publishing a 10 volume work Cours de Mathématique (1797-1799). These texts had an influence beyond France for it was through English translations of the texts that the 'new continental mathematics' entered universities in Britain. Lacroix held the view that algebra and geometry should be treated separately, as far apart as they can be; and that the results in each should serve for mutual clarification, corresponding, so to speak, to the text of a book and its translation. Article by: J J O'Connor and E F Robertson List of References (5 books/articles)

A Quotation

Mathematicians born in the same country Cross-references to Famous Curves

Devil's Curve

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lacroix.html (1 of 2) [2/16/2002 11:18:46 PM]

Lacroix

Other references in MacTutor

Chronology: 1800 to 1810

Honours awarded to Sylvestre Lacroix (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Lacroix

Paris street names

Rue Lacroix (17th Arrondissement)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lacroix.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lacroix.html (2 of 2) [2/16/2002 11:18:46 PM]

Lagny

Thomas Fantet de Lagny Born: 7 Nov 1660 in Lyon, France Died: 11 April 1734 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas De Lagny was taught first by his uncle, then he entered a Jesuit college in Lyon. After this he studied law in Toulouse. In 1686 De Lagny became a mathematics tutor to a family in Paris, a position he held for about 10 years. He collaborated with de L'Hôpital while in Paris. In 1695 he became a member of the Académie Royale des Sciences, then two years later he became professor of hydrography at Rochefort. He also held positions as librarian at Bibliothéque du Roi for a time and spent two years as deputy director of a bank. De Lagny is well known for his contributions to computational mathematics, calculating and also making useful comments on the convergence of the series he was using.

to 120 places

He constructed trigonometric tables and used binary arithmetic. In 1733 de Lagny examined the continued fraction expansion of the quotient of two integers and, as an example, considered adjacent Fibonacci numbers as the worst case expansion for the Euclidean algorithm. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of Thomas De Lagny

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lagny.html (1 of 2) [2/16/2002 11:18:48 PM]

Lagny

Cross-references to History Topics

1. Pi through the ages 2. A chronology of pi

Other references in MacTutor

The Fibonacci sequence

Honours awarded to Thomas De Lagny (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1718

Paris street names

Passage de Lagny and Rue de Lagny (20th Arrondissement)

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lagny.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lagny.html (2 of 2) [2/16/2002 11:18:48 PM]

Lagrange

Joseph-Louis Lagrange Born: 25 Jan 1736 in Turin, Sardinia-Piedmont (now Italy) Died: 10 April 1813 in Paris, France

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph-Louis Lagrange is usually considered to be a French mathematician, but the Italian Encyclopaedia [40] refers to him as an Italian mathematician. They certainly have some justification in this claim since Lagrange was born in Turin and baptised in the name of Giuseppe Lodovico Lagrangia. Lagrange's father was Giuseppe Francesco Lodovico Lagrangia who was Treasurer of the Office of Public Works and Fortifications in Turin, while his mother Teresa Grosso was the only daughter of a medical doctor from Cambiano near Turin. Lagrange was the eldest of their 11 children but one of only two to live to adulthood. Turin had been the capital of the duchy of Savoy, but become the capital of the kingdom of Sardinia in 1720, sixteen years before Lagrange's birth. Lagrange's family had French connections on his father's side, his great-grandfather being a French cavalry captain who left France to work for the Duke of Savoy. Lagrange always leant towards his French ancestry, for as a youth he would sign himself Lodovico LaGrange or Luigi Lagrange, using the French form of his family name. Despite the fact that Lagrange's father held a position of some importance in the service of the king of Sardinia, the family were not wealthy since Lagrange's father had lost large sums of money in unsuccessful financial speculation. A career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the College of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull. Lagrange's interest in mathematics began when he read a copy of Halley's 1693 work on the use of algebra in optics. He was also attracted to physics by the excellent teaching of Beccaria at the College of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html (1 of 8) [2/16/2002 11:18:50 PM]

Lagrange

Turin and he decided to make a career for himself in mathematics. Perhaps the world of mathematics has to thank Lagrange's father for his unsound financial speculation, for Lagrange later claimed:If I had been rich, I probably would not have devoted myself to mathematics. He certainly did devote himself to mathematics, but largely he was self taught and did not have the benefit of studying with leading mathematicians. On 23 July 1754 he published his first mathematical work which took the form of a letter written in Italian to Giulio Fagnano. Perhaps most surprising was the name under which Lagrange wrote this paper, namely Luigi De la Grange Tournier. This work was no masterpiece and showed to some extent the fact that Lagrange was working alone without the advice of a mathematical supervisor. The paper draws an analogy between the binomial theorem and the successive derivatives of the product of functions. Before writing the paper in Italian for publication, Lagrange had sent the results to Euler, who at this time was working in Berlin, in a letter written in Latin. The month after the paper was published, however, Lagrange found that the results appeared in correspondence between Johann Bernoulli and Leibniz. Lagrange was greatly upset by this discovery since he feared being branded a cheat who copied the results of others. However this less than outstanding beginning did nothing more than make Lagrange redouble his efforts to produce results of real merit in mathematics. He began working on the tautochrone, the curve on which a weighted particle will always arrive at a fixed point in the same time independent of its initial position. By the end of 1754 he had made some important discoveries on the tautochrone which would contribute substantially to the new subject of the calculus of variations (which mathematicians were beginning to study but which did not receive the name 'calculus of variations' before Euler called it that in 1766). Lagrange sent Euler his results on the tautochrone containing his method of maxima and minima. His letter was written on 12 August 1755 and Euler replied on 6 September saying how impressed he was with Lagrange's new ideas. Although he was still only 19 years old, Lagrange was appointed professor of mathematics at the Royal Artillery School in Turin on 28 September 1755. It was well deserved for the young man had already shown the world of mathematics the originality of his thinking and the depth of his great talents. In 1756 Lagrange sent Euler results that he had obtained on applying the calculus of variations to mechanics. These results generalised results which Euler had himself obtained and Euler consulted Maupertuis, the president of the Academy, about this remarkable young mathematician. Not only was Lagrange an outstanding mathematician but he was also a strong advocate for the principle of least action so Maupertuis had no hesitation but to try to entice Lagrange to a position in Prussia. He arranged with Euler that he would let Lagrange know that the new position would be considerably more prestigious than the one he held in Turin. However, Lagrange did not seek greatness, he only wanted to be able to devote his time to mathematics, and so he shyly but politely refused the position. Euler also proposed Lagrange for election to the Berlin Academy and he was duly elected on 2 September 1756. The following year Lagrange was a founding member of a scientific society in Turin, which was to become the Royal Academy of Science of Turin. One of the major roles of this new Society was to publish a scientific journal the Mélanges de Turin which published articles in French or Latin. Lagrange was a major contributor to the first volumes of the Mélanges de Turin volume 1 of which

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html (2 of 8) [2/16/2002 11:18:50 PM]

Lagrange

appeared in 1759, volume 2 in 1762 and volume 3 in 1766. The papers by Lagrange which appear in these transactions cover a variety of topics. He published his beautiful results on the calculus of variations, and a short work on the calculus of probabilities. In a work on the foundations of dynamics, Lagrange based his development on the principle of least action and on kinetic energy. In the Mélanges de Turin Lagrange also made a major study on the propagation of sound, making important contributions to the theory of vibrating strings. He had read extensively on this topic and he clearly had thought deeply on the works of Newton, Daniel Bernoulli, Taylor, Euler and d'Alembert. Lagrange used a discrete mass model for his vibrating string, which he took to consist of n masses joined by weightless strings. He solved the resulting system of n+1 differential equations, then let n tend to infinity to obtain the same functional solution as Euler had done. His different route to the solution, however, shows that he was looking for different methods than those of Euler, for whom Lagrange had the greatest respect. In papers which were published in the third volume, Lagrange studied the integration of differential equations and made various applications to topics such as fluid mechanics (where he introduced the Lagrangian function). Also contained are methods to solve systems of linear differential equations which used the characteristic value of a linear substitution for the first time. Another problem to which he applied his methods was the study the orbits of Jupiter and Saturn. The Académie des Sciences in Paris announced its prize competition for 1764 in 1762. The topic was on the libration of the Moon, that is the motion of the Moon which causes the face that it presents to the Earth to oscillate causing small changes in the position of the lunar features. Lagrange entered the competition, sending his entry to Paris in 1763 which arrived there not long before Lagrange himself. In November of that year he left Turin to make his first long journey, accompanying the Marquis Caraccioli, an ambassador from Naples who was moving from a post in Turin to one in London. Lagrange arrived in Paris shortly after his entry had been received but took ill while there and did not proceed to London with the ambassador. D'Alembert was upset that a mathematician as fine as Lagrange did not receive more honour. He wrote on his behalf [1]:Monsieur de la Grange, a young geometer from Turin, has been here for six weeks. He has become quite seriously ill and he needs, not financial aid, for the Marquis de Caraccioli directed upon leaving for England that he should not lack for anything, but rather some signs of interest on the part of his native country ... In him Turin possesses a treasure whose worth it perhaps does not know. Returning to Turin in early 1765, Lagrange entered, later that year, for the Académie des Sciences prize of 1766 on the orbits of the moons of Jupiter. D'Alembert, who had visited the Berlin Academy and was friendly with Frederick II of Prussia, arranged for Lagrange to be offered a position in the Berlin Academy. Despite no improvement in Lagrange's position in Turin, he again turned the offer down writing:It seems to me that Berlin would not be at all suitable for me while M Euler is there. By March 1766 d'Alembert knew that Euler was returning to St Petersburg and wrote again to Lagrange to encourage him to accept a post in Berlin. Full details of the generous offer were sent to him by Frederick II in April, and Lagrange finally accepted. Leaving Turin in August, he visited d'Alembert in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html (3 of 8) [2/16/2002 11:18:50 PM]

Lagrange

Paris, then Caraccioli in London before arriving in Berlin in October. Lagrange succeeded Euler as Director of Mathematics at the Berlin Academy of Science on 6 November 1766. Lagrange was greeted warmly by most members of the Academy and he soon became close friends with Lambert and Johann(III) Bernoulli. However, not everyone was pleased to see this young man in such a prestigious position, particularly Castillon who was 32 years older than Lagrange and considered that he should have been appointed as Director of Mathematics. Just under a year from the time he arrived in Berlin, Lagrange married his cousin Vittoria Conti. He wrote to d'Alembert:My wife, who is one of my cousins and who even lived for a long time with my family, is a very good housewife and has no pretensions at all. They had no children, in fact Lagrange had told d'Alembert in this letter that he did not wish to have children. Turin always regretted losing Lagrange and from time to time his return there was suggested, for example in 1774. However, for 20 years Lagrange worked at Berlin, producing a steady stream of top quality papers and regularly winning the prize from the Académie des Sciences of Paris. He shared the 1772 prize on the three body problem with Euler, won the prize for 1774, another one on the motion of the moon, and he won the 1780 prize on perturbations of the orbits of comets by the planets. His work in Berlin covered many topics: astronomy, the stability of the solar system, mechanics, dynamics, fluid mechanics, probability, and the foundations of the calculus. He also worked on number theory proving in 1770 that every positive integer is the sum of four squares. In 1771 he proved Wilson's theorem (John Wilson) (first stated without proof by Waring) that n is prime if and only if (n -1)! + 1 is divisible by n. In 1770 he also presented his important work Réflexions sur la résolution algébrique des équations which made a fundamental investigation of why equations of degrees up to 4 could be solved by radicals. The paper is the first to consider the roots of a equation as abstract quantities rather than having numerical values. He studied permutations of the roots and, although he does not compose permutations in the paper, it can be considered as a first step in the development of group theory continued by Ruffini, Galois and Cauchy. Although Lagrange had made numerous major contributions to mechanics, he had not produced a comprehensive work. He decided to write a definitive work incorporating his contributions and wrote to Laplace on 15 September 1782:I have almost completed a Traité de mécanique analytique, based uniquely on the principle of virtual velocities; but, as I do not yet know when or where I shall be able to have it printed, I am not rushing to put the finishing touches to it. Caraccioli, who was by now in Sicily, would have liked to see Lagrange return to Italy and he arranged for an offer to be made to him by the court of Naples in 1781. Offered the post of Director of Philosophy of the Naples Academy, Lagrange turned it down for he only wanted peace to do mathematics and the position in Berlin offered him the ideal conditions. During his years in Berlin his health was rather poor on many occasions, and that of his wife was even worse. She died in 1783 after years of illness and Lagrange was very depressed. Three years later Frederick II died and Lagrange's position in Berlin became a less happy one. Many Italian States saw their chance and attempts were made to entice him back to Italy.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html (4 of 8) [2/16/2002 11:18:50 PM]

Lagrange

The offer which was most attractive to Lagrange, however, came not from Italy but from Paris and included a clause which meant that Lagrange had no teaching. On 18 May 1787 he left Berlin to become a member of the Académie des Sciences in Paris, where he remained for the rest of his career. Lagrange survived the French Revolution while others did not and this may to some extent be due to his attitude which he had expressed many years before when he wrote:I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable. The Mécanique analytique which Lagrange had written in Berlin, was published in 1788. It had been approved for publication by a committee of the Académie des Sciences comprising of Laplace, Cousin, Legendre and Condorcet. Legendre acted as an editor for the work doing proof reading and other tasks. The Mécanique analytique summarised all the work done in the field of mechanics since the time of Newton and is notable for its use of the theory of differential equations. With this work Lagrange transformed mechanics into a branch of mathematical analysis. He wrote in the Preface:One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course. Lagrange was made a member of the committee of the Académie des Sciences to standardise weights and measures in May 1790. They worked on the metric system and advocated a decimal base. Lagrange married for a second time in 1792, his wife being Renée-Françoise-Adélaide Le Monnier the daughter of one of his astronomer colleagues at the Académie des Sciences. He was certainly not unaffected by the political events. In 1793 the Reign of Terror commenced and the Académie des Sciences, along with the other learned societies, was suppressed on 8 August. The weights and measures commission was the only one allowed to continue and Lagrange became its chairman when others such as the chemist Lavoisier, Borda, Laplace, Coulomb, Brisson and Delambre were thrown off the commission. In September 1793 a law was passed ordering the arrest of all foreigners born in enemy countries and all their property to be confiscated. Lavoisier intervened on behalf of Lagrange, who certainly fell under the terms of the law, and he was granted an exception. On 8 May 1794, after a trial that lasted less than a day, a revolutionary tribunal condemned Lavoisier, who had saved Lagrange from arrest, and 27 others to death. Lagrange said on the death of Lavoisier, who was guillotined on the afternoon of the day of his trial:It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like. The Ecole Polytechnique was founded on 11 March 1794 and opened in December 1794 (although it was called the Ecole Centrale des Travaux Publics for the first year of its existence). Lagrange was its first professor of analysis, appointed for the opening in 1794. In 1795 the Ecole Normale was founded with the aim of training school teachers. Lagrange taught courses on elementary mathematics there. We mentioned above that Lagrange had a 'no teaching' clause written into his contract but the Revolution changed things and Lagrange was required to teach. However, he was not a good lecturer as Fourier, who attended his lectures at the Ecole Normale in 1795 wrote:His voice is very feeble, at least in that he does not become heated; he has a very pronounced Italian accent and pronounces the s like z ... The students, of whom the majority are incapable of appreciating him, give him little welcome, but the professors make amends

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html (5 of 8) [2/16/2002 11:18:50 PM]

Lagrange

for it. Similarly Bugge who attended his lectures at the Ecole Polytechnique in 1799 wrote:... whatever this great man says, deserves the highest degree of consideration, but he is too abstract for youth. Lagrange published two volumes of his calculus lectures. In 1797 he published the first theory of functions of a real variable with Théorie des fonctions analytique although he failed to give enough attention to matters of convergence. He states that the aim of the work is to give:... the principles of the differential calculus, freed from all consideration of the infinitely small or vanishing quantities, of limits or fluxions, and reduced to the algebraic analysis of finite quantities. Also he states:The ordinary operations of algebra suffice to resolve problems in the theory of curves. Not everyone found Lagrange's approach to the calculus the best however, for example de Prony wrote in 1835:Lagrange's foundations of the calculus is assuredly a very interesting part of what one might call purely philosophical study: but when it is a case of making transcendental analysis an instrument of exploration for questions presented by astronomy, marine, geodesy, and the different branches of science of the engineer, the consideration of the infinitely small leads to the aim in a manner which is more felicitous, more prompt, and more immediately adapted to the nature of the questions, and that is why the Leibnizian method has, in general, prevailed in French schools. The second work of Lagrange on this topic Leçons sur le calcul des fonctions appeared in 1800. Napoleon named Lagrange to the Legion of Honour and Count of the Empire in 1808. On 3 April 1813 he was named grand croix of the Ordre Impérial de la Réunion. He died a week later. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (62 books/articles)

Some Quotations (11)

A Poster of Joseph-Louis Lagrange

Mathematicians born in the same country

Some pages from publications

Extract of Lagrange's Reflexions sur la resolution algebrique des equations from his collected works (1869).

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html (6 of 8) [2/16/2002 11:18:50 PM]

Lagrange

Cross-references to History Topics

1. The development of group theory 2. Matrices and determinants 3. Mathematical games and recreations 4. The fundamental theorem of algebra 5. Orbits and gravitation 6. Arabic mathematics : forgotten brilliance? 7. General relativity 8. An overview of the history of mathematics 9. The rise of the calculus

Other references in MacTutor

1. Chronology: 1740 to 1760 2. Chronology: 1760 to 1780 3. Chronology: 1780 to 1800

Honours awarded to Joseph-Louis Lagrange (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1791

Lunar features

Crater Lagrange

Paris street names

Rue Lagrange (5th Arrondissement)

Street name in Turin

Via Giuseppe Lodovico Lagrange

Commemorated on the Eiffel Tower Other Web sites

1. Rouse Ball 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html (7 of 8) [2/16/2002 11:18:50 PM]

Lagrange

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lagrange.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lagrange.html (8 of 8) [2/16/2002 11:18:50 PM]

Laguerre

Edmond Nicolas Laguerre Born: 9 April 1834 in Bar-le-Duc, France Died: 14 Aug 1886 in Bar-le-Duc, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edmond Laguerre attended the Ecole Polytechnique in Paris but only ranked 46th in his class. He was an artillery officer from 1854 to 1864 when he returned to the Ecole Polytechnique where he remained for the rest of his life. His most important work was in the areas of analysis and geometry. His work in geometry was important at the time but has been overtaken by Lie group theory, Cayley's work and Klein's work. Laguerre studied approximation methods and is best remembered for the special functions the Laguerre polynomials which are solutions of the Laguerre differential equations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Edmond Laguerre

Mathematicians born in the same country

Cross-references to History Topics

Abstract linear spaces

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laguerre.html (1 of 2) [2/16/2002 11:18:52 PM]

Laguerre

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Laguerre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laguerre.html (2 of 2) [2/16/2002 11:18:52 PM]

Lakatos

Imre Lakatos Born: 9 Nov 1922 in Hungary Died: 2 Feb 1974 in London, England

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Imre Lakatos was given the name Imre Lipschitz at birth, born into a Jewish family as his name clearly indicated. His life would be dominated by the chaos that resulted from the Nazi rise to power and World War II, the war breaking out when Imre was in his final years at school. It was a difficult period for Hungary, with the country unsure whether to side with Hitler or with the allies, but in many ways Hungary had few options. Hitler decided that he could not leave his vital communications at the mercy of an uncommitted Hungarian regime. In March 1944 Hitler offered Hungary the choice of either cooperating with Germany or the German armies would occupy the country. Hungary chose cooperation and appointed a government to collaborate with Hitler. The Germans did as they pleased, suppressing opponents and arresting anyone who spoke out against them. Jews were compelled to wear a yellow star and their property was taken away. Imre had spent the war years at the University of Debrecen and he graduated in 1944 with a degree in mathematics, physics and philosophy. To avoid the Nazi persecution of Jews he changed his name to Imre Molnár, and he survived while others of Jewish descent were deported to the gas chambers of German concentration camps. More than 550,000 of Hungary's 750,000 Jews were killed by the Nazis during the war, including Imre's mother and grandmother who both died in Auschwitz. After the war ended Imre, who by this time was an active communist, realised that he would have difficulty wearing his old shirts with "I. L." on them when his name was now Imre Molnár. Hungary was in grave financial trouble and getting a new collection of shirts was harder than changing one's name so he changed his name, not back to the Jewish Lipschitz but rather, in keeping with his political views, to the Hungarian working class name of Lakatos. He may have borrowed the name from the Hungarian general Géza Lakatos who headed a peace seeking Hungarian government for a short while before the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lakatos.html (1 of 3) [2/16/2002 11:18:54 PM]

Lakatos

Germans put their own man in charge. At least Imre Lakatos could now wear his "I. L." shirts again! In 1947 Imre Lakatos obtained a post in the Hungarian Ministry of Education. However he was not good at taking orders from Russian authorities without questioning them and Lakatos soon found that his views had put him in political trouble. In 1950 he was arrested and served three year in a Stalinist prison. On his release in 1953, the year of Stalin's death, Rényi helped Lakatos find work. Lakatos earned his living translating mathematics books into Hungarian. Among the books that he translated at this time was Pólya's book How to Solve it. In 1956 there was revolution in Hungary against the Russian regime which controlled the country. On 1 November 1956 Hungary withdrew from the Warsaw Pact and asked the United Nations to recognise it as a neutral state, under the protection of the United Nations. Two days later Russian tanks were in position and a puppet government was set up. Many people were sent to the Soviet Union and many of those never returned. Around 200,000 refugees escaped to the West, a substantial proportion being Hungary's educated classes. Lakatos realised that he was about to be arrested and fled to Vienna. Eventually Lakatos found his way to England and he began to study at the University of Cambridge for a doctorate in philosophy. His work was influenced by Popper and by Pólya and he went on to write his doctoral thesis Essays in the Logic of Mathematical Discovery submitted to Cambridge in 1961. At Pólya's suggestion his thesis took as its theme the history of the Euler-Descartes formula V - E + F = 2. In 1960 Lakatos was appointed to the London School of Economics and he taught there for 14 years until his death. Lakatos published Proofs and Refutations in 1963-64 in four parts in the British Journal for Philosophy of Science. This work was based on his doctoral thesis and is written in the form of a discussion between a teacher and a group of students. Worrall [16] describes the paper:... as well as having great philosophical and historical value, was circulated in offprint form in enormous numbers. During his lifetime Lakatos refused to publish the work as a book since he intended to improve it. However, in 1976, two years after his death, the work did appear as a book "J Worrall and E G Zahar (eds.), I Lakatos : Proofs and Refutations : The Logic of Mathematical Discovery ". Worrall [16] describes the work:The thesis of 'Proofs and Refutations' is that the development of mathematics does not consist (as conventional philosophy of mathematics tells us it does) in the steady accumulation of eternal truths. Mathematics develops, according to Lakatos, in a much more dramatic and exciting way - by a process of conjecture, followed by attempts to 'prove' the conjecture (i.e. to reduce it to other conjectures) followed by criticism via attempts to produce counter-examples both to the conjectured theorem and to the various steps in the proof. Hersh [8] says that Proofs and Refutations is:... an overwhelming work. The effect of its polemical brilliance, its complexity of argument and self-conscious sophistication, its sheer weight of historical learning, is to dazzle the reader. Lakatos wrote a number of papers on the philosophy of mathematics before moving on to write more

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lakatos.html (2 of 3) [2/16/2002 11:18:54 PM]

Lakatos

generally on the philosophy of science. However, like his doctoral thesis, he often used historical case studies to illustrate his arguments. I [EFR] would strongly recommend the article in The Mathematical Intelligencer (3) (1978), 151-161 by Lakatos. This article, Cauchy and the Continuum : The Significance of Non-Standard Analysis for the History and Philosophy of Mathematics is one of the most enjoyable that I have read. Hersh [8] explains the point of the approach to history that Lakatos uses in this article:The point is not merely to rethink the reasoning of Cauchy, not merely to use the mathematical insight available from Robinson's non-standard analysis to re-evaluate our attitude towards the whole history of the calculus and the notion of the infinitesimal. The point is to lay bare the inner workings of mathematical growth and change as a historical process, as a process with its own laws and its own 'logic', one which is best understood in its rational reconstruction, of which the actual history is perhaps only a parody. Lakatos died at a time when he was highly productive with many plans to publish new work, make replies to his critics and apply his ideas to new areas. Worrall [16] however points out that the achievement of which Lakatos would have been most proud was leaving:... a thriving research programme manned, at the London School of Economics and elsewhere, by young scholars engaged in developing and criticising his stimulating ideas and applying them to new areas. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (16 books/articles)

Some Quotations (2)

Mathematicians born in the same country Other references in MacTutor

Chronology: 1970 to 1980

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lakatos.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lakatos.html (3 of 3) [2/16/2002 11:18:54 PM]

Lalla

Lalla Born: about 720 in India Died: about 790 in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lalla's father was Trivikrama Bhatta and Trivikrama's father, Lalla's paternal grandfather, was named Samba. Lalla was an Indian astronomer and mathematician who followed the tradition of Aryabhata I. Lalla's most famous work was entitled Shishyadhividdhidatantra. This major treatise was in two volumes. The first volume, On the computation of the positions of the planets, was in thirteen chapters and covered topics such as: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; syzygies; risings and settings; the shadow of the moon; the moon's crescent; conjunctions of the planets with each other; conjunctions of the planets with the fixed stars; the patas of the moon and sun, and a final chapter in the first volume which forms a conclusion. The second volume was On the sphere. In this volume Lalla examined topics such as: graphical representation; the celestial sphere; the principle of mean motion; the terrestrial sphere; motions and stations of the planets; geography; erroneous knowledge; instruments; and finally selected problems. In Shishyadhividdhidatantra Lalla uses Sanskrit numerical symbols. Ifrah writes in [2]:... over the centuries, Sanskrit has lent itself admirably to the rules of prosody and versification. This explains why Indian astronomers [like Lalla] favoured the use of Sanskrit numerical symbols, based on a complex symbolism which was extraordinarily fertile and sophisticated, possessing as it did an almost limitless choice of synonyms. Despite writing the most famous treatise giving the views of Aryabhata I, Lalla did not accept his theory given in the Aryabhatiya that the earth rotated. Lalla argues in his commentary, like many other Indian astronomers before him such as Varahamihira and Brahmagupta, that if the earth rotated then the speed would have to be such that one would have to ask how do the bees or birds flying in the sky come back to their nests? In fact Lalla misinterpreted some of Aryabhata I's statements about the rotating earth. One has to assume that the idea appeared so impossible to him that he just could not appreciate Aryabhata I's arguments. As Chatterjee writes in [3], Lalla in his commentary:... did not interpret the relevant verses in the way meant by Aryabhata I. Astrology at this time was based on astronomical tables and often the horoscopes allow one to identify the tables used. Some Arabic horoscopes were based on astronomical tables calculated in India. The most frequently used tables were by Aryabhata I. Lalla improved on these tables and he produced a set of corrections for the Moon's longitude. One aspect of Aryabhata I's work which Lalla did follow was his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lalla.html (1 of 2) [2/16/2002 11:18:55 PM]

Lalla

value of place.

. Lalla uses

= 62832/20000, i.e.

= 3.1416 which is a value correct to the fourth decimal

Lalla also wrote a commentary on Khandakhadyaka, a work of Brahmagupta. Lalla's commentary has not survived but there is another work on astrology by Lalla which has survived, namely the Jyotisaratnakosa. This was a very popular work which was the main one on the subject in India for around 300 years. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR November 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Lalla.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lalla.html (2 of 2) [2/16/2002 11:18:55 PM]

Lamb

Horace Lamb Born: 29 Nov 1849 in Stockport, England Died: 4 Dec 1934 in Cambridge, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Horace Lamb's father was the foreman of a cotton mill who had become well known for his improvements to spinning machines. Horace was educated at the Grammar School in Stockport. In 1866, when Horace was only 17, he won a scholarship to read classics at Queen's College, Cambridge but declined the scholarship to spend a year studying at Owens College, Manchester. It was at Owens College that Lamb's interests turned firmly towards mathematics so that, when he entered Trinity College, Cambridge the following year it was to study mathematics. Lamb was taught by Stokes and Maxwell at Cambridge and graduated as Second Wrangler in 1872. The same year he was awarded a Smith's Prize and he was made a Fellow and Lecturer at Trinity College. In 1875 he was appointed to the chair of mathematics at Adelaide, Australia where he remained for 10 years. Adelaide was extremely fortunate in their choice of Lamb as their first professor of mathematics and he rapidly built the reputation of the mathematics department there. His own reputation as a teacher at Adelaide was very high and he was described as a wonderful teacher who gave very clear, very lucid lectures. Lamb left Australia in 1885, accepting a chair at Victoria University in England (now the University of Manchester). In [4] his influence on the mathematics department at Manchester is described:Under him it grew rapidly. His lecture courses were numerous, and his books provide a record of his methods. Many of his students were engineers, and they found in him a sympathetic guide, one who understood their difficulties and shared their interest in applications of mathematics to mechanics.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lamb.html (1 of 3) [2/16/2002 11:18:56 PM]

Lamb

Describing his own teaching at the celebrations for his eightieth birthday, Lamb said:I did try to make things clear, first to myself (an important point) and then to my students, and somehow to make these dry bones live. Lamb held the chair at Manchester until 1920 when, at the age of 70, he retired and moved to Cambridge. An honorary lectureship, the Rayleigh lectureship, was specially created for him and he continued his research. Lamb wrote important texts and made important contributions to applied mathematics, in particular to acoustics and fluid dynamics, for example Mathematical Theory of the Motion of Fluids (1878). His book Hydrodynamics (1895) was for many years the standard work on the subject. In his address to the British Association in 1904 he explained his reasons for writing these books:It is ... essential that from time to time someone should come forward to sort out and arrange the accumulated material, rejecting what has proved unimportant, and welding the rest into a connected system. Lamb's texts had a major role on teaching in British universities for many years. Other topics he worked on include wave propagation, electrical induction, earthquakes, and the theory of tides. He wrote important papers on the oscillations of a viscous spheroid, the vibrations of elastic spheres, waves in elastic solids, electric waves and the absorption of light. In a famous paper in the Proceedings of the London Mathematical Society he showed how Rayleigh's results on the vibrations of thin plates fitted with the general equations of the theory. Another paper reported on his study of the propagation of waves on the surface of an elastic solid where he tried to understand the way that earthquake tremors are transmitted around the surface of the Earth. Lamb wrote books in addition to those mentioned above, including Infinitesimal Calculus (1897), Dynamical Theory of Sound (1910), and Higher Mechanics (1920). Love, writing in [5], describes his writing:His writings call up before one the picture of an extremely acute and wonderful alert mind, endowed with a profound knowledge of the facts of physics, especially on its dynamical side, keenly interested in the work of others, particularly when it had a bearing on any matter of mechanics or wave transmission, equipped with an exceptionally varied and powerful mathematical technique, and ever on the look-out for topics on which his analysis could be employed for the promotion of natural knowledge. Elected to the Royal Society in 1884, he was a member of the Council of the Society and twice its Vice-President. He received the Royal Medal from the Royal Society in 1902 and in 1923 was further honoured with the award of its Copley Medal. An extremely strong supporter of the London Mathematical Society, he served that Society as President in 1902-04 and received its De Morgan Medal in 1911. He was honoured with memberships of many mathematical societies in Europe, and received the highest award of his country when he was knighted in 1931. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lamb.html (2 of 3) [2/16/2002 11:18:56 PM]

Lamb

List of References (5 books/articles)

A Quotation

Mathematicians born in the same country Honours awarded to Horace Lamb (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1884

Royal Society Copley Medal

Awarded 1923

Royal Society Royal Medal

Awarded 1902

London Maths Society President

1902 - 1904

LMS De Morgan Medal

Awarded 1911

Lunar features

Crater Lamb

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lamb.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lamb.html (3 of 3) [2/16/2002 11:18:56 PM]

Lambert

Johann Heinrich Lambert Born: 26 Aug 1728 in Mülhausen, Alsace, France Died: 25 Sept 1777 in Berlin, Prussia (now Germany)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Lambert was a colleague of Euler and Lagrange at the Berlin Academy of Sciences. In 1766 Lambert wrote Theorie der Parallellinien which was a study of the parallel postulate. By assuming that the parallel postulate was false, he managed to deduce a large number of non-euclidean results. He noticed that in this new geometry the sum of the angles of a triangle increases as its area decreases. Lambert is best known, however, for his work on . Euler had already established in 1737 that e and e2 are irrational. Lambert was the first to provide a rigorous proof that is irrational. In a paper presented to the Berlin Academy in 1768 Lambert showed that, if x is a nonzero rational number, then neither ex nor tan x can be rational. Since tan /4 = 1 then /4 must be irrational. Lambert conjectured that e and are transcendental. This was not proved for another century when Hermite proved that e is transcendental and Lindemann proved that is transcendental. Lambert also made the first systematic development of hyperbolic functions. A few years earlier they had been studied by Vincenzo Riccati. Lambert is also responsible for many innovations in the study of heat and light as well as working on the theory of probability. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lambert.html (1 of 2) [2/16/2002 11:18:58 PM]

Lambert

Click on this link to see a list of the Glossary entries for this page List of References (22 books/articles)

Some Quotations (2)

A Poster of Johann H Lambert

Mathematicians born in the same country

Some pages from publications

Title page of Freye perspective (1774) A page from Theorie der parallelinien (1895)

Cross-references to History Topics

1. Non-Euclidean geometry 2. Squaring the circle 3. The trigonometric functions 4. Pi through the ages 5. A history of group theory

Other references in MacTutor

Chronology: 1760 to 1780

Honours awarded to Johann H Lambert (Click a link below for the full list of mathematicians honoured in this way) Planetary features

Crater T Mayer on Mars

Paris street names

Rue Lambert (18th Arrondissement)

Other Web sites

1. Rouse Ball 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lambert.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lambert.html (2 of 2) [2/16/2002 11:18:58 PM]

Lame

Gabriel Lamé Born: 22 July 1795 in Tours, France Died: 1 May 1870 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gabriel Lamé was a student at the Ecole Polytechnique, entering in 1813 and graduating in 1817. Already during these undergraduate years Lamé was writing research papers, and he published his first paper Mémoire sur les intersections des lignes et des surfaces in Gergonne's Journal in 1816-17. After graduating from the Ecole Polytechnique, Lamé studied engineering at the Ecole des Mines in Paris, graduating from there in 1820. While at the Ecole des Mines Lamé published his second work, this time on a method he had invented to calculate the angles between faces of crystals. In 1820 Lamé, together with his colleague Emile Clapeyron, went to Russia. We should give some background to this event which, on the face of it, looks rather a strange career move for the two young mathematicians. Alexander I was emperor of Russia from 1801 to 1825. The French Revolution and events in France which followed it, had shown Alexander the importance of scientific knowledge and its applications to military techniques and industrial development. He understood that for Russia to be powerful it must follow suit. He looked towards Europe and European scientists and tried to introduce policies to encourage them to cooperate with Russian scientists. He encouraged teachers to go to Russia to teach the latest scientific theories and to create scientific contacts between Russia and Europe. In line with this policy, the Russian government made a request to France who responded by sending Lamé and Clapeyron to St Petersburg. Lamé was appointed professor and engineer at the Institut et Corps du Genie des Voies de Communication in St Petersburg. At first things were rather difficult for Lamé but later his visit proved highly productive. He lectured on analysis, physics, mechanics, chemistry, and engineering topics. He published papers in both Russian and French journals during his 12 years there, some jointly with http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lame.html (1 of 4) [2/16/2002 11:19:00 PM]

Lame

Clapeyron. They published in, for example, the Journal des voies de communications, the Journal du genie civil, the Bulletin des sciences mathematique, the Receuil des savants etrangers, and Crelle's Journal (Journal für die reine und angewandte Mathematik) after it began publication in 1826. In [6] an interesting episode which occurred during Lamé's time in St Petersburg is related. It concerns Lamé's attempt to spread Cauchy's new ideas of rigorous analysis. A professor at the Institute where Lamé taught had written a book which contained a proof of Taylor's theorem. Lamé produced a manuscript criticising the proof using Cauchy's arguments. Another side to Lamé's work in St Petersburg was his involvement in helping with plans that were being drawn up for building bridges and roads around the city. At this time he became more aware of the vast potential of railway development, and this would be a topic of great interest to him after his return to France. Before that, he was present when the Liverpool-Manchester line opened in England on 15 September 1830. Bradley [4] gives a lot more detail regarding Lamé's time in Russia. She concludes in her paper that:... the repressive atmosphere in France during the period of the Bourbon restoration had made work abroad seem more attractive for research and the application of new ideas. Lame and Clapeyron seized an opportunity offered to them by successful French engineers already established in Russia who had taken with them the spirit of the early years of the Ecole Polytechnique. Important engineers like Betancourt and Bazaine helped them to pursue their careers in a land of scientific opportunity where their ideological convictions were strengthened through contact and discussion with their compatriots. In 1832 Lamé returned to Paris and at first he formed part of an engineering firm set up jointly with Clapeyron and two others. After only a few months, and still in 1832, Lamé accepted the chair of physics at the Ecole Polytechnique. He did not restrict his interests to teaching and research, however, for in remained an engineer ready for consulting work in that area. In 1836 he was appointed chief engineer of mines and he was also involved in the building of the railway from Paris to Versailles and of the railway from Paris to St Germain, which was opened in 1837. Lamé was elected to the Académie des Sciences in 1843 when Louis Puissant died leaving a vacancy in the geometry section. In the following year he left his chair of physics at the Ecole Polytechnique and accepted a post at the Sorbonne in mathematical physics and probability. He was appointed to the chair of mathematical physics and probability at the Sorbonne in 1851. He worked on a wide variety of different topics. Often problems in the engineering tasks he undertook led him to study mathematical questions. For example his work on the stability of vaults and on the design of suspension bridges led him to work on elasticity theory. In fact this was not a passing interest, for Lamé made substantial contributions to this topic. Another example is his work on the conduction of heat which led him to his general theory of curvilinear coordinates. Curvilinear coordinates proved a very powerful tool in Lamé's hands. He used them to transform Laplace's equation into ellipsoidal coordinates and so separate the variables and solve the resulting equation. The trademark of Lamé's career was moving from one topic to another in a quite logical way but he often ended up studying problems very far removed from the original. This happened with curvilinear coordinates for he was led to study the equation (x/a)n + (y/b)n+ (z/c)n = 0

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lame.html (2 of 4) [2/16/2002 11:19:00 PM]

Lame

which, in non-homogeneous form he wrote as (x/a)n + (y/b)n= 1 which, with a = b is xn + yn = an so he was led to Fermat's last theorem. Although he was basically an applied mathematician, Lamé made a substantial contribution to the problem by solving the case n = 7. In fact he believed that he had solved the whole problem at one stage but he had overlooked the lack of unique factorisation in certain subrings of the complex numbers. He also did important work on differential geometry and, in another contribution to number theory, he showed that the number of divisions in the Euclidean algorithm never exceeds five times the number of digits in the smaller number. As we noted above, he worked on engineering mathematics and elasticity where two elastic constants are named after him. He studied diffusion in crystalline material. Lamé was considered the leading French mathematician of his time by many, in particular Gauss who was never one to give praise easily held this opinion. Rather strangely he was more highly thought of outside France than inside, for the French seemed to feel that he was too practical for a mathematician and yet too theoretical for an engineer. His own opinion was that curvilinear coordinates were his most important contribution, but there are strange twists and turns in the history of mathematics and very soon after Lamé introduced them curvilinear coordinates became obsolete through the generalisations introduced by Hermite, Klein, and Bôcher. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Gabriel Lamé

Mathematicians born in the same country

Cross-references to History Topics

Fermat's last theorem

Cross-references to Famous Curves

1. Ellipse 2. Hyperbola 3. Lamé Curves

Other references in MacTutor

Chronology: 1830 to 1840

Honours awarded to Gabriel Lamé (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Gabriel Lamé (12th Arrondissement)

Commemorated on the Eiffel Tower

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lame.html (3 of 4) [2/16/2002 11:19:00 PM]

Lame

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lame.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lame.html (4 of 4) [2/16/2002 11:19:00 PM]

Lamy

Bernard Lamy Born: 15 June 1640 in Le Mans, France Died: 29 Jan 1715 in Rouen, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bernard Lamy studied at the Oratorian college in Le Mans. At the age of 18 he went to Paris and entered the Maison d'Institution. After a year he went to Saumur where he studied philosophy. He was admitted to the Congregation of the Oratory in 1662. Malebranche was also a member of the Congregation of the Oratory and they met as students and remained friends for the rest of their lives. In 1661 Lamy was appointed professor of classics at Vendome. He held this post for two years before being appointed to a similar post in Juilly. Lamy was ordained in 1667 and studied theology, also at Saumur, from 1669 until 1671. After this he became professor of philosophy at the College of Saumur, moving to a similar post at the College of Angers two years later. Lamy taught Descartes's philosophy at Angers and for this he was exiled by order of the King in 1676. After four years his exile ended and he was able to teach in Grenoble. He published Traitez de Mechanique in 1679 in which the parallelogram of forces law is given. Varignon discovered the parallelogram of forces law independently, at about the same time, and he saw more consequences of it than did Lamy. Lamy also published Traité de la grandeur en general (1680) and Les éléments de géometrie (1685). In 1686 Lamy obtained permission to live in Paris but trouble over a theological work had him sent away in 1689 and he lived from 1690 in Rouen, remaining there for the rest of his life. He published several books while in Rouen, including Traité de perspective (1701). Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lamy.html (1 of 2) [2/16/2002 11:19:01 PM]

Lamy

Other Web sites

1. The Galileo Project 2. The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lamy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lamy.html (2 of 2) [2/16/2002 11:19:01 PM]

Lanczos

Cornelius Lanczos Born: 2 Feb 1893 in Székesfehérvár, Hungary Died: 25 June 1974 in Budapest, Hungary

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Cornelius Lanczos was born Kornél Löwy but when there was a reaction in Hungary against German names he, along with large numbers of his countrymen, changed his name to Kornél Lánczos (or rather to Lánczos Kornél since Hungarians put the family name first). His family were of Jewish origins, his father being a lawyer. Lanczos attended a Jewish elementary school where he learn several foreign languages, then he entered the local Gymnasium which was a Catholic school run by the Cistercians. He graduated from the gymnasium in 1910 and, the following September, he entered the University of Budapest. There he had several inspiring teachers who were to make a great impression on Lanczos. His physics teacher was Eötvös who first interested Lanczos in relativity. His mathematics teacher was Fejér and Lax writes in [2]:Lanczos was much influenced by Fejér; he learnt from him about Fourier series, orthogonal polynomials, and interpolation. It is likely that Lanczos was influenced by Fejér's style of lecturing... After graduating in 1915, Lanczos was appointed an assistant at the Technical University of Budapest. He worked on relativity theory and after writing his dissertation Relation of Maxwell's Aether Equations to Functional Theory he sent a copy to Einstein. It impressed Einstein who replied:I have read your work in as much detail as my present excess of work allows. I can say that it is sound and original thinking. It makes you worthy of the doctorate. I am pleased to give you my permission to honour me by dedicating it to me. Lanczos received his doctorate in 1921 and, because of laws in Hungary against Jews, he went to

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lanczos.html (1 of 3) [2/16/2002 11:19:03 PM]

Lanczos

Germany taking up a post at the University of Freiburg. He spent three years in Freiburg, then took up a post in Frankfurt am Main where he became a colleague of Paul Epstein. During the year 1928-29 Lanczos was Einstein's assistant in Berlin, returning to Frankfurt in 1929. In 1931 Lanczos spent a year as a visiting professor at the University of Purdue in Lafayette, Indiana. Returning to Germany he found that the political situation there becoming unacceptable for someone of Jewish origin and he returned to a Professorship at Purdue in 1932. As Gellai writes in [3]:Had he not been compelled to leave Germany for the United States, the field of applied mathematics might lack today several very useful and elegant mathematical methods. During his time at Purdue, Lanczos published mathematical physics papers at first but in 1938 he published his first work in numerical analysis. Two years later he published a matrix method of calculating Fourier coefficients which, over 25 years later, was recognised as the 'Fast Fourier Transform' algorithm described by Tukey. Lanczos continued to work on his first love of relativity and corresponded with Einstein both on a scientific level and as a friend. A new young man was appointed as Head of Physics at Purdue and Lanczos felt his work there was no longer appreciated. He wrote to Einstein:This brings me into a most difficult situation and I am trying desperately to get away from here. Lanczos spent the year of 1944 working for the Boeing Aircraft Company and, in 1946, he resigned his post in Purdue to take up a permanent appointment with Boeing. There he worked on applications of mathematics to aircraft design and was able to develop new numerical methods to solve the problems. In 1949 he moved to the Institute for Numerical Analysis of the National Bureau of Standards in Los Angeles. Here he worked on developing digital computers and was able to produce versions of the numerical methods he had developed earlier to program on the digital computers. At the Institute for Numerical Analysis he had Otto Szász, Taussky-Todd and her husband John Todd as colleagues. However during the early 1950s United States senator Joseph R McCarthy whipped up strong feelings against communism in the USA. At the Institute for Numerical Analysis, as in many other institutions, there were investigations and suspicions and the atmosphere became unpleasant. Lanczos was therefore delighted to receive an offer from Schrödinger to head the Theoretical Physics Department at the Dublin Institute for Advance Study in Ireland. He took up the post in 1952 and, as Gellai writes in [3]:A very special, and in some ways the most beautiful, period of his life started there when he returned again to his 'first love' in science, devoting himself primarily to the study of the theory of relativity. Indeed, this position offered the support and the circumstances which he had longed for his entire life. Lanczos published over 120 papers and books in a career spanning over 40 years. About half of these papers and books were published after he took up the position in Dublin. He certainly travelled widely during this period, most often in the United states. While on a visit to the Eötvös Lóránd University in Budapest in 1974 he had a sudden heart attack and died in hospital the following day. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lanczos.html (2 of 3) [2/16/2002 11:19:03 PM]

Lanczos

List of References (15 books/articles)

Some Quotations (2)

A Poster of Cornelius Lanczos

Mathematicians born in the same country

Other Web sites

●

North Carolina (A description of the Complete Works of Lanczos)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lanczos.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lanczos.html (3 of 3) [2/16/2002 11:19:03 PM]

Landau

Edmund Georg Hermann Landau Born: 14 Feb 1877 in Berlin, Germany Died: 19 Feb 1938 in Berlin, Germany

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edmund Landau attended the French Lycée in Berlin, graduating at the age of 16 which is two years earlier than was normal. He then studied mathematics at the University of Berlin. His doctoral work there was supervised by Frobenius and Landau received his doctorate in 1899 for a work on number theory. Landau was always interested in mathematical puzzles and even before he received his doctorate he had published two books on mathematical problems in chess. He submitted this Habilitation thesis in 1901, only two years after his doctorate, again on analytic number theory. Frobenius was somewhat critical of the area Landau worked in, and remarked at times that Landau's work would cease to become important if the Riemann hypothesis were proved. Landau taught at the University of Berlin from 1899 until 1909. During this period his publication list rapidly grew so that by 1904 his publications exceeded his age of 27. While at Berlin his ability to teach became clearly evident. He taught beginners courses, which he did not have to take, and also lectured on his own speciality of number theory and, in addition, other lecture courses on foundations of mathematics, irrational numbers and set theory. Schappacher notes in [7] however:... it should also be said that he tended not to have cordial relationships with his students, being rather an aloof person. In 1909 he was appointed to Göttingen as successor to Minkowski. He had Hilbert and Klein as colleagues at Göttingen until Klein retired in 1913. The successor to Klein was not easily found. Carathéodory came and went in less than three years while he was followed by Hecke who also left quickly in 1919. Landau worked hard to have Schur fill the chair but, against Landau's wishes, Courant http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Landau.html (1 of 3) [2/16/2002 11:19:06 PM]

Landau

was appointed. Landau was to remain at Göttingen until the National Socialist regime forced him out of office in 1933. Teichmüller, as leader of the students, organised a boycott of Landau's lectures. After this Landau moved to Berlin and only lectured outside Germany, spending some time in Cambridge and in Holland. Landau's main work was in analytic number theory and the distribution of primes. He gave a proof of the prime number theorem in 1903 which was considerably simpler that the ones given in 1896 by Vallée Poussin and Hadamard. His work of 1909 gave the first systematic presentation of analytic number theory. He also wrote important works on the theory of analytic functions of a single variable. Landau wrote over 250 papers on number theory which had a major influence on the development of the subject. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles)

Some Quotations (2)

A Poster of Edmund Landau

Mathematicians born in the same country

Cross-references to History Topics

Pi through the ages

Other references in MacTutor

1. Prime Number Theorem 2. Chronology: 1900 to 1910 3. Chronology: 1910 to 1920

Other Web sites

1. Landau-Ramanujan Constant 2. Bloch-Landau Constants 3. Landau-Kolmogorov Constants

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Landau.html (2 of 3) [2/16/2002 11:19:06 PM]

Landau

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Landau.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Landau.html (3 of 3) [2/16/2002 11:19:06 PM]

Landau_Lev

Lev Davidovich Landau Born: 22 Jan 1908 in Baku, Azerbaijan, Russian Empire Died: 1 April 1968 in Moscow, USSR

Click the picture above to see five larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lev Landau studied physics and chemistry at Baku University and then went in 1924 to the Leningrad State University, graduating in 1927. He continued research at the Leningrad Physico- Technical Institute. In 1929 Landau travelled to Göttingen, Leipzig and Copenhagen where he worked in Niels Bohr's Institute for Theoretical Physics. From that time he considered himself a pupil of Bohr's whose influence was to dictate the direction of Landau's work. In 1932 Landau become the head of the Theory Division of the Ukrainian Technical Institute in Kharkov. In addition, in 1935, he was made head Physics at the Kharkov Gorky State University. Landau soon made his School in Kharkov the centre of theoretical physics in the USSR. In 1937 Landau went to Moscow to become Head of the Theory Division of the Physical Institute of the Academy of Sciences. Landau worked on low-temperature physics, atomic and nuclear physics and plasma physics. The work he did on the theory to explain why liquid helium was super-fluid earned him the 1962 Nobel Prize for Physics. In 1962, Landau was involved in a car accident after which he was unconscious for six weeks. Several times doctors declared him clinically dead. Remarkably Landau regained consciousness and although in most ways he returned to normal, he could never again perform creative work. He died six years later having never completely recovered. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Landau_Lev.html (1 of 2) [2/16/2002 11:19:08 PM]

Landau_Lev

List of References (8 books/articles) Mathematicians born in the same country Honours awarded to Lev Landau (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1962

Fellow of the Royal Society

Elected 1960

Lunar features

Crater Landau

Other Web sites

1. Nobel prizes site (A biography of Lev Landau and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Landau_Lev.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Landau_Lev.html (2 of 2) [2/16/2002 11:19:08 PM]

Landen

John Landen Born: 23 Jan 1719 in Peakirk (near Peterborough), England Died: 15 Jan 1790 in Milton (near Peterborough), England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

A trained surveyor, John Landen's interest in mathematics was his leisure activity. He acted as a land-agent from 1762 to 1788. He wrote Mathematical lubrications in 1755 and he sent his results on making the differential calculus into a purely algebraic theory to the Royal Society. This work appeared as Residual analysis in 1764. These ideas were taken up by Lagrange. Landen investigated the dilogarithm in 1760, about the same time as Euler, and introduced the trilogarithm. The trilogarithm was studied further by Spence around 1809 and Kummer around 1840. Landen wrote on dynamics, summation of series and an important transformation giving a relation between elliptic functions. This last result, known by his name, expressed a hyperbolic arc in terms of two elliptic ones. It was published in the Philosophical Transactions of the Royal Society in 1775. Applications of this result appeared in his Mathematical memoirs of 1780. He also solved the problem of the spinning top and explained Newton's error in calculating the precession. Landen was elected a Fellow of the Royal Society in 1766. He corrected Stewart's result on the distance of the Sun from the Earth in 1771. In [5] his achievements are summed up as follows:Though foreigners gave him a high rank among English analysts, he failed to develop and combine his discoveries. He led a retired life, chiefly at Walton in Northamptonshire. Though humane and honourable, he was too dogmatic in society. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Landen.html (1 of 2) [2/16/2002 11:19:09 PM]

Landen

Honours awarded to John Landen (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1766

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Landen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Landen.html (2 of 2) [2/16/2002 11:19:09 PM]

Landsberg

Georg Landsberg Born: 30 Jan 1865 in Breslau, Germany (now Wroclaw, Poland) Died: 14 Sept 1912 in Kiel, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Landsberg attended school at Breslau. He then studied at the Universities of Breslau and Leipzig between 1883 and 1889. His doctorate was awarded by the University of Breslau in 1890. Landsberg joined the teaching staff at the University of Heidelberg in 1893 and he taught there being promoted to professor in 1897. In 1904 he returned to Breslau as extraordinary professor of mathematics but he was only there for two years accepting an offer of a post at the University of Kiel. At Kiel he was promoted to ordinary professor of mathematics in 1911 but sadly he was only to hold this post for a short while since he died the following year. Landsberg studied the theory of functions of two variables and also the theory of higher dimensional curves. In particular he studied the role of these curves in the calculus of variations and in mechanics. He worked with ideas related to those of Weierstrass, Riemann and Heinrich Weber on theta functions and Gaussian sums. His most important work, however was his contribution to the development of the theory of algebraic functions of a single variable. Here he studied the Riemann-Roch theorem. He was able to combine Riemann's function theoretic approach with the Italian geometric approach and with the Weierstrass arithmetical approach. His arithmetic setting of this result led eventually to the modern abstract theory of algebraic functions. One of his most important works was Theorie der algebraischen Funktionen einer Varaiblen (Leipzig, 1902) which he wrote jointly with Kurt Hensel. This work remained the standard text on the subject for many years. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Landsberg.html (1 of 2) [2/16/2002 11:19:10 PM]

Landsberg

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Landsberg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Landsberg.html (2 of 2) [2/16/2002 11:19:10 PM]

Langlands

Robert Phelan Langlands Born: 6 Oct 1936 in New Westminster, British Columbia, Canada

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert P Langlands' father was Robert Langlands while his mother was Kathleen Phelan. He married Charlotte Lorraine Cheverie on 13 August 1956 while he was a nineteen year old undergraduate at the University of British Columbia. He was awarded his B.A. in the following year and continued to study at the University of British Columbia for his master's degree which was awarded in 1958. Langlands then studied at Yale University for his doctorate. He submitted his thesis Semi-groups and representations of Lie groups to Yale in 1960 and received the degree of Ph.D. Langlands wrote:There are two, related parts to this thesis: one on representations of Lie semi-groups and one on operators associated to representations of Lie groups. The first part was published in the Canadian Journal of Mathematics, but the second was published only as an announcement in the Proceedings of the National Academy of Sciences of the USA. It nevertheless had the good fortune to be taken seriously by Derek Robinson, who incorporated some of the results into his book on Elliptic Operators and Lie Groups. Also writing of his doctoral thesis Langlands regretted that it remains:... my only active encounter with partial differential equations, a subject to which I had always hoped to return but in a different vein. Appointed to Princeton as an instructor after completing his doctoral studies, Langlands taught there for seven years and was promoted to associate professor. He spent 1964-65 at the University of California, Berkeley as a Miller Foundation Fellow and an Alfred P Sloan Fellow. Then in 1967 he returned to Yale University as a full professor. However Langlands spent 1967-68 visiting in Ankara, Turkey having an office next to that of Cahit Arf. After five years at Yale he returned again to Princeton, this time as professor of mathematics at the Institute for Advanced Study. He has remained at the Institute for Advanced Study since his appointment there in 1972.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Langlands.html (1 of 3) [2/16/2002 11:19:12 PM]

Langlands

In 1988 Langlands received the National Academy of Sciences Award in Mathematics. He was the first recipient of this award which was established by the American Mathematical Society. The citation for the award to Langlands recognises his:... extraordinary vision that has brought the theory of group representations into a revolutionary new relationship with the theory of automorphic forms and number theory. Let us explain a little of Langlands' work which led to this award. As soon as he had completed his doctoral work, Langlands began to work on automorphic forms. In a remarkable paper he applied recent results by Harish-Chandra to obtain a formula for the dimension of certain spaces of automorphic forms. Then, over the next couple of years, he produced deep results on Eisenstein series and went on to apply Eisenstein series to prove a number theory conjecture due to Weil. In 1967 he wrote a letter to Weil which contains profound mathematical ideas which continue to drive a whole area of mathematical research. The letter was 17 pages hand-written and sent in January 1967. It sketched what soon became known as "the Langlands conjectures". Weil had the letter typed and this typed version circulated widely among mathematicians interested in the topics. Casselman writes in [3] that the letter contained:... a collection of far-reaching and uncannily accurate conjectures relating number theory, automorphic forms, and representation theory. Theses have formed the core of a program still being carried out, and have come to play a central role in all three subjects. Other letters of Langlands also proved remarkably important. While he was in Ankara in 1967-68 he wrote to Serre with ideas which would eventually be formulated as the Deligne-Langlands conjecture; this was proved eventually by Kazhdan and Lusztig. The National Academy of Sciences Award in Mathematics which we referred to above is certainly not the only award which Langlands has received for his work. In 1975 he was awarded the Wilbur Cross Medal from Yale University. He received the Cole Prize in Number Theory from the American Mathematical Society in 1982 for his pioneering work on automorphic forms Eisenstein series, and product formulas. More recently he shared the 1995-96 Wolf Prize in Mathematics with Wiles. The Prize was awarded to Langlands for his:... path-blazing work and extraordinary insights in the fields of number theory, automorphic forms, and group representation. Elected a Fellow of the Royal Society of Canada in 1972, he was elected a Fellow of the Royal Society of London in 1981. He has received honorary doctorates from the University of British Columbia, McMaster University, The City University of New York, the University of Waterloo, the University of Paris VII, McGill University, and the University of Toronto. Casselman, in [3], ends with the following:[Langlands'] astounding insight has provided a whole generation of mathematicians working in automorphic forms and representation theory with a seemingly unlimited expanse of deep, interesting, and above all approachable problems to work away on. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Langlands.html (2 of 3) [2/16/2002 11:19:12 PM]

Langlands

List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1980 to 1990

Honours awarded to Robert P Langlands (Click a link below for the full list of mathematicians honoured in this way) AMS Cole Prize winner

1982

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Langlands.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Langlands.html (3 of 3) [2/16/2002 11:19:12 PM]

Lansberge

Philip van Lansberge Born: 25 Aug 1561 in Ghent, Netherlands (now Belgium) Died: 8 Dec 1632 in Middelburg, Netherlands

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Philippe van Lansberge was born in the Netherlands but his family left there in 1566 going first to France, then to England. Lansberge studied mathematics and theology in France and England. He then went to Flanders where, in 1579, he became a minister at a Protestant church. The following year he moved to become minister at an Antwerp church but, after five years in Antwerp, the Spanish armies came and Lansberge returned to the Netherlands. He enrolled as a theology student at Leiden in 1585 but moved to be minister at a church in Goes the following year. Lansberge remained in Goes from 1586 until 1613 when he ran into political problems and left to go to Middleburg. Here he spent the last 20 years of his life and here, as well as publishing mathematical works, he practised medicine. In 1591 he wrote a 4 volume work on mathematics. Volume 1 was on trigonometric functions; volume 2 gave methods of constructing tables for sines, tangents, and secants derived from Viète and Fincke and giving the tables themselves; volume 3 contained plane geometry following Regiomontanus while volume 4 contained spherical trigonometry. In 1616 Lansberge wrote on

calculating it to 28 places using a new method.

Lansberge's work on astronomy followed Copernicus. He wrote works supporting Copernicus's theories in both 1619 and 1629. However he did not accept Kepler's ellipse theories and he published astronomical tables which he hoped would support Copernicus over Kepler. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lansberge.html (1 of 2) [2/16/2002 11:19:14 PM]

Lansberge

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Some pages from publications

Portrait from Tabulae motum coelestium (1632) Title page of Tabulae motum coelestium (including a small portrait of Lansberge with other astronomers)

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lansberge.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lansberge.html (2 of 2) [2/16/2002 11:19:14 PM]

Laplace

Pierre-Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre-Simon Laplace's father, Pierre Laplace, was comfortably well off in the cider trade. Laplace's mother, Marie-Anne Sochon, came from a fairly prosperous farming family who owned land at Tourgéville. Many accounts of Laplace say his family were 'poor farming people' or 'peasant farmers' but these seem to be rather inaccurate although there is little evidence of academic achievement except for an uncle who is thought to have been a secondary school teacher of mathematics. This is stated in [1] in these terms:There is little record of intellectual distinction in the family beyond what was to be expected of the cultivated provincial bourgeoisie and the minor gentry. Laplace attended a Benedictine priory school in Beaumont-en-Auge, as a day pupil, between the ages of 7 and 16. His father expected him to make a career in the Church and indeed either the Church or the army were the usual destinations of pupils at the priory school. At the age of 16 Laplace entered Caen University. As he was still intending to enter the Church, he enrolled to study theology. However, during his two years at the University of Caen, Laplace discovered his mathematical talents and his love of the subject. Credit for this must go largely to two teachers of mathematics at Caen, C Gadbled and P Le Canu of whom little is known except that they realised Laplace's great mathematical potential. Once he knew that mathematics was to be his subject, Laplace left Caen without taking his degree, and went to Paris. He took with him a letter of introduction to d'Alembert from Le Canu, his teacher at Caen. Although Laplace was only 19 years old when he arrived in Paris he quickly impressed d'Alembert. Not only did d'Alembert begin to direct Laplace's mathematical studies, he also tried to find him a position to earn enough money to support himself in Paris. Finding a position for such a talented young man did not http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html (1 of 8) [2/16/2002 11:19:16 PM]

Laplace

prove hard, and Laplace was soon appointed as professor of mathematics at the Ecole Militaire. Gillespie writes in [1]:Imparting geometry, trigonometry, elementary analysis, and statics to adolescent cadets of good family, average attainment, and no commitment to the subjects afforded little stimulus, but the post did permit Laplace to stay in Paris. He began producing a steady stream of remarkable mathematical papers, the first presented to the Académie des Sciences in Paris on 28 March 1770. This first paper, read to the Society but not published, was on maxima and minima of curves where he improved on methods given by Lagrange. His next paper for the Academy followed soon afterwards, and on 18 July 1770 he read a paper on difference equations. Laplace's first paper which was to appear in print was one on the integral calculus which he translated into Latin and published at Leipzig in the Nova acta eruditorum in 1771. Six years later Laplace republished an improved version, apologising for the 1771 paper and blaming errors contained in it on the printer. Laplace also translated the paper on maxima and minima into Latin and published it in the Nova acta eruditorum in 1774. Also in 1771 Laplace sent another paper Recherches sur le calcul intégral aux différences infiniment petites, et aux différences finies to the Mélanges de Turin. This paper contained equations which Laplace stated were important in mechanics and physical astronomy. The year 1771 marks Laplace's first attempt to gain election to the Académie des Sciences but Vandermonde was preferred. Laplace tried to gain admission again in 1772 but this time Cousin was elected. Despite being only 23 (and Cousin 33) Laplace felt very angry at being passed over in favour of a mathematician who was so clearly markedly inferior to him. D'Alembert also must have been disappointed for, on 1 January 1773, he wrote to Lagrange, the Director of Mathematics at the Berlin Academy of Science, asking him whether it might be possible to have Laplace elected to the Berlin Academy and for a position to be found for Laplace in Berlin. Before Lagrange could act on d'Alembert's request, another chance for Laplace to gain admission to the Paris Academy arose. On 31 March 1773 he was elected an adjoint in the Académie des Sciences. By the time of his election he had read 13 papers to the Academy in less than three years. Condorcet, who was permanent secretary to the Academy, remarked on this great number of quality papers on a wide range of topics. We have already mentioned some of Laplace's early work. Not only had he made major contributions to difference equations and differential equations but he had examined applications to mathematical astronomy and to the theory of probability, two major topics which he would work on throughout his life. His work on mathematical astronomy before his election to the Academy included work on the inclination of planetary orbits, a study of how planets were perturbed by their moons, and in a paper read to the Academy on 27 November 1771 he made a study of the motions of the planets which would be the first step towards his later masterpiece on the stability of the solar system. Laplace's reputation steadily increased during the 1770s. It was the period in which he [1]:... established his style, reputation, philosophical position, certain mathematical techniques, and a programme of research in two areas, probability and celestial mechanics, in which he worked mathematically for the rest of his life.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html (2 of 8) [2/16/2002 11:19:16 PM]

Laplace

The 1780s were the period in which Laplace produced the depth of results which have made him one of the most important and influential scientists that the world has seen. It was not achieved, however, with good relationships with his colleagues. Although d'Alembert had been proud to have considered Laplace as his protégé, he certainly began to feel that Laplace was rapidly making much of his own life's work obsolete and this did nothing to improve relations. Laplace tried to ease the pain for d'Alembert by stressing the importance of d'Alembert's work since he undoubtedly felt well disposed towards d'Alembert for the help and support he had given. It does appear that Laplace was not modest about his abilities and achievements, and he probably failed to recognise the effect of his attitude on his colleagues. Lexell visited the Académie des Sciences in Paris in 1780-81 and reported that Laplace let it be known widely that he considered himself the best mathematician in France. The effect on his colleagues would have been only mildly eased by the fact that Laplace was right! Laplace had a wide knowledge of all sciences and dominated all discussions in the Academy. As Lexell wrote:... in the Academy he wanted to pronounce on everything. It was while Lexell was in Paris that Laplace made an excursion into a new area of science [2]:Applying quantitative methods to a comparison of living and nonliving systems, Laplace and the chemist Antoine Lavoisier in 1780, with the aid of an ice calorimeter that they had invented, showed respiration to be a form of combustion. Although Laplace soon returned to his study of mathematical astronomy, this work with Lavoisier marked the beginning of a third important area of research for Laplace, namely his work in physics particularly on the theory of heat which he worked on towards the end of his career. In 1784 Laplace was appointed as examiner at the Royal Artillery Corps, and in this role in 1785, he examined and passed the 16 year old Napoleon Bonaparte. In fact this position gave Laplace much work in writing reports on the cadets that he examined but the rewards were that he became well known to the ministers of the government and others in positions of power in France. Laplace served on many of the committees of the Académie des Sciences, for example Lagrange wrote to him in 1782 saying that work on his Traité de mécanique analytique was almost complete and a committee of the Académie des Sciences comprising of Laplace, Cousin, Legendre and Condorcet was set up to decide on publication. Laplace served on a committee set up to investigate the largest hospital in Paris and he used his expertise in probability to compare mortality rates at the hospital with those of other hospitals in France and elsewhere. Laplace was promoted to a senior position in the Académie des Sciences in 1785. Two years later Lagrange left Berlin to join Laplace as a member of the Académie des Sciences in Paris. Thus the two great mathematical geniuses came together in Paris and, despite a rivalry between them, each was to benefit greatly from the ideas flowing from the other. Laplace married on 15 May 1788. His wife, Marie-Charlotte de Courty de Romanges, was 20 years younger than the 39 year old Laplace. They had two children, their son Charles-Emile who was born in 1789 went on to a military career. Laplace was made a member of the committee of the Académie des Sciences to standardise weights and measures in May 1790. This committee worked on the metric system and advocated a decimal base. In 1793 the Reign of Terror commenced and the Académie des Sciences, along with the other learned

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html (3 of 8) [2/16/2002 11:19:16 PM]

Laplace

societies, was suppressed on 8 August. The weights and measures commission was the only one allowed to continue but soon Laplace, together with Lavoisier, Borda, Coulomb, Brisson and Delambre were thrown off the commission since all those on the committee had to be worthy:... by their Republican virtues and hatred of kings. Before the 1793 Reign of Terror Laplace together with his wife and two children left Paris and lived 50 km southeast of Paris. He did not return to Paris until after July 1794. Although Laplace managed to avoid the fate of some of his colleagues during the Revolution, such as Lavoisier who was guillotined in May 1794 while Laplace was out of Paris, he did have some difficult times. He was consulted, together with Lagrange and Laland, over the new calendar for the Revolution. Laplace knew well that the proposed scheme did not really work because the length of the proposed year did not fit with the astronomical data. However he was wise enough not to try to overrule political dogma with scientific facts. He also conformed, perhaps more happily, to the decisions regarding the metric division of angles into 100 subdivisions. In 1795 the Ecole Normale was founded with the aim of training school teachers and Laplace taught courses there including one on probability which he gave in 1795. The Ecole Normale survived for only four months for the 1200 pupils, who were training to become school teachers, found the level of teaching well beyond them. This is entirely understandable. Later Laplace wrote up the lectures of his course at the Ecole Normale as Essai philosophique sur les probabilités published in 1814. A review of the Essai states:... after a general introduction concerning the principles of probability theory, one finds a discussion of a host of applications, including those to games of chance, natural philosophy, the moral sciences, testimony, judicial decisions and mortality. In 1795 the Académie des Sciences was reopened as the Institut National des Sciences et des Arts. Also in 1795 the Bureau des Longitudes was founded with Lagrange and Laplace as the mathematicians among its founding members and Laplace went on to lead the Bureau and the Paris Observatory. However although some considered he did a fine job in these posts others criticised him for being too theoretical. Delambre wrote some years later:... never should one put a geometer at the head of an observatory; he will neglect all the observations except those needed for his formulas. Delambre also wrote concerning Laplace's leadership of the Bureau des Longitudes:One can reproach [Laplace] with the fact that in more than 20 years of existence the Bureau des Longitudes has not determined the position of a single star, or undertaken the preparation of the smallest catalogue. Laplace presented his famous nebular hypothesis in 1796 in Exposition du systeme du monde, which viewed the solar system as originating from the contracting and cooling of a large, flattened, and slowly rotating cloud of incandescent gas. The Exposition consisted of five books: the first was on the apparent motions of the celestial bodies, the motion of the sea, and also atmospheric refraction; the second was on the actual motion of the celestial bodies; the third was on force and momentum; the fourth was on the theory of universal gravitation and included an account of the motion of the sea and the shape of the Earth; the final book gave an historical account of astronomy and included his famous nebular hypothesis. Laplace states his philosophy of science in the Exposition as follows:If man were restricted to collecting facts the sciences were only a sterile nomenclature and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html (4 of 8) [2/16/2002 11:19:16 PM]

Laplace

he would never have known the great laws of nature. It is in comparing the phenomena with each other, in seeking to grasp their relationships, that he is led to discover these laws... In view of modern theories of impacts of comets on the Earth it is particularly interesting to see Laplace's remarkably modern view of this:... the small probability of collision of the Earth and a comet can become very great in adding over a long sequence of centuries. It is easy to picture the effects of this impact on the Earth. The axis and the motion of rotation have changed, the seas abandoning their old position..., a large part of men and animals drowned in this universal deluge, or destroyed by the violent tremor imparted to the terrestrial globe. Exposition du systeme du monde was written as a non-mathematical introduction to Laplace's most important work Traité du Mécanique Céleste whose first volume appeared three years later. Laplace had already discovered the invariability of planetary mean motions. In 1786 he had proved that the eccentricities and inclinations of planetary orbits to each other always remain small, constant, and self-correcting. These and many other of his earlier results formed the basis for his great work the Traité du Mécanique Céleste published in 5 volumes, the first two in 1799. The first volume of the Mécanique Céleste is divided into two books, the first on general laws of equilibrium and motion of solids and also fluids, while the second book is on the law of universal gravitation and the motions of the centres of gravity of the bodies in the solar system. The main mathematical approach here is the setting up of differential equations and solving them to describe the resulting motions. The second volume deals with mechanics applied to a study of the planets. In it Laplace included a study of the shape of the Earth which included a discussion of data obtained from several different expeditions, and Laplace applied his theory of errors to the results. Another topic studied here by Laplace was the theory of the tides but Airy, giving his own results nearly 50 years later, wrote:It would be useless to offer this theory in the same shape in which Laplace has given it; for that part of the Mécanique Céleste which contains the theory of tides is perhaps on the whole more obscure than any other part... In the Mécanique Céleste Laplace's equation appears but although we now name this equation after Laplace, it was in fact known before the time of Laplace. The Legendre functions also appear here and were known for many years as the Laplace coefficients. The Mécanique Céleste does not attribute many of the ideas to the work of others but Laplace was heavily influenced by Lagrange and by Legendre and used methods which they had developed with few references to the originators of the ideas. Under Napoleon Laplace was a member, then chancellor, of the Senate, and received the Legion of Honour in 1805. However Napoleon, in his memoirs written on St Hélène, says he removed Laplace from the office of Minister of the Interior, which he held in 1799, after only six weeks:... because he brought the spirit of the infinitely small into the government. Laplace became Count of the Empire in 1806 and he was named a marquis in 1817 after the restoration of the Bourbons. The first edition of Laplace's Théorie Analytique des Probabilités was published in 1812. This first edition was dedicated to Napoleon-le-Grand but, for obvious reason, the dedication was removed in later editions! The work consisted of two books and a second edition two years later saw an increase in the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html (5 of 8) [2/16/2002 11:19:16 PM]

Laplace

material by about an extra 30 per cent. The first book studies generating functions and also approximations to various expressions occurring in probability theory. The second book contains Laplace's definition of probability, Bayes's rule (so named by Poincaré many years later), and remarks on moral and mathematical expectation. The book continues with methods of finding probabilities of compound events when the probabilities of their simple components are known, then a discussion of the method of least squares, Buffon's needle problem, and inverse probability. Applications to mortality, life expectancy and the length of marriages are given and finally Laplace looks at moral expectation and probability in legal matters. Later editions of the Théorie Analytique des Probabilités also contains supplements which consider applications of probability to: errors in observations; the determination of the masses of Jupiter, Saturn and Uranus; triangulation methods in surveying; and problems of geodesy in particular the determination of the meridian of France. Much of this work was done by Laplace between 1817 and 1819 and appears in the 1820 edition of the Théorie Analytique. A rather less impressive fourth supplement, which returns to the first topic of generating functions, appeared with the 1825 edition. This final supplement was presented to the Institute by Laplace, who was 76 years old by this time, and by his son. We mentioned briefly above Laplace's first work on physics in 1780 which was outside the area of mechanics in which he contributed so much. Around 1804 Laplace seems to have developed an approach to physics which would be highly influential for some years. This is best explained by Laplace himself:... I have sought to establish that the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule, and that the consideration of these actions must serve as the basis of the mathematical theory of these phenomena. This approach to physics, attempting to explain everything from the forces acting locally between molecules, already was used by him in the fourth volume of the Mécanique Céleste which appeared in 1805. This volume contains a study of pressure and density, astronomical refraction, barometric pressure and the transmission of gravity based on this new philosophy of physics. It is worth remarking that it was a new approach, not because theories of molecules were new, but rather because it was applied to a much wider range of problems than any previous theory and, typically of Laplace, it was much more mathematical than any previous theories. Laplace's desire to take a leading role in physics led him to become a founder member of the Société d'Arcueil in around 1805. Together with the chemist Berthollet, he set up the Society which operated out of their homes in Arcueil which was south of Paris. Among the mathematicians who were members of this active group of scientists were Biot and Poisson. The group strongly advocated a mathematical approach to science with Laplace playing the leading role. This marks the height of Laplace's influence, dominant also in the Institute and having a powerful influence on the Ecole Polytechnique and the courses that the students studied there. After the publication of the fourth volume of the Mécanique Céleste, Laplace continued to apply his ideas of physics to other problems such as capillary action (1806-07), double refraction (1809), the velocity of sound (1816), the theory of heat, in particular the shape and rotation of the cooling Earth (1817-1820), and elastic fluids (1821). However during this period his dominant position in French science came to an end and others with different physical theories began to grow in importance. The Société d'Arcueil, after a few years of high activity, began to become less active with the meetings http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html (6 of 8) [2/16/2002 11:19:16 PM]

Laplace

becoming less regular around 1812. The meetings ended completely the following year. Arago, who had been a staunch member of the Society, began to favour the wave theory of light as proposed by Fresnel around 1815 which was directly opposed to the corpuscular theory which Laplace supported and developed. Many of Laplace's other physical theories were attacked, for instance his caloric theory of heat was at odds with the work of Petit and of Fourier. However, Laplace did not concede that his physical theories were wrong and kept his belief in fluids of heat and light, writing papers on these topics when over 70 years of age. At the time that his influence was decreasing, personal tragedy struck Laplace. His only daughter, Sophie-Suzanne, had married the Marquis de Portes and she died in childbirth in 1813. The child, however, survived and it is through her that there are descendants of Laplace. Laplace's son, Charles-Emile, lived to the age of 85 but had no children. Laplace had always changed his views with the changing political events of the time, modifying his opinions to fit in with the frequent political changes which were typical of this period. This way of behaving added to his success in the 1790s and 1800s but certainly did nothing for his personal relations with his colleagues who saw his changes of views as merely attempts to win favour. In 1814 Laplace supported the restoration of the Bourbon monarchy and caste his vote in the Senate against Napoleon. The Hundred Days were an embarrassment to him the following year and he conveniently left Paris for the critical period. After this he remained a supporter of the Bourbon monarchy and became unpopular in political circles. When he refused to sign the document of the French Academy supporting freedom of the press in 1826, he lost the remaining friends he had in politics. On the morning of Monday 5 March 1827 Laplace died. Few events would cause the Academy to cancel a meeting but they did on that day as a mark of respect for one of the greatest scientists of all time. Surprisingly there was no quick decision to fill the place left vacant on his death and the decision of the Academy in October 1827 not to fill the vacant place for another 6 months did not result in an appointment at that stage, some further months elapsing before Puissant was elected as Laplace's successor. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (76 books/articles)

Some Quotations (11)

A Poster of Pierre-Simon Laplace

Mathematicians born in the same country

Some pages from publications

The title page of Théorie analytique des probabilités (1812). Translation of a page of Essai philosophique sur les probabilités (1812).

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html (7 of 8) [2/16/2002 11:19:16 PM]

Laplace

Cross-references to History Topics

1. Orbits and gravitation 2. The fundamental theorem of algebra 3. General relativity 4. An overview of the history of mathematics 5. Matrices and determinants 6. An overview of Indian mathematics 7. Indian numerals

Cross-references to Famous Curves

Frequency Curve

Other references in MacTutor

1. Chronology: 1780 to 1800 2. Chronology: 1810 to 1820

Honours awarded to Pierre-Simon Laplace (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1789

Lunar features

Promontorium Laplace

Paris street names

Rue Laplace (5th Arrondissement)

Commemorated on the Eiffel Tower Other Web sites

1. The Catholic Encyclopedia 2. Rouse Ball 3. D Matthieu (A page about Beaumont en Auge) 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Laplace.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laplace.html (8 of 8) [2/16/2002 11:19:16 PM]

Larmor

Sir Joseph Larmor Born: 11 July 1857 in Magheragall, County Antrim, Ireland Died: 19 May 1942 in Holywood, County Down, Ireland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Larmor's father was Hugh Larmor and his mother was Anna Wright. Now Anna Wright was the daughter of Joseph Wright, and Joseph Larmor was named after his maternal grandfather. Hugh Larmor was a farmer at the time Joseph was born but he gave up farming when Joseph was around six or seven years old to become a trader with a grocer's shop in Belfast. Joseph was the eldest of a large family. By the time Joseph was of an age to attend school his parents had moved to Belfast so it was in that city that he attended the Royal Belfast Academical Institution. At this time he was [1]:A shy, delicate [and] precocious boy ... After leaving school Larmor continued his education in Belfast, studying for his B.A. and M.A. at Queen's University, Belfast. In 1877, having graduated from the Queen's University, he went to St John's College, Cambridge where he studied the mathematical Tripos. In 1880 he graduated as Senior Wrangler (the top First Class student) and he was first Smith's prizeman. It is interesting to note that J J Thomson, who like Larmor would make an important contribution to the understanding of the electron, was Second Wrangler (taking second place in the Tripos examinations to Larmor). After graduating Larmor was elected a Fellow of St John's College. Soon after this, still in 1880, he returned to Ireland when he was appointed as professor of Natural Philosophy at Queen's College, Galway. He spent five years, 1880 to 1885, teaching in Galway before he returned to St John's College Cambridge as a lecturer in 1885. He went on to become Lucasian Professor of Mathematics at Cambridge in 1903, the chair becoming vacant on the death of Stokes in February of that year. Larmor's contributions came at a time when there were major revolutions in physics with the passing of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Larmor.html (1 of 4) [2/16/2002 11:19:18 PM]

Larmor

classical physics to be replaced by quantum theory and relativity. His contributions can be seen as a bridge between the old and the new physics. In [4] and [5] Buchwald describes the impact of Larmor's major contributions. The main underlying idea was the principle of least action, which would be of fundamental importance in all Larmor's work throughout his career, and its implications were first set out by him in his paper Least action as the fundamental formulation in dynamics and physics which he published in the Proceedings of the London Mathematical Society in 1884. He published three papers all entitled A dynamical theory of the electric and luminiferous medium between 1894 and 1897. These papers presented his theory of the electron, which of course gained further weight in 1897 when J J Thomson experimentally identified the electron. Buchwald puts these papers by Larmor in context:Between 1873 and 1894 British and American physicists were proponents of a theory which they almost all learned directly from J C Maxwell's book Treatise on electricity and magnetism (1873). After 1897 only a few among them, including Heaviside, still adhered to that theory. During these three years (1894-97) the most basic principles of Maxwell's theory of electromagnetism were abandoned, and the entire subject was reconstructed on a new foundation - the electron - by Joseph Larmor in consultation with George FitzGerald. ... [He proposed that] the only source of charge is a particle, that the flow of such particles uniquely constitutes the current of conduction, and that the ether must be strictly separated from matter ... Warwick in [13] explains in detail how Larmor developed his theory. He summarises the process as follows:... Larmor initially tackled the problem of the earth's motion through physical optics and thermodynamics, but ... as he made contact with other Maxwellians beyond Cambridge especially with George FitzGerald - he came increasingly to make electromagnetic theory fundamental to his work. Indeed, following the introduction of the electron, he began to approach the problem of motion through the ether as one in the electrodynamics of moving bodies. In this specifically electromagnetic context, Larmor confronted the problem of the null result of the Michelson-Morley experiment, adopted the famous FitzGerald-Lorentz contraction hypothesis, and became the first physicist to employ what are now called the 'Lorentz' transformations. Larmor wrote Aether and Matter in 1900 (renamed by Lamb Aether and no matter ) which was a winning entry for the Adams Prize at Cambridge in 1898. It incorporated much of the work of the three major papers of 1894-1897 we referred to above. Warwick writes in [14]:His book of 1900, Aether and matter, Cambridge University Press, Cambridge, 1900, helped to establish a research school that guided the development of mathematical electromagnetic theory in Cambridge until the end of World War I. However Warwick [14] also writes:Today, however, Larmor is widely remembered by scientists for just two formulae and one theorem which, although correctly attributed to him, have been seen by historians of science as tangential to his main research interests. Indeed, none of the recent scholarly studies of Larmor's scientific work even mention the now famous formulae and theorem. We should take Warwick's lead and make sure that we mention those concepts to which Larmor's name is attached today. These are the 'Larmor precession', the 'Larmor frequency', 'Larmor's theorem' and 'Larmor's formula'. The first explains the splitting and polarisation of the spectral lines in a magnetic http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Larmor.html (2 of 4) [2/16/2002 11:19:18 PM]

Larmor

field. The Larmor frequency relates to electrons orbiting in a magnetic field and led him to postulate electrons as orbiting around some centre. He appears to have been the first to predict this behaviour. Larmor's theorem is a related result concerning how a certain transformation can negate the magnetic field for a charged particle subject to electric and magnetic fields. He was the first to calculate the rate of energy radiation from an accelerating electron and for this he gave Larmor's formula which gives the power radiated in terms of the electron's charge and acceleration. The formula breaks down for velocities close to the speed of light due to relativistic effects. When George Stokes and William Thomson (Lord Kelvin) died, Larmor acted as an editor for their complete works. He also brought out a new version of Henry Cavendish's works in 1921, Maxwell had been the editor for the original publication. Larmor also put considerable effort into writing obituaries of Stokes (1903), Gibbs (1905), and Thomson (1908). Larmor retired from the Lucasian Chair of Mathematics at Cambridge in 1932. He was succeeded in this position by Dirac. With his health deteriorating, Larmor returned to Ireland where he spent his final years at Holywood, County Down. He never married and was described by those close to him as [1]:... an unassuming, diffident man who did not readily form close friendships and whose numerous acts of generosity were performed without publicity. D'Arcy Thompson wrote:Larmor made few friends, perhaps; but while he lived, and they lived, he lost none. He became a member of the London Mathematical Society in 1884 and he contributed much to the Society being a council member from 1887 until 1912. During his period on the council he was vice president in 1890 and 1891 and also served as treasurer of the Society for over twenty years from 1892 until 1914. He was president of the Society in 1914 and, in the same year, he was awarded the De Morgan Medal by the Society. The Royal Society of London elected Larmor as a Fellow 1892 and he served as secretary from 1901 to 1912. The Royal Society awarded him its Royal Medal in 1915 and its Copley Medal in 1921. He was honoured by various universities who awarded him honorary degrees: Dublin, Oxford, Belfast, Glasgow, Aberdeen, Birmingham, St Andrews, Durham and Cambridge. He was also elected to membership of many learned societies including the Royal Irish Academy, the American Academy of Arts and Sciences, and the Accademia dei Lincei. Knighted in 1909, Larmor served as MP for the University of Cambridge from 1911 to 1922. He made his maiden speech in Parliament in 1912 when, not surprisingly given his background, he supported the Unionists in a debate on Irish home rule. His main contributions in Parliament, however, were to give strong support to universities in particular, and education in general. Larmor was active in college affairs, being a member of the council of St John's College for many years. In [8] a nice story is told of his involvement in College affairs:... he was conservative in temperament, questioning modern trends even in such matters as the installation of baths in the College (1920). "We have done without them for 400 years, why begin now?", he once said at a College meeting. Yet once the innovation were made he was a regular user. Morning by morning in a mackintosh and cap, in which he was not seen at other times, he found his way across the bridge to the New Court baths.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Larmor.html (3 of 4) [2/16/2002 11:19:18 PM]

Larmor

Article by: J J O'Connor and E F Robertson List of References (14 books/articles) A Poster of Joseph Larmor

Mathematicians born in the same country

Cross-references to History Topics

1. Special relativity 2. Chrystal and the RSE

Honours awarded to Joseph Larmor (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1892

Royal Society Copley Medal

Awarded 1921

Royal Society Royal Medal

Awarded 1915

Royal Society Bakerian lecturer

1909

London Maths Society President

1914 - 1916

LMS De Morgan Medal

Awarded 1914

Lucasian Professor of Mathematics

1903

Lunar features

Crater Larmor

Other Web sites

1. Bob Bruen 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Larmor.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Larmor.html (4 of 4) [2/16/2002 11:19:18 PM]

Lasker

Emanuel Lasker Born: 24 Dec 1868 in Berlinchen, Prussia (now Barlinek, Poland) Died: 11 Jan 1941 in New York, USA

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Emanuel Lasker studied mathematics and philosophy at several German universities and was a student of Hilbert's receiving a doctorate in 1902. In 1905 he introduced the notion of a primary ideal, which corresponds to an irreducible variety, and plays a role similar to prime powers in the prime decomposition of an integer. In 1894 Lasker became World Chess Champion and is considered by many to be the greatest chess player ever. He held the World Chess Championship until 1921. When Hitler came to power he was forced to leave Germany and had to come out of retirement in chess to make enough money to live. Gareth Williams, writing in Chess Monthly, describes Lasker's last few years: ... the Laskers were forced out of their comfortable retirement. The regime confiscated the Laskers' Berlin appartment, their farm at Thyrow and their lifetime savings. Emanuel and Martha Lasker, in their old age, suddenly found themselves destitute, without money home or homeland. In order to survive Lasker had once again to build a career in chess. The first tournament he was invited to after nine years retirement was Zurich. ..... Lasker was invited to Moscow in 1936 to participate in another great international tournament. ... The Laskers were encouraged to stay on in Moscow after the tournament and Dr Emanuel Lasker, mathematician, was invited to become a member of the Moscow Academy of Science. The offer was accepted and the Laskers took up permanent residence in Moscow. Emanuel

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lasker.html (1 of 2) [2/16/2002 11:19:19 PM]

Lasker

became absorbed with his mathematical studies at Moscow Academy. While on a visit to the USA in 1937 Martha Lasker took ill and they were advised not to travel. Lasker gave lectures and demonstrations over the next couple of years but, in 1939, during a lecture, he became dizzy. This was the start of an illness which slowly worsened until his death. Lasker, in addition to his algebraic results and his chess genius, also introduced a number of interesting mathematical games. For example he devised Laska, and produced an interesting modification to the rules of Nim. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles)

A Quotation

A Poster of Emanuel Lasker

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1900 to 1910

Other Web sites

1. Barnet Chess Club 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lasker.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lasker.html (2 of 2) [2/16/2002 11:19:19 PM]

Laurent_Hermann

Matthieu Paul Hermann Laurent Born: 2 Sept 1841 in Echternach, Luxembourg Died: 19 Feb 1908 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hermann Laurent's doctoral thesis at Nancy was on continuity of functions of a complex variable. He wrote 30 books and a fair number of papers on infinite series, equations, differential equations and geometry. He also developed statistical and interpolation formulas for calculating actuarial tables. His textbooks and his teaching of mathematics were particularly important. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Laurent_Hermann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laurent_Hermann.html [2/16/2002 11:19:20 PM]

Laurent_Pierre

Pierre Alphonse Laurent Born: 18 July 1813 in Paris, France Died: 2 Sept 1854 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Laurent was in the engineering corps and spent six years directing operations for the enlargement of the port of Le Havre. He submitted a work for the Grand Prize of 1842, unfortunately after the final date for submission. Cauchy reported on his work, which gives the Laurent series for a complex function, saying that it should be approved but it was not. After Laurent's death his widow arranged for two more of his memoirs to be presented to the Academy. One was never published, the second appeared in 1863. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Laurent_Pierre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Laurent_Pierre.html [2/16/2002 11:19:21 PM]

Lavanha

Joao Baptista Lavanha Born: 1550 in Portugal Died: 31 March 1624 in Madrid, Spain Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joao Baptista Lavanha is said to have studied in Rome. He was appointed by Philip II of Spain to be professor of mathematics in Madrid in 1582. Philip had sent the Duke of Alba with an army to conquer Portugal in 1580 and soon realised that Portugal was more advanced in studies of navigation than Spain. In an attempt to correct this, Philip founded an Academy of Mathematics in Madrid with Lavanha as its first professor. From 1587 Lavanha became chief engineer to Philip II. He was appointed cosmographer to the king in 1596 and about the same time he moved to Lisbon where he taught mathematics to sailors and navigators. Lavanha is best known for his contributions to navigation. His book Regimento nautico gives rules for determining latitude and tables of declination of the Sun. He also worked on maps, producing some interesting new ideas. He produced a map of Aragon in about 1615. Among his publications was a translation of Euclid. Lavanha also studied instruments used in navigation, constructing astrolabes, quadrants and compasses. He also devised a new navigational instrument. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lavanha.html (1 of 2) [2/16/2002 11:19:22 PM]

Lavanha

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lavanha.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lavanha.html (2 of 2) [2/16/2002 11:19:22 PM]

Lavrentev

Mikhail Alekseevich Lavrentev Born: 19 Nov 1900 in Kazan, Russia Died: 15 Oct 1980 in Moscow, Russia

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mikhail Lavrentev studied at the University of Moscow, graduating in 1922. His doctorate was awarded by Moscow University in 1933. From 1933 he held the chair of Analysis and Theory of Functions at Moscow State University. In 1934 the Steklov Mathematical Institute, together with some other institutions of the Academy of Sciences of the USSR, was moved from Leningrad to Moscow. Lavrentev headed the Department of the Theory of Functions of a Complex Variable at the Steklov Mathematical Institute. From 1939 to 1941 and then again from 1944 to 1949, Lavrentev was the Director of the Institute of Mathematics of the Academy of Sciences of the Ukraine in Kiev. In 1945 he was also appointed vice-president of the Academy of Sciences of the Ukraine. Then, in 1950, he became director of the Institute of Mechanics and Computational Technology of the Ukraine. Lavrentev moved to Novosibirsk when he was vice-president of the Academy of Sciences of the USSR between 1957 snd 1975. During this time he was Head of the Siberian Division of the Academy of Sciences of the USSR. Then, in 1975, Lavrentev returned to Moscow where he worked at the Institute of Physics and Technology until his death. Lavrentev is remembered for an outstanding book on conformal mappings and he made many important contributions to that topic. In the 1940s he developed the theory of quasi-conformal mappings which gave a new geometrical approach to partial differential equations. One of the major areas to which he applied this work was to hydrodynamics.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lavrentev.html (1 of 2) [2/16/2002 11:19:24 PM]

Lavrentev

The 1940s was a period of industrialisation and construction and, after 1945, Lavrentev founded new areas of research in mechanics and applied physics which were aimed at laying the theoretical foundation necessary for the large contruction projects of building dams, canals and bridges on the Volga, Dnieper and Don rivers. He also applied the theory of complex variables to other topics, in particular to non-linear waves. Other topics where he made substantial contributions where the theory of sets, the general theory of functions, and the theory of differential equations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Other Web sites

Akron (More pictures)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lavrentev.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lavrentev.html (2 of 2) [2/16/2002 11:19:24 PM]

Lax

Gaspar Lax Born: 1487 in Sariñena, Aragón, Spain Died: 23 Feb 1560 in Zaragoza, Spain Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

After studying in the University of Zaragoza, where he obtained a first degree, Gaspar Lax went to Paris where he took further degrees including a divinity degree taken at the Sorbonne. Lax remained in Paris and taught there at the College de Calvi and then transferred to the Collège de Montaigu where he taught and studied under Maior. In 1524 Lax left France and returned to Spain. He was appointed professor at the University of Zaragoza and remained there for the rest of his life. He did achieve certain higher posts within the University of Zaragoza, eventually becoming Vice Chancellor. Lax published several good mathematics books based on works of Boethius, Euclid, Jordanus and Campanus. He was also known as a philosopher, often called the Prince of Sophists. Perhaps Maior's influence was one of the reasons he studied logical subtleties. Lax also published Quaestiones phisicales in 1527. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lax.html (1 of 2) [2/16/2002 11:19:25 PM]

Lax

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lax.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lax.html (2 of 2) [2/16/2002 11:19:25 PM]

Le_Fevre

Jean Le Fèvre Born: 9 April 1652 in Lisieux, France Died: 1706 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean Le Fèvre is thought to have started work as a weaver. We know nothing of the first thirty years of Le Fèvre's life but some biographers suggest that he came from a humble background (which is almost certain to be true). The first definite mention of him in the scientific record is as a friend of an amateur astronomer by the name of Father Pierre. Father Pierre was Professor of rhetoric at the Collège de Lisieux in Paris and as such he was a colleague of Jean Picard and de La Hire. Picard needed help with carrying out calculations for his work Connaissances des temps and looked for an assistant. Father Pierre recommended his friend Le Fèvre to Picard as a possible assistant and, after Picard had tested Le Fèvre's abilities, he employed him to assist in the production of Connaissances des temps. Le Fèvre thus became an associate of Picard and de La Hire, and these two recommended him for membership of the Académie des Sciences shortly after this. Indeed Le Fèvre was elected to membership and began work with Picard on calculating astronomical tables. In October 1682 Jean Picard died. Le Fèvre continued with the task he had been set of calculating the tables, and he published Connaissances des temps during the period 1685 to 1701. There is some evidence that in fact he may have published an earlier edition during 1682 to 1684 but this is not certain. He also appears to have assisted de La Hire with the surveys of France that he was carrying out. Le Fèvre accused La Hire of stealing his astronomical tables after La Hire published Tabulae astronomicae in 1687. This was a strange episode for one might have expected Le Fèvre to be well disposed towards La Hire. Despite the ill-feeling which now grew up between these two men, nothing dramatic happened during the years from 1687 to 1700. Then in that year La Hire's son, Gabriel-Philippe de La Hire who had been commissioned by the Académie des Sciences to draw up new astronomical tables, published in Paris Ephemerides ad annum 1701. With this publication by Gabriel-Philippe de La Hire, Le Fèvre's anger exploded. It is likely that he felt that he should have been named the official publisher of ephemerides by the Académie des Sciences and he was angry that this commission had been given to Gabriel-Philippe de La Hire when he felt that this he should have been given to him. In 1701 Le Fèvre vented his anger by publishing a preface to

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Le_Fevre.html (1 of 2) [2/16/2002 11:19:26 PM]

Le_Fevre

Connaissances des temps which attacked both de La Hire and his son. The preface was replaced on government orders by one which praised both de La Hire and his son. Le Fèvre was excluded from the Academy soon after this, almost certainly because of his preface, but the official reason given was his failure to attend regularly meetings of the Académie des Sciences (which was indeed a rule of the Academy). Le Fèvre continued to publish ephemerides under the pseudonym J de Beaulieu, these appearing in 1702, 1703, and 1706. Lalande identifies J de Beaulieu as Charles Desforges, but McKeon in [1] argues convincingly that J de Beaulieu was Le Fèvre's pseudonym. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Le_Fevre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Le_Fevre.html (2 of 2) [2/16/2002 11:19:26 PM]

Le_Paige

Constantin Marie Le Paige Born: 9 March 1852 in Liège, Belgium Died: 26 Jan 1929 in Liège, Belgium

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Constantin Le Paige's secondary education was at Spa and at Liège. He entered the University of Liège to study mathematics in 1869 where he attended lectures by Catalan. After graduating in 1875 he began teaching at the University of Liège. He taught courses on the theory of determinants, which is not surprising since this was Catalan's speciality, and he also taught higher analysis. He was appointed professor at the University of Liège in 1882 and remained there the whole of his career, retiring in 1922. Le Paige worked on the theory of algebraic forms, a topic whose study was initiated by Boole in 1841 and then developed by Cayley, Sylvester, Hermite, Clebsch and Aronhold. In particular Le Paige studied the geometry of algebraic curves and surfaces, building on this earlier work. He is best known for his construction of a cubic surface given by 19 points. Le Paige studied the generation of plane cubic and quartic curves, developing further Chasles's work on plane algebraic curves and Steiner's results on the intersection of two projective pencils. The history of mathematics was another topic which interested Le Paige. He published Sluze's correspondence with Pascal, Huygens, Oldenburg and Wallis. In 1897 Le Paige was appointed director of the Institut d'Astrophysique de Cointe-Sclession. After this he wrote a number of astronomical texts. He also wrote on the history of mathematical notation in Sur l'origine de certains signes d'opération (1891). His interesting history of mathematics Notes pour servir

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Le_Paige.html (1 of 2) [2/16/2002 11:19:28 PM]

Le_Paige

à l'histoire des mathématique dans l'ancien pays de Liège contains much information on the Belgium astronomer Wendelin. Le Paige received many honours. He was elected to the Royal Academy of Sciences of Belgium in 1885, to the Royal Society of Sciences of Liège in 1878, to the Royal Society of Bohemia in 1881, to the Royal Academy of Sciences of Lisbon in 1883 and to the Mathematical Society of Amsterdam in 1886. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Le_Paige.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Le_Paige.html (2 of 2) [2/16/2002 11:19:28 PM]

Le_Tenneur

Jacques Alexandre Le Tenneur Born: 1610 in Paris, France Died: 1660 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Little is known of Jacques Le Tenneur (even the dates given for his birth and death are uncertain) except that he was a friend of Mersenne and that he corresponded with Gassendi. It is thought that he spent the first 30 or so years of his life in Paris where he was almost certainly educated. It is known for certain that by 1646 he was in Clermont in the Auvergne region of central France. Clermont was the town that Pascal was born in 23 years before. In 1651 he was counsellor to the provincial senate of Guyenne. At this time the Fronde, a civil war in France, was taking place and 1651 is the year Louis XIV lifted the siege of Cognac and assured the obedience of Guyenne. It is highly likely that Le Tenneur was involved with the political feuding of the Fronde. Le Tenneur's most important work De motu naturaliter accelerato was published in 1649 where he showed that he understood Galileo's arguments for free falling objects while Fabri and others did not. Most people at that time believed that the speed of a body in free fall was proportional to the distance it had fallen. Le Tenneur also published Traité des quantités incommensurables which is a work on the foundations of algebra. He clearly was trying to argue against the notions current at the time on using algebra to study geometry. He wished geometry to be Greek style, not in the style of Descartes and his followers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Le_Tenneur.html (1 of 2) [2/16/2002 11:19:29 PM]

Le_Tenneur

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Le_Tenneur.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Le_Tenneur.html (2 of 2) [2/16/2002 11:19:29 PM]

Le_Verrier

Urbain Jean Joseph Le Verrier Born: 11 March 1811 in Saint-Lô, France Died: 23 Sept 1877 in Paris, France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Urbain Le Verrier was appointed to teach astronomy at the Ecole Polytechnique in 1837 and abandoned his first subject of chemistry. He worked at the Paris Observatory for most of his life where his drive for efficiency was to made him very unpopular. His main work was in celestial mechanics. Working independently of Adams, Le Verrier calculated the position of Neptune from irregularities in Uranus's orbit. As one of his colleagues said:... he discovered a star with the tip of his pen, without any instruments other than the strength of his calculations alone. Le Verrier was better served by the German astronomer Galle (who found the planet in 1 hour) than Adams was by Airy who gave the task to Challis, the director of the Cambridge Observatory. He observed the planet first but did not recognise it. Arago, who had first suggested that Le Verrier work on this problem, said In the eyes of all impartial men, this discovery will remain one of the most magnificent triumphs of theoretical astronomy, one of the glories of the Académie and one of the most beautiful distinctions of our country. Le Verrier received many honours and widespread recognition for his achievement. The London Times carried the headline on the 1 October 1846:Le Verrier's planet found. He was awarded the Copley Medal of the Royal Society of London and, in France, became an officer in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Le_Verrier.html (1 of 3) [2/16/2002 11:19:30 PM]

Le_Verrier

the Legion of Honour. In 1854 Le Verrier became director of the Paris Observatory but his unpopularity, mentioned above, led to him being removed from the post in 1870. A successor was appointed but died in 1873. At this time Le Verrier again was given the post but his authority was severely restricted as he was supervised by a council. Le Verrier discovered of a discrepancy in the motion in the perihelion of Mercury in 1855, soon after his appointment as director of the Paris Observatory. The advance of the perihelion of Mercury by more than Newtonian theory predicted was to become important evidence for Einstein's general theory of relativity. Le Verrier, however, attributed this to a planet, which he called Vulcan, closer to the Sun than Mercury or to a second asteroid belt so close to the Sun as to be invisible. He spent much effort searching for asteroids inside the orbit of Mercury in an attempt to prove his theory but it was long after his death, in 1915, that Einstein's general theory of relativity explained the orbit of Mercury without the need for perturbing bodies. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) A Poster of Urbain Le Verrier

Mathematicians born in the same country

Cross-references to History Topics

1. Orbits and gravitation 2. Mathematical discovery of planets 3. General relativity

Honours awarded to Urbain Le Verrier (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1847

Royal Society Copley Medal

Awarded 1846

Fellow of the Royal Society of Edinburgh Lunar features

Crater Le Verrier

Paris street names

Rue Le Verrier (6th Arrondissement)

Commemorated on the Eiffel Tower Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Le_Verrier.html (2 of 3) [2/16/2002 11:19:30 PM]

Le_Verrier

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Le_Verrier.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Le_Verrier.html (3 of 3) [2/16/2002 11:19:30 PM]

Lebesgue

Henri Léon Lebesgue Born: 28 June 1875 in Beauvais, Oise, Picardie, France Died: 26 July 1941 in Paris, France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henri Lebesgue studied at Ecole Normale Supérieure. He taught in the Lycée at Nancy from 1899 to 1902. Building on the work of others, including that of the French mathematicians Emile Borel and Camille Jordan, Lebesgue formulated the theory of measure in 1901 and the following year he gave the definition of the Lebesgue integral that generalises the notion of the Riemann integral by extending the concept of the area below a curve to include many discontinuous functions. This is one of the achievements of modern analysis which greatly expands the scope of Fourier analysis. This outstanding piece of work appears in Lebesque's dissertation, Intégrale, longueur, aire, presented to the University of Nancy in 1902. In addition to about 50 papers he wrote two major books Leçons sur l'intégration et la recherché des fonctions primitives (1904) and Leçons sur les séries trigonométriques (1906). He also made major contributions in other areas of mathematics, including topology, potential theory, and Fourier analysis. In 1905 he gave a deep discussion of the various conditions Lipschitz and Jordan had used in order to ensure that f(x) is the sum of its Fourier series. He was appointed to the Sorbonne in 1910 but he did not concentrate on the field he had himself started. This was because his work was a striking generalisation, yet Lebesgue himself was fearful of generalisations. He wrote Reduced to general theories, mathematics would be a beautiful form without content. It would quickly die.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lebesgue.html (1 of 2) [2/16/2002 11:19:32 PM]

Lebesgue

Although future developments showed his fears to be groundless, they do allow us to understand the course his own work followed. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (16 books/articles)

A Quotation

A Poster of Henri Lebesgue

Mathematicians born in the same country

Cross-references to History Topics

The beginnings of set theory

Other references in MacTutor

Chronology: 1900 to 1910

Honours awarded to Henri Lebesgue (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1934

Lunar features

Crater Lebesgue

Other Web sites

1. Lebesgue Constants 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lebesgue.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lebesgue.html (2 of 2) [2/16/2002 11:19:32 PM]

Ledermann

Walter Ledermann Born: 18 March 1911 in Berlin, Germany

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Walter Ledermann was born in Berlin into a Jewish family. His father was William Ledermann and his mother was Charlotte Apt. He entered the Köllnnisches Gymnasium in Berlin in 1917, progressing to the Leibniz Gymnasium in the same city in 1920. There he learnt classics, studying Latin for nine years and Greek for six years. The school also taught French but, as was usual at this time, not much science. Although very little mathematics was taught in German schools in general, Walter had the advantage that the Leibniz Gymnasium taught more mathematics than other schools as a mark of respect for Leibniz after whom the school was named. He writes [2]:Although I was fond of the classics, especially Greek with its wonderful literature, I was fascinated by mathematics immediately after my first lesson at the age of eleven, and I decided there and then to make mathematics my career. In 1928, when he was seventeen years old, Ledermann graduated from the Leibniz Gymnasium and received the necessary certificate which allowed him to study at any German university. He remained in Berlin and entered the University there (now the Humboldt University) to study for the State Examination which was the qualification necessary to enter secondary school teaching. At the University of Berlin Ledermann was taught by many famous mathematicians including Schur, Schmidt, von Mises, Planck, Schrödinger, Hopf, Feigl, and others. As one might imagine given this array of famous names, he found it a stimulating experience. Ledermann was most influenced by Schur and, in [3], he describes attending Schur's lectures:Schur was a superb lecturer. His lectures were meticulously prepared... [and] were exceedingly popular. I remember attending his algebra course which was held in a lecture theatre filled with about 400 students. Sometimes, when I had to be content with a seat at the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ledermann.html (1 of 5) [2/16/2002 11:19:35 PM]

Ledermann

back of the lecture theatre, I used a pair of opera glasses to get at least a glimpse of the speaker. His main subjects were mathematics and physics, but he also had to study chemistry to a lower level and to take an oral examination on philosophy. Ledermann spent one semester in 1931 in Marburg but other than this all his courses were taken in Berlin. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. However, there was an exemption clause which exempted non-Aryans who had civil service appointments before the end of World War I. Schur qualified under this clause and this allowed him to keep his lecturing post in Berlin in 1933. He would examine Ledermann before the end of 1933. Ledermann was nearing the end of his studies for the State Examination when Hitler came to power and began passing the anti-Jewish legislation. In order to complete the course he had to write a dissertation and be given an oral examination. Schur gave him the topic for his dissertation On the various ways of expressing an orthogonal matrix in terms of parameters and conducted his oral examination in November 1933. At the oral he was examined by Schur and also by Bieberbach who was wearing Nazi uniform. It was quite clear to Ledermann that he had to leave Germany to escape the Nazi persecution of the Jews. He had already made many strenuous efforts to find a way to leave Germany before his oral. He won a scholarship from the International Student Service in Geneva to study at the University of St Andrews in Scotland and they supplied him with the necessary papers to allow him to travel to St Andrews in January 1934. Ledermann writes [5]:I have a deep affection for St Andrews. For it was the students and the people of St Andrews who saved my life by helping me to escape from the Nazis. It became my home in a real sense. At first it appeared that he might get caught up in the University of St Andrews' regulations. The problem was that the State Examination was just that, in that it was awarded by the Ministry of Education, not by the University of Berlin. Therefore it was not a degree but a diploma. Fortunately Ledermann was spared the stupidity of having to take the undergraduate courses at St Andrews, and, as Ledermann writes in [2]:... for the first time in the five hundred years history of the university a person with a German state examination was admitted as a research student. Ledermann's doctoral studies were supervised by Turnbull and he was awarded his Ph.D. in 1936. Under Turnbull, Ledermann studied the problem of finding the canonical form for a pair of real or complex n n matrices under simultaneous equivalence. He also worked on the problem of classifying the stabiliser of the pencil which is a linear combination of the two matrices. In 1937 he became a temporary lecturer in Dundee but the most fruitful work he undertook during this period was as a private assistant to Professor Sir Godfrey Thomson at the University of Edinburgh. Thomson headed the Moray House Group at Edinburgh which was undertaking research into intelligence testing. Ledermann was able to use his expert knowledge of matrix theory to put the work of this group http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ledermann.html (2 of 5) [2/16/2002 11:19:35 PM]

Ledermann

onto a sound mathematical footing. As well as matrix theory he was involved in using statistical methods and he retained this interest in his later research publications. The quality of the work Ledermann undertook at this time is clearly shown from the fact that Edinburgh awarded him a D.Sc. for it in 1940. While in Edinburgh Ledermann also worked with Max Born and A C Aitken. In 1938 Ledermann returned to St Andrews. The letter [6] from Turnbull to Ledermann offers him a one year post as an assistant, capable of renewal, at St Andrews for the academic year 1938-39 at a salary of £350. Turnbull writes [6]:It will involve from 8 to 10 lectures or tutorials a week in mathematics pure and applied; and the Court understand that I have looked out for one who is competent to teach particularly in Analysis and Applied Mathematics. Your special knowledge of the applications of mathematics to problems involving matrices and statistics is an advantage; but you will realise, Walter, in view of the needs of this Department, that proficiency in teaching Applied Mathematics and the functions connected therewith will be important. In fact Ledermann remained at St Andrews from 1938 until 1946. During this time he became a British citizen (1940) and also undertook some war work. In [5] he describes his association with Freundlich during this period:... Freundlich was very close to me. He was a fatherly friend of whom I have many fond memories, most especially because he introduced me to my wife [Ruth Stefanie (Rushi) Stadler], whom I married in 1946. During the war Freundlich and I taught navigation at the Initial Training Wing of the RAF which was stationed in St Andrews. We also published a joint paper in the Monthly Notices of the Royal Astronomical Society in 1944 ... . We had other interests in common apart from mathematics: Freundlich was a keen cellist, and we frequently played chamber music where I played the violin or viola. Once we went on holiday together to the West coast of Scotland, when Mrs Freundlich was unable to come. ... he was a tall impressive man, and when we walked side by side through the streets of St Andrews people would say: "Here come the Sun and Moon". Ledermann's wife Ruth (known as Rushi) was a psychotherapist and the newly married couple decided to move to a bigger city so that Ruth might be able to pursue her work. Ledermann accepted a lectureship at the University of Manchester in 1946. He writes [4]:The sixteen years we spent in Manchester were stimulating and fruitful. Particularly of note during his time in Manchester was the fact that he was secretary to the first British Mathematical Colloquium which he organised in Manchester at the request of Hodge, Henry Whitehead, and Max Newman. By the time he left Manchester, Ledermann had been promoted to a Senior lecturer. In 1962 he accepted a readership at the newly established University of Sussex and three years later he was promoted to professor. He retired in 1978 and was made Emeritus Professor. For many years after retiring he continued to teach at Sussex, giving both tutorials and seminars. As well as work in matrix theory which we have commented on above, Ledermann was especially known for his work in homology theory, group theory, and number theory. As a result of Schur's teaching he [4]:... developed a liking for "concrete" mathematics and a distaste for "abstraction" for its own http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ledermann.html (3 of 5) [2/16/2002 11:19:35 PM]

Ledermann

sake. This is evident, even in his work in what is usually thought of as one of the most abstract of topics, homology theory. First I [EFR] will make some comments on Ledermann's book Introduction to the Theory of Finite Groups (1949). It is the book from which I learnt group theory and, although it was not the only influence on my choice of research topic, it was a major factor in my decision to work on group theory problems for my doctoral dissertation. The topics covered in the book look fairly standard but one has to remember that in the 1940s there were few group theory texts and the concept of standard material for such courses did not exit. The little book (152 pages) discusses the group axioms, isomorphisms, cyclic groups, coset decompositions, Lagrange's theorem, permutation groups, normal subgroups, quotient groups, homomorphisms, the first and second isomorphism theorems, and the Jordan-Hölder theorem. The simplicity of the alternating groups is proved and the Sylow theorems, p-groups and finitely generated abelian groups are discussed. Ledermann succeeds admirably in meeting his own aims in that he:... never hesitated to sacrifice completeness for breadth or to reject more modern methods when [he] considered alternative presentations to be more intelligible. Other books which Ledermann has written for undergraduates include Complex numbers (1960), Integral calculus (1964), Multiple integrals (1966), Introduction to group theory (1973), and Introduction to group characters (1977). This last volume, which still shows Schur's influence, strikes a good balance between the abstract approach to representation theory emphasising modules, and the concrete approach built around matrices. It is an outstanding text from which to teach the topic. Among the editorial work undertaken by Ledermann is his editorship of the Journal of the London Mathematical Society (1968-71) and of the Bulletin of the London Mathematical Society (1974-77). He is the chief editor of the Handbook of Applicable Mathematics which consists of nine volumes, an index volume and a number of guide books. This project, again very much in line with Ledermann's approach to mathematics, is designed for the "professional adults" who:... find themselves needing to understand a particular mathematical idea... will then be able to turn to the appropriate article in the core volume ... and find out just what they want to know. Ledermann has received honours for his work which include election to the Royal Society of Edinburgh (1944) and an honorary doctorate from the Open University (1993). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

Ledermann's St Andrews interview

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ledermann.html (4 of 5) [2/16/2002 11:19:35 PM]

Ledermann

Other references in MacTutor

Some anecdotes about Ledermann's book, Introduction to the theory of finite groups are at this link

Honours awarded to Walter Ledermann (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ledermann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ledermann.html (5 of 5) [2/16/2002 11:19:35 PM]

Leech

John Leech Born: 21 July 1926 in Weybridge, Surrey, England Died: 28 Sept 1992 in At sea between Rothesay and Largs, Scotland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Leech was educated at Trent College in Derbyshire. He entered King's College Cambridge, graduating as a wrangler with a B.A. in 1950. After graduating Leech was appointed to a post with Ferranti in Manchester where he worked on the construction of an early digital computer. In 1954 Leech left Ferranti to return to Cambridge, becoming a research student in the mathematical laboratory. He was appointed as a lecturer in the Computing Laboratory of Glasgow University in 1959. He spent the academic year 1967/68 as a research fellow at the Atlas Computer Laboratory near Harwell in England. In 1968 a new university was created in Stirling, 40 km from Glasgow. Leech was appointed as Reader and first Head of Computing Science at Stirling. Two years later he was promoted to a Personal Chair, the first awarded by the University of Stirling. He took early retirement in 1980, having worked part-time for a few years before this due to ill-health. Douglas Munn, writing in [2], describes Leech's mathematical work as follows:By inclination he was a pure mathematician, with a taste for number theory, geometry and combinatorial group theory, his interests tending towards the particular rather than the general. ... he developed one of the first programs to implement the Todd-Coxeter coset enumeration algorithm - a pioneering achievement in the application of computers to algebra. Leech is, however, best known for the Leech lattice which gives rise to three sporadic simple groups. In 1964 he published a paper on sphere packing in eight or more dimensions. It contained a lattice packing http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leech.html (1 of 3) [2/16/2002 11:19:36 PM]

Leech

in 24 dimensions. In 1965 he submitted a supplement to the paper giving a packing in 24 dimensions with a lattice now known as the Leech lattice. A few months later he found a packing of twice the density and his paper was rewritten appearing in 1967. Leech knew that the symmetry group would be interesting, and he worked on it for some time giving a lower bound for its order (which later proved to be the actual order of the group). Knowing that he did not have the group theory skills necessary to prove his conjectures he tried to interest others, see [1]:I dangled the problem under various noses, including those of Coxeter, Todd, and Graham Higman, but Conway was the first to swallow the bait... A detailed description of this discovery is given in [1]. A few weeks before his death, Leech visited us in St Andrews on each day for about a week and four of us worked on a number of problems which had been left unsolved by Leech 30 years before. To his great delight we succeeded in solving some (but not all) of the problems. It came as a great shock to me [EFR] to hear of his death. In [3] this is described as follows:One of the most avid supporters of the preservation of the paddle steamer Waverley, John Leech ... collapsed and died on board the ship during the final cruise of the season. ... He was standing between the funnels on the return leg ... between Rothsay and Largs when he suffered a heart attack. As the steamer made its way back up river to Glasgow, the red Ensign was lowered to half mast as a tribute... Leech died almost exactly one month after Gorenstein who had overseen the classification of finite simple groups. The three sporadic groups which Conway deduced from Leech's lattice play an important role in the classification. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of John Leech

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leech.html (2 of 3) [2/16/2002 11:19:36 PM]

Leech

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Leech.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leech.html (3 of 3) [2/16/2002 11:19:36 PM]

Lefschetz

Solomon Lefschetz Born: 3 Sept 1884 in Moscow, Russia Died: 5 Oct 1972 in Princeton, New Jersey, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Solomon Lefschetz was a Russian born, Jewish mathematician who was the main source of the algebraic aspects of topology. His father Alexander Lefschetz and his mother Vera were both Turkish citizens but since Alexander Lefschetz worked as an importer he was required to travel a great deal. As a consequence the family decided to make a base for themselves in France where their children could be educated. Shortly after Solomon was born his family set up home in Paris. Since Lefschetz was educated in France from a young age, French was his first language. He trained to be an engineer at the Ecole Centrale in Paris from 1902 to 1905 and there attended lectures by Emile Picard and Paul Appell. However, not being a French citizen he would have found great difficulty obtaining an academic post in France. Fully understanding this, in November 1905 at the age of 21, Lefschetz went to the United States. For a few months he worked at the Baldwin Locomotive works, then from 1907 to 1910 he worked for Westinghouse Electric Company in Pittsburgh. He had the misfortune to lose both his hands in a laboratory accident in November 1907 when they were burnt off in a transformer explosion. He also lost his forearms and spent a while in hospital. As one might imagine this accident had a major mental impact on Lefschetz in addition to the physical disability he suffered. The result was a deep depression, but the tragedy eventually pushed him towards mathematics, which was the right subject for him and, of course, it was fortunate for topology that he found his true love of mathematics. After teaching mathematics to apprentices at the Westinghouse Electric Company in 1910, he enrolled as a doctoral student in mathematics at Clark University in Worcester, Massachusetts where he was a fellow from 1910 to 1911. While on the graduate program at Clark, Lefschetz met one of the mathematics students Alice Berg Hayes. They married on 3 July 1913, a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lefschetz.html (1 of 5) [2/16/2002 11:19:38 PM]

Lefschetz

year after he became an American citizen on 17 June 1912. Alice [1]:... helped him to overcome his handicap, encouraging him in his work and moderating his combative ebullience. They had no children. Lefschetz received his Ph.D. in mathematics in 1911 with a thesis on algebraic geometry entitled On the existence of loci with given singularities. That same year, 1911, Lefschetz was appointed an instructor in mathematics at the University of Nebraska in Lincoln, then, two years later, he was appointed to the University of Kansas in Lawrence. There he was promoted to assistant professor in 1916, to associate professor in 1919 and full professor in 1923. During these years he wrote a series of important papers on topology despite being out the mainstream of mathematical research. Lefschetz wrote of the importance of these years in his mathematical development:My years in the west with totally hermetic isolation played in my development the role of 'a job in a lighthouse' which Einstein would have every young scientist assume so that he may develop his own ideas in his own way. His most important results from this period are contained in On certain numerical invariants of algebraic varieties with application to abelian varieties, which he had published in the Transactions of the American Mathematical Society in 1921, and in his famous monograph of 1924 L'analysis situs et la géométrie algébrique. Lefschetz explained with his famous quote:It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry. Hodge wrote (see [9] or [10]):Our greatest debt to Lefschetz lies in the fact that he showed us that a study of topology was essential for all algebraic geometers. Poincaré had studied curves on a surface but Lefschetz pushed the ideas into much more general settings by building a theory of subvarieties of an algebraic variety. For his remarkable contributions during this period he was awarded the Prix Bordin by the Académie des Sciences in Paris in 1919 and the Bôcher Memorial Prize from the American Mathematical Society in 1923 for his 1921 paper we mentioned above. In Continuous transformations of manifolds published in the Proceedings of the U.S. National Academy of Sciences in 1923 Lefschetz announced that he had:... new and far reaching methods... for investigating continuous maps of manifolds and, in particular, their fixed points. He published his fixed point theorem for compact orientable manifolds in 1923, giving a fuller account of his famous fixed point theorem in Intersections and transformations of complexes and manifolds published in the Transactions of the American Mathematical Society in 1926. In this paper he promised more than the:... far reaching theory of the intersection of complexes on a manifold. He wrote:With suitable restrictions the formulas derived are susceptible of extension to a wider range of manifolds, but this will be reserved for a later occasion. In 1927 he fulfilled his promise extending his fixed point theorems to manifolds with boundary, and by http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lefschetz.html (2 of 5) [2/16/2002 11:19:38 PM]

Lefschetz

this stage Brouwer's fixed point theorem became a special case. This paper is more than an extension, however, for in it he used matrices, in particular the trace of a matrix, to greatly simplify the formulas he had presented in his 1926 paper. He did further work on fixed point theorems studying the case of any finite complex in 1927 and any locally connected space in 1936. On Alexander's recommendation, in 1924 Lefschetz went to Princeton as a visiting professor for a year. At the end of his visit in 1925 he was offered a permanent post at Princeton as associate professor which he happily accepted. He became Henry Fine research professor in 1933 filling the position left vacant when Veblen moved from Princeton University to the Institute for Advanced Study. Sylvia Nasar writes [16]:When he first came to Princeton in the 1920s, he often said he was an "invisible man". He was one of the first Jews on the faculty, loud, rude, and badly dressed to boot. People pretended not to see him in the hallways and gave him a wide berth at faculty parties. But Lefschetz had overcome far more formidable obstacles in his life than a bunch of prissy Wasp snobs. Lefschetz worked on results which provided a deep generalisation of Emile Picard's theorems in function theory to several complex variables. Lefschetz was able to go further than Emile Picard and incorporate Poincaré's ideas. In doing this he developed a theory of algebraic topology of algebraic varieties of higher dimension. The word 'topology' comes from the title of a monograph written by Lefschetz in 1930. Another text which would have a huge influence on the development of the field was Algebraic topology which was published in 1942. In the course of his work he introduced many of what would be considered today the basic tools of algebraic topology [1]:He made extensive use of product spaces; he developed intersection theory, including the theory of the intersection ring of a manifold; and he made essential contributions to various kinds of homology theory, notably relative homology, singular homology, and cohomology. Lefschetz had two artificial hands over which he always wore a shiny black glove. First thing every morning a graduate student had to push a piece of chalk into his hand and remove it at the end of the day. The students at Princeton made up a ditty about Lefschetz:Here's to Lefschetz, Solomon L. Irrepressible as hell When he's at last beneath the sod He'll then begin to heckle God. Saunders MacLane has added that:In my experience, Lefschetz was both obstreperous and enthusiastic - about research in mathematics. For Lefschetz, independent thinking and originality were what mattered in mathematical research. Unlike most mathematicians he had no respect for elegance and if something was to him clearly true, he would consider it at best a waste of time producing a rigorous argument to verify it. When a student proudly showed him a clever argument that he had produced to give a short proof of one of Lefschetz's theorems, rather than compliment the student, he is claimed to have retorted:Don't come to me with your pretty proofs. We don't bother with that baby stuff around here. Even if there is little truth in a joke which circulated about Lefschetz, namely that he never wrote a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lefschetz.html (3 of 5) [2/16/2002 11:19:38 PM]

Lefschetz

correct proof or stated an incorrect theorem, there is an underlying truth in it reflecting on his style of mathematics. Sylvia Nasar gives this vivid description of the impact Lefschetz had on Princeton [16]:Entrepreneurial and energetic, Lefschetz was the supercharged human locomotive that ... pulled the Princeton department out of genteel mediocrity right to the top. He recruited mathematicians with only one criterion in mind: research. His high-handed and idiosyncratic editorial policies made the Annals of Mathematics, Princeton's once-tired monthly, into the most revered mathematical journal in the world. He was sometimes accused of caving in to anti-Semitism for refusing to admit many Jewish students (his rationale being that nobody would hire them when they completed their degrees), but no one denies that he had brilliant snap judgement. He exhorted, bossed, and bullied, but with the aim of making the department great and turning his students into real mathematicians, tough like himself. He was the editor of the Annals of Mathematics from 1928 to 1958, bringing it up to the standard of one of the very best world class journals. Steenrod wrote:The importance to American mathematicians of a first-class journal is that it sets high standards for them to aim at. In this somewhat indirect manner, Lefschetz profoundly affected the development of mathematics in the United States. Lefschetz was a strong supporter of the American Mathematical Society serving as its President from 1935 to 1936. During World War II, Lefschetz undertook applied mathematical work directed at the war effort. At the same time he heard of work being undertaken in the Soviet Union on applications of modern mathematical tools to applied mathematical problems. In 1944 these two influences inspired Lefschetz to return to his interest in engineering but now he had deep mathematical skills which he could bring to bear on the problems. He tackled problems related to dissipative nonlinear ordinary differential equations but did not take the usual approach of using linear theory to tackle nonlinear differential equations. Although initially his work was rather concrete, over the years it became more abstract as Lefschetz developed it further. In the end the ideas he developed became the foundation of a new branch of mathematics, namely global analysis. Lefschetz had many students working in this area and, between 1950 and 1960, a series of important publications Contributions to the theory of nonlinear oscillations appeared in the Annals of Mathematics Studies, published by Princeton University Press. From this work by Lefschetz's school, came the two important concepts of structural stability and genericity. Lefschetz did remarkable work even well into his 80's. In his last book, he wrote on recent ideas outside his own area, namely the topology of Feynman integrals. During the 1920s and 1930s Lefschetz was able to indulge his love of travel with many trips to European countries. In particular he loved to visit France, Italy and the Soviet Union. However the outbreak of World War II made European travel virtually impossible so Lefschetz had to find new places to visit. He chose Mexico, visiting the National University of Mexico for the first time in 1944. It was the first of many visits and he eventually fell into the habit of spending the summer months there every year. His contribution to mathematics in Mexico was, as a result of these visits, of major importance and he helped build a flourishing school there. His contributions were recognised when he was awarded the Order of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lefschetz.html (4 of 5) [2/16/2002 11:19:38 PM]

Lefschetz

the Aztec Eagle in 1964. This was one of many honours which were awarded to Lefschetz. He received the Antonio Feltrinelli International Prize from the Accademia Nazionale dei Lincei in 1956. He was awarded honorary degrees from the universities of Paris, Prague, Mexico, Clark, Brown, and Princeton. In 1964 he received the National Medal of Science:... for indomitable leadership in developing mathematics and training mathematicians, for fundamental publications in algebraic geometry and topology, and for stimulating needed research in nonlinear control processes. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (18 books/articles)

A Quotation

A Poster of Solomon Lefschetz

Mathematicians born in the same country

Honours awarded to Solomon Lefschetz (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1961

American Maths Society President

1935 - 1936

AMS Colloquium Lecturer

1930

AMS Bôcher Prize

Awarded 1924

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lefschetz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lefschetz.html (5 of 5) [2/16/2002 11:19:38 PM]

Legendre

Adrien-Marie Legendre Born: 18 Sept 1752 in Paris, France Died: 10 Jan 1833 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adrien-Marie Legendre would perhaps have disliked the fact that this article contains details of his life for Poisson wrote of him in [12]:Our colleague has often expressed the desire that, in speaking of him, it would only be the matter of his works, which are, in fact, his entire life. It is not surprising that, given these views of Legendre, there are few details of his early life. We have given his place of birth as Paris, as given in [1] and [2], but there is some evidence to suggest that he was born in Toulouse and the family moved to Paris when he was very young. He certainly came from a wealthy family and he was given a top quality education in mathematics and physics at the Collège Mazarin in Paris. In 1770, at the age of 18, Legendre defended his thesis in mathematics and physics at the Collège Mazarin but this was not quite as grand an achievement as it sounds to us today, for this consisted more of a plan of research rather than a completed thesis. In the thesis he listed the literature that he would study and the results that he would be aiming to prove. With no need for employment to support himself, Legendre lived in Paris and concentrated on research. From 1775 to 1780 he taught with Laplace at Ecole Militaire where his appointment was made on the advice of d'Alembert. He then decided to enter for the 1782 prize on projectiles offered by the Berlin Academy. The actual task was stated as follows:Determine the curve described by cannonballs and bombs, taking into consideration the resistance of the air; give rules for obtaining the ranges corresponding to different initial

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Legendre.html (1 of 5) [2/16/2002 11:19:40 PM]

Legendre

velocities and to different angles of projection. His essay Recherches sur la trajectoire des projectiles dans les milieux résistants won the prize and launched Legendre on his research career. In 1782 Lagrange was Director of Mathematics at the Academy in Berlin and this brought Legendre to his attention. He wrote to Laplace asking for more information about the prize winning young mathematician. Legendre next studied the attraction of ellipsoids. He gave a proof of a result due to Maclaurin, that the attractions at an external point lying on the principal axis of two confocal ellipsoids was proportional to their masses. He then introduced what we call today the Legendre functions and used these to determine, using power series, the attraction of an ellipsoid at any exterior point. Legendre submitted his results to the Académie des Sciences in Paris in January 1783 and these were highly praised by Laplace in his report delivered to the Académie in March. Within a few days, on 30 March, Legendre was appointed an adjoint in the Académie des Sciences filling the place which had become vacant when Laplace was promoted from adjoint to associé earlier that year. Over the next few years Legendre published work in a number of areas. In particular he published on celestial mechanics with papers such as Recherches sur la figure des planètes in 1784 which contains the Legendre polynomials; number theory with, for example, Recherches d'analyse indéterminée in 1785; and the theory of elliptic functions with papers on integrations by elliptic arcs in 1786. The 1785 paper on number theory contains a number of important results such as the law of quadratic reciprocity for residues and the results that every arithmetic series with the first term coprime to the common difference contains an infinite number of primes. Of course today we attribute the law of quadratic reciprocity to Gauss and the theorem concerning primes in an arithmetic progression to Dirichlet. This is fair since Legendre's proof of quadratic reciprocity was unsatisfactory, while he offered no proof of the theorem on primes in an arithmetic progression. However, these two results of are great importance and credit should go to Legendre for his work on them, although he was not the first to state the law of quadratic reciprocity since it occurs in Euler's work of 1751 and also of 1783 (see [15]). Legendre's career in the Académie des Sciences progressed in a satisfactory manner. He became an associé in 1785 and then in 1787 he was a member of the team whose task it was to work with the Royal Observatory at Greenwich in London on measurements of the Earth involving a triangulation survey between the Paris and Greenwich observatories. This work resulted in his election to the Royal Society of London in 1787 and also to an important publication Mémoire sur les opérations trigonométriques dont les résultats dépendent de la figure de la terre which contains Legendre's theorem on spherical triangles. On 13 May 1791 Legendre became a member of the committee of the Académie des Sciences with the task to standardise weights and measures. The committee worked on the metric system and undertook the necessary astronomical observations and triangulations necessary to compute the length of the metre. At this time Legendre was also working on his major text Eléments de géométrie which he had been encouraged to write by Condorcet. However the Académie des Sciences was closed due to the Revolution in 1793 and Legendre had special difficulties since he lost the capital which provided him with a comfortable income. He later wrote to Jacobi explaining his personal circumstances around this time (see [1]):-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Legendre.html (2 of 5) [2/16/2002 11:19:40 PM]

Legendre

I married following a bloody revolution that had destroyed my small fortune; we had great problems and some very difficult moments, but my wife staunchly helped me to put my affairs in order little by little and gave me the tranquillity necessary for my customary work and for writing new works which have steadily increased my reputation. Following the work of the committee on the decimal system on which Legendre had served, de Prony in 1792 began a major task of producing logarithmic and trigonometric tables, the Cadastre. Legendre and de Prony headed the mathematical section of this project along with Carnot and other mathematicians. They had between 70 to 80 assistants and the work was undertaken over a period of years, being completed in 1801. In 1794 Legendre published Eléments de géométrie which was the leading elementary text on the topic for around 100 years. The work is described in [2]:In his "Eléments" Legendre greatly rearranged and simplified many of the propositions from Euclid's "Elements" to create a more effective textbook. Legendre's work replaced Euclid's "Elements" as a textbook in most of Europe and, in succeeding translations, in the United States and became the prototype of later geometry texts. In "Eléments" Legendre gave a simple proof that is irrational, as well as the first proof that 2 is irrational, and conjectured that is not the root of any algebraic equation of finite degree with rational coefficients. In 1795 the Académie des Sciences was reopened as the Institut National des Sciences et des Arts and from then until 1806 it met in the Louvre. Each section of the Institut contained six places, and Legendre was one of the six in the mathematics section. In 1803 Napoleon reorganised the Institut and a geometry section was created and Legendre was put into this section. Legendre published a book on determining the orbits of comets in 1806. In this he wrote:I have thought that what there was better to do in the problem of comets was to start out from the immediate data of observation, and to use all means to simplify as much as possible the formulas and the equations which serve to determine the elements of the orbit. His method involved three observations taken at equal intervals and he assumed that the comet followed a parabolic path so that he ended up with more equations than there were unknowns. He applied his methods to the data known for two comets. In an appendix Legendre gave the least squares method of fitting a curve to the data available. However, Gauss published his version of the least squares method in 1809 and, while acknowledging that it appeared in Legendre's book, Gauss still claimed priority for himself. This greatly hurt Legendre who fought for many years to have his priority recognised. In 1808 Legendre published a second edition of his Théorie des nombres which was a considerable improvement on the first edition of 1798. For example Gauss had proved the law of quadratic reciprocity in 1801 after making critical remarks about Legendre's proof of 1785 and Legendre's much improved proof of 1798 in the first edition of Théorie des nombres. Gauss was correct, but one could understand how hurtful Legendre must have found an attack on the rigour of his results by such a young man. Of course Gauss did not state that he was improving Legendre's result but rather claimed the result for himself since his was the first completely rigorous proof. Legendre later wrote (see [20]):This excessive impudence is unbelievable in a man who has sufficient personal merit not to have need of appropriating the discoveries of others. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Legendre.html (3 of 5) [2/16/2002 11:19:40 PM]

Legendre

To his credit Legendre used Gauss's proof of quadratic reciprocity in the 1808 edition of Théorie des nombres giving proper credit to Gauss. The 1808 edition of Théorie des nombres also contained Legendre's estimate for (n) the number of primes n of (n) = n/(log(n) - 1.08366). Again Gauss would claim that he had obtained the law for the asymptotic distribution of primes before Legendre, but certainly it was Legendre who first brought these ideas to the attention of mathematicians. Legendre's major work on elliptic functions in Exercises du Calcul Intégral appeared in three volumes in 1811, 1817, and 1819. In the first volume Legendre introduced basic properties of elliptic integrals and also of beta and gamma functions. More results on beta and gamma functions appeared in the second volume together with applications of his results to mechanics, the rotation of the Earth, the attraction of ellipsoids and other problems. The third volume was largely devoted to tables of elliptic integrals. In November 1824 he decided to reprint a new edition but he was not happy with this work by September 1825 publication began of his new work Traité des Fonctions Elliptiques again in three volumes of 1825, 1826, and 1830. This new work covered similar material to the original but the material was completely reorganised. However, despite spending 40 years working on elliptic functions, Legendre never gained the insight of Jacobi and Abel and the independent work of these two mathematicians was making Legendre's new three volume work obsolete almost as soon as it was published. Legendre's attempt to prove the parallel postulate extended over 30 years. However as stated in [1] his attempts:... all failed because he always relied, in the last analysis, on propositions that were "evident" from the Euclidean point of view. In 1832 (the year Bolyai published his work on non-euclidean geometry) Legendre confirmed his absolute belief in Euclidean space when he wrote:It is nevertheless certain that the theorem on the sum of the three angles of the triangle should be considered one of those fundamental truths that are impossible to contest and that are an enduring example of mathematical certitude. In 1824 Legendre refused to vote for the government's candidate for the Institut National. Abel wrote in October 1826:Legendre is an extremely amiable man, but unfortunately as old as the stones. As a result of Legendre's refusal to vote for the government's candidate in 1824 his pension was stopped and he died in poverty. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (20 books/articles)

A Quotation

A Poster of Adrien-Marie Legendre

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Legendre.html (4 of 5) [2/16/2002 11:19:40 PM]

Legendre

Cross-references to History Topics

1. Fermat's last theorem 2. Non-Euclidean geometry 3. Prime numbers

Other references in MacTutor

1. Prime Number Theorem 2. Legendre's estimates for the density of primes 3. Chronology: 1780 to 1800 4. Chronology: 1800 to 1810 5. Chronology: 1830 to 1840

Honours awarded to Adrien-Marie Legendre (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Legendre

Paris street names

Passage Legendre and Rue Legendre (17th Arrondissement)

Commemorated on the Eiffel Tower Other Web sites

1. Rouse Ball 2. The Prime Pages (The Prime Number Theorem) 3. Kevin Brown (The Prime Number Theorem) 4. Daily Telegraph, London (A problem posed by Legendre: See question 9) and its solution. 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Legendre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Legendre.html (5 of 5) [2/16/2002 11:19:40 PM]

Leger

Emile Léger Born: 15 Aug 1795 in Lagrange-aux-Bois, France Died: 15 Dec 1838 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Emile Léger's father was Claude Léger who was a professor of rhetoric at the Lycée de Mayence (the city of Mainz, now in Germany, which was occupied by the French at that time). Emile Léger was admitted to the Ecole Polytechnique in 1813. He was still a student at the Ecole Polytechnique in March 1815 when Napoleon Bonaparte escaped for his Hundred Days. The students were told to defend Paris and Léger was decorated for his bravery defending the capital. In 1816 he left the Ecole Polytechnique and joined his family in Montmorency where he father had set up his own educational establishment. He taught at his father's school taking charge after his father retired. The school was very successful in training students for the entrance examinations for university, in particular training students to enter the Ecole Polytechnique. Léger only published four mathematical papers but one contains possibly the first mention of what today is a well known fact about the Euclidean algorithm, see [1]:Emile Léger appears to have been the first (or second, if the work of de Lagny ... is counted) to recognise that the worst case of the Euclidean algorithm occurs when the inputs are consecutive Fibonacci numbers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leger.html (1 of 2) [2/16/2002 11:19:41 PM]

Leger

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Leger.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leger.html (2 of 2) [2/16/2002 11:19:41 PM]

Leibniz

Gottfried Wilhelm von Leibniz Born: 1 July 1646 in Leipzig, Saxony (now Germany) Died: 14 Nov 1716 in Hannover, Hanover (now Germany)

Click the picture above to see eight larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gottfried Leibniz was the son of Friedrich Leibniz, a professor of moral philosophy at Leipzig. Friedrich Leibniz [3]:...was evidently a competent though not original scholar, who devoted his time to his offices and to his family as a pious, Christian father. Leibniz's mother was Catharina Schmuck, the daughter of a lawyer and Friedrich Leibniz's third wife. However, Friedrich Leibniz died when Leibniz was only six years old and he was brought up by his mother. Certainly Leibniz learnt his moral and religious values from her which would play an important role in his life and philosophy. At the age of seven, Leibniz entered the Nicolai School in Leipzig. Although he was taught Latin at school, Leibniz had taught himself far more advanced Latin and some Greek by the age of 12. He seems to have been motivated by wanting to read his father's books. As he progressed through school he was taught Aristotle's logic and theory of categorising knowledge. Leibniz was clearly not satisfied with Aristotle's system and began to develop his own ideas on how to improve on it. In later life Leibniz recalled that at this time he was trying to find orderings on logical truths which, although he did not know it at the time, were the ideas behind rigorous mathematical proofs. As well as his school work, Leibniz studied his father's books. In particular he read metaphysics books and theology books from both Catholic and Protestant writers. In 1661, at the age of fourteen, Leibniz entered the University of Leipzig. It may sound today as if this were a truly exceptionally early age for anyone to enter university, but it is fair to say that by the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html (1 of 9) [2/16/2002 11:19:44 PM]

Leibniz

standards of the time he was quite young but there would be others of a similar age. He studied philosophy, which was well taught at the University of Leipzig, and mathematics which was very poorly taught. Among the other topics which were included in this two year general degree course were rhetoric, Latin, Greek and Hebrew. He graduated with a bachelors degree in 1663 with a thesis De Principio Individui (On the Principle of the Individual) which:... emphasised the existential value of the individual, who is not to be explained either by matter alone or by form alone but rather by his whole being. In this there is the beginning of his notion of "monad". Leibniz then went to Jena to spend the summer term of 1663. At Jena the professor of mathematics was Erhard Weigel but Weigel was also a philosopher and through him Leibniz began to understand the importance of the method of mathematical proof for subjects such as logic and philosophy. Weigel believed that number was the fundamental concept of the universe and his ideas were to have considerable influence of Leibniz. By October 1663 Leibniz was back in Leipzig starting his studies towards a doctorate in law. He was awarded his Master's Degree in philosophy for a dissertation which combined aspects of philosophy and law studying relations in these subjects with mathematical ideas that he had learnt from Weigel. A few days after Leibniz presented his dissertation, his mother died. After being awarded a bachelor's degree in law, Leibniz worked on his habilitation in philosophy. His work was to be published in 1666 as Dissertatio de arte combinatoria (Dissertation on the combinatorial art). In this work Leibniz aimed to reduce all reasoning and discovery to a combination of basic elements such as numbers, letters, sounds and colours. Despite his growing reputation and acknowledged scholarship, Leibniz was refused the doctorate in law at Leipzig. It is a little unclear why this happened. It is likely that, as one of the younger candidates and there only being twelve law tutorships available, he would be expected to wait another year. However, there is also a story that the Dean's wife persuaded the Dean to argue against Leibniz, for some unexplained reason. Leibniz was not prepared to accept any delay and he went immediately to the University of Altdorf where he received a doctorate in law in February 1667 for his dissertation De Casibus Perplexis (On Perplexing Cases). Leibniz declined the promise of a chair at Altdorf because he had very different things in view. He served as secretary to the Nuremberg alchemical society for a while (see [188]) then he met Baron Johann Christian von Boineburg. By November 1667 Leibniz was living in Frankfurt, employed by Boineburg. During the next few years Leibniz undertook a variety of different projects, scientific, literary and political. He also continued his law career taking up residence at the courts of Mainz before 1670. One of his tasks there, undertaken for the Elector of Mainz, was to improve the Roman civil law code for Mainz but [3]:Leibniz was also occupied by turns as Boineburg's secretary, assistant, librarian, lawyer and advisor, while at the same time a personal friend of the Baron and his family. Boineburg was a Catholic while Leibniz was a Lutheran but Leibniz had as one of his lifelong aims the reunification of the Christian Churches and [30]:... with Boineburg's encouragement, he drafted a number of monographs on religious topics, mostly to do with points at issue between the churches...

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html (2 of 9) [2/16/2002 11:19:44 PM]

Leibniz

Another of Leibniz's lifelong aims was to collate all human knowledge. Certainly he saw his work on Roman civil law as part of this scheme and as another part of this scheme, Leibniz tried to bring the work of the learned societies together to coordinate research. Leibniz began to study motion, and although he had in mind the problem of explaining the results of Wren and Huygens on elastic collisions, he began with abstract ideas of motion. In 1671 he published Hypothesis Physica Nova (New Physical Hypothesis). In this work he claimed, as had Kepler, that movement depends on the action of a spirit. He communicated with Oldenburg, the secretary of the Royal Society of London, and dedicated some of his scientific works to The Royal Society and the Paris Academy. Leibniz was also in contact with Carcavi, the Royal Librarian in Paris. As Ross explains in [30]:Although Leibniz's interests were clearly developing in a scientific direction, he still hankered after a literary career. All his life he prided himself on his poetry (mostly Latin), and boasted that he could recite the bulk of Virgil's "Aeneid" by heart. During this time with Boineburg he would have passed for a typical late Renaissance humanist. Leibniz wished to visit Paris to make more scientific contacts. He had begun construction of a calculating machine which he hoped would be of interest. He formed a political plan to try to persuade the French to attack Egypt and this proved the means of his visiting Paris. In 1672 Leibniz went to Paris on behalf of Boineburg to try to use his plan to divert Louis XIV from attacking German areas. His first object in Paris was to make contact with the French government but, while waiting for such an opportunity, Leibniz made contact with mathematicians and philosophers there, in particular Arnauld and Malebranche, discussing with Arnauld a variety of topics but particularly church reunification. In Paris Leibniz studied mathematics and physics under Christiaan Huygens beginning in the autumn of 1672. On Huygens' advice, Leibniz read Saint-Vincent's work on summing series and made some discoveries of his own in this area. Also in the autumn of 1672, Boineburg's son was sent to Paris to study under Leibniz which meant that his financial support was secure. Accompanying Boineburg's son was Boineburg's nephew on a diplomatic mission to try to persuade Louis XIV to set up a peace congress. Boineburg died on 15 December but Leibniz continued to be supported by the Boineburg family. In January 1673 Leibniz and Boineburg's nephew went to England to try the same peace mission, the French one having failed. Leibniz visited the Royal Society, and demonstrated his incomplete calculating machine. He also talked with Hooke, Boyle and Pell. While explaining his results on series to Pell, he was told that these were to be found in a book by Mouton. The next day he consulted Mouton's book and found that Pell was correct. At the meeting of the Royal Society on 15 February, which Leibniz did not attend, Hooke made some unfavourable comments on Leibniz's calculating machine. Leibniz returned to Paris on hearing that the Elector of Mainz had died. Leibniz realised that his knowledge of mathematics was less than he would have liked so he redoubled his efforts on the subject. The Royal Society of London elected Leibniz a fellow on 19 April 1673. Leibniz met Ozanam and solved one of his problems. He also met again with Huygens who gave him a reading list including works by Pascal, Fabri, Gregory, Saint-Vincent, Descartes and Sluze. He began to study the geometry of infinitesimals and wrote to Oldenburg at the Royal Society in 1674. Oldenburg replied that Newton and Gregory had found general methods. Leibniz was, however, not in the best of favours with the Royal Society since he had not kept his promise of finishing his mechanical calculating machine. Nor was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html (3 of 9) [2/16/2002 11:19:44 PM]

Leibniz

Oldenburg to know that Leibniz had changed from the rather ordinary mathematician who visited London, into a creative mathematical genius. In August 1675 Tschirnhaus arrived in Paris and he formed a close friendship with Leibniz which proved very mathematically profitable to both. It was during this period in Paris that Leibniz developed the basic features of his version of the calculus. In 1673 he was still struggling to develop a good notation for his calculus and his first calculations were clumsy. On 21 November 1675 he wrote a manuscript using the f(x) dx notation for the first time. In the same manuscript the product rule for differentiation is given. By autumn 1676 Leibniz discovered the familiar d(xn) = nxn-1dx for both integral and fractional n. Newton wrote a letter to Leibniz, through Oldenburg, which took some time to reach him. The letter listed many of Newton's results but it did not describe his methods. Leibniz replied immediately but Newton, not realising that his letter had taken a long time to reach Leibniz, thought he had had six weeks to work on his reply. Certainly one of the consequences of Newton's letter was that Leibniz realised he must quickly publish a fuller account of his own methods. Newton wrote a second letter to Leibniz on 24 October 1676 which did not reach Leibniz until June 1677 by which time Leibniz was in Hanover. This second letter, although polite in tone, was clearly written by Newton believing that Leibniz had stolen his methods. In his reply Leibniz gave some details of the principles of his differential calculus including the rule for differentiating a function of a function. Newton was to claim, with justification, that ..not a single previously unsolved problem was solved ... by Leibniz's approach but the formalism was to prove vital in the latter development of the calculus. Leibniz never thought of the derivative as a limit. This does not appear until the work of d'Alembert. Leibniz would have liked to have remained in Paris in the Academy of Sciences, but it was considered that there were already enough foreigners there and so no invitation came. Reluctantly Leibniz accepted a position from the Duke of Hanover, Johann Friedrich, of librarian and of Court Councillor at Hanover. He left Paris in October 1676 making the journey to Hanover via London and Holland. The rest of Leibniz's life, from December 1676 until his death, was spent at Hanover except for the many travels that he made. His duties at Hanover [30]:... as librarian were onerous, but fairly mundane: general administration, purchase of new books and second-hand libraries, and conventional cataloguing. He undertook a whole collection of other projects however. For example one major project begun in 1678-79 involved draining water from the mines in the Harz mountains. His idea was to use wind power and water power to operate pumps. He designed many different types of windmills, pumps, gears but [3]:... every one of these projects ended in failure. Leibniz himself believed that this was because of deliberate obstruction by administrators and technicians, and the worker's fear that technological progress would cost them their jobs. In 1680 Duke Johann Friedrich died and his brother Ernst August became the new Duke. The Harz

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html (4 of 9) [2/16/2002 11:19:44 PM]

Leibniz

project had always been difficult and it failed by 1684. However Leibniz had achieved important scientific results becoming one of the first people to study geology through the observations he compiled for the Harz project. During this work he formed the hypothesis that the Earth was at first molten. Another of Leibniz's great achievements in mathematics was his development of the binary system of arithmetic. He perfected his system by 1679 but he did not publish anything until 1701 when he sent the paper Essay d'une nouvelle science des nombres to the Paris Academy to mark his election to the Academy. Another major mathematical work by Leibniz was his work on determinants which arose from his developing methods to solve systems of linear equations. Although he never published this work in his lifetime, he developed many different approaches to the topic with many different notations being tried out to find the one which was most useful. An unpublished paper dated 22 January 1684 contains very satisfactory notation and results. Leibniz continued to perfect his metaphysical system in the 1680s attempting to reduce reasoning to an algebra of thought. Leibniz published Meditationes de Cognitione, Veritate et Ideis (Reflections on Knowledge, Truth, and Ideas) which clarified his theory of knowledge. In February 1686, Leibniz wrote his Discours de métaphysique (Discourse on Metaphysics). Another major project which Leibniz undertook, this time for Duke Ernst August, was writing the history of the Guelf family, of which the House of Brunswick was a part. He made a lengthy trip to search archives for material on which to base this history, visiting Bavaria, Austria and Italy between November 1687 and June 1690. As always Leibniz took the opportunity to meet with scholars of many different subjects on these journeys. In Florence, for example, he discussed mathematics with Viviani who had been Galileo's last pupil. Although Leibniz published nine large volumes of archival material on the history of the Guelf family, he never wrote the work that was commissioned. In 1684 Leibniz published details of his differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus... in Acta Eruditorum, a journal established in Leipzig two years earlier. The paper contained the familiar d notation, the rules for computing the derivatives of powers, products and quotients. However it contained no proofs and Jacob Bernoulli called it an enigma rather than an explanation. In 1686 Leibniz published, in Acta Eruditorum, a paper dealing with the integral calculus with the first appearance in print of the

notation.

Newton's Principia appeared the following year. Newton's 'method of fluxions' was written in 1671 but Newton failed to get it published and it did not appear in print until John Colson produced an English translation in 1736. This time delay in the publication of Newton's work resulted in a dispute with Leibniz. Another important piece of mathematical work undertaken by Leibniz was his work on dynamics. He criticised Descartes' ideas of mechanics and examined what are effectively kinetic energy, potential energy and momentum. This work was begun in 1676 but he returned to it at various times, in particular while he was in Rome in 1689. It is clear that while he was in Rome, in addition to working in the Vatican library, Leibniz worked with members of the Accademia. He was elected a member of the Accademia at this time. Also while in Rome he read Newton's Principia. His two part treatise Dynamica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html (5 of 9) [2/16/2002 11:19:44 PM]

Leibniz

studied abstract dynamics and concrete dynamics and is written in a somewhat similar style to Newton's Principia. Ross writes in [30]:... although Leibniz was ahead of his time in aiming at a genuine dynamics, it was this very ambition that prevented him from matching the achievement of his rival Newton. ... It was only by simplifying the issues... that Newton succeeded in reducing them to manageable proportions. Leibniz put much energy into promoting scientific societies. He was involved in moves to set up academies in Berlin, Dresden, Vienna, and St Petersburg. He began a campaign for an academy in Berlin in 1695, he visited Berlin in 1698 as part of his efforts and on another visit in 1700 he finally persuaded Friedrich to found the Brandenburg Society of Sciences on 11 July. Leibniz was appointed its first president, this being an appointment for life. However, the Academy was not particularly successful and only one volume of the proceedings were ever published. It did lead to the creation of the Berlin Academy some years later. Other attempts by Leibniz to found academies were less successful. He was appointed as Director of a proposed Vienna Academy in 1712 but Leibniz died before the Academy was created. Similarly he did much of the work to prompt the setting up of the St Petersburg Academy, but again it did not come into existence until after his death. It is no exaggeration to say that Leibniz corresponded with most of the scholars in Europe. He had over 600 correspondents. Among the mathematicians with whom he corresponded was Grandi. The correspondence started in 1703, and later concerned the results obtained by putting x = 1 into 1/(1+x) = 1 - x + x2 -x3 + .... Leibniz also corresponded with Varignon on this paradox. Leibniz discussed logarithms of negative numbers with Johann Bernoulli, see [156]. In 1710 Leibniz published Théodicée a philosophical work intended to tackle the problem of evil in a world created by a good God. Leibniz claims that the universe had to be imperfect, otherwise it would not be distinct from God. He then claims that the universe is the best possible without being perfect. Leibniz is aware that this argument looks unlikely - surely a universe in which nobody is killed by floods is better than the present one, but still not perfect. His argument here is that the elimination of natural disasters, for example, would involve such changes to the laws of science that the world would be worse. In 1714 Leibniz wrote Monadologia which synthesised the philosophy of his earlier work, the Théodicée. Much of the mathematical activity of Leibniz's last years involved the priority dispute over the invention of the calculus. In 1711 he read the paper by Keill in the Transactions of the Royal Society of London which accused Leibniz of plagiarism. Leibniz demanded a retraction saying that he had never heard of the calculus of fluxions until he had read the works of Wallis. Keill replied to Leibniz saying that the two letters from Newton, sent through Oldenburg, had given:... pretty plain indications... whence Leibniz derived the principles of that calculus or at least could have derived them. Leibniz wrote again to the Royal Society asking them to correct the wrong done to him by Keill's claims. In response to this letter the Royal Society set up a committee to pronounce on the priority dispute. It was totally biased, not asking Leibniz to give his version of the events. The report of the committee, finding in favour of Newton, was written by Newton himself and published as Commercium epistolicum near the beginning of 1713 but not seen by Leibniz until the autumn of 1714. He learnt of its contents in 1713 in a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html (6 of 9) [2/16/2002 11:19:44 PM]

Leibniz

letter from Johann Bernoulli, reporting on the copy of the work brought from Paris by his nephew Nicolaus(I) Bernoulli. Leibniz published an anonymous pamphlet Charta volans setting out his side in which a mistake by Newton in his understanding of second and higher derivatives, spotted by Johann Bernoulli, is used as evidence of Leibniz's case. The argument continued with Keill who published a reply to Charta volans. Leibniz refused to carry on the argument with Keill, saying that he could not reply to an idiot. However, when Newton wrote to him directly, Leibniz did reply and gave a detailed description of his discovery of the differential calculus. From 1715 up until his death Leibniz corresponded with Samuel Clarke, a supporter of Newton, on time, space, freewill, gravitational attraction across a void and other topics, see [4], [62], [108] and [202]. In [2] Leibniz is described as follows:Leibniz was a man of medium height with a stoop, broad-shouldered but bandy-legged, as capable of thinking for several days sitting in the same chair as of travelling the roads of Europe summer and winter. He was an indefatigable worker, a universal letter writer (he had more than 600 correspondents), a patriot and cosmopolitan, a great scientist, and one of the most powerful spirits of Western civilisation. Ross, in [30], points out that Leibniz's legacy may have not been quite what he had hoped for:It is ironical that one so devoted to the cause of mutual understanding should have succeeded only in adding to intellectual chauvinism and dogmatism. There is a similar irony in the fact that he was one of the last great polymaths - not in the frivolous sense of having a wide general knowledge, but in the deeper sense of one who is a citizen of the whole world of intellectual inquiry. He deliberately ignored boundaries between disciplines, and lack of qualifications never deterred him from contributing fresh insights to established specialisms. Indeed, one of the reasons why he was so hostile to universities as institutions was because their faculty structure prevented the cross-fertilisation of ideas which he saw as essential to the advance of knowledge and of wisdom. The irony is that he was himself instrumental in bringing about an era of far greater intellectual and scientific specialism, as technical advances pushed more and more disciplines out of the reach of the intelligent layman and amateur. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (229 books/articles)

Some Quotations (13)

A Poster of Gottfried Leibniz

Mathematicians born in the same country

Some pages from publications

A page from Leibnizens mathematische schriffen (published in 1850)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html (7 of 9) [2/16/2002 11:19:44 PM]

Leibniz

Cross-references to History Topics

1. Matrices and determinants 2. Quadratic, cubic and quartic equations 3. Longitude and the Académie Royale 4. The fundamental theorem of algebra 5. An overview of the history of mathematics 6. Pi through the ages 7. The rise of the calculus 8. Abstract linear spaces

Cross-references to Famous Curves

1. Astroid 2. Catenary 3. Cycloid 4. Epicycloid 5. Epitrochoid 6. Hypocycloid 7. Hypotrochoid 8. semi cubical parabola 9. Tractrix

Other references in MacTutor

1. Leibniz's calculating machine 2. Another picture of it. 3. Chronology: 1650 to 1675 4. Chronology: 1675 to 1700

Honours awarded to Gottfried Leibniz (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1673

Lunar features

Crater Leibnitz

Paris street names

Rue Leibnitz and Square Leibnitz (18th Arrondissement)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html (8 of 9) [2/16/2002 11:19:44 PM]

Leibniz

Other Web sites

1. The Galileo Project 2. Rouse Ball 3. G Don Allen 4. Gregory Brown 5. Don Rutherford 6. Kevin Brown (Leibniz on computers) 7. The Catholic Encyclopedia 8. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Leibniz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leibniz.html (9 of 9) [2/16/2002 11:19:44 PM]

Lemoine

Emile Michel Hyacinthe Lemoine Born: 22 Nov 1840 in Quimper, France Died: 21 Dec 1912 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Emile Lemoine was educated at the Ecole Polytechnique graduating in 1860. He remained there teaching for six years when he retired through ill health. Changing career he became a civil engineer but worked as an amateur mathematician and musician. His serious interest in music while at Ecole Polytechnique led him to join a chamber orchestra "La Trompette" which was good enough to have Saint-Saëns write music especially for it. As a civil engineer he rose to the rank of chief inspector and in that capacity he was responsible for the gas supply to Paris. He worked in the gas inspection service from 1886 until 1896. Lemoine work in mathematics was mainly on geometry. He founded a new study of properties of a triangle in a paper of 1873 where he studied the point of intersection of the symmedians of a triangle. He had been a founder member of the Association Française pour l'Avancement des Sciences and it was at a meeting of the Association in 1873 in Lyon that he presented his work on the symmedians. A symmedian of a triangle from vertex A is obtained by reflecting the median from A in the bisector of the angle A. He proved that the symmedians are concurrent, the point where they meet now being called the Lemoine point. Among other results on symmedians in Lemoine's 1873 paper is the result that the symmedian from the vertex A cuts the side BC of the triangle in the ratio of the squares of the sides AC and AB. He also proved that if parallels are drawn through the Lemoine point parallel to the three sides of the triangle then the six points lie on a circle, now called the Lemoine circle. Its centre is at the mid-point of the line joining the Lemoine point to the circumcentre of the triangle.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lemoine.html (1 of 3) [2/16/2002 11:19:45 PM]

Lemoine

These results are interesting but Lemoine's next venture failed to interest many mathematicians. He produced a classification of geometry according to five operations:i. ii. iii. iv. v.

placing a compass end on a given point placing a compass end on a given line drawing a circle with the compass so placed placing a straight edge on a given line drawing a line with the straight edge so placed.

Lemoine then classified the "simplicity" of a construction according to how many times these five operations had to be used. As an example of the types of results that he obtained was to study the problem of constructing a circle tangent to three given circles: the Apollonius problem. The usual construction required over 400 of Lemoine's operations but he was able to reduce the number to 199. He presented these results to the meeting of the Association Française pour l'Avancement des Sciences in 1888 at Oran in Algeria. One would have to say that these results were not thought to be particularly interesting by mathematicians at the meeting and there has been a similar lack of interest ever since. It is perhaps worth asking what is interesting in mathematics. Why are these results of Lemoine not found interesting? All I [EFR] can add is that I agree with the mathematicians of the time who preferred a construction with a large number of easily understood steps to a shorter one with sophisticated, rather obscure, steps. Let me add that I do find Lemoine's results on symmedians of a triangle to be very interesting and beautiful! Lemoine gave up active mathematical research in 1895 but continued to support the subject. He had helped found a mathematical journal in 1894 and he became its first editor, a role he held for many years. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles)

A Quotation

Mathematicians born in the same country Cross-references to Famous Curves

You can see a picture of the Lemoine point configuration and of the Lemoine circle configuration

Other Web sites

Clark Kimberling

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lemoine.html (2 of 3) [2/16/2002 11:19:45 PM]

Lemoine

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Lemoine.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lemoine.html (3 of 3) [2/16/2002 11:19:45 PM]

Leonardo

Leonardo da Vinci Born: 15 April 1452 in Vinci (near Empolia), Italy Died: 2 May 1519 in Cloux, Amboise, France

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leonardo da Vinci was educated in his father's house receiving the usual elementary education of reading, writing and arithmetic. In 1467 he became an apprentice learning painting, sculpture and acquiring technical and mechanical skills. He was accepted into the painters' guild in Florence in 1472 but he continued to work as an apprentice until 1477. From that time he worked for himself in Florence as a painter. Already during this time he sketched pumps, military weapons and other machines. Between 1482 and 1499 Leonardo was in the service of the Duke of Milan. He was described in a list of the Duke's staff as a painter and engineer of the duke. As well as completing six paintings during his time in the Duke's service he also advised on architecture, fortifications and military matters. He was also consider as a hydraulic and mechanical engineer. During his time in Milan, Leonardo became interested in geometry. He read Leon Battista Alberti's books on architecture and Piero della Francesca's On Perspective in Painting. He illustrated Pacioli's Divina proportione and he continued to work with Pacioli and is reported to have neglected his painting because he became so engrossed in geometry. Leonardo studied Euclid and Pacioli's Suma and began his own geometry research, sometimes giving mechanical solutions. He gave several methods of squaring the circle, again using mechanical methods. He wrote a book, around this time, on the elementary theory of mechanics which appeared in Milan around 1498. Leonardo certainly realised the possibility of constructing a telescope and in Codex Atlanticus written in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leonardo.html (1 of 3) [2/16/2002 11:19:47 PM]

Leonardo

1490 he talks of ... making glasses to see the Moon enlarged. In a later work, Codex Arundul written about 1513, he says that ... in order to observe the nature of the planets, open the roof and bring the image of a single planet onto the base of a concave mirror. The image of the planet reflected by the base will show the surface of the planet much magnified. See [28] for more details of this quotation and more of Leonardo's ideas about the Universe. He understood the fact that the Moon shone with reflected light from the Sun and he correctly explained the 'old Moon in the new Moon's arms' as the Moon's surface illuminated by light reflected from the Earth. He thought of the Moon as being similar to the Earth with seas and areas of solid ground. In 1499 the French armies entered Milan and the Duke was defeated. Some months later Leonardo left Milan together with Pacioli. He travelled to Mantua, Venice and finally reached Florence. Although he was under constant pressure to paint, mathematical studies kept him away from his painting activity much of the time. He was for a time employed by Cesare Borgia as a senior military architect and general engineer. By 1503 he was back in Florence advising on the project to divert the River Arno behind Pisa to help with the siege of the city which the Florentines were engaged in. He then produced plans for a canal to allow Florence access to the sea. The canal was never built nor was the River Arno diverted. In 1506 Leonardo returned for a second period in Milan. again his scientific work took precedence over his painting and he was involved in hydrodynamics, anatomy, mechanics, mathematics and optics. In 1513 the French were removed from Milan and Leonardo moved again, this time to Rome. However he seems to have led a lonely life in Rome again more devoted to mathematical studies and technical experiments in his studio than to painting. After three years of unhappiness Leonardo accepted an invitation from King Francis I to enter his service in France. The French King gave Leonardo the title of first painter, architect, and mechanic of the King but seems to have left him to do as he pleased. This means that Leonardo did no painting except to finish off some works he had with him, St. John the Baptist, Mona Lisa and the Virgin and Child with St Anne. Leonardo spent most of his time arranging and editing his scientific studies. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (29 books/articles)

Some Quotations (11)

A Poster of Leonardo da Vinci

Mathematicians born in the same country

Cross-references to History Topics

Squaring the circle

Honours awarded to Leonardo da Vinci (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leonardo.html (2 of 3) [2/16/2002 11:19:47 PM]

Leonardo

Lunar features

Crater da Vinci

Planetary features

Crater da Vinci on Mars

Paris street names

Rue Leonardo da Vinci (16th Arrondissement)

Other Web sites

1. Leonardo Museum at Vinci 2. A calculating machine devised by Leonardo 3. MIT (A bibliography on Leonardo) 4. Illinois (Some of Leonardo's inventions) 5. Some Leonardo links 6. The Catholic Encyclopedia 7. Mark Harden's Artchive (Some of Leonardo's pictures) 8. George W Hart (Leonardo's polyhedra) 9. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Leonardo.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leonardo.html (3 of 3) [2/16/2002 11:19:47 PM]

Leray

Jean Leray Born: 7 Nov 1906 in Nantes, Loire-Inférieure, France Died: 10 Nov 1998 in La Baule, France

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean Leray's father was Francis Leray, who was a professor, and his mother was Baptistine Pineau. Jean attended the Lycée at Nantes, then moving to the Lycée at Rennes before completing his education at the école Normale Supérieure where he was awarded his doctorate. In Paris he worked on hydrodynamics. He married Marguerite Trumier on 20 October 1932. They had three children, Jean-Claude, Françoise, and Denis. In 1933 Juliusz Schauder arrived in Paris on a Rockefeller scholarship to work with Hadamard. This led to a collaboration between Leray and Schauder and their joint work led to a paper Topologie et équations fonctionelles published in the Annales scientifiques de l'Ecole normale Supérieure. This 1934 paper on topology and partial differential equations is of major importance:In this paper what is now known as Leray-Schauder degree (a homotopy invariant) is defined. This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations. After his 1934 paper with Schauder, Leray published a paper on algebraic topology in the following year on the topology of Banach spaces. He then returned to work on analysis, in particular studying differential equations arising from hydrodynamics. He studied solutions of the initial value problem for three-dimensional Navier-Stokes equations. He examined not only the existence and uniqueness of solutions but he showed that the solutions remained smooth for only a finite time after which turbulent solutions arise. In producing this theory Leray introduced many ideas of functional analysis which have today become standard tools. In 1936 Leray was appointed Professor at the Faculty of Science at Nancy. World War II began in 1939 and Leray served as an army officer. He was captured in 1940 and sent to a prisoner of war camp in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leray.html (1 of 3) [2/16/2002 11:19:50 PM]

Leray

Austria where he remained until the end of the war in 1945. While at the camp Leray and some of his fellow captives organised a "université en captivité" and Leray became its rector. Not wishing the Germans to know that he was an expert in hydrodynamics, since he feared that if they found out he would be forced to undertake war work for them, Leray claimed to be a topologist. He worked only on topological problems for the years he was held captive in the camp. Although he had undertaken some topological work it was not easy for Leray to work on the topic without reading topological literature. He was able to obtain some papers through Hopf who was at this time in Zurich but much of Leray's work was done independently of the developments which had taken place in the subject. After his release in 1945 Leray published a three part work Algebraic topology taught in captivity. Leray continued to work on topological questions after his return to Paris where he became professor at the Collège de France in 1947. For Leray [2]:... algebraic topology should not only study the topology of a space, i.e. algebraic objects attached to a space, invariant under homomorphisms, but also the topology of a representation (continuous map), i.e. topological invariants of a similar nature for continuous maps. Following this line he published papers which introduced sheaves, and the spectral sequence of a continuous map. In the 1950s Leray worked in a number of areas. He studied time dependent hyperbolic partial differential equations and also began to work on the Cauchy problem. In particular he published a paper on the Cauchy problem for equations with variable coefficients in 1956. In 1957 he explained the aims of his work in this area:We propose to study globally the linear Cauchy problem in the complex case, then in the real hyperbolic case, assuming that the given data is analytic. He was able to generalise results in the theory of ordinary linear analytic differential equations to obtain similar results for partial differential equations. Leray's work on the Cauchy problem led him to study residues theory. In 1959 he [2]:... developed a general residue theory on complex manifolds and applied it to the investigation of concrete integrals depending on parameters arising from solving the Cauchy problem. In [5] Ekeland sums up Leray's achievements as follows:Leray was so far ahead of his time because of his tremendous technical capability and geometrical insight. In his hands, energy estimates for partial differential equations became combined with ideas from algebraic topology (such as fixed point theorems) in a highly original combination which cracked open the toughest problems. He was the first to adopt the modern point of view, whereby a function is not a complicated relation between two sets of variables, but a point in some infinite dimensional space ... Leray [can be said to have been] the first modern analyst. Leray received many honours. He was a member of the Academy of Sciences from 1953 and he was elected to the National Academy of Sciences in the United States in 1965. The following year he was elected to the Academy of Sciences of the USSR. He was also a member of the Royal Academy of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leray.html (2 of 3) [2/16/2002 11:19:50 PM]

Leray

Belgium, a fellow of the Royal Society of London, and a member of the academies of Milan, Boston, Göttingen, Turin, Palermo, Warsaw, and Lincei. In 1967 he was awarded an honorary doctorate from the University of Chicago. The citation stated:Mathematician of penetration and originality, whose inventions revolutionized partial differential equations and algebraic topology. He was awarded the Malaxa prize in 1938, the Feltrinelli prize in 1971, the Wolf prize in 1979 and the M V Lomonosov Gold Medal in 1988. He was also made Commandeur de la Légion d'honneur. We should end this biography with some comments on Leray's lecturing style [5]:He was a mild mannered, dapper man with a grey moustache, who squinted at his audience and lost it rather quickly; but he continued to write on the blackboard amidst a respectful silence, confident that the mathematics were there for all to see and needed no further explanation. Article by: J J O'Connor and E F Robertson List of References (19 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1930 to 1940

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Leray.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leray.html (3 of 3) [2/16/2002 11:19:50 PM]

Lerch

Mathias Lerch Born: 20 Feb 1860 in Milinov (near Susice), Bohemia (now Czech Republic) Died: 3 Aug 1922 in Susice, Czechoslovakia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mathias Lerch studied first in Prague. After leaving Prague he went to the University of Berlin where he studied during 1884 -85 and was taught by Weierstrass, Kronecker and Fuchs. In 1886 Lerch joined the teaching staff at the Czech Technical Institute in Prague. Then in 1896 he was appointed to a chair in Germany when he accepted a professorship at the University of Freiburg. He returned to the Czech Republic in 1906 where he was appointed professor of mathematics at the Czech Technical Institute in Brno. Tomás Masaryk was a political leader who liberated the Czechs and Slovaks from Austrian rule. In 1918 Masaryk was elected president of Czechoslovakia. A year after the founding of Czechoslovakia, a new university, named the Masaryk University after the first president, was founded in Brno and Lerch became the first professor of mathematics there in 1920. Lerch wrote 238 papers, listed in [7], mostly on analysis (about 150 papers) and number theory (about 40 papers). Some of his work is fundamental in modern operator calculus. He also wrote on geometry and numerical methods. Matti Jutila, in his review of [2], describes Lerch's work in number theory as follows:Matyas Lerch (1860-1922) was a remarkable Czech mathematician who published about 250 papers, some fifty of which were devoted to number theory. His favourite topics in number theory included binary quadratic forms, quadratic residues, Gauss sums and Fermat

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lerch.html (1 of 2) [2/16/2002 11:19:51 PM]

Lerch

quotients. Also, the Lerch zeta-function is a well-known generalization of Riemann's zeta-function due to him ... this contribution to analytic number theory [is] published in Acta Mathematica in 1887... In [2] there is a short biography of Lerch and the reviews of his number-theoretic papers given in the Jahrbuch über die Fortschritte der Mathematik are reproduced. Also quoted in full in [2] is the first paper Lerch wrote on number theory and two other papers which he wrote on Fermat quotients. It also contains a summary of a paper on quadratic residues and forms which was published after his death. Lerch won the Grand Prize of the Paris Academy in 1900 with a work on number theory, a great honour for any mathematician and an even greater achievement for a mathematician from outside France. He is also well known, however, for his work in analysis. In that topic he studied infinite series, and the gamma function as well as other special functions. He also studied elliptic functions and integral equations. Often the importance of his work is in the methods which he introduced rather than the specific results themselves. He introduced an auxiliary parameter for meromorphic functions. He also studied the principle of most rapid convergence of a series. He is remembered today for his solution of integral equations in operator calculus and for the 'Lerch formula' for the derivative of Kummer's trigonometric expansion for log G(v). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lerch.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lerch.html (2 of 2) [2/16/2002 11:19:51 PM]

Leshniewski

Stanislaw Leshniewski Born: 30 March 1886 in Serpukhov (near Ivanovo-Vosniesiensk), Russia Died: 13 May 1939 in Warsaw, Poland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stanislaw Leshniewski's name is sometimes transliterated as Lesniewski but we will used the form Leshniewski throughout this article. Leshniewski's father was Isydor Leshniewski, a Polish railway engineer. It was a job which involved Isydor Leshniewski being sent to places where railways were being constructed and at the stage when Stanislaw was attending secondary school the family were living in Siberia. Stanislaw attended school there in the town of Irkutsk. He studied at several universities, spending some time in Munich where he attended lecturers by Hans Cornelius, before taking his doctorate at the Polish University of Lwów (now Lvov, Ukraine but then under the control of Austria). In Lwów he studied mainly philosophy and also took mathematics courses, attending mathematics lectures by Jozef Puzyna and Waclaw Sierpinski. Leshniewski, whose doctoral supervisor was Kazimierz Twardowski, published the two papers A contribution to the analysis of existential propositions and An attempt at a proof of the ontological principle of contradiction while still undertaking his doctoral research. These papers were published in Leshniewski's mother tongue of Polish but in 1913 a Russian translation of the two papers was published under the single title Logical Studies. His doctorate was awarded in 1912. At that time Jan Lukasiewicz was teaching at Lwów, being promoted from Privatdozent to extraordinary professor in 1911, and he greatly influenced Leshniewski in the first course on mathematical logic which he gave there. One way in particular that this influence was exerted was over the law of the excluded middle. One of Leshniewski's first projects had been to attempt to disprove this principle, but Lukasiewicz had published an appendix to his 1910 publication On the principle of contraction in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leshniewski.html (1 of 4) [2/16/2002 11:19:53 PM]

Leshniewski

Aristotle which caused Leshniewski to change to direction of his research. He began to study formal logic and began to make strenuous attempts to understand Russell's paradox which he had learnt through Lukasiewicz. In 1913 Leshniewski published an article on the law of the excluded middle, then in the following year a publication on Russell's paradox. He left Lwów to take up a teaching position at a Warsaw school but after the start of World War I he made the decision to return to Russia, He spent 1914-18 in Moscow where he taught at a Polish High School. Although he had presented his first ideas on a new theory of classes which would avoid the paradoxes while he was in Lwów, it was during his time in Moscow that Leshniewski published his formal theory called mereology. We gave some further technical details of this theory below. As soon as Poland was liberated at the end of the war, Leshniewski returned to Warsaw. There he began to get more involved in the study of mathematical logic. Janiszewski and Mazurkiewicz had created in Warsaw by the end of the war one of the strongest schools of mathematics in the world. Led by Janiszewski this school was particularly interested in set theory, and the foundations of mathematics. Leshniewski enthusiastically joined Janiszewski's school of mathematics. In 1919, he accepted the chair of the philosophy of mathematics at Warsaw where Lukasiewicz was already teaching. Various mathematicians in Warsaw, including Janiszewski, Mazurkiewicz and Leshniewski, played a major role in the setting up of the journal Fundamenta Mathematicae. It was Janiszewski who proposed the name of the journal in 1919 but Leshniewski was a member of the editorial board influencing policy. The first volume appeared in 1920 and, although the intention was for a truly international journal, the editors had quite deliberately decided to make the first volume contain papers by Polish authors only. Janiszewski wrote:... it is my intention to present, if possible, all Polish mathematicians working in the field of set theory, to which the journal is devoted. Lukasiewicz who as we mentioned was also on the staff at Warsaw University at this time, began collaborating with Leshniewski. Lukasiewicz had considerable influence on Polish education over this periods for he served as Polish Minister of Education in 1919 and was twice rector of Warsaw University. During this time Lukasiewicz and Leshniewski founded the Warsaw School of Logic. They gathered round them a group of impressive students. Tarski was a student of Leshniewski who helped make this school internationally famous as he progressed from student to colleague of Leshniewski and Lukasiewicz. In 1927 Leshniewski published his first important work on the foundations of mathematics. From then until 1939 he published a series of twelve papers giving his theories of logic and mathematics. His theories overcame the paradoxes of Russell in set theory. The editors of [7] write in the Introduction to that work:For Leshniewski his publications were not the only way of publishing his new system of the foundations of mathematics. He attached great importance to his university lectures and he lectured almost entirely about his own work. These lectures are given in [5]. Fortunately Tarski was able to make known the unpublished results of Leshniewski which were destroyed in World War II. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leshniewski.html (2 of 4) [2/16/2002 11:19:53 PM]

Leshniewski

The author of [15] argues that the importance of Leshniewski's work is in providing an alternative to the classical approach to logic and the foundations of mathematics. Leshniewski's contributions to logic concentrate on the structure of a sentence, and he argues for the traditional idea of a sentence as consisting of a subject, an object and a copula. His mathematical work concentrates on set theory, where his concern is the nature of a set. Leshniewski puts the main emphasis on the distinction between sets in the distributive sense and sets in the collective sense. Leshniewski's views developed from his analysis of Russell's paradox which he concluded confused two different notions of class. The three major logical systems which Leshniewski developed were: Protothetic, a theory of propositions and propositional functors, similar in power to a theory of propositional types, providing an extended propositional calculus with quantified functional variables; Ontology, which is an axiomatised theory of common names based on protothetic which may be characterised as a cross between traditional term logic and modern type theory, containing, besides singular terms, also empty and plural terms and a host of other interesting features; and Mereology, which is an axiomatic extension of ontology for a theory of classes quite different from set theory providing a formal theory of part and whole similar to the calculus of individuals. Surma, Srzednicki, Barnett and Rickey as the editors of [7] sum up Leshniewski's contributions:Stanislaw Leshniewski was one of the co-founders of the Polish School of Logic and an author of a new and wholly original system of the foundations of logic and mathematics. He was also the forerunner and originator of many ideas included as a matter of course in modern textbooks of logic and the foundations of mathematics. Although Leshniewski played a considerable role during the period of development of modern mathematical logic and of the foundations of mathematics, his systems are not as well known as they deserve to be and the fact remains that his systems are not generally accepted as a tool in the foundational practice. Nevertheless, they have greatly influenced the very philosophy of mathematics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (30 books/articles) Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leshniewski.html (3 of 4) [2/16/2002 11:19:53 PM]

Leshniewski

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Leshniewski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leshniewski.html (4 of 4) [2/16/2002 11:19:53 PM]

Leslie

John Leslie Born: 16 April 1766 in Largo, Fife, Scotland Died: 3 Nov 1832 in Coates (near Largo), Fife, Scotland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Leslie studied at St Andrews and Edinburgh, then became tutor to the Wedgwood family (1790-1804). Appointed as professor of mathematics at Edinburgh in 1805 (after a bitter dispute, since he was not ordained by the Church), Leslie became professor in Natural Philosophy in 1819. He was elected a Corresponding Member of the French Academy of Sciences in 1820 and was knighted in 1832. He published 10 books and several journal and encyclopaedia articles. His mathematical works include texts on geometry, trigonometry and The Philosophy of Arithmetic, but he is best known for his physical researches on heat. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Honours awarded to John Leslie (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leslie.html (1 of 2) [2/16/2002 11:19:55 PM]

Leslie

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Leslie.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leslie.html (2 of 2) [2/16/2002 11:19:55 PM]

Leucippus

Leucippus of Miletus Born: about 480 BC in (possibly) Miletus, Asia Minor Died: about 420 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leucippus of Miletus carried on the scientific philosophy which had begun to become associated with Miletus. We know little of his life but it is thought that he founder the School at Abdera on the coast of Thrace near the mouth of the Nestos River. Today the town is in Greece and is called Avdhira. At the time that Leucippus would have lived in Abdera it was a prosperous town which politically was a member of the Delian League. The philosopher Protagoras was born in Abdera and he was a contemporary of Leucippus but Protagoras, the first of the Sophists, spent most of his life in Athens and may have left Abdera before Leucippus arrived there. Although now there seems little doubt that Leucippus existed, it is worth remarking that Epicurus, at the end of the fourth century BC, actually believed that Leucippus had never existed since so little was known of him. However we now know enough in the way of independent evidence to be sure that Leucippus did exist. Aristotle refers to Leucippus as a philosopher with rather different views to those of Parmenides. Aristotle refers to him several times and quotes from his works on a number of occasions. For example in De caelo Aristotle writes:... of those who have maintained the existence of indivisibles, some, as for example Leucippus and Democritus, believe in indivisible bodies, others, like Xenocrates, in indivisible lines. Unfortunately Aristotle is not entirely consistent in his references to Leucippus. Some quotes suggest that atomism began with Leucippus, other quotes such as the one above bracket Leucippus and Democritus, while in a few places Aristotle seems to imply that Democritus alone invented atomism. Certainly it seems that Leucippus was much influenced in his thinking by Zeno of Elea and by Parmenides, but it seems relatively unlikely that there is any truth in the later claim that he was a pupil of Zeno of Elea. More likely here is that later writers realised that Leucippus followed Zeno's ideas and 'pupil' was intended in this sense. It is thought that Democritus was a pupil of Leucippus, where this time 'pupil' really does have its standard meaning. Together they are considered as the joint founders of atomic theory. Leucippus stated http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leucippus.html (1 of 3) [2/16/2002 11:19:56 PM]

Leucippus

that atoms are [7]:... imperceptible, individual particles that differ only in shape and position. The mixing of these particles gives rise to the world we experience. The reason that some early writers did not believe in the existence of Leucippus seems to be because his views and those of Democritus became completely entwined. Quite soon the whole became attributed to Democritus who was the more famous of the pair. It seems likely that Democritus as a pupil of Leucippus, developed the ideas of his teacher but it is quite beyond us to disentangle the contributions of each to this important doctrine. Two works, almost certainly written by Leucippus, are The Great World System and On the Mind. The first of these is attributed to Leucippus by Theophrastus. Theophrastus (about 372 BC - 287 BC) was a pupil of Aristotle who had studied at Athens under Aristotle. Theophrastus became head of the Lyceum in Athens after Aristotle in 323 BC. He was in a position to be able to distinguish the works of Leucippus from those of Democritus and we shall describe his views on this matter. Theophrastus claimed that the basic ideas of atomism were present in the philosophy of Leucippus according to which [1]:Both matter and void have real existence. The constituents of matter are elements infinite in number and always in motion, with an infinite variety of shapes, completely solid in composition. According to Diogenes Laertius, the cosmology put forward by Leucippus in The Great World System is a creation of worlds by agglomerations of atoms by chance collisions. There is then differentiation with the smaller atoms being sent off into the infinity of space while the rest form into a spherical structure with the larger atoms at the centre and the smaller atoms further away from the centre. From the treatise On the Mind we have the only quotation of the words of Leucippus which have survived. In this work he writes (see for example [8]):Nothing happens in vain, but everything from reason and of necessity. Leucippus also contributed to the method of exhaustion. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Pi through the ages 2. The rise of the calculus

Honours awarded to Leucippus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Leucippus

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leucippus.html (2 of 3) [2/16/2002 11:19:56 PM]

Leucippus

Other Web sites

1. Internet Encyclopedia of Philosophy 2. S M Cohen (Atomism) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Leucippus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Leucippus.html (3 of 3) [2/16/2002 11:19:56 PM]

Levi-Civita

Tullio Levi-Civita Born: 29 March 1873 in Padua, Veneto, Italy Died: 29 Dec 1941 in Rome, Italy

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Tullio Levi-Civita took his degree at the University of Padua where one of his teachers was Ricci with whom Levi-Civita was to collaborate. Levi-Civita was appointed to the Chair of Mechanics at Padua in 1898, a post which he was to hold for 20 years. In 1918 he was appointed to the Chair of Mechanics at Rome where he spent another 20 years until removed from office by the discrimination policies of the government (he was of Jewish descent). Levi-Civita had very great command of pure mathematics, his geometric intuition was particularly strong, which he applied to a variety of problems of applied mathematics. One of his papers in 1895 improved on Riemann's contour integral formula for the number of primes in a given interval. Levi-Civita is best known for his work on the absolute differential calculus with its applications to the theory of relativity. In 1887 he published a famous paper in which he developed the calculus of tensors, following on the work of Christoffel, including covariant differentiation. In 1900 he published, jointly with Ricci, the theory of tensors Méthodes de calcul differential absolu et leures applications in a form which was used by Einstein 15 years later. Weyl was to take up Levi-Civita's ideas and make them into a unified theory of gravitation and electromagnetism. Levi-Civita's work was of extreme importance in the theory of relativity, and he produced a series of papers treating elegantly the problem of a static gravitational field. Analytic dynamics was another topic studied by Levi-Civita, many of his papers examining special cases

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levi-Civita.html (1 of 3) [2/16/2002 11:19:58 PM]

Levi-Civita

of the Three Body Problem. He also wrote on hydrodynamics and the theory of systems of partial differential equations. He added to the theory of Cauchy and Kovalevskaya and wrote up this work in an excellant book written in 1931. In 1933 he contributed to Dirac's equations of quantum theory. The Royal Society conferred the Sylvester medal on Levi-Civita in 1922, while in 1930 he was elected a foreign member. He was also an honorary member of the London Mathematical Society, the Royal Society of Edinburgh, and the Edinburgh Mathematical Society. He attended a meeting of the Edinburgh Mathematical Society in St Andrews. Levi-Civita, like Volterra and many other Italian scientists, were strongly and actively opposed to Fascism. After he was dismissed from his post the blow soon told on his health and he developed severe heart problems. He died of a stroke. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (21 books/articles) A Poster of Tullio Levi-Civita

Mathematicians born in the same country

Cross-references to History Topics

1. The quantum age begins 2. General relativity

Other references in MacTutor

1. Chronology: 1880 to 1890 2. Chronology: 1900 to 1910

Honours awarded to Tullio Levi-Civita (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1930

Royal Society Sylvester Medal

Awarded 1922

Fellow of the Royal Society of Edinburgh Honorary Fellow of the Edinburgh Maths Society

Elected 1930

Lunar features

Crater Levi-Civita

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levi-Civita.html (2 of 3) [2/16/2002 11:19:58 PM]

Levi-Civita

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Levi-Civita.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levi-Civita.html (3 of 3) [2/16/2002 11:19:58 PM]

Levi

Levi ben Gerson Born: 1288 in Bagnols, Gard, France Died: 1344 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Levi ben Gerson wrote Book of Numbers in 1321 dealing with arithmetical operations, including extraction of roots. Also, in 1342, he wrote On Sines, Chords and Arcs which examined trigonometry, in particular proving the sine theorem for plane triangles and giving 5 figure sine tables. One year later, at the request of the bishop of Meaux, he wrote The Harmony of Numbers which is a commentary on the first 5 books of Euclid. He also invented Jacob's staff, an instrument to measure the angular distance between celestial objects. It is described as consisting ... of a staff of 4 1/2 feet long and about one inch wide, with six or seven perforated tablets which could slide along the staff, each tablet being an integral fraction of the staff length to facilitate calculation, used to measure the distance between stars or planets, and the altitudes and diameters of the Sun, Moon and stars. Levi observed a solar eclipse in 1337. After he had observed this event he proposed a new theory of the sun which he proceeded to test by further observations. Another eclipse observed by Levi was the eclipse of the Moon on 3 October 1335. He described a geometrical model for the motion of the Moon and made other astronomical observations of the Moon, Sun and planets using a camera obscura. Some of his beliefs were well wide of the truth such as his belief that the Milky Way was on the sphere of the fixed stars and shines by the reflected light of the Sun. His other work was philosophical and he wrote complex Biblical commentaries. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles)

A Quotation

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levi.html (1 of 2) [2/16/2002 11:19:59 PM]

Levi

Mathematicians born in the same country Cross-references to History Topics

Longitude and the Académie Royale

Other references in MacTutor

Chronology: 1300 to 1500

Honours awarded to Levi ben Gerson (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Rabbi Levi

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Levi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levi.html (2 of 2) [2/16/2002 11:19:59 PM]

Levinson

Norman Levinson Born: 11 Aug 1912 in Boston, Massachesetts, USA Died: 10 Oct 1975 in Boston, Massachesetts, USA

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Norman Levinson's family were very poor Russian Jewish immigrants to the United States. Norman, born in Lynn, Massachusetts just before World War I, was the son of a shoe factory worker there, who earned three dollars a week and whose education consisted of attending a yeshiva for a few years. Norman's mother was illiterate but, despite the poverty, she managed to feed Norman and his sister Pauline. Zipporah (Fagi) Levinson, Norman's wife, wrote in [5]:Relatives and friends would give [Norman's mother] worn hand-me-down clothes. She would take them apart, and using the undersides which didn't show any wear, would make clothes for her children and husband. Despite a childhood of desperate poverty, and an education that consisted of attending rundown vocation schools, Norman said of his childhood:We were very poor, but we didn't think of ourselves as poor. Norman's father changed jobs, working for Forbes Lithograph, and the family bought a little house in Revere. It had no bathroom, and was heated by a single oil stove in the kitchen. Norman's mother did a remarkable job of providing for the family, planting fruit trees and vegetables behind the house. With little else for food, she would walk over a kilometre each day to buy stale bread for the family which was sold at half price. Norman attended Revere High School but the family doctor said that as he suffered from rheumatic fever he should not take any physical exercise. These school years were not easy for Norman but he helped his fellow pupils with their homework which put him on good terms with them. These years had a lasting effect on Norman, however. They may have been the cause for him being very shy, and they certainly http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levinson.html (1 of 6) [2/16/2002 11:20:02 PM]

Levinson

were the reason that he became a hypochondriac. To try to help the family finances, Norman worked in the evenings in a grocery store while studying at Revere High School. Levinson entered the Massachusetts Institute of Technology in 1929 but he did not register for a mathematics degree, rather he studied for a degree in Electrical Engineering. In June 1934 he was awarded a Bachelor of Science degree and a Master of Science degree, both in electrical engineering. However he had by this time taken twenty graduate courses in mathematics at MIT, almost all the graduate courses the mathematics department offered. In fact by this time, while still an electrical engineering student, he had written a thesis with Norbert Wiener and, according to the Head of Mathematics, had:... results sufficient for a doctoral thesis of unusual excellence. The turning point in Levinson's studies had come when he signed up for Wiener's graduate course on Fourier series and integrals in 1933-34. Levinson described what happened (see for example [6]):I became acquainted with Wiener in September 1933, while still a student of electrical engineering, when I enrolled in his graduate course. It was at that time really a seminar course. At that level he was a most stimulating teacher. He would actually carry on his research at the blackboard. As soon as I displayed a slight comprehension of what he was doing, he handed me the manuscript of Paley-Wiener for revision. I found a gap in a proof and proved a lemma to set it right. Wiener thereupon sat down at his typewriter, typed my lemma, affixed my name and sent it off to a journal. A prominent professor does not often act as secretary for a young student. he convinced me to change my course from electrical engineering to mathematics. After the award of his Master's degree in electrical engineering, Levinson applied to MIT to begin studying for his doctorate in mathematics. However the Mathematics Department were convinced that he had already done sufficient for the Ph.D. before starting the course! Instead Wiener together with Phillips, the Head of Mathematics, arranged for Levinson to receive an MIT Redfield Proctor Traveling Fellowship so that he could spend the year at Cambridge in England. He was assured that he would receive his doctorate on his return to MIT irrespective of any work he did in Cambridge. The attraction of Cambridge for Levinson was the fact the G H Hardy, one of the world's most respected mathematicians, taught there. Arriving in Cambridge at the end of August he wrote back to MIT in January:I have been researching along here at quite a rate in sheets of paper consumed per day. However, usually the end product of my effort warms my soul only by blazing up in the fireplace. So far my stay here has produced two completed papers and several semi-completed ones. This is rather a remarkable statement, if we look beyond the modesty and the joke; for a young student four months into his doctoral studies to have written two papers and have several more underway is truly remarkable. Rota, in [9], makes a rather surprising statement about Levinson's year at Cambridge which we reproduce without comment:Norman told me that the first thing he did when he arrived in England was to buy a new wardrobe for himself. His year in Cambridge was uneventful: he never even met Hardy, as a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levinson.html (2 of 6) [2/16/2002 11:20:02 PM]

Levinson

matter of fact he never had a high opinion of Hardy, he thought Littlewood to be a stronger mathematician. When Levinson returned to MIT in 1935 he was awarded a Doctor of Science Degree for a thesis entitled Non-vanishing of a function. He then was awarded a National Research Council Fellowship which enabled him to spend two years at the Institute for Advanced Study at Princeton. At the Institute for Advanced Study, Levinson was attached to von Neumann who was to act as a supervisor, but Levinson was a fully independent research worker by this time and certainly did not need a supervisor. The Great Depression began in 1929, the year Levinson entered MIT, and by 1932 one quarter of the workers in the United States were unemployed. As Levinson undertook his research at Princeton he felt that he had little prospects of gaining a university job, partly because of the high unemployment, but also because anti-Semitism in the United States at this time meant that Jewish mathematicians found it much harder than others to get posts. There was no shortage of high quality Jewish mathematicians, too, since by 1937 many such people were fleeing from Germany and surrounding countries and emigrating to the United States. Levinson formed a plan to train as an actuary and try for a job with an insurance company after his Fellowship at Princeton ended. If Levinson really had not met Hardy during his year at Cambridge, it was certainly Hardy who fought for Levinson to get a permanent job when he visited the United States. In the autumn of 1936 Jesse Douglas, who was awarded one of the first Fields' Medals that year, became ill and could not teach his courses at MIT. Levinson was an obvious person for them to hire and was recommended to them by Wiener but anti-Semitism at MIT tried to prevent such a move. The university's provost, Vannevar Bush, turned down Wiener's recommendation that Levinson be offered a position as Instructor but Hardy, on a visit to MIT, went with Wiener to the provost's office to protest against the decision. Hardy is reported to have said:Tell me, Mr Bush, do you think you're running an engineering school or a theological seminar? Is this the Massachusetts Institute of Theology? If it isn't, why not hire Levinson. Levinson was appointed as an Instructor at MIT in February 1937 having been released from his Fellowship by Princeton before its term was complete. He married Zipporah Wallman (known to all as Fagi) on 11 February 1938 and their two daughters, Sylvia and Joan (Zorza) were born in 1939 and 1941. Levinson was promoted to Assistant Professor in the year that his first daughter was born. In 1940 Levinson published Gap and density theorems in the American Mathematical Society Colloquium Publication Series. It was a great tribute to the young mathematician that he had been invited to write a book in a series which was reserved for distinguished senior mathematicians. The book contains his early researches on Fourier transforms in the complex domain which had developed from his study of the Paley-Wiener book which began his research career. Gap and density theorems [2]:... subsumes much of Levinson's brilliant early research in harmonic and complex analysis. Martin explains in [6] how Levinson changed the direction of his research after the publication of this treatise in 1940:Norman decided to shift his field from gap and density theorems to non-linear differential equations, both ordinary and partial. I recall our talking about this decision in 1940, and how difficult is was to more into this new field, and how hard Norman worked over a period of two or three years before he felt that he had enough mastery to obtain substantial results http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levinson.html (3 of 6) [2/16/2002 11:20:02 PM]

Levinson

in this field; but this mastery he did achieve, and his outstanding contributions to non-linear differential equations were recognised officially in 1954 when the American Mathematical Society awarded Norman the Bôcher Prize. By 1954 Levinson had been a professor for five years. He had been promoted to Associate Professor in 1944, and to full Professor at MIT in 1949. However his career nearly came to a premature end in the McCarthy era. Levinson believed in employment for all, fighting anti-Semitism, and fighting discrimination against blacks. He found that these were exactly the views of the American Communist Party which he therefore joined. When he learnt of the direction that Stalin had taken Communism in Russia, Levinson left the American Communist Party. However in 1953 he, and two other colleagues at MIT, were forced to testify to the House Un-American Activities Committee. The Committee demanded that Levinson name other members of the American Communist Party but, although he agreed to talk freely about what he did as a member of the Party, he refused to name others since he knew the consequences that would have. He had already worked out in his own mind what he would do if fired by MIT, but a skilful lawyer saved him from this fate. It did mean some difficult years for Levinson and his family, but they came through them. Let us return to discuss further Levinson's mathematical contributions. In 1955 he published another text which quickly became a classic. This was Theory of ordinary differential equations (written jointly with Earl Coddington) which [2]:... has literally become the bible for students that helped train several generations of mathematicians, scientists and engineers since it was published in 1955. Levinson wrote only two papers on time series, but these had a large impact. They had implications for geophysical signal processing (and signal processing in general) and they contributed to improved methods of oil exploration, particularly in off-shore oil fields. To gain a full appreciation of Levinson's mathematical contribution we quote at length from the Preface to [2]:The deep and original ideas of Norman Levinson have had a lasting impact on fields as diverse as differential and integral equations, harmonic, complex and stochastic analysis, and analytic number theory during more than half a century. Yet, the extent of his contributions has not always been fully recognized in the mathematics community. For example, the horseshoe mapping constructed by Stephen Smale in 1960 played a central role in the development of the modern theory of dynamical systems and chaos. The horseshoe map was directly stimulated by Levinson's research on forced periodic oscillations of the Van der Pol oscillator, and specifically by his seminal work initiated by Cartwright and Littlewood. In other topics, Levinson provided the foundation for a rigorous theory of singularly perturbed differential equations. He also made fundamental contributions to inverse scattering theory by showing the connection between scattering data and spectral data, thus relating the famous Gelfand-Levitan method to the inverse scattering problem for the Schrodinger equation. He was the first to analyze and make explicit use of wave functions, now widely known as the Jost functions. Near the end of his life, Levinson returned to research in analytic number theory and made profound progress on the resolution of the Riemann hypothesis. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levinson.html (4 of 6) [2/16/2002 11:20:02 PM]

Levinson

For a paper on number theory Levinson received the 1971 Chauvenet Prize of the Mathematical Association of America. Shortly before his death he wrote a series of important papers on the Riemann hypothesis arising from this fundamental number theory paper. E A Robinson, see [1], writes:Levinson ... was on the threshold of perhaps his greatest achievements in mathematics at the time of his death. Although Levinson had feared all his life that he would die of a heart attack, and was extremely careful with his diet in an attempt to avoid this fate, it was a brain tumour which lead to his death. Rota writes:One day shortly after his paper on the Riemann zeta function appeared, he knocked at the door, came in, and sat down. He looked pale and ill. He complained of a strong headache. ... Shortly afterwards, he entered Massachusetts General Hospital. ... in the August of that summer [I] visited him ... His head was shaven, and red and black lines were drawn on it. ... I never saw him again. After his death the MIT Faculty prepared a tribute to Levinson. It read:Norman Levinson was the heart of mathematics at MIT, a man who combined creative intellect of the highest order with human compassion and unremitting dedication to science and to excellence in its pursuit. Throughout the mathematical world the name of MIT and the name of Norman Levinson have been synonymous for many years. To those of us who were fortunate to have him as a friend and colleague, this is entirely fitting, because we are aware that, with extraordinary effectiveness and caring, he devoted forty six years of his life to mathematics and to this Institute. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Honours awarded to Norman Levinson (Click a link below for the full list of mathematicians honoured in this way) AMS Bôcher Prize

Awarded 1953

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levinson.html (5 of 6) [2/16/2002 11:20:02 PM]

Levinson

Glossary index

Quotations index

Mathematicians of the day

Poster index Anniversaries for the year

JOC/EFR September 2001 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Levinson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levinson.html (6 of 6) [2/16/2002 11:20:02 PM]

Levy_Hyman

Hyman Levy Born: 28 Feb 1889 in Edinburgh, Scotland Died: 27 Feb 1975 in Wimbledon, London, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hyman Levy's father was a dealer in paintings in Edinburgh. Hyman was the third of eight children in the orthodox Jewish family. The actual date of his birth seems a little hard to determine, given as 7 March in Who's Who and one has to assume that Levy himself supplied this information. Yet Barnard in [1] claims that although 7 March appears on his birth certificate this is incorrect and his actual date of birth was 28 February. Levy attended George Heriot's School in Edinburgh, then entered the University of Edinburgh to study mathematics and physics. He graduated from Edinburgh in 1911 with an M.A. with First Class honours and then won a Ferguson Scholarship, a 1851 Exhibition and a Carnegie Research Fellowship which enabled him to undertake research at the University of Göttingen. In 1914 when World War I broke out, Levy was in Göttingen working with Hilbert and Runge. Escaping from Germany, he returned to England where he worked at the University of Oxford with A E H Love until 1916. In 1916 Levy was honoured by being elected a fellow of the Royal Society of Edinburgh. Then, between 1916 and 1920, he worked as a member of the aeronautics research staff of the National Physical Laboratory. Levy left the National Physical Laboratory in 1920 and became an assistant professor of mathematics at the Royal College of Science of the Imperial College of Science and Technology, London. He was promoted to Professor of Mathematics three years later, then, in 1946, he became Head of the Mathematics and Mechanics Department. Under his leadership the Department expanded greatly and he also provided leadership to the whole of the Royal College of Science, being its Dean from 1946 to 1952. In 1954 Levy retired and became Professor Emeritus but he agreed to continue to act as head of the mathematics department at Imperial College for one further year until 1955. Levy's main work was in numerical methods, numerical solution of differential equations, finite difference equations and statistics. As Barnard writes in [1]:He published books on these topics long before their importance, later taken for granted, was recognised in Britain. Among Levy's most important mathematical books was Aeronautics in Theory and Experiment (1918) which was [1]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levy_Hyman.html (1 of 3) [2/16/2002 11:20:03 PM]

Levy_Hyman

... perhaps the earliest text covering, at advanced level, the whole theory of aeroplane design and operation... Among other mathematical works he published were Numerical Studies in Differential Equations (1934), Elements of the Theory of Probability (1936), and Finite Difference Equations (1958). However, Levy was more than a mathematician. He was a philosopher of science and also a political activist who [1]:... strongly influenced by poverty and degradation he saw around him in childhood, ... devoted his life to the idea that science should be used to provide the basis of a full life for all humanity. He expounded his materialistic philosophy in a number of books, the first being The Universe of Science (1932). He was a political activist in the Labour Party in the 1920s. His advice was followed and the Labour Party set up a Science Advisory Committee which Levy chaired from 1924 to 1930. In 1931 he moved further to the political left, becoming a member of the British Communist Party. He visited Moscow in 1956 as a member of a British Communist Party Delegation with the remit of investigating the Russian persecution of Jews. The persecution of Jewish intellectuals in Russia appalled Levy. Rather than resign from the British Communist Party, Levy attacked the leadership of his own Party demanding to know whether they had been aware of the treatment of the Jews in Russia. He wrote Jews and the National Question in 1957. Then in 1958 he was expelled from the Communist Party. Levy had also suffered family problems because of his Jewish upbringing. In 1918 he married Marion Aitken, the daughter of the headmaster of a school in Selkirk. Marion was a devout Presbyterian and this caused a breakdown of relations between Levy and his own family. A strong supporter of the London Mathematical Society during his career in London, Levy served the Society on its Council from 1929 to 1933, being vice-president of the Society during 1931-32. He is described in [1] as follows:With the soft Scottish accent that he retained throughout his life, Levy's warmth, human kindness, and ready wit won him many friends and much respect amongst those who strongly disagreed with his politics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levy_Hyman.html (2 of 3) [2/16/2002 11:20:03 PM]

Levy_Hyman

Honours awarded to Hyman Levy (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Levy_Hyman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levy_Hyman.html (3 of 3) [2/16/2002 11:20:03 PM]

Levy_Paul

Paul Pierre Lévy Born: 15 Sept 1886 in Paris, France Died: 15 Dec 1971 in Paris, France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Lévy was born into a family containing several mathematicians. His grandfather was a professor of mathematics while Paul's father, Lucien Lévy, was an examiner with the Ecole Polytechnique and wrote papers on geometry. Paul attended the Lycée Saint Louis in Paris and he achieved outstanding success winning prizes not only in mathematics but also in Greek, chemistry and physics. He was placed first for entry to the Ecole Normale Supérieur and second for entry to the Ecole Polytechnique in the Concours d'entrée for the two institutions. He chose to attend the Ecole Polytechnique and he while still an undergraduate there published his first paper on semiconvergent series in 1905. After graduating in first place, Lévy took a year doing military service before entering the Ecole des Mines in 1907. While he studied at the Ecole des Mines he also attended courses at the Sorbonne given by Darboux and Emile Picard. In addition he attended lectures at the Collège de France by Georges Humbert and Hadamard. It was Hadamard who was the major influence in determining the topics on which Lévy would undertake research. Finishing his studies at the Ecole des Mines in 1910 he began research in functional analysis. His thesis on this topic was examined by Emile Picard, Poincaré and Hadamard in 1911 and he received his Docteur ès Sciences in 1912. Lévy became professor Ecole des Mines in Paris in 1913, then professor of analysis at the Ecole Polytechnique in Paris in 1920 where he remained until he retired in 1959. During World War I Lévy served in the artillery and was involved in using his mathematical skills in solving problems concerning defence against attacks from the air. A young mathematician R Gateaux was killed near the beginning of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levy_Paul.html (1 of 3) [2/16/2002 11:20:05 PM]

Levy_Paul

the war and Hadamard asked Lévy to prepare Gateaux's work for publication. He did this but he did not stop at writing up Gateaux's results, rather he took Gateaux's ideas and developed them further publishing the material after the war had ended in 1919. As we indicated above Lévy first worked on functional analysis [12]:... done in the spirit of Volterra. This involved extending the calculus of functions of a real variable to spaces where the points are curves, surfaces, sequences or functions. In 1919 Lévy was asked to give three lectures at the Ecole Polytechnique on (see [9]):... notions of calculus of probabilities and the role of Gaussian law in the theory of errors. Taylor writes in [12]:At that time there was no mathematical theory of probability - only a collection of small computational problems. Now it is a fully-fledged branch of mathematics using techniques from all branches of modern analysis and making its own contribution of ideas, problems, results and useful machinery to be applied elsewhere. If there is one person who has influenced the establishment and growth of probability theory more than any other, that person must be Paul Lévy. Loève, in [9], gives a very colourful description of Lévy's contributions:Paul Lévy was a painter in the probabilistic world. Like the very great painting geniuses, his palette was his own and his paintings transmuted forever our vision of reality. ... His three main, somewhat overlapping, periods were: the limit laws period, the great period of additive processes and of martingales painted in pathtime colours, and the Brownian pathfinder period. Not only did Lévy contribute to probability and functional analysis but he also worked on partial differential equations and series. In 1926 he extended Laplace transforms to broader function classes. He undertook a large-scale work on generalised differential equations in functional derivatives. He also studied geometry. His main books are Leçons d'analyse fonctionnelle (1922), Calcul des probabilités (1925), Théorie de l'addition des variables aléatoires (1937-54), and Processus stochastiques et mouvement brownien (1948). In 1963 Lévy was elected to honorary membership of the London Mathematical Society. In the following year he was elected to the Académie des Sciences. Loève sums up his article [9] in these words:He was a very modest man while believing fully in the power of rational thought. ... whenever I pass by the Luxembourg gardens, I still see us there strolling, sitting in the sun on a bench; I still hear him speaking carefully his thoughts. I have known a great man. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levy_Paul.html (2 of 3) [2/16/2002 11:20:05 PM]

Levy_Paul

Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Levy_Paul.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levy_Paul.html (3 of 3) [2/16/2002 11:20:05 PM]

Levytsky

Volodymyr Levytsky Born: 31 Dec 1872 in Ternopil, Galicia (now Ukraine) Died: 13 Aug 1956 in Lvov, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Volodymyr Levytsky attended the University of Lvov, receiving his doctorate in 1901. After this he taught both mathematics and physics at high schools. The University of Lvov, which Levytsky studied at, was an ancient educational establishment which was founded in 1784. However problems arose between the Polish and Ukrainian populations after World War I. Ukrainian students were not permitted to enrol at the University in 1919 and the following year Ukrainian lecturers were banned from the University, only Polish citizens being allowed as lecturers. The Ukrainian students who could no longer enrol at Lvov University set up their own University, the Lvov (Underground) Ukrainian University, in July 1921. Levytsky taught mathematics at this new university from its foundation. The Underground Ukrainian University was financed by private donations and was able to survive for a few years but, when it was denied official recognition, it was forced to close in 1925. Levytsky headed the mathematics-physics section of the Shevchenko Scientific Society in Lvov. He served for two terms as the President of the Society from 1931 to 1935 and also was editor of the Journal of the Society. From 1940, until his death in 1956, Levytsky taught at the Lvov Pedagogical Institute. Levytsky wrote the first mathematical paper in Ukrainian and was the editor of the first Ukrainian mathematical journal. Most of his publications were in the area of functions of a complex variable. However he was also very active in applications of mathematics to theoretical physics. He is known for his work on Ukrainian mathematical, physical and chemical terminology, which was

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levytsky.html (1 of 2) [2/16/2002 11:20:07 PM]

Levytsky

one of the main areas of study of the Shevchenko Scientific Society in Lvov. Levytsky also wrote important historical works on mathematics at the Lvov (Underground) Ukrainian University and mathematics at the Shevchenko Scientific Society. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Levytsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Levytsky.html (2 of 2) [2/16/2002 11:20:07 PM]

Lexell

Anders Johan Lexell Born: 24 Dec 1740 in Äbo, Sweden (now Turku, Finland) Died: 11 Dec 1784 in St Petersburg, Russia Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Anders Lexell is sometimes known by the Russian version of his name which is Andrei Ivanovich Lexell. His father, Jonas Lexell, was a jeweller by trade but was also involved in politics as a local councillor. Anders' mother was Magdalena Catharina Björckegren. He was educated in Abo, attending the university there and graduating in 1760. Three years later he was appointed assistant professor at Uppsala Nautical School and in 1766 he became professor of mathematics there. In 1768 Lexell was invited to St Petersburg. The St Petersburg Academy of Science had been founded by Catherine I, the wife of Peter the Great, in 1725 and Euler had worked there since 1727. By this time Euler was getting quite old, being 62 years of age when the young mathematician Lexell arrived in 1769. However, working in the same Academy as Euler and other high quality scientists was something which Lexell found exciting and enjoyable. Euler discussed research plans with Lexell and the other mathematicians at the Academy. They shared ideas while Lexell sometimes developed further ideas suggested by Euler, sometimes calculating tables, and compiling examples. For example Lexell is given full credit on the title page for his help with Euler's 1772 publication Theoria motuum lunae, nova methodo pertractata. In 1771 Lexell was appointed professor of astronomy at the St Petersburg Academy of Science and a few years later he was approached by the Swedish government trying to persuade him to return to Sweden. By this time Lexell had achieved quite a fine reputation as both a mathematician and astronomer and he was highly involved in the exciting work at the Academy. Knowing this, the Swedish government tried too attract him with back with a cleverly worked out offer. He would be appointed to a chair at the University of Abo immediately (this was in 1775) but since he was so involved at work being undertaken at the St Petersburg Academy he would be allowed to remain there for five years to complete the work before returning to Abo. Despite the attractive proposition, Lexell was having none of it and turned it down in favour of staying permanently in St Petersburg. Despite wanting to remain in St Petersburg after 1780, Lexell did in fact spend two years travelling through to the mathematical centres of excellence throughout Europe, in particular visiting Germany, France and England. He returned to St Petersburg in 1782 and, following Euler's death in 1783, Lexell was appointed to succeed him to the chair of mathematics at the St Petersburg Academy of Science. He did not hold this chair for very long since he died in the following year. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lexell.html (1 of 3) [2/16/2002 11:20:08 PM]

Lexell

Lexell's work in mathematics is mainly in the area of analysis and geometry. Lexell made a detailed investigation of exact equations differential equations. His work here extended a necessary condition which had been discovered earlier by Condorcet and Euler. He also gave a proof which was not based on using the calculus of variations. In addition Lexell developed a theory of integrating factors for differential equations at the same time as Euler but, although it has often been thought that he learnt of the technique from Euler, the author of [2] argues that he independently discovered original methods to solve problems investigated by Euler. Lexell did work in analysis on topics other than differential equations, for example he suggested a classification of elliptic integrals and he worked on the Lagrange series. He was also the first to develop a general system of polygonometry. This is a study of polygons similar to earlier work on triangles. It involves the solution of polygons given certain sides and angles between them, their mensuration, division by diagonals, circumscribing polygons around circles and inscribing polygons in circles. His work on this topic was continued by Lhuilier. Lexell made major contributions to spherical geometry and trigonometry. In fact trigonometry was the main tool used by Lexell in his work on polygonometry. Spherical geometry was a major tool in his astronomical studies. Specific problems which Lexell studied in astronomy were his calculation of the solar parallax and his calculation of the orbits of several comets. One comet for which he calculated an orbit had been discovered by Messier. Lexell found that it had a period of five and a half years which made it the first comet to be discovered with a short period. He observed it pass close to Jupiter and its moons and since the moons were unaffected Lexell deduced that, despite the large size of comets, their mass was extremely low. When William Herschel discovered a new body in the solar system on 13 March 1781, Lexell computed its orbit which showed that it was a planet (later named Uranus) twice as far from the sun as Saturn, rather than a comet as had been thought at first. Although he did not predict the position of Neptune, as did Adams and Le Verrier, Lexell's initial calculations of the orbit of Uranus showed that it was being perturbed and he deduced that the perturbations were due to another more distant planet. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

Orbits and gravitation

Honours awarded to Anders Lexell (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lexell.html (2 of 3) [2/16/2002 11:20:08 PM]

Lexell

Lunar features

Crater Lexell

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lexell.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lexell.html (3 of 3) [2/16/2002 11:20:08 PM]

Lexis

Wilhelm Lexis Born: 17 July 1837 in Eschweiler (near Aachen), Germany Died: 25 Oct 1914 in Göttingen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Lexis attended the University of Bonn where he was awarded a degree in mathematics and then went on to write a thesis on analytical mechanics. He graduated in 1859 and then went to Heidelberg where he undertook research in Bunsen's chemistry laboratories. Despite the fact that his training up to this point had been a broad one, in 1861 Lexis went to Paris to study social sciences. These studies led to his first publication in 1870 which was on the export policies of France. Heiss writes in [4]:This work displays the feature that characterises his later economic writings: a scepticism towards "pure economics" and towards the application of supposedly descriptive mathematical models which have no reference to economic reality. Even in this early work he insisted that economic theory should be founded on quantitative economic data. In 1872 Lexis was appointed as professor of economics at the University of Strassburg. He held this post for two years and during this time he worked on his second major publication on the theory of population which was published in 1875. By this time, however, he was in Dorpat having been appointed to the chair of geography, ethnology and statistics in 1874. Again he held this post for only two years before he was appointed to the chair of economics at Freiburg. It was in Freiburg that Lexis made his major contributions to statistics. Zabell writes in [8]:Dissatisfied with the uncritical and usually unsupported assumption of statistical homogeneity in sampling, often made by Quetelet and his followers, Lexis devised a statistic Q, now called the 'Lexis ratio', to test this assumption and demonstrate its frequent

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lexis.html (1 of 3) [2/16/2002 11:20:10 PM]

Lexis

invalidity. From 1876-79 Lexis studied data presented as a series over time. He initiated the study of time series with his 1879 article On the theory of the stability of statistical series. Lexis required a binomial dispersion for his series to be stable and this ruled out most interesting series. However it applied to the problem of year to year fluctuation of the sex ratio among children born in a city. It posed the question of whether an empirical index of dispersion is consistent with the assumption that sex is governed by a simple chance mechanism. As Stigler writes in [5]:Many scientists attempted to adapt probability-based methods to social science problems, including Quetelet and Lexis, but in the end they were frustrated, Quetelet because his methods were too insensitive to segregate his data into categories amenable to statistical analysis, Lexis because his binomial models were insufficiently rich for interesting applications. After what is not much more than a three year period working on statistics in Freiburg, he began to produce a series of papers on economics. Lexis left Freiburg in 1884 to take up the chair of economics at the University of Breslau, then in 1887 he made his final career move when he accepted the chair of political science at Göttingen. Rather than concentrate on any one of his diverse interests as his career progressed, Lexis studied and wrote on a wider and wider collection of topics. In addition to studying economics, the theory of population, and statistics, he also worked on the production of an economic encyclopaedia, edited a series of works on education in general and university education in particular, and was the director of the first institute of actuarial science in Germany. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) A Poster of Wilhelm Lexis

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1870 to 1880

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lexis.html (2 of 3) [2/16/2002 11:20:10 PM]

Lexis

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lexis.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lexis.html (3 of 3) [2/16/2002 11:20:10 PM]

Lhuilier

Simon Antoine Jean Lhuilier Born: 24 April 1750 in Geneva, Switzerland Died: 28 March 1840 in Geneva, Switzerland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Simon Lhuilier (sometimes written Simon L'Huilier) was the son of Laurent Lhuilier, a jeweller and goldsmith. His mother, Suzanne-Constance Matte, was Laurent Lhuilier's second wife and there were three older children in the family. The Lhuilier's were a Huguenot family, originally from Mâcon, but after the Edict of Nantes (which had granted religious liberty to the Huguenots) was revoked by Louis XIV in 1685, they had to flee. They settled in Geneva in 1691. There was a strange episode in Lhuilier's life when he was still young. A wealthy relation proposed that he would leave Lhuilier a large fortune if he followed a career in the church. However, Lhuilier had already found the attraction of mathematics and money was not going to make him give up the attractions of the topic so he refused his relative's offer. Lhuilier was an exceptional secondary school pupil and he went on to study mathematics at a Calvin Academy where he was taught mathematics by one of Euler's former pupils, Louis Bertrand, and physics by Georges-Louis Le Sage. It was through Le Sage that Lhuilier obtained his first post as tutor to the Rilliet-Plantamour family, a post he held for two years. The next career move by Lhuilier was also as a result of him knowing Le Sage. Another of Le Sage's students, Christoph Pffeiderer, had been appointed to the position of professor of mathematics and physics at the Military Academy in Warsaw. Pffeiderer was put in charge of a competition to find the best authors to write texts for Polish schools and in 1775 he sent details to his old teacher Le Sage. Le Sage tried to persuade Lhuilier to submit an entry to write a physics text but Lhuilier preferred to enter the competition to write a mathematics text. Lhuilier's proposal won the competition giving him the right to write a mathematics textbook to be used in Polish schools. Adam Kazimierz Czartoryski was a Polish prince who had been educated in England and prepared to take over the Polish throne but he refused it in 1763. He later became the first minister of education in a European country and his palace at Pulawy became an important centre of culture providing an excellent school for his sons and for the sons of other important people in the neighbourhood. Czartoryski had been involved in the competition to find authors of Polish school texts and he was so impressed by Lhuilier's entry that he offered him a position as tutor at Pulawy in 1777, in particular as a tutor to his son Adam Jerzy Czartoryski who was seven years old at the time. Lhuilier spent eleven years at Pulawy. Adam Jerzy Czartoryski proved an extremely bright and gifted http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lhuilier.html (1 of 3) [2/16/2002 11:20:12 PM]

Lhuilier

pupil and, in addition to his tutoring duties, Lhuilier found time to write his mathematics text, undertake research in mathematics which resulted in several fine publications, and enjoy a busy social life with hunting parties. He also submitted an entry for the prize topic proposed in 1784 by the Berlin Academy of Science. The Academy sought the best article on the theory of the mathematical infinity and they designed the competition to encourage mathematicians to seek a sound basis for the new differential calculus. Lhuilier submitted the paper Exposition elementaire des principes des calculs superieurs and his essay won the prize and was published in Berlin in 1786. The standard concepts and notation for derivatives, and the standard elementary theorems on limits which appear in an undergraduate calculus text today appear in a remarkably similar form in Lhuilier's prize winning essay. Lhuilier introduced the notation "lim", and was the first to allow two-sided limits. The topic of limits was a particularly fortunate one for Lhuilier since he had been thinking about limits before the topic was ever proposed for the prize. In fact his Polish school textbook which was published in 1780 contains a section on limits. In 1789 Lhuilier returned to Switzerland but the political situation there seemed fragile and he feared that there would soon be a revolution. Pffeiderer, who had become a friend through the Polish episode, was by this time teaching mathematics in Tübingen and Lhuilier went to be with Pffeiderer there. He would stay with Pffeiderer in Tübingen until 1794. In the following year Lhuilier was offered a chair of mathematics in Leiden, but he preferred to compete for the chair in Geneva which had been held by his former teacher Louis Bertrand. Having won the competition, Lhuilier was appointed in 1795 and held this chair until he retired in 1823. Not only was 1795 the year of his appointment to the Academy in Geneva but it was a year marked by two other important events in Lhuilier's life. In that year an improved version of his prize-winning essay on limits was published in Latin in Tübingen. Also in that year Lhuilier married Marie Cartier and they would have two children, a son and a daughter. Lhuilier was also involved in politics in Geneva, being President of the Legislative Council there in 1796. He also achieved a high position in the Academy at Geneva, becoming its rector. He enjoyed many academic honours too, being elected a corresponding member of the Berlin Academy, of the Göttingen Academy, of the St Petersburg Academy, and of the Royal Society of London. His work on Euler's polyhedra formula, and exceptions to that formula, were important in the development of topology. Lhuilier also corrected Euler's solution of the Konigsberg bridge problem. He also wrote four important articles on probability during the years 1796 and 1797. One further work by Lhuilier is worth commenting on. This is the two volume work Eléments raisonnés d'algèbre that he published in 1804 for his students in Geneva. This work was really a sequel to the text which he wrote for Polish schools many years before. Speziali writes in [1]:The main value of these two volumes lay in the author's clear exposition and judicious selection of exercises ... Also in [1] his character is described as follows:Whereas the Poles found Lhuilier distinctly puritanical, his fellow citizens of Geneva reproached him for his lack of austerity and his whimsicality, although the latter quality http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lhuilier.html (2 of 3) [2/16/2002 11:20:12 PM]

Lhuilier

never went beyond putting geometric theorems into verse and writing ballads on the number three and on the square root of minus one. His most famous pupil was Charles-Francois Sturm who studied under Lhuilier during the last few years of his career in Geneva. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Topology enters mathematics 2. An overview of Indian mathematics

Honours awarded to Simon Lhuilier (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1791

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lhuilier.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lhuilier.html (3 of 3) [2/16/2002 11:20:12 PM]

Libri

Count Guglielmo Libri Carucci dalla Sommaja Born: 1 Jan 1803 in Florence, Italy Died: 28 Sept 1869 in Fiesole, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Guglielmo Libri came from one of the oldest of the families of Florence. Educated in Italy he was appointed to the chair of Mathematical Physics at Pisa in 1823. The following year he visited Paris and was well received by the top mathematicians of the day. He returned to Italy but, in 1830, he was involved in political problems there when he was implicated in a conspiracy. He chose to flee to France where he felt he had many scientific friends who would support him. Indeed he became a French citizen three years later and was, in that same year of 1833, elected to the Académie des Sciences to succeed Legendre. A further appointment at the Collège de France followed and he was then appointed Inspector of the Libraries of France. However it was soon reported that precious books and manuscripts went missing from libraries and all these losses coincided with a visit to the library by Libri. This merited an investigation but eventually nothing came of it. There was a Revolution in France in 1848 and, shortly after this, Libri was informed that a warrant was about to be issued for his arrest on suspicion of stealing precious books. He did not wait to be arrested but fled as quickly as possible to London where he claimed to be a political refugee of the French Revolution. He was well received in London and treated as a hero. However in 1850 he was convicted in France in his absence on the charges of stealing valuable books which had been brought against him and he was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Libri.html (1 of 2) [2/16/2002 11:20:13 PM]

Libri

sentenced to 10 years in prison. Certainly Libri could not return to France. Although Libri arrived in England without any money he was not poor for long. Where did his new wealth come from? Perhaps not surprisingly it came from the sale of many precious books and manuscripts which Libri happened to have with him when he arrived in London. In fact he made over one million francs from his sales of documents and books. In 1888 the French government requested that the precious books and manuscripts which Libri had stolen, and then sold, be made available for them to buy back. Indeed many of the precious documents were returned to France at this time. Libri's early work was on the theory of heat. He produced a major work on the history of Italian mathematics Histoire des sciences mathématique en Italie published in 4 volumes between 1838 and 1841. It concerned the history from Roman times up to Galileo. The book was written in French. Article by: J J O'Connor and E F Robertson List of References (7 books/articles)

A Quotation

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Libri.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Libri.html (2 of 2) [2/16/2002 11:20:13 PM]

Lie

Marius Sophus Lie Born: 17 Dec 1842 in Nordfjordeide, Norway Died: 18 Feb 1899 in Kristiania (now Oslo), Norway

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sophus Lie's father was Johann Herman Lie, a Lutheran minister. His parents had six children and Sophus was the youngest of the six. Sophus first attended school in the town of Moss, which is a port in south-eastern Norway, on the eastern side of the Oslo Fjord. In 1857 he entered Nissen's Private Latin School in Christiania (the city which became Kristiania, then Oslo in 1925) . While at this school he decided to take up a military career, but his eyesight was not sufficiently good so he gave up the idea and entered University of Christiania. At university Lie studied a broad science course. There was certainly some mathematics in this course, and Lie attended lectures by Sylow in 1862. Although not on the permanent staff, Sylow taught a course, substituting for Broch, in which he explained Abel's and Galois' work on algebraic equations. Lie also attended lectures by Carl Bjerknes on mathematics, so he certainly had teachers of considerable quality, yet he graduated in 1865 without having shown any great ability for the subject, or any great liking for it. There followed a period when Lie could not decide what subject to pursue and he taught pupils while trying to make his decision. The one thing knew he wanted was an academic career and he thought for a while that astronomy might be the right topic. He learnt some mechanics, wondered whether botany or zoology or physics might be the right subjects and in general became rather confused. However, there are signs that from 1866 he began to read more and more mathematics and the library records in the University of Christiania show clearly that his interests were steadily turning in that direction. It was during the year 1867 that Lie had his first brilliant new mathematical idea. It came to him in the middle of the night and, filled with excitement, he rushed to see his friend Ernst Motzfeldt, woke him up

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html (1 of 6) [2/16/2002 11:20:16 PM]

Lie

and shouted:I have found it, it is quite simple! This was not the end of Lie's problems of course (far from it for Lie would always have problems), but at least in his own mind he now knew the career he wanted and it would be fair to say that from that moment on Lie became a mathematician. The type of mathematics that Lie would study became more clearly defined during 1868 when he avidly read papers on geometry by Plücker and Poncelet. Plücker's [1]:... monumental idea to create new geometries by choosing figures other than points - in fact straight lines - as elements of space pervaded all of Lie's work. Lie wrote a short mathematical paper in 1869, which he published at his own expense, based on the inspiration which had struck him in 1867. He wrote up a more detailed exposition, but the world of mathematics was too cautious to quickly accept Lie's revolutionary notions. The Academy of Science in Christiania was reluctant to publish his work, and at this stage Lie began to despair that he would become accepted in the mathematical world. His friend Motzfeldt did a superb job of encouraging Lie to press on with his mathematical ideas and the breakthrough came later in 1869 when Crelle's Journal accepted his paper. He sent letters to two Prussian mathematicians, Reye and Clebsch, still attempting to gain recognition for his ideas. The paper in Crelle's Journal, however, proved vital for, on the strength of the paper, Lie was awarded a scholarship to travel and meet the leading mathematicians. Setting off near the end of the year 1869, Lie went to Prussia and visited Göttingen and then Berlin. In Berlin he met Kronecker, Kummer and Weierstrass. Lie was not attracted to the style of Weierstrass's mathematics which dominated Berlin. His interests fitted more closely with Kummer, and Lie lectured on his own results in Kummer's seminar and was able to correct some errors that Kummer had made in his work on line congruences of degree 3. Most important to Lie, however, was the fact that in Berlin he met Felix Klein. It was easy to see that these two would instantly find common ground in mathematics since Klein had been a student of Plücker, and Lie, although he never met Plücker, always said that he felt like Plücker's student. Despite the common link through Plücker's line geometry, Lie and Klein were rather different in character as Freudenthal points out in [1]:Lie and Klein had quite different characters as humans and mathematicians: the algebraist Klein was fascinated by the peculiarities of charming problems; the analyst Lie, parting from special cases, sought to understand a problem in its appropriate generalisation. It was in Berlin that Lie developed a new self-confidence in his mathematical ability. He received high praise from Kummer, and he received replies from Reye and Clebsch to his earlier letters which greatly encouraged him. Lie wrote to his friend Motzfeldt in Christiania saying (see for example [34]):... in the years 1864-68, I really underestimated my own mental power. In the spring of 1870 Lie and Klein were together again in Paris. There they met Darboux, Chasles and Camille Jordan. Jordan seems to have succeeded in a way that Sylow did not, for Jordan made Lie realise how important group theory was for the study of geometry. Lie started develop ideas which would later appear in his work on transformation groups. He began to discuss with Klein these new ides on groups and geometry and he would collaborate later with Klein in publishing several papers. This joint work had as one of its outcomes Klein's characterisation of geometry in his Erlangen Program of 1872 as properties invariant under a group action. While in Paris Lie discovered contact transformations. These http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html (2 of 6) [2/16/2002 11:20:16 PM]

Lie

transformations allowed a 1-1 correspondence between lines and spheres in such a way that tangent spheres correspond to intersecting lines. While Lie and Klein thought deeply about mathematics in Paris, the political situation between France and Prussia was deteriorating. The popularity of Napoleon III, the French emperor, was declining in France and he thought a war with Prussia might change his political fortunes since his advisers having told him that the French Army could defeat Prussia. Bismarck, the Prussian chancellor, saw a war with France as an opportunity to unite the South German states. With both sides feeling that a war was to their advantage, the Franco-Prussian War became inevitable. On 14 July, Bismarck sent a telegram which infuriated the French government and on the 19 July France declared war on Prussia. For Klein, a Prussian citizen who happened to be in Paris when war was declared, there was only one possibility: he had to return quickly to Berlin. However, Lie was a Norwegian and he was finding mathematical discussions in Paris very stimulating. He decided to remain but became anxious as the German offensive met with only an ineffective French reply. In August, the German army trapped part of the French army in Metz and Lie decided it was time for him to leave and he planned to hike to Italy. He reached Fontainebleau but there he was arrested as a German spy, his mathematics notes being assumed to be top secret coded messages. Only after the intervention of Darboux was Lie released from prison. The French army had surrendered on 1 September, and on 19 September the German army began to blockade Paris. Lie fled again to Italy, then from there he made his way back to Christiania via Germany so that he could meet and discuss mathematics with Klein. In 1871 Lie became an assistant at Christiania, having obtained a scholarship, and he also taught at Nissen's Private Latin School in Christiania where he had been a pupil himself. He submitted a dissertation On a class of geometric transformations (written in Norwegian) for his doctorate which was duly awarded in July 1872. The dissertation contained ideas from his first results published in Crelle's Journal and also the work on contact transformations, a special case of these transformations being a transformation which maps a line into a sphere, which he had discovered while in Paris. It was clear that Lie was a remarkable mathematician and the University of Christiania reacted in a very positive way, creating a chair for him in 1872. The famous Norwegian mathematician Abel had died more than 40 years before this (some 14 years before Lie was born) but, despite Abel's short career, his complete works had not been published at that time. It was natural that Norwegian mathematicians would undertake the task, and between 1873 and 1881 Sylow and Lie prepared an edition of Abel's complete works. Lie, however, always claimed that most of the work was done by Sylow. Another event which took place within two years of Lie being appointed to his chair was his marriage. He married Anna Birch and they would have three children, one daughter and two sons. Lie had started examining partial differential equations, hoping that he could find a theory which was analogous to the Galois theory of equations. He wrote:... the theory of differential equations is the most important discipline in modern mathematics. He examined his contact transformations considering how they affected a process due to Jacobi of generating further solutions of differential equations from a given one. This led to combining the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html (3 of 6) [2/16/2002 11:20:16 PM]

Lie

transformations in a way that Lie called an infinitesimal group, but which is not a group with our definition, rather what is today called a Lie algebra. It was during the winter of 1873-74 that Lie began to develop systematically what became his theory of continuous transformation groups, later called Lie groups leaving behind his original intention of examining partial differential equations. Later Killing was to examine the Lie algebras associated with Lie groups. He did this quite independently of Lie (and not it would appear in a manner which Lie found satisfactory), and it was Cartan who completed the classification of semisimple Lie algebras in 1900. Although Lie was producing highly innovative mathematics, he became increasingly sad at the lack of recognition he was receiving in the mathematical world. One reason was undoubtedly was his isolation in Christiania but a second reason was that his papers were not easily understood, partly through his style of writing and partly because his geometrical intuition greatly exceeded that of other mathematicians. Klein, realising the problems, had the excellent idea of sending Friedrich Engel to Christiania to help Lie. Engel had received his doctorate from Leipzig in 1883 having studied under Adolph Mayer writing a thesis on contact transformations. Klein recognised that he was the right man to assist Lie and, at Klein's suggestion, Engel went to work with Lie in Christiania starting in 1884. He worked with Lie for nine months leaving in 1885. Engel then was appointed to Leipzig and, when Klein left the chair at Leipzig in 1886, Lie was appointed to succeed him. The collaboration between Engel and Lie continued for nine years culminating with their joint major publication Theorie der Transformationsgruppen in three volumes between 1888 and 1893. This was Lie's major work on continuous groups of transformations. In Leipzig, life for Lie was rather different from that in Christiania. He was now in the mainstream of mathematics and students came from many countries to study under him. He had a much heavier teaching load, however [38]:Lie's lectures on his own research were highly rated by the students, in contrast to his somewhat unpopular obligatory lectures on standard topics. ... he preferred to draw a picture instead of giving rigorous proofs. However all was not well, he still felt unrecognised and, as Svare writes in [38]:In Leipzig Lie was troubled by constant homesickness. A keen outdoor man, he missed the forests and mountains of Norway. Towards the end of the 1880s Lie's relationship with Engel broke down. In 1892 the lifelong friendship between Lie and Klein broke down and the following year Lie publicly attacked Klein saying:I am no pupil of Klein, nor is the opposite the case, although this might be closer to the truth. It is difficult for any biographer to represent these events, and the events which followed, fairly since there is a great deal of contradictory material in the literature. The reason for this is not hard to understand, for information about Lie was for many years based on [13] which Engel wrote on Lie's death. The position is complicated by the mental difficulties which Lie suffered in 1889. Klein's [34]:... "defence" of Lie's behaviour by referring to the close relationship between genius and madness really created a generally accepted explanation which has survived up to the present. By this act of "defence" Klein did his old friend an incredible injustice. The truth is that Lie's behaviour was not totally irrational as it has been portrayed, but was indeed http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html (4 of 6) [2/16/2002 11:20:16 PM]

Lie

motivated by the way that both Engel and Klein had behaved. Purkert in [26] discusses the breakdown of relations between Lie and Engel. He has studied material from the University of Leipzig and believes that Lie changed his attitude toward Engel because Lie still felt a lack of recognition yet he knew that he was in a different class as a creative mathematician to Engel. Lie returned to Christiania in 1898 to take up a post specially created for him. He produced a report about who should fill his chair, and this is given in full in [26]. Despite Engel being one of the leading workers in Lie's own research field, Purkert believes that Lie's assessment that he lacked creativity was entirely fair. In [15] Fritzsche comments on Lie's illness. He writes:Through information about Sophus Lie's illness it is possible to trace consequences that shed light on certain biographical aspects of his life; for example, his break with Friedrich Engel and Felix Klein. Furthermore, this evidence contradicts the oft-stated opinion that Lie's sickness was brought about by overwork. Straume in [34] points out why Lie's behaviour towards Klein, with the final breakdown in 1892, was not irrational:Klein's Erlangen Program from 1872 had not attracted much attention; in fact, it was Lie rather than Klein himself who had influenced the mathematical development envisioned in this Program. ... Klein decided to republish the Program and also write about its origins (in which Lie was much involved), but Lie disagreed strongly with Klein's views on what had happened in the past. It also turned out that Klein burned all the letters he had received from Lie up to 1877 (and thus breaking a previous mutual agreement between them). Lie reacted by publicly attacking Klein in the Preface to the third volume of his Theorie der Transformationsgruppen in 1893. Certainly Lie was an angry man but he was attacking someone holding such a leading role on the world scene of mathematics that the attack was always more likely to rebound on Lie rather than hurt Klein. Already current research is showing Lie in a much better light over this affair (and therefore Klein in a less good one) than previously reported and all the indications are that further research will prove even more favourable to Lie. Perhaps an indication of Lie's love for his homeland is the fact that he continued to hold his chair in Christiania from his first appointment in 1872, being officially on leave while holding the chair in Leipzig. However his health was already deteriorating when he returned to a chair in Christiania in 1898, and he died of pernicious anaemia in February 1899 soon after taking up the post. Let us end by quoting from Robert Hermann's preface to [4]:In reading Lie's work in preparation for my commentary on these translations, I was overwhelmed by the richness and beauty of the geometric ideas flowing from Lie's work. Only a small part of this has been absorbed into mainstream mathematics. He thought and wrote in grandiose terms, in a style that has now gone out of fashion, and that would be censored by our scientific journals! The papers translated here and in the succeeding volumes of our translations present Lie in his wildest and greatest form. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html (5 of 6) [2/16/2002 11:20:16 PM]

Lie

List of References (40 books/articles)

A Quotation

A Poster of Sophus Lie

Mathematicians born in the same country

Cross-references to History Topics

An overview of the history of mathematics

Other references in MacTutor

Chronology: 1880 to 1890

Honours awarded to Sophus Lie (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1895

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lie.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lie.html (6 of 6) [2/16/2002 11:20:16 PM]

Lifshitz

Evgenni Mikhailovich Lifshitz Born: 21 Feb 1915 in Kharkov, Russia (now Ukraine) Died: 29 Oct 1985 in Moscow, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Evgenii M Lifshitz wrote a curriculum vitae in 1976 which provides a better description of his life than might come from other sources. We quote this CV in full, particularly as Lifshitz wrote it very much in the style of the other entries in this History Archive:I was born in 1915 in Kharkov. My father was a doctor and a professor at the Institute of Medicine. My mother is now a pensioner. After finishing secondary school in 1929, I studied for two years at the chemical college, and went in 1931 to the physics and mechanics faculty of Kharkov Mechanics and Machine Building Institute where I graduated in 1933, having completed the examinations and had a diploma thesis accepted. In 1933 I began as a graduate student at the Ukrainian Physicotechnical Institute, under Lev Landau. I completed the course and took the PhD examination in 1934. I worked at the Institute until 1938 as a senior research scientist. In 1939 my thesis for the DSc examination of Leningrad State University was accepted. Since 1939 I have worked entirely at the Academy of Sciences Institute of Physical Problems in Moscow. As well as doing scientific work, I taught at various educational institutions: Kharkov University, Kharkov Mechanics and Machine Building Institute, Kharkov Chemical Technology Institute, Moscow University and the Pedagogical Institute. For more than twenty years I have been the deputy chief editor of Zhurnal eksperimentAl-noi i teoreticheskoi fiziki. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lifshitz.html (1 of 2) [2/16/2002 11:20:17 PM]

Lifshitz

In 1966 I was elected a corresponding member of the USSR Academy of Sciences. In 1954 I was awarded a State Prize, and in 1962 the Lenin Prize jointly with Lev Landau for our Course of Theoretical Physics. The Academy of Sciences awarded me the Lomonosov Prize in 1958 and The Lev Landau Prize in 1974. Lifshitz is perhaps best remembered for his ten volume Course of Theoretical Physics written jointly with Lev Landau. They began work on this in the 1930's and the first parts of the book are based on lecture notes. Lifshitz continued to work on the book after Lev Landau's death and it was not completed until 1979. The work includes many of the results of Lifshitz's research over many years including in particular the results of many research papers written jointly with Lev Landau. The chapters of the book indicates the main topics of Lifshitz's research: Mechanics, theory of fields, quantum mechanics, quantum electrodynamics, classical statistical physics, quantum statistical physics, fluid mechanics, theory of elasticity, electrodynamics of continuous media, physical kinetics. In the latter part of his research activities, particularly after the death of Lev Landau, he work on singularities in the cosmological solutions of the equations of general relativity. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Evgenii M Lifshitz (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1982

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lifshitz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lifshitz.html (2 of 2) [2/16/2002 11:20:17 PM]

Lighthill

Michael James Lighthill Born: 23 Jan 1924 in Paris, France Died: 17 July 1998 in Sark, Channel Islands

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sir James Lighthill was known as Michael Lighthill when he was a young man. He was educated at Winchester College and, at the age of 15 he won a scholarship to Trinity College Cambridge. However, he chose to wait until he was 17 years old before entering Trinity College which he did in 1941. He graduated with a BA in 1943, after taking a course shortened because of World War II. While at Cambridge, Lighthill met Nancy Dumaresq who was studying mathematics at Newnham College. Lighthill tried to get a job in the Royal Aircraft Establishment at Farnborough after he graduated, since Nancy already had a job there. However, he was offered a job in the Aerodynamics Division of the National Physical Laboratory at Teddington. Lighthill married Nancy in 1945, the year he finished his job at the National Physical Laboratory. Lighthill was elected a fellow of Trinity College in 1945 and he held this fellowship until 1949. In 1946 he was appointed as a Senior Lecturer at Manchester University and there he set up a very strong fluid dynamics group which soon dominated research in fluids. In 1950 Lighthill was promoted to Beyer Professor of Applied Mathematics at Manchester University. In 1959 Lighthill moved from Manchester to become director of the Royal Aircraft Establishment at Farnborough. In the early 1960s he formed links between the Royal Aircraft Establishment and the Post Office to develop commercial television and communications satellites. He was also involved in plans for a manned space craft which would return to earth. His work at this time on supersonic aircraft proved to be vital in the development of the joint French-UK project for the Concorde supersonic passenger aircraft.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lighthill.html (1 of 5) [2/16/2002 11:20:20 PM]

Lighthill

In 1953 Lighthill had been elected as a fellow of the Royal Society of London and, in 1964, he became Royal Society Research Professor attached to Imperial College in London. Also at this time Lighthill, who had become unhappy with the support given to applied mathematics from government sources, founded the Institute of Mathematics and its Applications, becoming its first president in 1965-67. In 1969 Paul Dirac retired as Lucasian professor of Mathematics at the University of Cambridge and Lighthill was appointed to succeed him. Lighthill held the Lucasian chair for 10 years and was proud to hold the chair once held by Newton. He became Provost of University College London in 1979, Stephen Hawking succeeding him as Lucasian Professor of Mathematics, and Lighthill held this administrative post for 10 years until he retired in 1989. In this post Lighthill was much involved in fund raising but, despite a heavy administrative load, he continued his mathematical work studying chaotic systems, methods of extracting wave energy, and human hearing on which topic he gave the lecture Acoustic streaming in the ear itself at a conference on fluid dynamics in biology at Seattle in 1991. After Lighthill retired in 1989 he took on the position as chairman of the Special Committee on the International Decade for Natural Disaster Reduction which was sponsored by the International Council of Scientific Unions. He held this position from 1990 to 1995. He spoke on a topic retated to this Large scale hazards - tropical cyclones, earthquakes, risk, mathematics at the ICIAM 95 Conference in Hamburg in 1995. Lighthill's mathematical publications began in 1944 with publications such as Two-dimensional supersonic aerofoil theory, The conditions behind the trailing edge of the supersonic aerofoil, and Supersonic flow past bodies of revolution. Crighton, in [1], describes his work during his time at Manchester where he:... worked extensively on gas dynamics, including effects important at very high speed, in his studies of ionisation processes, and the diffraction of shock and blast waves. He also launched two major new fields in fluid mechanics. The first of these new fields was aeroacoustics which proved to be of vital importance in the reduction of noise from jet engines. He introduced this topic in two fundamental papers On sound generated aerodynamically. I. General theory and On sound generated aerodynamically. II. Turbulence as a source of sound which appeared in the Proceedings of the Royal Society of London in 1952 and 1954 respectively. On this topic he gave Lighthill's eighth power law which states that the acoustic power radiated by a jet is proportional to the eighth power of the jet speed. The second new field introduced by Lighthill during his time at Manchester was nonlinear acoustics which [1]:... was initiated by a famous 100-page article written in 1956 in honour of the 70th birthday of another great mechanics scientist Sir Geoffrey Taylor. This field is again represented now by many thousands of papers, and applications include kidney-stone-crushing lithotripsy machines and, with the same mathematics, flood waves in rivers and traffic flow on highways. Another new field introduced by Lighthill during his time as Royal Society Research Professor at Imperial College London was mathematical biofluiddynamics. In his important text Mathematical biofluiddynamics (1975) he writes:The present author as a lifelong devotee of fluid dynamics has attempted in this book to

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lighthill.html (2 of 5) [2/16/2002 11:20:20 PM]

Lighthill

demonstrate that during the past two decades there has come of age a new major division of the subject: biofluiddynamics. The first part of the book covers topics such as: swimming and flying of animals for high Reynolds number, and ciliary and flagellar propulsion in low Reynolds number. It studies the theory of fish locomotion and the flight of birds and insects. The second part of the book deals with respiratory flow and pulse propagation. It also considers blood flow, arterial disease, and microcirculation. During his time as Lucasian Professor of Mathematics, Lighthill [1]:... widened his range yet further with work on control systems; on active control of sound, or antisound; more and more on waves; on oceanography and atmospheric dynamics, including monsoon prediction and propagation; and on biological mechanics at the microscopic level. Lighthill's classic text Waves in fluids was published in 1978. In it he writes:This book is designed as a comprehensive introduction to the science of wave motions in fluids (that is, in liquids and gases), an area of knowledge which forms an essential part of the dynamics of fluids, as well as a significant part of general wave science, and, also has important applications to the sciences of the environment and of engineering. The [book] has two principal aims. First ... it allows an analysis in depth of four important and representative types of waves in fluids (sound waves, one-dimensional waves in fluids, water waves, internal waves)... At the same time, the subject matter ... is chosen so that ... all the most generally useful fundamental ideas of the science of waves in fluids can be developed at length, one after another. It should not be thought from this brief summary of Lighthill's work that he was interested only in applications of standard mathematical techniques. He did considerable work developing new mathematical tools particularly in the area of Fourier analysis and generalised functions. Lighthill certainly attracted attention in many ways such as in 1959 when he was fined 1 in a very public court case in which he was accused of jumping off a moving train. He had discovered that the train he was on did not stop at Crewe and he persuaded the guard to have the train slow down enough for him to jump out! Several times he was accused for speeding in his car. Many others would plead guilty to such an offence and pay the fine but not so Lighthill. He would successfully contest the charge by telling the magistrate that [1]:... as Lucasian Professor, he was fully seized both of the laws of mechanics and of his duty to society not to waste energy, the latter compelling him to desist from applying the brake on any downhill section of road. Swimming was one of Lighthill's joys in life. In the early 1970s he was a main speaker at the British Theoretical Mechanics Colloquium in St Andrews and on the afternoon off he chose not to go on the conference bus trip. Instead he went swimming in St Andrews bay where he was spotted far out to sea. The rescue helicopter was called out but when one of the crew was winched down to rescue him, he refused to be rescued saying that he was only out for a few miles swim and not in any trouble. In 1973 Lighthill became the first person to swim round the Channel Island of Sark [3]:He spent two weeks studying the hazardous currents before setting off one sunny morning at http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lighthill.html (3 of 5) [2/16/2002 11:20:20 PM]

Lighthill

10am. Using a 'two-arm, two-leg backstroke, thrusting with the arms and legs alternately' he reached Grande Grève after two and a half hours, and shared a picnic lunch there with Lady Lighthill. He then continued the swim, completing it by 7pm. He modestly called the nine mile swim 'a pleasant way to see the scenery'. He repeated the achievement half a dozen times before the accident that claimed his life. The accident which claimed his life was another attempt to swim around Sark. The accident was reported in [2]:Sir James Lighthill was found in rough seas off the island's rocky coast more than nine hours after he stepped into the waves for the nine mile swim. ... Before his death, he was staying at a hotel on Sark with his wife of 53 years, Nancy, and their son. He had nearly completed his swim around the island when people on the shore realised he had stopped swimming and alerted someone with a boat. Many honours from all parts of the world were bestowed on Lighthill during his distinguished career for his outstanding mathematical contributions. We noted above that he was elected a fellow of the Royal Society of London in 1953, at the age of only 29. He was awarded the Royal Medal of the Royal Society in 1964, then, between 1965 and 1969, he served the Society first as its Secretary and then as its Vice-President. He also served as president of the International Union of Theoretical and Applied Mechanics from 1984 to 1988. Among other medals and prizes he was awarded are the Gold Medal of the Royal Aeronautical Society in 1965, the Harvey Prize for Science and Technology, Israel Institute of Technology in 1981, and the Gold Medal of the Institute of Mathematics and its Applications in 1982. In 1961 Lighthill was elected a fellow of the Royal Aeronautical Society. He was also elected to the American Academy of Arts and Sciences (1958), the American Institute of Aeronautics and Astronautics (1961), the American Philosophical Society (1970), the French Academy of Sciences (1976), the US National Academy of Science (1976), and the US National Academy of Engineering (1977). Many universities have awarded Lighthill honorary doctorates including Liverpool (1961), Leicester (1965), Strathclyde (1966), Essex (1967), Princeton (1967), East Anglia (1968), Manchester (1968), Bath (1969), St Andrews (1969), Surrey (1969), Cranfield (1974), Paris (1975), Aachen (1975), Rensselaer (1980), Leeds (1983), Brown (1984), Southern California (1984), Lisbon (1986), Rehovot (1987), London (1993), Compiègne (1994), Kiev (1994), St Petersburg (1996), and Tallahassee (1996). Lighthill received the Commander Order of Léopold in 1963 and was knighted in 1971. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Honours awarded to James Lighthill (Click a link below for the full list of mathematicians honoured in this way) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lighthill.html (4 of 5) [2/16/2002 11:20:20 PM]

Lighthill

Fellow of the Royal Society

Elected 1953

Royal Society Royal Medal

Awarded 1964

Royal Society Bakerian lecturer

1961

Lucasian Professor of Mathematics

1969

Royal Society Copley Medal winner

1998

Other Web sites

1. Bob Bruen 2. Bob Bruen (An interview with Lighthill) 3. AMS (Obituary) 4. Obituary from The Independent 5. Contribution to Biological Fluid Dynamics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 1998 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Lighthill.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lighthill.html (5 of 5) [2/16/2002 11:20:20 PM]

Lindelof

Ernst Leonard Lindelöf Born: 7 March 1870 in Helsingfors, Sweden (now Helsinki, Finland) Died: 4 June 1946 in Helsinki, Finland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernst Lindelöf's father Leonard Lorenz Lindelöf was professor of mathematics in Helsingfors from 1857 to 1874. Helsingfors, today Helsinki, was controlled by Sweden and Russia at various times in its history. Finland had been ceded to Russia in 1809. At the time that time Lindelöf's father was appointed professor of mathematics at the university, the main building of the university on Senate Square had recently been completed. Helsingfors was a town of only 20,000 at this time and under Russian control. By the time that Lindelöf went to study mathematics at Helsingfors University in 1887 his father was no longer the professor there. The city was still under Russian control but it had undergone a rapid expansion and by then had a population of 60,000. Lindelöf spent the year 1891 in Stockholm, and the years 1893-94 in Paris returning to Helsingfors where he graduated in 1895. He then taught there as a docent, visiting Göttingen in 1901. He returned to Helsingfors where he became assistant professor in 1902, becoming a full professor the following year. Helsinki was still under Russian control and indeed the Russians had implemented a policy of Russification in reply to the national movements which had arisen. By 1904 the rapidly growing city had a population of 111,000 and was the centre of activists working for an independent Finland. This was proclaimed in 1917. Lindelöf remained as professor of mathematics in Helsinki until he retired in 1938. It was a time of rapid economic growth for the new country and the university flourished and rapidly expanded. Lindelöf supported his new country undertaking his university duties with great enthusiasm. From 1907 he served on the editorial board of Acta Mathematica. Lindelöf's first work in 1890 was on the existence of solutions for differential equations. It is an http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lindelof.html (1 of 3) [2/16/2002 11:20:21 PM]

Lindelof

outstanding paper. Then he worked on analytic functions, applying results of Mittag-Leffler in a study of the asymptotic investigation of Taylor series. In particular he was interested in the behaviour of such functions in the neighbourhood of singular points. He considered analogues of Fourier series and applied them to gamma functions. He also wrote on conformal mappings. His work on analytic continuation is explained in a well-written book Le calcul des résidus et ses applications à la théorie des fonctions (Paris, 1905). Oettel describes the contents of this treatise in [1]:In it he examines the role which residue theory (Cauchy) plays in function theory as a means of access to modern analysis. In this endeavour he applies the results of Mittag-Leffler. Moreover he considers series analogous to Fourier summation formulas and applications to the gamma function and the Riemann function. In addition, new results concerning the Stirling series and analytic continuation are presented. The book concludes with an asymptotic investigation of series defined by Taylor's formula. This work was translated into several different languages, including German and Finnish and Swedish and ran to several editions. Later in his life Lindelöf gave up research to devote himself to teaching and writing his excellent textbooks. In addition to the 1905 work referred to above which is largely on his own research, he wrote the textbook Differential and integral calculus and their applications which was published in four volumes between 1920 and 1946. Another fine textbook Introduction to function theory was published in 1936. Another important role which Lindelöf played in Finland was the encouragement of the study of the history of mathematics in that country. For his outstanding contributions to Scandinavian mathematics he was honoured by the universities of Uppsala, Oslo, Stockholm, and Helsinki. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lindelof.html (2 of 3) [2/16/2002 11:20:21 PM]

Lindelof

JOC/EFR March 2001

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lindelof.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lindelof.html (3 of 3) [2/16/2002 11:20:21 PM]

Lindemann

Carl Louis Ferdinand von Lindemann Born: 12 April 1852 in Hannover, Hanover (now Germany) Died: 6 March 1939 in Munich, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Ferdinand von Lindemann was the first to prove that any algebraic equation with rational coefficients.

Next

Main index

is transcendental, that is,

is not the root of

His father, also named Ferdinand Lindemann, was a modern language teacher at the Gymnasium in Hannover at the time of his birth. His mother was Emilie Crusius, the daughter of the headmaster of the Gymnasium. When Ferdinand (the subject of this biography) was two years old his father was appointed as director of a gasworks in Schwerin. The family moved to that town where Ferdinand spent his childhood years and he attended school in Schwerin. As was the standard practice of students in Germany in the second half of the 19th century, Lindemann moved from one university to another. He began his studies in Göttingen in 1870 and there he was much influenced by Clebsch. He was fortunate to be taught by Clebsch for he had only been appointed to Göttingen in 1868 and sadly he died in 1872. Later Lindemann was able to make use of the lecture notes he had taken attending Clebsch's geometry lectures when he edited and revised these note for publication in 1876. Lindemann also studied at Erlangen and at Munich. At Erlangen he studied for his doctorate and, under Klein's direction, he wrote a dissertation on non-Euclidean line geometry and its connection with non-Euclidean kinematics and statics. The degree was awarded in 1873 for the dissertation Uber unendlich kleine Bewegungen und über Kraftsysteme bei allgemeiner projektivischer Massbestimmung. After the award of his doctorate Lindemann set off to visit important mathematical centres in England

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lindemann.html (1 of 3) [2/16/2002 11:20:23 PM]

Lindemann

and France. In England he made visits to Oxford, Cambridge and London, while in France he spent time at Paris where he was influenced by Chasles, Bertrand, Jordan and Hermite. Returning to Germany Lindemann worked for his habilitation. This was awarded by the University of Würzburg in 1877 and later that year he was appointed as extraordinary professor at the University of Freiburg. He was promoted to ordinary professor at Freiburg in 1879. Lindemann became professor at the University of Königsberg in 1883. Hurwitz and Hilbert both joined the staff at Königsberg while he was there. While in Königsberg he married Elizabeth Küssner, an actress, and daughter of a local school teacher. In 1893 Lindemann accepted a chair at the University of Munich where he was to remain for the rest of his career. Lindemann's main work was in geometry and analysis. He is famed for his proof that is transcendental. The problem of squaring the circle, namely constructing a square with the same area as a given circle using ruler and compasses alone, had been one of the classical problems of Greek mathematics. In 1873, the year in which Lindemann was awarded his doctorate, Hermite published his proof that e is transcendental. Shortly after this Lindemann visited Hermite in Paris and discussed the methods which he had used in his proof. Using methods similar to those of Hermite, Lindemann established in 1882 that was also transcendental. In fact his proof is based on the proof that e is transcendental together with the fact that e i = -1. Many historians of science regret that Hermite, despite doing most of the hard work, failed to make the final step to prove the result concerning which would have brought him fame outside the world of mathematics. This fame was instead heaped on Lindemann but many feel that he was a mathematician clearly inferior to Hermite who, by good luck, stumbled on a famous result. Although there is some truth in this, it is still true that many people make their own luck and in Lindemann's case one has to give him much credit for spotting the trick which Hermite had failed to see. Lambert had proved in 1761 that was irrational but this was not enough to prove the impossibility of squaring the circle with ruler and compass since certain algebraic numbers can be constructed with ruler and compass. Lindemann's proof that is transcendental finally established that squaring the circle with ruler and compasses is insoluble. He published his proof in the paper über die Zahl in 1882. Physics was also an area of interest for Lindemann. He worked on the theory of the electron, and came into conflict with Arnold Sommerfeld on this subject. Eckert, in [4], looks at Lindemann's contributions to physics, using manuscript materials, including correspondence with Sommerfeld. Another research interest of Lindemann was the history of mathematics. He also undertook, in collaboration with his wife, translating work. In particular they translated and revised some of Poincaré's writings. Lindemann was elected to the Bavarian Academy of Sciences in 1894 as an associate member, becoming a full member in the following year. He given an honorary degree by the University of St Andrews in 1912. Wussing writes in [1]:Lindemann was one of the founders of the modern German educational system. He

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lindemann.html (2 of 3) [2/16/2002 11:20:23 PM]

Lindemann

emphasised the development of the seminar and in his lectures communicated the latest research results. He also supervised more than sixty doctoral students, including David Hilbert. Hilbert was Lindemann's doctoral student in Königsberg. Another of his doctoral students was Oskar Perron who studied under him in Munich.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) A Poster of Ferdinand von Lindemann Cross-references to History Topics

Mathematicians born in the same country 1. The beginnings of set theory 2. Squaring the circle 3. Pi through the ages

Other references in MacTutor

Chronology: 1880 to 1890

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Lindemann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lindemann.html (3 of 3) [2/16/2002 11:20:23 PM]

Linnik

Yuri Vladimirovich Linnik Born: 21 Jan 1915 in Belaya Tserkov, Ukraine Died: 30 June 1972 in Leningrad (now St Petersburg), Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yuri Linnik's father was Vladimir Pavlovitch Linnik who was himself a famous scientist making major contributions to optics. Like his son, Vladimir was elected to the USSR Academy of Sciences. After studying at secondary school Yuri Linnik worked as a laboratory assistant for a year in 1931. Then he entered Leningrad University (St Petersburg before 1914 and now St Petersburg again) to study mathematics and theoretical physics. He graduated in 1938, obtaining a doctorate there in 1940. In the same year he joined the Leningrad branch of the Steklov Institute for Mathematics. From 1944 Linnik was professor of mathematics at Leningrad University in addition to his position in the Steklov Institute. He organised the chair of probability theory there and founded the Leningrad school of probability and mathematical statistics. His main research topics were number theory, probability theory and mathematical statistics. He introduced ergodic methods into number theory in his first work. In a 1941 paper he introduced the large sieve method in number theory. In 1950 he introduced the ideas of probability into number theory and introduced the dispersion method in number theory. Later Linnik made major contributions to probability with his work on limit theorems and was the first to use powerful techniques from analysis in mathematical statistics. He solved the Behrens-Fisher problem and many other difficult problems of mathematical statistics. Linnik wrote several important texts including Characterisation Problems in Mathematical Statistics.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Linnik.html (1 of 2) [2/16/2002 11:20:25 PM]

Linnik

Linnik was President of the Moscow Mathematical Society for many years. He was honoured by the national mathematical society of Sweden and was awarded an honorary degree from Paris. He was elected to the USSR Academy of Sciences in 1964. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1940 to 1950

Other Web sites

Linnik's Constant

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Linnik.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Linnik.html (2 of 2) [2/16/2002 11:20:25 PM]

Lions

Pierre-Louis Lions Born: 11 Aug 1956 in Grasse, Alpes-Maritimes, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre-Louis Lions was born in Grasse in the Provence- Alpes- Côte- d'Azur region of France, northwest of Cannes. Grasse is not very far from Draguignan where Alain Connes, who won a Fields Medal 12 years before Lions won a Fields Medal, was born. Lions studied at the Ecole Normale Supérieure from 1975 to 1979. His thesis, supervised by H Brézis, was presented to the University of Pierre and Marie Curie in 1979. From 1979 to 1981 Lions held a research post at the Centre National de la Recherche Scientifique in Paris. Then, in 1981, he was appointed professor at the University of Paris-Dauphine. While still holding this post he was attached to the Centre National de la Recherche Scientifique as Director of Research in 1995. He has also held the position of Professor of Applied Mathematics at the Ecole Polytechnique from 1992. Lions has made some of the most important contributions to the theory of nonlinear partial differential equations through the 1980s and 1990s. Evans, in [2], writes:He has made truly fundamental discoveries cutting across many disciplines, pure and applied, and his publications are so numerous and varied as to defy easy classification. Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many physical, probalistic, geometric etc. - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods. Pierre-Louis Lions is unique in his unbelievable ability to transcend these boundaries and to solve pressing problems throughout the field.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lions.html (1 of 3) [2/16/2002 11:20:27 PM]

Lions

The references quoted [2], [3] and [4] decribe some important aspects of Lions work which led to the award of a Fields Medal at the International Congress of Mathematicians in Zurich in 1994. The first area of Lions work that is highlighted by both [1] and [3] is his work on "viscosity solutions" for nonlinear partial differential equations. The method was first introduced by Lions in joint work with M G Crandall in 1983 in which they studied Hamilton-Jacobi equations. Lions and others have since applied the method to a wide class of partial differential equations, the so-called "fully nonlinear second order degenerate elliptic partial differential equations." The problem that arises is decribed in [2]:... such nonlinear partial differential equation simply do not have smooth or even C1 solutions existing after short times. ... The only option is therefore to search for some kind of "weak" solution. This undertaking is in effect to figure out how to allow for certain kinds of "physically correct" singularities and how to forbid others. ... Lions and Crandall at last broke open the problem by focusing attention on viscosity solutions, which are defined in terms of certain inequalities holding wherever the graph of the solution is touched on one side or the other by a smooth test function. Another equally innovative piece of work by Lions was his work on the Boltzmann equation and other kinetic equations. The Boltzmann equation keeps track of interactions between colliding particles, not individually but in terms of a density. In 1989 Lions, in joint work with DiPerma, was the first to give a rigorous solution with arbitrary initial data. Another major contribution by Lions, in a long series of important papers, is to variational problems. Varadhan, speaking at the Congress of Mathematicians in Zurich in 1994 about Lions' work [4], said:There are many nonlinear PDEs that are Euler equations for variational problems. The first step in solving such equations by the variational method is to show that the extremum is attained. This requires some coercivity or compactness. If the quantity to be minimised has an "energy"-like term involving derivatives, then one has control on local regularity along a minimising sequence. Lions's clever idea was to introduce "concentration compactness" techniques which look at energy concentrations and so avoid problems which occur when examining the minimising sequences without compactness. He introduced certain measures to handle the concentrations. Lions has received many awards for his outstanding contributions to mathematics. He in a member of the French Academy of Sciences and he was awarded prizes by the Academy, the Doistau-Blutet Foundation Prize in 1986 and the Ampère Prize in 1992. He also received the IBM Prize in 1987 and the Philip Morris Prize in 1991. In addition to the Paris Academy, Lions has been elected a member of the Naples Academy and the European Academy. He is also Chevalier of the Légion d'Honneur. He is on the editorial board of around 25 journals world-wide. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lions.html (2 of 3) [2/16/2002 11:20:27 PM]

Lions

Mathematicians born in the same country Other references in MacTutor

Chronology: 1990 to 2000

Honours awarded to Pierre-Louis Lions (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1994

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lions.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lions.html (3 of 3) [2/16/2002 11:20:27 PM]

Liouville

Joseph Liouville Born: 24 March 1809 in Saint-Omer, France Died: 8 Sept 1882 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Liouville's father was an army captain in Napoleon's army so Joseph had to spend the first few years of his life with his uncle. His father was certainly fortunate to survive the wars and after Napoleon was defeated he retired to live with his family. The family then settled in Toul where Joseph attended school. From Toul he went to the Collège St Louis in Paris where he studied mathematics at the highest levels. After reading articles in Gergonne's Journal he proved some geometrical results which he wrote up as papers although they were never published. Liouville entered the Ecole Polytechnique in 1825 and attended Ampère's Cours d'analyse et de mécanique in session 1825-26. He also attended courses by Arago at the Ecole Polytechnique as well as a second course by Ampère at the Collège de France. Although Liouville does not seen to have attended any of Cauchy's courses, it is clear that Cauchy must have had a strong influence on him. Liouville graduated in 1827 with de Prony and Poisson among his examiners. After graduating from the Ecole Polytechnique Liouville entered the Ecole des Ponts et Chaussées. However his health suffered when he had to undertake engineering projects and he spent some time at his home in Toul recovering. By now Liouville was set on an academic career and he found it impossible to study away from Paris. After a number of periods of leave, one of which allowed him to marry and have a few days honeymoon, it became clear to him that he must resign from the Ecole des Ponts et Chaussées. This he did in October of 1830 but even at this stage he had written a number of papers which he had submitted to the Academy on electrodynamics, partial differential equations and the theory of heat.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Liouville.html (1 of 5) [2/16/2002 11:20:29 PM]

Liouville

In 1831 Liouville was appointed to his first academic post, as assistant to Claude Mathieu who had been appointed to Ampère's chair at the Ecole Polytechnique. He was also appointed to a number of private schools and to the Ecole Centrale. It is remarkable that during this period of his life Liouville taught between 35 and 40 hours a week at the different institutions. Perhaps with a schedule this heavy it is not surprising that some courses would not go particularly well and it appears that he lectured at too high a level for some of the less able students. In 1836 Liouville founded a mathematics journal Journal de Mathématiques Pures et Appliquées. This journal, sometimes known as Journal de Liouville, did much for mathematics in France throughout the 19th century. Liouville had already gained an international reputation with papers published in Crelle's Journal but at the same time the quality of Crelle's Journal made him aware of deficiencies in the avenues for mathematical publications which there were in France. Certainly he was unhappy with the style of the Paris Journals for he wrote in 1836:.. a peculiar spirit of emigration has seized some critics and we have seen them heap abuse on one after the other of the men who in various fields of science have honoured France with great dignity. ... this sharp and peremptory style ... will never be mine, for it dishonours both the character and talent of those who adopt it. Liouville became favourite to fill the chair at the Ecole Polytechnique which fell vacant when Navier died in 1836. However, after a close competition, Duhamel was appointed. In 1837 Liouville was appointed to lecture at the College de France as a substitute for Biot. In 1838 Liouville was appointed Professor of Analysis and Mechanics at the Ecole Polytechnique. The following year he was elected to the astronomy section of the Académie des Sciences but this was only after strong opposition from Libri. In fact the quarrel between Liouville and Libri intensified after his election to the Academy. In 1840, after a vacancy resulting from the death of Poisson, Liouville was elected to the Bureau des Longitudes. In many ways 1840 was a turning point in Liouville's career. As Lützen writes in [4]:Before 1840, Liouville had pursued some clear paths to secure his own career; during the following twenty years, the promotion and development of other mathematicians' ideas became the central issue. Life for Liouville developed into a year with two distinct parts. During the long summer period, spent at Toul, he undertook research, wrote papers and carried out editing duties. From November to July he lived in Paris and carried out his teaching and administrative duties. Not everything went Liouville's way however. When Lacroix died in 1843, Liouville applied for his chair at the Collège de France where he lectured only as a substitute for Biot. However after a close election Libri was appointed. Liouville immediately resigned from the Collège de France, writing in his resignation letter:I am profoundly humiliated as a person and as a geometer by the events that took place yesterday at the Collège de France. From this moment it is impossible for me to lecture at this institution. Another aspect of Liouville's life was his involvement in politics. One of his friends, and mathematical colleagues, was Arago who entered the Chamber of Deputies in 1831 and became leader of the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Liouville.html (2 of 5) [2/16/2002 11:20:29 PM]

Liouville

Republican Party. Other mathematical colleagues had also become involved with the political events of the time, for example Catalan, whose political views were similar to the republican views of Liouville, had damaged his mathematical career. Liouville certainly never let his political views hold him back as he advanced his mathematical career, unlike Cauchy who had refused to swear the oaths of allegiance to the King that Liouville and even Arago had been prepared to do. Encouraged by Arago, Liouville stood for election to the Constituting Assembly in 1848. In his recommendation of Liouville as a candidate Arago wrote:... Mr Liouville is one of my best friends. He is a very eminent man, a patriot, an experienced republican. God grant that the National Assembly will contain many members of this calibre. Elected on 23 April 1848, Liouville took his seat among the moderate republican majority. However there was unrest in Paris as workers felt that their revolution had been taken over by the bourgeoisie. When Arago tried to address the crowds a heckler shouted the telling comment:Mr Arago, you have never been hungry. Liouville continued his political career by being renominated for the Assembly elections in 1849 but the tide had turned against the moderate republicans and he was not elected. The election defeat proved another turning point in Liouville's life. As Lützen writes in [4]:The political defeat changed Liouville's personality. In earlier letters, he was often depressed because of illness, and could vent his anger towards his enemies such as Libri, but he always fought for what he believed was right. After the election in 1849, he resigned and became bitter, even towards his old friends. When he sat down at his desk, he did not only work, ... he also pondered his ill fate. ... his mathematical notes were interrupted with quotes from poets and philosophers... Libri escaped from France during the 1848 revolution, not for political reasons, but to avoid a prison sentence for stealing precious books and manuscripts. His chair at the Collège de France was declared vacant in 1850 and Cauchy and Liouville competed for the post. In a close contest Liouville triumphed and began his lectures at the Collège de France in 1851. Although Liouville's mathematical output had been greatly reduced while he was involved with politics, it picked up again in the 1850s despite health problems. In fact 1856 and 1857 were two of Liouville's most productive years. However after being appointed to the chair of mechanics at the Faculté des Sciences in 1857 his teaching load began to take its toll on him. Not only did he have a high teaching load but Liouville was a perfectionist which meant that when he felt that he could not devote all the time necessary to give the best possible lectures he began to suffer. He continued to publish but mostly results he had discovered during his highly productive year of 1856 and these without the proofs which now he could not find time to polish. Another blow to Liouville was the death of Dirichlet in 1859. This was a serious blow to him mathematically for, as well as losing a close friend, he lost his main mathematical correspondent. Liouville's mathematical work was extremely wide ranging, from mathematical physics to astronomy to pure mathematics. One of the first topics he studied, which developed from his early work on electromagnetism, was a new topic, now called the fractional calculus. He defined differential operators http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Liouville.html (3 of 5) [2/16/2002 11:20:29 PM]

Liouville

of arbitrary order Dê. Usually t is an integer but in this theory developed by Liouville in papers between 1832 and 1837, t could be a rational, an irrational or most generally of all a complex number. Liouville investigated criteria for integrals of algebraic functions to be algebraic during the period 1832-33. Having established this in four papers, Liouville went on to investigate the general problem of integration of algebraic functions in finite terms. His work at first was independent of that of Abel, but later he learnt of Abel's work and included several ideas into his own work. Another important area which Liouville is remembered for today is that of transcendental numbers. Liouville's interest in this stemmed from reading a correspondence between Goldbach and Daniel Bernoulli. Liouville certainly aimed to prove that e is transcendental but he did not succeed. However his contributions were great and led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions. In 1851 he published results on transcendental numbers removing the dependence on continued fractions. In particular he gave an example of a transcendental number, the number now named the Liouvillian number 0.1100010000000000000000010000... where there is a 1 in place n! and 0 elsewhere. His work on boundary value problems on differential equations is remembered because of what is called today Sturm-Liouville theory which is used in solving integral equations. This theory, which has major importance in mathematical physics, was developed between 1829 and 1837. Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions. Liouville contributed to differential geometry studying conformal transformations. He proved a major theorem concerning the measure preserving property of Hamiltonian dynamics. The result is of fundamental importance in statistical mechanics and measure theory. In 1842 Liouville began to read Galois's unpublished papers. In September of 1843 he announced to the Academy that he had found deep results in Galois's work and promised to publish Galois's papers together with his own commentary. Liouville was therefore a major influence in bringing Galois's work to general notice when he published this work in 1846 in his Journal. However he had waited three years before publishing the papers and, rather strangely, he never published his commentary although he certainly wrote a commentary which filled in the gaps in Galois's proofs. Liouville also lectured on Galois's work and Serret, possibly together with Bertrand and Hermite, attended the course. In number theory Liouville wrote around 200 papers, working on quadratic reciprocity and many other topics. He wrote over 400 papers in total. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (16 books/articles) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Liouville.html (4 of 5) [2/16/2002 11:20:29 PM]

Liouville

A Poster of Joseph Liouville

Mathematicians born in the same country

Cross-references to History Topics

1. A comment from Thomas Hirst's diary 2. Fermat's last theorem 3. The development of group theory 4. Orbits and gravitation

Other references in MacTutor

1. Chronology: 1830 to 1840 2. Chronology: 1840 to 1850 3. Chronology: 1850 to 1860

Honours awarded to Joseph Liouville (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1850

Lunar features

Crater Liouville

Paris street names

Rue Joseph Liouville (15th Arrondissement)

Other Web sites

1. Liouville-Roth Constants 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Liouville.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Liouville.html (5 of 5) [2/16/2002 11:20:29 PM]

Lipschitz

Rudolf Otto Sigismund Lipschitz Born: 14 May 1832 in Königsberg, Germany (now Kaliningrad, Russia) Died: 7 Oct 1903 in Bonn, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Rudolf Lipschitz's father was a landowner and Rudolf was born his father's estate at Bönkein which was near Königsberg. He began his university studies at a young age, entering the University of Königsberg and studying there under Franz Neumann. Following the custom of that time to study at different universities, Lipschitz went from Königsberg to Berlin where he studied under Dirichlet. This was not a particularly easy time for Lipschitz whose health was rather poor and caused him to take a year away from his studies to recover. However, he completed his doctoral studies with the award of a doctorate on 9 August 1853. There was no immediate university teaching post for Lipschitz who spent four years teaching at the Gymnasium in Königsberg and at the Gymnasium in Elbing. In 1857, however, Lipschitz became a Privatdozent at the University of Berlin. In this same year he married Ida Pascha, the daughter of one of the landowners with an estate near to his father's. Then in 1862 he became an extraordinary professor at Breslau. During his two years in Breslau, Lipschitz wrote two not very important papers. Jointly with Heinrich Schroeter and M Frankenheim, he founded a seminar in mathematics and mathematical physics. The paper [5] looks at Lipschitz's career during these two years. He was nominated an ordinary professor by the University of Bonn and he left Breslau at Easter 1864. The University of Bonn was where Lipschitz spent the rest of his career. This was not because he did not have the opportunity to move. Quite the reverse, after Clebsch died in November 1872 he was offered his

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lipschitz.html (1 of 3) [2/16/2002 11:20:31 PM]

Lipschitz

chair at Göttingen in the following year. Lipschitz was quite happy at Bonn, however, and he turned down the offer from Göttingen. Klein received his doctorate from the University of Bonn in 1868. He was supervised by Plücker, and examined by Lipschitz. Perhaps if Klein had still been in Göttingen when Lipschitz was offered the chair there, he may have been more inclined to accept. Perhaps the most remarkable fact about Lipschitz's work was the widely different topics on which he contributed [1]:He carried out many important and fruitful investigations in number theory, in the theory of Bessel functions and of Fourier series, in ordinary and partial differential equations, and in analytical mechanics and potential theory. He worked on quadratic differential forms and mechanics. In the paper [4] the author shows convincingly how Lipschitz mechanical interpretation of Riemann's differential geometry would prove to be a vital step in the road towards Einstein's special theory of relativity. Lipschitz showed that [4]:... the geometrical statements could be interpreted as mechanical laws [but these were] the very mechanical concepts that made it possible to deepen the corresponding geometrical relations. Lipschitz's work on the Hamilton-Jacobi method for integrating the equations of motion of a general dynamical system led to important applications in celestial mechanics. Lipschitz is remembered for the 'Lipschitz condition', an inequality that guarantees a unique solution to the differential equation y' = f(x,y). Peano gave an existence theorem for this differential equation, giving conditions which guarantee at least one solution. His work in algebraic number theory led him to study the quaternions and generalisations such as Clifford algebras. In fact Lipschitz rediscovered Clifford algebras and was the first to apply them to represent rotations of Euclidean spaces, thus introducing the spin groups Spin(n). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Rudolf Lipschitz

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lipschitz.html (2 of 3) [2/16/2002 11:20:31 PM]

Lipschitz

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lipschitz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lipschitz.html (3 of 3) [2/16/2002 11:20:31 PM]

Lissajous

Jules Antoine Lissajous Born: 4 March 1822 in Versailles, France Died: 24 June 1880 in Plombières, France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jules Lissajous entered Ecole Normale Supérieure in 1841. Afterwards he became professor of mathematics at the Lycée Saint-Louis. In 1850 he was awarded a doctorate for a thesis on vibrating bars using Chladni's sand pattern method to determine nodal positions. In 1874 Lissajous became rector of the Academy at Chambéry, then in 1875 he was appointed rector of the Academy at Besançon. Lissajous was interested in waves and developed an optical method for studying vibrations. At first he studied waves produced by a tuning fork in contact with water. In 1855 he described a way of studying acoustic vibrations by reflecting a light beam from a mirror attached to a vibrating object onto a screen. Duhamel had tried to demonstrate these vibrations with a mechanical linkage but Lissajous wanted to avoid the problems caused by the linkage. He obtained Lissajous figures by successively reflecting light from mirrors on two tuning forks vibrating at right angles. The curves are only seen because of persistence of vision in the human eye. Lissajous studied beats seen when his tuning forks had slightly different frequencies, in this case a rotating ellipse is seen. Lissajous was awarded the Lacaze Prize in 1873 for his work on the optical observation of vibration. Helmholtz used Lissajous' instruments in his study of string vibrations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lissajous.html (1 of 2) [2/16/2002 11:20:33 PM]

Lissajous

List of References (2 books/articles) Mathematicians born in the same country Cross-references to Famous Curves

Lissajous Curves

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lissajous.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lissajous.html (2 of 2) [2/16/2002 11:20:33 PM]

Listing

Johann Benedict Listing Born: 25 July 1808 in Frankfurt am Main, Germany Died: 24 Dec 1882 in Göttingen, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Benedict Listing's family were of Czech descent. His father, also named Johann Benedict Listing, was a maker of brushes while his mother, Caroline Friederike Listing, came from a poor peasant background. Listing was an only child and brought up in a family which struggled with financial difficulty. He was a bright boy and his talents were such that he received help with his education from several benefactors including the Städel foundation, supporters of art and museums. He received a good foundation for his education at the Musterschule which he attended from the age of eight. In this school he first became interested in science and mathematics, mainly because of his talented teacher Müller, but he had a real talent for art. Listing pleased his parents by helping out with the family's finances by earning a little money from drawing and calligraphy from the age of thirteen. In 1825 Listing entered a Gymnasium where he studied for five years. He mastered English, French, Italian and Latin and well as increasing his knowledge of mathematics and science at this school. He had already decided to pursue an academic career and his talents were recognised by the award of a generous scholarship by the Städel foundation. Being an art foundation they could not support Listing to undertake a degree in mathematics, which would have been his preferred option, but instead awarded him the scholarship for the study of mathematics and architecture. Although there was a time when architecture was considered as a branch of mathematics, it certainly was not considered such in Listing's day so the combination was a rather peculiar one dictated by a compromise between Listing's wishes and the remit of the Städel foundation.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Listing.html (1 of 4) [2/16/2002 11:20:35 PM]

Listing

Listing entered Göttingen University in 1830 and attended a remarkably broad range of courses, much broader than the mathematics and architecture specified by his scholarship. In addition to these two topics he also took courses on astronomy, anatomy, physiology, botany, mineralogy, geology and chemistry. Soon Listing was attending mathematics courses given by Gauss and he was quickly spotted by Gauss as being both a very able and a very hard working student. Gauss invited him to join his circle of friends who included Weber. Listing was not the only student invited into close friendship with Gauss. Another student was Walterhausen who was one year younger than Listing. Walterhausen, who was interested in geology, became a lifelong friend of Listing. The influence of Gauss on Listing was, however, very marked. It was from Gauss that Listing began to learn topological concepts. He also was a talented experimenter and collaborated with Gauss in physics experiments, particularly those relating to terrestrial magnetism. Gauss became the supervisor of Listing's dissertation De superficiebus secundi ordinis which was on surfaces of the second degree and ternary forms. He received his doctorate in June 1834. Sartorius, a geologist at Göttingen, was setting out on a trip to Scilly to study volcanoes. He had also agreed to collect data on terrestrial magnetism for Gauss on this trip. Listing was an obvious choice for an assistant to accompany him and the pair set off less than a month after Listing had obtained his doctorate. It was a journey on which Listing maintained his interests in many different topics and as well as the geology and physics which were the purpose of the trip, he worked in his spare time on mathematics, in particular working on the topological ideas which had first been suggested by Gauss. He decided to summarise his thoughts on the topic and did so in a long letter to his old school teacher Müller. It is in this letter that the word "topology" appears for the first time. He disliked the term "geometria situs", then used for topological ideas, and [4]:The entire doctrine being rather new, he felt justified to give it a new name and therefore called it "topology", which he though more appropriate. While on his travels Listing was approached by the Höhere Gewerbeschule Hannover asking if he would be interested in a post as a teacher of applied mathematics. Without saying yes or no he politely thanked them for the offer. Because of a cholera epidemic Sartorius and Listing delayed their return to Göttingen. In fact they took a ship to England where they spent a short while before returning to Germany. Listing went to Hannover to be interviewed for the post in the Gewerbeschule and was appointed, starting his teaching career in November 1837. When Victoria became Queen of Britain in 1837 her uncle became ruler of Hanover and revoked the liberal constitution. Weber, the professor of physics at Göttingen and a close collaborator of Gauss, was one of seven professors at the university to sign a protest and all seven were dismissed. Weber remained at Göttingen without a position until 1843. However his professorship of physics had to be filled and after a while Gauss was asked to suggest possible candidates. He produced a list of three names with Listing in third place. When the first two turned down the offer, Listing was appointed in 1839 despite never having published a paper. As professor of physics Listing could choose his area for research. A versatile scientist, he chose yet another area to the ones which he had already worked in, and began to study the optics of the human eye. He published Beitrage zur physiologischen Optik in 1845 which became a classic. As well as containing data from careful experimental work, the book was beautifully illustrated by Listing using all his skills in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Listing.html (2 of 4) [2/16/2002 11:20:35 PM]

Listing

drawing and calligraphy. In September 1846 Listing married Pauline Elvers. Almost immediately the couple were in financial problems as Pauline seemed unable to keep her spending within the family's income. Listing continued to think about topological ideas, however, and he wrote the book Vorstudien zur Topologie in 1847. It was the first published use of the word topology although, as we mentioned above, it was first used in Listing's letter of 1836. The subject was known as analysis situs for many years and only in the late 1920's was the English word topology used by Lefschetz. In this book, really an extended essay meant only as the title states as a preliminary work, Listing writes:By topology we mean the doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity. In 1848 the revolution which swept Europe had its consequences. The most marked effect was that Weber was reappointed to the chair at Göttingen which he had lost ten years before. Of course Listing had been appointed to fill Weber's chair so a compromise had to be reached. This was that Listing was promoted to ordinary professor of mathematical physics while Weber became professor of experimental physics. Although little changed in the research that Listing undertook, he had always gone in the direction which interested him, he did have to give up a large part of his laboratory to Weber. However, Listing's family life did not go well. The couple had two daughters - one in 1848 and the second in the following year. However Pauline Listing's [4]:... treatment of servants brought her before the magistrates any number of times, while her relations with landlords led to many moves for the family. Neither Listing nor his wife seemed capable of managing the family finances [4]:Listing borrowed frequently and heavily, sometimes from usurers; Pauline habitually abused credit, and again ended up in court with some regularity. they tended to live beyond their means, and on one occasion barely avoided bankruptcy. The near bankruptcy came around the time that Listing was publishing another remarkable contribution to topology. In 1858 he had discovered the properties of the Möbius band at almost the same time, and independently of, Möbius. In 1862 he published Der Census raumlicher Complexe oder Verallgemeinerung des Euler'schen Satzes von den Polyedern which discusses extensions of Euler's formula for the Euler characteristic of oriented three- dimensional polyhedra to the case of certain four-dimensional simplicial complexes. This second work on topology by Listing is discussed by detail in [7] where Tripodi compares it with some work by Cayley. In fact bankruptcy was avoided due to the ministry in Hannover saving him. It was Sartorius who had arranged for the rescue act by the ministry and here he was repaying a debt to Listing who had saved his life by nursing him through a serious illness during their journey of nearly thirty years earlier. However, Listing did not find favour with his colleagues because of the behaviour of himself and his wife and it is certain that he received less recognition for his scientific achievements because of it. Yet he was [4]:... industrious and inquisitive, kind and helpful, gregarious and witty, good-natured to a degree, a true friend to many, and an enemy to none. The range of contributions made by Listing is quite remarkable. Breitenberger in [4] writes:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Listing.html (3 of 4) [2/16/2002 11:20:35 PM]

Listing

... he studied the figure of the earth in minute detail; he made observations in meteorology, terrestrial magnetism, and spectroscopy; he wrote on the quantitative determination of sugar in the urine of diabetics; he promoted the nascent optical industry in Germany and better street lighting in Göttingen; he travelled to the world exhibitions in London 1851, Vienna 1873 an London 1876 as an observer for his government; he assisted in geodetic surveys; ... he invented a good many terms [other than topology], some of which have became current: "entropic phenomenona", "nodal points", "homocentric light", "telescopic system", " geoid" ...he coined "one micron" for the millionth of a metre ... Among the honours which Listing did receive were election to the Göttingen Academy and the Royal Society of Edinburgh. He was awarded an honorary doctorate by Tübingen. He died of a stroke. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) A Poster of Johann Benedict Listing

Mathematicians born in the same country

Cross-references to History Topics

Topology enters mathematics

Other references in MacTutor

Chronology: 1860 to 1870

Honours awarded to Johann Benedict Listing (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Listing.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Listing.html (4 of 4) [2/16/2002 11:20:35 PM]

Littlewood

John Edensor Littlewood Born: 9 June 1885 in Rochester, Kent, England Died: 6 Sept 1977 in Cambridge, Cambridgeshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John E Littlewood studied at Trinity College Cambridge. From 1907 to 1910 he lectured at the University of Manchester. He became a fellow of the Trinity College (1908) returning there in 1910. He was to become Rouse Ball professor of mathematics there in 1928. In World War I Littlewood served in the Royal Garrison Artillery. For 35 years he collaborated with G H Hardy working on the theory of series, the Riemann zeta function, inequalities and the theory of functions. The collaboration led to a series of papers Partito numerorum using the Hardy- Littlewood- Ramanujan analytical method. Hardy wrote of Littlewood that he knew of no one else who could command such a combination of insight, technique and power. Littlewood was seldom seen outside Cambridge, in fact there were jokes around that he was the invention of Hardy. In the late 1930's the Department of Scientific and Industrial Research tried to interest pure mathematicians in nonlinear differential equations which were important for radio engineers and scientists because they described the behaviour of electric circuits. The impending war motivated the interest. Littlewood, working jointly with Mary Cartwright, spent 20 years working on equations of this type such as van der Pol's equation. Littlewood was elected a Fellow of the Royal Society in 1915. He received the Royal Medal of the Society in 1929, the Sylvester Medal of the Society in 1943:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Littlewood.html (1 of 3) [2/16/2002 11:20:37 PM]

Littlewood

... in recognition of his mathematical discoveries and supreme insoght in the analytic theory of numbers. He also received the Copley Medal of the Society in 1958:... in recognition of his distinguished contributions to many branches of analysis, including Tauberian theory, the Riemann zeta-function, and non-linear differential equations. In 1953 he wrote the book A Mathematician's Miscellany which continues to be one of the best and most accessible popular books written by a mathematician. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles)

Some Quotations (24)

A Poster of John E Littlewood

Mathematicians born in the same country

Honours awarded to John E Littlewood (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1916

Royal Society Copley Medal

Awarded 1929

Royal Society Royal Medal

Awarded 1929

Royal Society Sylvester Medal

Awarded 1943

London Maths Society President

1941 - 1943

LMS De Morgan Medal

Awarded 1938

LMS Berwick Prize winner

1960

Other Web sites

1. The Prime Pages (The Prime Number Theorem) 2. Hardy-Littlewood Constants

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Littlewood.html (2 of 3) [2/16/2002 11:20:37 PM]

Littlewood

JOC/EFR November 1997

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Littlewood.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Littlewood.html (3 of 3) [2/16/2002 11:20:37 PM]

Littlewood_Dudley

Dudley Ernest Littlewood Born: 7 Sept 1903 in London, England Died: 6 Oct 1979 in Bangor, Wales

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dudley Littlewood was educated at Trinity College, Cambridge where his undergraduate tutor was Littlewood (J E) who was not related. He graduated in 1925, the same year as Hall and Hodge also graduated from Cambridge. He began research in analysis at Cambridge but it appears that he was neither sufficiently good or interested in analysis and, lacking financial support, decided to give up research. Dudley's first appointment was as a school teacher, but, in 1928, he found a post as a lecturer at University College Swansea. He worked for a short time at Queen's College, Dundee (at that time part of the University of St Andrews) but returned to Swansea where he worked until 1947. Dudley was keen to return to Cambridge and, when the chance came in 1947, he accepted a post as College lecturer. It was not a College appointment so he only had an office through the kindness of Hodge. Littlewood's family were not happy with the move to Cambridge but soon he was appointed to the chair of mathematics at Bangor in 1948. Until Littlewood's appointment to Swansea he had no definite research interests. However at Swansea the professor, A R Richardson, was an algebraist and he introduced Littlewood to research in algebra. His first work was on quaternion algebras and some of his first papers were written jointly with A R Richardson. During this period, developments of his first papers led to further work in which he laid the foundations of invariant theory of forms in non-commutative algebra. Invariant theory was at its height in the 19th Century with the work of Cayley, Sylvester, Clebsch,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Littlewood_Dudley.html (1 of 3) [2/16/2002 11:20:39 PM]

Littlewood_Dudley

Gordan and others. Littlewood claimed that interest in invariant theory had flagged somewhat, one reason for this being the introduction of tensors. Another reason was certainly the work of Hilbert, but Littlewood tried to remedy the "tensor reason" in a series of papers on tensors and invariant theory. Littlewood's main work, however, began in 1934 when he began an investigation of group characters, in particular the characters of the symmetric group. He examined S-functions (named after Schur) and applied these to invariant theory. He also studied quantum mechanics and some of the problems in representation theory he considered were motivated from this. He published three books perhaps the first The theory of group characters and matrix representations of groups (1940) being the most famous. J A Green, a student of Littlewood's, summed up his approach to mathematics writing:Littlewood's mathematical strength lay in his extraordinary insight into the way certain algebraic processes worked. In [1] the authors write:He clearly had a strong intuitive grasp of formal mathematics and when he felt a result to be true he could be perfunctory about its proof. Littlewood had a great love for the works of Frobenius, Schur and Weyl - these were mathematicians who produced the kind of usable formulae which he could and did appreciate. But he did not appreciate their mathematical rigour, he grasped their methods and results and he proceeded to develop them in his own fashion. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Littlewood_Dudley.html (2 of 3) [2/16/2002 11:20:39 PM]

Littlewood_Dudley

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Littlewood_Dudley.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Littlewood_Dudley.html (3 of 3) [2/16/2002 11:20:39 PM]

Livsic

Mikhail Samuilovich Livsic Born: 4 July 1917 in Pokotilova (near Uman), Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Moshe Livsic's family moved to Odessa when he was four years old since his father had been appointed professor of mathematics at the Academy there. The family were Jewish and Livsic brought up in a very religious household. He was introduced to mathematics from a young age at his home where Chebotaryov, Kagan and Shatunovsky were frequent visitors. Livsic graduated from school in 1931 and, with a school friend, planned his future studies. His aim was to study philosophy but first he would have to study physical sciences. However to be successful in the physical sciences Livsic and his friend would first have to become knowledgeable in mathematics. However, as he wrote, see [1]:I succeeded in thoroughly studying some fields in mathematics. I succeeded less in studying some fields in physics. There was no time for philosophy. Livsic began studies at Odessa technical College aimed at his becoming a radio technician but, in 1933, he entered the Department of Physics and Mathematics at Odessa State University. This department had only recently opened. There he was taught by M G Krein, M A Naimark and, a couple of years later, by B Ya Levin. He was particularly interested in the courses in complex variable, integral equations and differential equations. These courses presented the latest research results. Livsic took part in the functional analysis seminar and studied analytic functions. In fact [1]:He used the ideas, techniques and methods of analytic function theory throughout all his research. He remained at Odessa working for his doctorate under Krein's supervision. However his interests changed over time. His Master's Degree was achieved with a thesis in quasianalytic functions, then he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Livsic.html (1 of 3) [2/16/2002 11:20:41 PM]

Livsic

became interested in operator theory which came out of earlier work he had done on the moment problem. At this time Livsic's studies were much influenced from his reading works by M H Stone, von Neumann, N I Akhiezer and A Y Plessner. During World War II Livsic avoided military service since his eyesight was poor. However he had to leave Odessa as the German armies advanced and his studies were somewhat disrupted. It was 1942 before he was able to defend his doctoral thesis on Hermitian operator theory and the generalised moment problem. His habilitation thesis on generalisations of von Neumann's extension theory was examined in 1945 by a powerful groups of mathematicians, namely Banach, Gelfand, Naimark and Plessner at the Steklov Institute. At the end of World War II, Livsic as not invited to return to his post at Odessa since:... he was not suited to represent the Ukrainian culture... Krein was dismissed from his post and the school of functional analysis closed. After working at Kirovograd, Livsic returned to Odessa to teach at the Hydrometerological Institute which was a minor institution. He remained there until 1957, publishing results on applications of his functional analysis results to quantum theory. From 1957 until 1962 Livsic was head of mathematics at Kharkov Mining Institute, joining Akhiezer at Kharkov State University in 1962. In 1975 Kharkov made active moves to enable him to go to Israel. He first took a post in the Institute of Agricultural Machines in Tbilisi, then after three years he was able to go to Israel where he was appointed to Ben Gurion University in 1978. Again he began building a research school in operator theory. He is described in [2] as follows:Moshe has always been very active and dedicated to his work. His recent breakthroughs in the theory of characteristic functions for several commuting operators indicate that in spite of his seventy years, mathematically Moshe is still a young man. He is loved by his friends, colleagues and students for his wonderful qualities of integrity, honour and benevolence, and respected as a great mathematician. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Livsic.html (2 of 3) [2/16/2002 11:20:41 PM]

Search Form

Livsic

Glossary index

Quotations index

Mathematicians of the day JOC/EFR June 1997

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Livsic.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Livsic.html (3 of 3) [2/16/2002 11:20:41 PM]

Llull

Ramon Llull Born: 1235 in Majorca, Spain Died: 1316 in (probably) Tunis, Tunisia Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ramon Llull was brought up at the royal court of Majorca. There he learnt Arabic from the Moorish population. Llull left Majorca in about 1265 after he experienced mystical visions of Christ on the Cross. He then undertook missionary work in North Africa and Asia Minor where he taught the pacifist doctrines on St Francis of Assisi. Llull's main work is the Ars magna (1305-08), comprising of a number of treatises including The Tree of Knowledge and The Book of the Ascent and Descent of the Intellect. This work was inspired by another mystical experience related in [1]:About 1272, after another mystical experience on Majorca's Mount Randa in which Llull related seeing the whole universe reflecting the divine attributes, he conceived of reducing all knowledge to first principles and determining their convergent point of unity. Llull used logic and mechanical methods involving symbolic notation and combinatorial diagrams to relate all forms of knowledge. This work makes him a precursor of combinatorics and this aspect is fully discussed in [3]. It is unclear exactly how Llull died. In [1] it is related that:... according to legend, Llull was stoned in North Africa at Bejaia (Bougie) or Tunis and died a martyr at sea before reaching Majorca, where he was buried. Llull's reduction of Christianity to rational discussion, in which he attempted to prove the dogmas of the Church by logical argument, did not find favour after his death. In 1376 Pope Gregory XI charged Llull with confusing faith with reason and condemned his teachings. The Roman Catholic Church did however pardon Llull more quickly than Galileo, since he was venerated in the 19th century. Llull's work is important for a number of reasons, one certainly being the great influence it had on Leibniz. One of Llull's early works on chivalry was translated into English and thought important enough by the early printer William Caxton that he edited and printed the work. Llull's mystical writings such as The Book of the Lover and the Beloved are still popular and his work has an important role in Catalan culture. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Llull.html (1 of 2) [2/16/2002 11:20:42 PM]

Llull

List of References (4 books/articles)

Some Quotations (3)

Mathematicians born in the same country Other Web sites

1. More about Llull. 2. Geocities (Ramon Llull's Home page) 3. Aalborg, Denmark 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Llull.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Llull.html (2 of 2) [2/16/2002 11:20:42 PM]

Lobachevsky

Nikolai Ivanovich Lobachevsky Born: 1 Dec 1792 in Nizhny Novgorod (was Gorky from 1932-1990), Russia Died: 24 Feb 1856 in Kazan, Russia

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nikolai Ivanovich Lobachevskii's father Ivan Maksimovich Lobachevskii, worked as a clerk in an office which was involved in land surveying while Nikolai Ivanovich's mother was Praskovia Alexandrovna Lobachevskaya. Nikolai Ivanovich was one of three sons in this poor family. When Nikolai Ivanovich was seven years of age his father died and, in 1800, his mother moved with her three sons to the city of Kazan in western Russia on the edge of Siberia. There the boys attended Kazan Gymnasium, financed by government scholarships, with Nikolai Ivanovich entering the school in 1802. In 1807 Lobachevskii graduated from the Gymnasium and entered Kazan University as a free student. Kazan State University had been founded in 1804, the result of one of the many reforms of the emperor Alexander I, and it opened in the following year, only two years before Lobachevskii began his undergraduate career. His original intention was to study medicine but he changed to study a broad scientific course involving mathematics and physics. Vinberg writes [44]:In the first years the atmosphere in the Department was quite favourable. The students were full of enthusiasm. They studied day and night to compensate for lack of knowledge. The professors, mainly invited from Germany, turned out to be excellent teachers, which was not common. Lobachevskii was highly successful in all courses he took ... One of the excellent professors who had been invited from Germany was Martin Bartels (1769 - 1833) who had been appointed as Professor of Mathematics. Bartels was a school teacher and friend of Gauss, and the two corresponded. We shall return later to discuss ideas of some historians, for example M Kline, that Gauss may have given Lobachevskii hints regarding directions that he might take in his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lobachevsky.html (1 of 5) [2/16/2002 11:20:45 PM]

Lobachevsky

mathematical work through the letters exchanged between Bartels and Gauss. A skilled teacher, Bartels soon interested Lobachevskii in mathematics. We do know that Bartels lectured on the history of mathematics and that he gave a course based on the text by Montucla. Since Euclid's Elements and his theory of parallel lines are discussed in detail in Montucla's book, it seems likely that Lobachevskii's interest in the Fifth Postulate was stimulated by these lectures. Laptev, see [29], has established that Lobachevskii attended this history course given by Bartels. Lobachevskii received a Master's Degree in physics and mathematics in 1811. In 1814 he was appointed to a lectureship and in 1816 he became an extraordinary professor. In 1822 he was appointed as a full professor [1]:... the same year in which he began an administrative career as a member of the committee formed to supervise the construction of new university buildings. Lobachevskii had experienced difficulties during this period at the University of Kazan. Struik writes in [2] that the administration, led by the curator M L Magnitskii:... reflected the spirit of the later years of Tsar Alexander I, who was distrustful of modern science and philosophy, particularly that of the German philosopher Immanuel Kant, as evil products of the French Revolution and a menace to orthodox religion. The results at Kazan during the years 1819-26 were factionalism, decay of academic standards, dismissals, and departure of some of the best professors, including ... Bartels ... Despite these difficulties, many brought on according to Vinberg in [44] by Lobachevskii's "upright and independent character", he achieved many things. As well as his vigorous mathematical research, which we shall talk about later in this article, he taught a wide range of topics including mathematics, physics and astronomy. His lectures [44]:... were detailed and clear, so that they could be understood even by poorly prepared students. Lobachevskii bought equipment for the physics laboratory, and he purchased books for the library in St Petersburg. He was appointed to important positions within the university such as the dean of the Mathematics and Physics Department between 1820 and 1825 and head librarian from 1825 to 1835. He also served as Head of the Observatory and was clearly strongly influencing policy within the University. However [30]:... the clashes with the curator [Magnitskii] continued. In 1826 Tsar Nicholas I became ruler and introduced a more tolerant regime. In that year Magnitskii was dismissed as curator of Kazan University and a new curator M N Musin-Pushkin was appointed. The atmosphere now changed markedly and Musin-Pushkin found in Lobachevskii someone who could work with him in bringing important changes to the university. In 1827 Lobachevskii became rector of Kazan University, a post he was to hold for the next 19 years. The following year he made a speech (which was published in 1832) On the most important subjects of education and this gives clearly what were the ideas in his educational philosophy. Laptev writes in that Lobachevskii [30]:... outlined the ideal of the harmonious development of the personality, emphasised the social significance of upbringing and education, and discussed the role of the sciences and the scientist's duty to his country and people. The University of Kazan flourished while Lobachevskii was rector, and this was largely due to his influence. There was a vigorous programme of new building with a library, an astronomical observatory, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lobachevsky.html (2 of 5) [2/16/2002 11:20:45 PM]

Lobachevsky

new medical facilities, and physics, chemistry, and anatomy laboratories being constructed. He pressed strongly for higher levels of scientific research and he equally encouraged research in the arts, particularly developing a leading centre for Oriental Studies. There was a marked increase in the number of students and Lobachevskii invested much effort in raising not only the standards of education in the university, but also in the local schools. Two natural disasters struck the university while he was Rector of Kazan [44]:... a cholera epidemic in 1830 and a big fire in 1842. Owing to resolute and reasonable measures taken by Lobachevskii the damage to the University was reduced to a minimum. for his activity during the cholera epidemic Lobachevskii received a message of thanks from the Emperor. The book [5] contains some yearly reports Lobachevskii wrote as rector of Kazan University. Those published are only a small sample taken from the hundreds of pages of manuscript:... written in [Lobachevskii's] full, firm hand, with hardly an error, let alone a crossing-out, reports which were an obstacle to real work in the path of all academics then as now. Despite this heavy administrative load, Lobachevskii continued to teach a variety of different topics such as mechanics, hydrodynamics, integration, differential equations, the calculus of variations, and mathematical physics. He even found time to give lectures on physics to the general public during the years 1838 to 1840 but the heavy work-load was to eventually take its toll on his health. In 1832 Lobachevskii married Lady Varvara Alexivna Moisieva who came from a wealthy family. At the time of his marriage his wife was a young girl while Lobachevskii was forty years old. The marriage gave them seven children and it is claimed in [1] that the children:... and the cost of technological improvements for his estate left him with little money upon his retirement. In [44] Vinberg writes:The couple lived in a big three-storey house and received a lot of guests with lavish hospitality. However Lobachevskii was not lucky in his marriage. After Lobachevskii retired in 1846 (essentially dismissed by the University of Kazan), his health rapidly deteriorated. Matveev, in his article [34], quotes many records concerning Lobachevskii's estate which he purchased at Slobodka. There are many claims by biographers that:Lobachevskii was an impractical manager who jeopardised his financial position by purchasing the estate while living on a pension; that he had no time to look after the estate and took little interest in it; that he was left in poverty and ignored by the local officials, etc. but Matveev shows that these claims are totally unjustified. Soon after he retired, however, his favourite eldest son died and Lobachevskii was hit hard by this tragedy. The illness was he suffered from became progressively worse and led to blindness. These and financial difficulties added to the heavy burdens he had to bear over his last years. His great mathematical achievements, which we shall now discuss, were not recognised in his lifetime and he died without having any notion of the fame and importance that his work would achieve. Since Euclid's axiomatic formulation of geometry mathematicians had been trying to prove his fifth postulate as a theorem deduced from the other four axioms. The fifth postulate states that given a line and a point not on the line, a unique line can be drawn through the point parallel to the given line. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lobachevsky.html (3 of 5) [2/16/2002 11:20:45 PM]

Lobachevsky

Lobachevskii did not try to prove this postulate as a theorem. Instead he studied geometry in which the fifth postulate does not necessarily hold. Lobachevskii categorised euclidean as a special case of this more general geometry. His major work, Geometriya completed in 1823, was not published in its original form until 1909. On 11 February 1826, in the session of the Department of Physico-Mathematical Sciences at Kazan University, Lobachevskii requested that his work about a new geometry was heard and his paper A concise outline of the foundations of geometry was sent to referees. The text of this paper has not survived but the ideas were incorporated, perhaps in a modified form, in Lobachevskii's first publication on hyperbolic geometry. He published this work on non-euclidean geometry, the first account of the subject to appear in print, in 1929. It was published in the Kazan Messenger but rejected by Ostrogradski when it was submitted for publication in the St Petersburg Academy of Sciences. In 1834 Lobachevskii found a method for the approximation of the roots of algebraic equations. This method of numerical solution of algebraic equations, developed independently by Gräffe to answer a prize question of the Berlin Academy of Sciences, is today a particularly suitable for methods for using computers to solve such problems. This method is today called the Dandelin-Gräffe method since Dandelin also independently investigated it, but only in Russia does the method appear to be named after Lobachevskii who is the third independent discoverer. See [24] for a discussion of the method and its three discoverers. In 1837 Lobachevskii published his article Géométrie imaginaire and a summary of his new geometry Geometrische Untersuchungen zur Theorie der Parellellinien was published in Berlin in 1840. This last publication greatly impressed Gauss but much has been written about Gauss's role in the discovery of non-euclidean geometry which is just simply false. There is a coincidence which arises from the fact that we know that Gauss himself discovered non-euclidean geometry but told very few people, only his closest friends. Two of his friends were Farkas Bolyai, the father of János Bolyai (an independent discoverer of non-euclidean geometry), and Bartels who was Lobachevskii's teacher. This coincidence has prompted speculation that both Lobachevskii and Bolyai were led to their discoveries by Gauss. M Kline has put forward this theory but it has been refuted in several works; see for example [28]. Also Laptev in [29] has examined the correspondence between Bartels and Gauss and shown that Bartels did not know about Gauss's results in non-euclidean geometry. There are other claims made about Lobachevskii and the discovery of non-euclidean geometry which have been recently refuted. For example in [25] the claims that Lobachevskii was in correspondence with Gauss ( Gauss appreciated Lobachevskii's works very highly but had no personal correspondence with him), that Gauss studied Russian to read Lobachevskii's Russian papers as claimed for example in [1] (actually, Gauss had studied Russian before he had even heard of Lobachevskii), and that Gauss was a "good propagandist" of Lobachevskii's works in Germany (Gauss never commented publicly on Lobachevskii's work) are shown to be false. The story of how Lobachevskii's hyperbolic geometry came to be accepted is a complex one and this biography is not the place in which to go into details, but we shall note the main events. In 1866, ten years after Lobachevskii's death, Hoüel published a French translation of Lobachevskii's Geometrische Untersuchungen together with some of Gauss's correspondence on non-euclidean geometry. Beltrami, in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lobachevsky.html (4 of 5) [2/16/2002 11:20:45 PM]

Lobachevsky

1868, gave a concrete realisation of Lobachevskii's geometry. Weierstrass led a seminar on Lobachevskii's geometry in 1870 which was attended by Klein and, two years later, after Klein and Lie had discussed these new generalisations of geometry in Paris, Klein produced his general view of geometry as the properties invariant under the action of some group of transformations in the Erlanger Programm. There were two further major contributions to Lobachevskii's geometry by Poincaré in 1882 and 1887. Perhaps these finally mark the acceptance of Lobachevskii's ideas which would eventually be seen as vital steps in freeing the thinking of mathematicians so that relativity theory had a natural mathematical foundation. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (45 books/articles)

Some Quotations (2)

A Poster of Nikolai Ivanovich Lobachevskii

Mathematicians born in the same country 1. Non-Euclidean geometry

Cross-references to History Topics

2. The development of group theory 3. An overview of the history of mathematics Other references in MacTutor

Chronology: 1820 to 1830

Honours awarded to Nikolai Ivanovich Lobachevskii (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Lobachevskiy

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lobachevsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lobachevsky.html (5 of 5) [2/16/2002 11:20:45 PM]

Loewner

Charles Loewner Born: 29 May 1893 in Lany, Bohemia (now Czech Republic) Died: 8 Jan 1968 in Stanford, California, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Loewner has several versions of his name. As we shall explain below, he was a Czech whose education was in German. His name in Czech was Karel Löwner but he was known as Karl Löwner (using the German version of his first name). He adopted the English version of his name, Charles Loewner, later in life after going to the United States. We shall refer to him in this article as 'Charles' or as 'Loewner' even during the period when he certainly was not using these versions. Charles was born into a Jewish family who lived in the village of Lany, about 30 km from Prague. His father was Sigmund Löwner, who owned a store in the village. Although Jewish, and living near Prague, Sigmund was a lover of German culture and believed strongly in education, particularly German style education. Charles was brought up in a large family, having four brothers and five sisters; only eight of the nine children survived childhood however. Although he would be educated in German, the family spoke Czech at home. In keeping with his father's wish to have his children educated in the German tradition, Charles was sent to a German Gymnasium in Prague where not only the tradition but also the language was German. He graduated from the school in 1912 and, in that year, he began his studies in the German section of the Charles University of Prague. He embarked on a university course which would lead directly to a doctorate, rather than the somewhat lower level course which would lead to a qualification as a school teacher. In Prague Loewner's research supervisor was Georg Pick who was himself a student of Leo Königsberger. Loewner worked on geometric function theory for his doctorate and after submitting his

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Loewner.html (1 of 5) [2/16/2002 11:20:47 PM]

Loewner

thesis he received a Ph.D. in 1917. He was then appointed as an Assistant at the German Technical University in Prague and he worked there for four and a half years, from late 1917 to 1922. It was not a mathematically stimulating environment for Loewner who found that his colleagues were not involved in deep research. When Loewner was offered a position at the University of Berlin in 1922 he took up the offer enthusiastically, even though it meant that he was still going to be an Assistant. Although it was a lowly position, Loewner now had colleagues such as Schmidt, Schur, Alfred Brauer and his brother Richard Brauer, Hopf, von Neumann, and Szego. This is a stunning array of talent which the provided the mathematically stimulating environment which Loewner had lacked at the German Technical University in Prague. Bers writes [2]:Loewner often spoke of his time in Berlin, clearly a happy period of his life. After Prague, the cosmopolitan capital of the Weimar republic must have felt like another world. ... Mathematical life was at a high pitch; for the first time in his life Loewner was surrounded by his mathematical equals. Loewner began to move up the academic hierarchy from being an assistant to teaching as a Privatdozent in Berlin. Then in 1928 he was appointed as extraordinary professor at Cologne, a position he held for two years before returning to the Charles University of Prague in 1939. His initial appointment at Prague was as an extraordinary professor but he was soon promoted to full professor. On 30 January 1933 Hitler came to power in Germany and on 7 April 1933 the Civil Service Law was passed which provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Of course this did not affect Loewner in Prague, but as he watched the suffering of his Jewish colleagues in Germany he began to become increasingly uneasy. He did all he could to help the Jewish mathematicians who were dismissed from their posts in Germany. If the political situation gave him cause for concern, his own life was filled with happiness at this time. He married Elisabeth Alexander in 1934. She came from Breslau and was a trained singer. Loewner took piano lessons so that he could accompany his wife as she practised her singing. In 1936 their daughter Marion was born and they lived happily in Prague, seeing the reality of what was happening in Germany and fearing the inevitable outcome of events there. Preparing for the inevitable after the Germans marched into Austria, Loewner took English lessons so that he would be ready for the day he had to leave his homeland. Among the students Loewner supervised in Prague was Lipa Bers. Bers said that at first he failed to understand Loewner, since he felt he took little interest in his work. However Bers soon came to understand Loewner's methods which were to give his students as much help and encouragement as he felt they required - and Bers was a talented student who needed comparatively little support. Before moving on to describe the next stage of Loewner's life we should comment of the mathematics he had produced up to this time. It was mathematics of the highest quality, but Loewner had a policy which meant that he only published results he felt were significant. He only published six papers during the 25 years following the time that he began his research activities. However it is not an exaggeration to describe some of these as masterpieces. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Loewner.html (2 of 5) [2/16/2002 11:20:47 PM]

Loewner

As we have already mentioned, Loewner's research was on geometric function theory. He wrote a series of papers on this topic, culminating in one where he proved a special case of the Bieberbach conjecture in 1923. The Bieberbach conjecture states that if f is a complex function given by the series f(z) = aq + a1z + az2 + a3z3 + ... which maps the unit disc conformally in a one-one way then, if |z| < 1, |an| n for each n. It can be expressed as: The nth coefficient of a univalent function can be no more than n. Loewner proved that for such functions f, |a3| 3. We should also note that Loewner's proof uses the Loewner differential equation which has been studied extensively since he introduced it, and was used by de Branges in his celebrated proof of the Bieberbach conjecture. Another important paper written by Loewner during this period is devoted to properties of n-monotonic functions. The notion of an n-monotonic function is a generalisation of the usual idea of monotonicity. A function f:(a,b) R is said to be n-monotonic if, for all symmetric real positive n n-matrices X, Y with spectrum in (a,b), then X Y (in the sense of quadratic forms) implies f(X) f(Y). Bers writes:... both the problem posed and the answer given are totally unexpected. The functions which Loewner called n-monotonic turned out to be of importance for electrical engineering and for quantum physics ... Although aware of the increasing danger that he and his family were in, Loewner was still in Prague when the Nazis occupied the city. Loewner was immediately put in jail and he spent a week there trying to leave the country. After paying the 'emigration tax' twice over he was allowed to leave the country with his family. Bers believes that most of the credit for this escape must go to Loewner's wife who worked tirelessly to achieve it [2]:The Loewners arrived in America penniless, but managed to bring their furniture and books. It was at this point that he changed his name to Charles Loewner, a definite signal that he wished his family to make a new start in a new country. Von Neumann arranged a position for him at Louisville University and the committee set up in the United States to deal with refugees such as the Loewners agreed to pay his salary for the first year. It was not easy for the 46 year old, highly respected mathematician, to start from the bottom again, but that is what he had to do. Times were hard for Loewner [2]:... teaching many hours of elementary courses and having to grade staggering piles of homework. Some students asked him to teach an advanced course, but when he agreed to do so, without additional remuneration, he was told, first, that this would take his mind off his primary duties, and then, that there was no free classroom. Finally Loewner taught his advanced course in a local brewery before the arrival of the morning shift. He worked at Brown University from 1944 on a program related to war work. His contributions here were to work on fluid dynamics where he produced some deep results about critical subsonic flows. Other results arose from his study of how to defend against kamikaze bombers. In 1946 he went to Syracuse University where he remained for five years before he moved to Stanford [2]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Loewner.html (3 of 5) [2/16/2002 11:20:47 PM]

Loewner

This was the right place for him and his family. He loved the California weather and the California nature. The house in Los Altos was the first real home the Loewners had since Prague. Among the distinguished mathematicians there were his old friends Bergman and Szego, and he always knew how to make new friends. He had people to make music with and people to hike with (he said that he got his best mathematics; ideas while walking). He was a magnificent lecturer and students flocked to his courses and to his famous problem seminar. Only the untimely death of Elisabeth Loewner in 1956 darkened the California years. Loewner was described by Bers as follows (see [2]):Loewner was a man whom everybody liked, perhaps because he was a man at peace with himself. He conducted a life-long passionate love affair with mathematics, but was neither competitive, nor jealous, nor vain. His kindness and generosity in scientific matters, to students and colleagues alike, were proverbial. He seemed to be incapable of malice. His manners were mild and even diffident, but those hid a will of steel. Without being religious he strongly felt his Jewish identity. Without forgetting his native Czech he spoke pure and precise German ... Without having any illusions about Soviet Russia he was a man of the left. He was a good storyteller, with a sense of humour which was at once Jewish and humanistic. But first and foremost he was a mathematician. Finally we should add a few words about the direction Loewner's research took. We have already mention his brilliant concept of n-monotonic functions. In this context he studied order preserving mappings and semigroups of such mappings. Later on he looked at such semigroups in more abstract settings and produced some further beautiful results characterising projective mappings and certain geometric objects. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Loewner.html (4 of 5) [2/16/2002 11:20:47 PM]

Loewner

JOC/EFR September 2001

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Loewner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Loewner.html (5 of 5) [2/16/2002 11:20:47 PM]

Loewy

Alfred Loewy Born: 20 June 1873 in Rawitsch, Germany (now Rawicz, Poznan, Poland) Died: 25 Jan 1935 in Freiburg im Breisgau, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alfred Loewy studied at Breslau, Munich, Berlin and Göttingen between 1891 and 1895. He was awarded a doctorate by Munich in 1894 then, from 1897 he taught at the University of Freiburg. He was appointed professor at Freiburg from 1919 His eyesight began to deteriorate from about 1920 and he became totally blind before his death. Shortly before his death he was forced to retire by the Nazi regime since he was Jewish. Loewy worked on linear groups, the algebraic theory of differential equations and actuarial mathematics. He published 70 papers and a number of books. He edited German translations of works by Abel, Fourier, Charles Sturm and Etienne Pascal. Among Loewy's most famous works are Lehrbuch der Algebra (1915) and Mathematik des Geld- und Zahlungsverkehrs (1920). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Alfred Loewy

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Loewy.html (1 of 2) [2/16/2002 11:20:49 PM]

Loewy

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Loewy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Loewy.html (2 of 2) [2/16/2002 11:20:49 PM]

Lopatynsky

Yaroslav Borisovich Lopatynsky Born: 9 Nov 1906 in Tbilisi, Georgia, Russia Died: 10 March 1981 in Donetske, USSR

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yaroslav Lopatynsky attended the Azerbaijan University, graduating in 1926. He then taught at a number of different higher educational establishments in Baku, the capital of Azerbaijan. In 1945 he moved to Lvov where he was appointed to the chair of differential equations at Lvov University. In 1963, Lopatynsky left Lvov and moved to Moscow's Industrial Institute. In 1966 he became head of the partial differential equations Section of the Institute of Applied Mathematics and Mechanics of the Academy of Sciences of the Ukraine in Donetske. Lopatynsky's contributions to the theory of differential equations are particularly important. He worked on linear and nonlinear partial differential equations. He worked on the general theory of boundary value problems for linear systems of partial differential equations of elliptic type, finding general methods of solving boundary value problems. Petryshyn writes in [3]:Lopatynsky was the first person to formulate a condition on the relation between the coefficients of the system and the coefficients of the boundary operators which is necessary and sufficient for the normal solvability of boundary value problems. This is now known as the Lopatynsky Condition. He also obtained some basic results in the solvability of the Cauchy problem for operator equations in Banach spaces. In 1980 Lopatynsky published an important book Introduction to the Contemporary Theory of Partial Differential Equations. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lopatynsky.html (1 of 2) [2/16/2002 11:20:50 PM]

Lopatynsky

Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lopatynsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lopatynsky.html (2 of 2) [2/16/2002 11:20:50 PM]

Lorentz

Hendrik Antoon Lorentz Born: 18 July 1853 in Arnhem, Netherlands Died: 4 Feb 1928 in Haarlem, Netherlands

Click the picture above to see eight larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hendrik Lorentz attended primary school in Arnhem until he was 13 years of age when he entered the new High School there. He entered the University of Leiden in 1870 but, in 1872, he returned to Arnhem to take up teaching evening classes. He worked for his doctorate while holding the teaching post. Lorentz refined Maxwell's electromagnetic theory in his doctoral thesis The theory of the reflection and refraction of light presented in 1875. He was appointed professor of mathematical physics at Leiden University in 1878. He remained in this post until he retired in 1912 when Ehrenfest was appointed to his chair. After retiring from this chair, Lorentz was appointed director of research at the Teyler Institute, Haarlem. He retained an honorary position at Leiden, where he continued to lecture. Before the existence of electrons was proved, Lorentz proposed that light waves were due to oscillations of an electric charge in the atom. Lorentz developed his mathematical theory of the electron for which he received the Nobel Prize in 1902. The Nobel prize was awarded jointly to Lorentz and Pieter Zeeman, a student of Lorentz. Zeeman had verified experimentally Lorentz's theoretical work on atomic structure, demonstrating the effect of a strong magnetic field on the oscillations by measuring the change in the wavelength of the light produced. Lorentz is also famed for his work on the FitzGerald- Lorentz contraction, which is a contraction in the length of an object at relativistic speeds. Lorentz transformations, which he introduced in 1904, form the basis of Einstein's special theory of relativity. They describe the increase of mass, the shortening of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lorentz.html (1 of 3) [2/16/2002 11:20:52 PM]

Lorentz

length, and the time dilation of a body moving at speeds close to the velocity of light. Lorentz was chairman of the first Solvay Conference held in Brussels in the autumn of 1911. This conference looked at the problems of having two approaches, namely the classical physics and quantum theory. However Lorentz never fully accepted quantum theory and always hoped that it would be possible to incorporate it back into the classical approach. He said in his presidential address at the opening ceremony of the conference:In this stage of affairs there appeared to us like a wonderful ray of light the beautiful hypothesis of energy elements which was first expounded by Planck and then extended by Einstein and Nernst, and others to many phenomena. It has opened for us unexpected vistas, even those, who consider it with a certain suspicion, must admit its importance and fruitfulness. In [6] O W Richardson describes Lorentz as:... a man of remarkable intellectual powers ... . Although steeped in his own investigation of the moment, he always seemed to have in his immediate grasp its ramifications into every corner the universe. ... The singular clearness of his writings provides a striking reflection of his wonderful powers in this respect. .... He possessed and successfully employed the mental vivacity which is necessary to follow the interplay of discussion, the insight which is required to extract those statements which illuminate the real difficulties, and the wisdom to lead the discussion among fruitful channels, and he did this so skillfully taught the process was hardly perceptible. Lorentz received a great many honours for his outstanding work. He was elected a Fellow of the Royal Society in 1905. The Society awarded him their Rumford Medal in 1908 and their Copley Medal in 1918. The respect that Lorentz was held in in The Netherlands is seen in Richardson's description of his funeral [6]:The funeral took place at Haarlem at noon on Friday, 10 February. At the stroke of twelve the State telegraph and telephone services of Holland were suspended for three minutes as a revered tribute to the greatest man Holland has produced in our time. It was attended by many colleagues and distinguished physicists from foreign countries. The President, Sir Ernest Rutherford, represented the Royal Society and made an appreciative oration by the graveside. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Hendrik Lorentz

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lorentz.html (2 of 3) [2/16/2002 11:20:52 PM]

Lorentz

Cross-references to History Topics

1. The quantum age begins 2. General relativity 3. Special relativity

Other references in MacTutor

1. Chronology: 1880 to 1890 2. Chronology: 1900 to 1910

Honours awarded to Hendrik Lorentz (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1902

Fellow of the Royal Society

Elected 1905

Royal Society Copley Medal

Awarded 1918

Lunar features

Crater Lorentz

Other Web sites

1. Nobel prizes site (A biography of Lorentz and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lorentz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lorentz.html (3 of 3) [2/16/2002 11:20:52 PM]

Love

Augustus Edward Hough Love Born: 17 April 1863 in Weston-super-Mare, England Died: 5 June 1940 in Oxford, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Augustus Love graduated from Cambridge and held the Sedleian chair of natural philosophy at Oxford from 1899. He worked on the mathematical theory of elasticity (on which he wrote the two volume work A Treatise on the Mathematical Theory of Elasticity (1892-93) ) and on waves. His work on the structure of the Earth Some Problems in Geodynamics won the Adams Prize at Cambridge in 1911. An expert on spherical harmonics, Love discovered the existence of waves of short wavelength in the Earth's crust. The ideas in this work are still much used in geophysical research and the short wavelength earthquake waves he discovered are called 'Love waves'. He received many honours, the Royal Society awarded him its Royal Medal in 1909 and its Sylvester Medal in 1937, while the London Mathematical Society awarded him its De Morgan Medal in 1926. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) Mathematicians born in the same country Honours awarded to Augustus E H Love (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Love.html (1 of 2) [2/16/2002 11:20:54 PM]

Love

Fellow of the Royal Society

Elected 1894

Royal Society Royal Medal

Awarded 1909

Royal Society Sylvester Medal

Awarded 1937

London Maths Society President

1912 - 1914

LMS De Morgan Medal

Awarded 1926

Lunar features

Crater Love

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Love.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Love.html (2 of 2) [2/16/2002 11:20:54 PM]

Lovelace

Augusta Ada King, Countess of Lovelace Born: 10 Dec 1815 in Piccadilly, Middlesex (now in London), England Died: 27 Nov 1852 in Marylebone, London, England

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lady Ada Lovelace's father was Lord Byron, the famous poet. Her parents separated soon after her birth and she never knew either of them. She was educated by private tutors, advanced study in mathematics being provided by De Morgan. She became Countess of Lovelace when her husband William King, whom she married in 1835, was created an Earl in 1838. In 1833 Augusta became interested in Babbage's analytic engine. Ten years later she produced an annotated translation of Menabrea's Notions sur la machine analytique de Charles Babbage (1842). In the annotations she describes how the Analytical Engine could be programmed to compute Bernoulli numbers. She described the Analytical Engine saying the Analytical Engine weaves algebraic patterns, just as the Jacquard-loom weaves flowers and leaves. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles)

A Quotation

A Poster of Ada Lovelace

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lovelace.html (1 of 2) [2/16/2002 11:20:56 PM]

Lovelace

Other Web sites

1. Agnes Scott College 2. Simon Fraser University 3. The Ada Project 4. WISE project 5. AWC (Many other links) 6. Babbage pages 7. San Diego 8. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lovelace.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lovelace.html (2 of 2) [2/16/2002 11:20:56 PM]

Lowenheim

Leopold Löwenheim Born: 26 June 1878 in Krefeld, Germany Died: 5 May 1957 in Berlin, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leopold Löwenheim's father was a mathematics teacher and he revised and edited his father's unfinished work on Democritus. Despite war service in France, Hungary and Serbia he published a series of important papers on mathematical logic during this time, extending work by Charles Peirce, Schröder and Whitehead. Löwenheim is remembered for the Löwenheim-Skolem paradox (which Skolem pointed out is not a paradox!) which produces non-standard models, for example a denumerable model of the reals. Forced to retire in 1934 (he was a quarter non-Aryan !) he lost his mathematical manuscripts, 1000 drawings and models and much more in the 1943 bombing of Berlin. Löwenheim survived and taught mathematics again after the War. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lowenheim.html (1 of 2) [2/16/2002 11:20:57 PM]

Lowenheim

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lowenheim.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lowenheim.html (2 of 2) [2/16/2002 11:20:57 PM]

Loyd

Samuel Loyd Born: 31 Jan 1841 in Philadelphia, Pennsylvania , USA Died: 10 April 1911 in New York, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Always known as Sam, Loyd was the creator of famous mathematical puzzles and recreations. He studied engineering and intended to become a steam and mechanical engineer but he soon made his living from his puzzles and chess problems. From 1860 he was problem editor of the magazine Chess Monthly and, in 1878, he published a book of problems, Chess Strategy. Loyd's most famous puzzle was the 15 Puzzle which he produced in 1878. The craze swept America where employers put up notices prohibiting playing the puzzle during office hours. In Germany it was played by Deputies in the Reichstag while in France it was described as a greater scourge than alcohol or tobacco. Loyd produced over 10 000 puzzles in his lifetime many involving sophisticated mathematical ideas. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics Other Web sites

Mathematical games and recreations 1. B R Clarke 2. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Loyd.html (1 of 2) [2/16/2002 11:20:59 PM]

Loyd

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Loyd.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Loyd.html (2 of 2) [2/16/2002 11:20:59 PM]

Lucas

François Edouard Anatole Lucas Born: 4 April 1842 in Amiens, France Died: 3 Oct 1891 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edouard Lucas was educated at the Ecole Normale in Amiens. After this he worked at the Paris Observatory under Le Verrier. During the Franco-Prussian War (1870-1871) Lucas served as an artillery officer. After the French were defeated, Lucas became professor of mathematics at the Lycée Saint Louis in Paris. He later became professor of mathematics at the Lycée Charlemagne, also in Paris. Lucas is best known for his results in number theory: in particular he studied the Fibonacci sequence and the associated Lucas sequence is named after him. He gave the well-known formula for the Fibonacci numbers 5 fn = ((1 + 5)/2)n - ((1 - 5)/2)n. Lucas also devised methods of testing primality, essentially those used today. In 1876 he used his methods to prove that the Mersenne number 2127 - 1 is prime. This remains the largest prime number discovered without the aid of a computer. The Lucas test for primes was refined by Lehmer in 1930. It works as follows. Define the sequence S2 = 4, S3 = 14, S4 = 194, . . . where for n >2, Sn is defined inductively by Sn = Sn-12 - 2.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lucas.html (1 of 3) [2/16/2002 11:21:01 PM]

Lucas

The Lucas-Lehmer test states that a Mersenne number Mp = 2p -1, with p > 2, is prime if and only if Mp divides Sp. Lucas showed that S127 is divisible by M127 thus showing that M127 is prime. This was a extremely difficult calculation since M127 is a big number and S127 is unbelievably large. In fact M127 = 170141183460469231731687303715884105727 and Lucas was only able to perform the calculation since he showed that S127 is divisible by M127 without calculating S127. Lucas is also well known for his invention of the Tower of Hanoi puzzle and other mathematical recreations. The Tower of Hanoi puzzle appeared in 1883 under the name of M. Claus. Notice that Claus is an anagram of Lucas! His four volume work on recreational mathematics Récréations mathématiques (1882-94) has become a classic. Lucas died as the result of a freak accident at a banquet when a plate was dropped and a piece flew up and cut his cheek. He died of erysipelas a few days later. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of Edouard Lucas

Mathematicians born in the same country

Cross-references to History Topics

1. Mathematical games and recreations 2. Prime numbers

Other Web sites

1. The Prime Pages (The Lucas/Lehmer test for primes) 2. Clark Kimberling

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lucas.html (2 of 3) [2/16/2002 11:21:01 PM]

Lucas

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lucas.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lucas.html (3 of 3) [2/16/2002 11:21:01 PM]

Lueroth

Jacob Lueroth Born: 18 Feb 1844 in Mannheim, Germany Died: 14 Sept 1910 in Munich, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jacob Lueroth was interested in astronomy when he was at school and at one stage his heart was set on that subject. He began to make astronomical observations but he soon realised that his poor eyesight was not going to give him a future in that subject so his interest turned to mathematics. After leaving school he studied at a number of universities as was the usual practice of German students at that time. Between the years 1863 and 1866 he attended three universities, Heidelberg, Berlin and Giessen. He received a doctorate in 1865 for a thesis on the Pascal configuration. Lueroth was appointed to the teaching staff at Heidelberg and he taught there for two years before being appointed professor of mathematics at the Technische Hochschule in Karlsruhe. He spent 11 years at Karlsruhe, then in 1880 he moved to Munich teaching at the Technische Hochschule there for three years. In 1883 Lueroth moved again, this time being appointed to the University of Freiburg. He was to spend the rest of his career at Freiburg. Lueroth was taught by Hesse and Clebsch and continued to develop their work on geometry and invariants. He published results in the areas of analytic geometry, linear geometry and continued the directions of his teachers in his publications on invariant theory. In 1869 Lueroth discovered the "Lueroth quartic". This came out of an investigation he was carrying out into when a ternary quartic form could be represented as the sum of five fourth powers of linear forms. Some of his work on rational curves, published in Mathematische Annalen in 1876, was extended to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lueroth.html (1 of 2) [2/16/2002 11:21:02 PM]

Lueroth

surfaces by Castelnuovo in 1895. In 1883 Lueroth published his method on constructing a Riemann surface for a given algebraic curve. Lueroth also worked on the big problem of the topological invariance of dimension. He made some useful progress but this difficult problem was not completely solved until the work of Brouwer in 1911. Among his other work, Lueroth undertook editing. He was an editor of the complete works of Hesse and of Grassmann. He also has some fine results on logic, a topic he worked on in collaboration with his friend Ernst Schröder. Von Staudt's ideas of geometry interested Lueroth and he further developed von Staudt's complex geometry. He published Grundriss der Mechanik in 1881. This mechanics book makes heavy use of the vector calculus. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1860 to 1870

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lueroth.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lueroth.html (2 of 2) [2/16/2002 11:21:02 PM]

Lukacs

Eugene Lukacs Born: 14 Aug 1906 in Szombathely, Hungary Died: 21 Dec 1987 in Washington D.C., USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eugene Lukacs was born into a Jewish family. The family lived in Vienna where Eugene's father worked in a bank. However Eugene was born in his grandmother's house in Szombathely, then moved to Vienna a few weeks later and he was educated in that city. After primary and the Realgymnasium secondary school, Lukacs entered the Technical University in Vienna in 1925 where he studied mechanical engineering. However, he soon transferred to the University of Vienna to study mathematics. Among his teacher at the University of Vienna were H Hahn, E Helly, W Meyer, L Vietoris and W Wirtinger. After taking his first degree, Lukacs continued to study at Vienna for his doctorate. His doctoral dissertation on a geometry topic was supervised by W Meyer and, in 1930, he was awarded his doctorate. He continued to study, taking an actuarial degree in 1931. By May 1931 Austria was close to financial and economic disaster. This was no time to get an academic position in a university, so Lukacs took a post as a secondary school teacher in Vienna. However, the poor economic state and the rise of the National Socialists in Germany, resulted in considerable support being given to the Nazis in Austria. By spring 1933 Adolf Hitler was in power in Germany, and Nazi propaganda for the incorporation of Austria was greatly increased. Austria turned to Italy for help. All this was bad news for Lukacs, particularly since he was Jewish. Lukacs left school teaching in 1933 and took up a position as an actuary at an insurance company, having E Helly, another of Jewish origin, amongst his colleagues in the insurance company. The political situation in Austria continued to deteriorate. A civil war followed with four days of fighting. All political

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lukacs.html (1 of 4) [2/16/2002 11:21:04 PM]

Lukacs

parties were abolished except the Fatherland Front but, in July 1934, a group of Nazis attempted to seize power but they were compelled to surrender, and their leaders were executed. Hitler and Mussolini allied themselves in 1936 and Austria became totally isolated. During this time Lukacs had married Elizabeth Weisz. They had met at the University of Vienna in 1927 while they were both studying mathematics and physics. In 1937 Lukacs left the insurance company. He also taught mathematics at the Volkshochschule Wien Volksheim but his position was becoming increasingly difficult. In March 1938, German troops, accompanied by Hitler himself, entered Austria. Austria was absorbed into Germany and the Nazis arrested the leaders of the Austrian political parties. Many Austrians, especially those of Jewish origin, went into exile and Lukacs and his wife decided that he had to take this route. Elizabeth Lukacs left for the United States late in 1938 while Eugene some time later, reaching the United States in February 1939. While at the University of Vienna, Lukacs had met Wald who was also a Hungarian Jew. Wald had reached the United States in 1938 and now the two made contact. Wald was working on statistics and probability and he persuaded Lukacs to take an interest in this topic too. The job position in the United States was extremely bad, since this was the time following the great depression. However Elizabeth Lukacs found a post teaching at Garrison High School in Baltimore and she encouraged Eugene to apply for a position at Friends High School in Baltimore. He was appointed to teach Latin and advanced mathematics but, after a short time he moved to another college. He taught physics and mathematics at several colleges before being appointed to Our Lady of Cincinnati College in 1945. In 1942 Lukacs had made an important contribution to mathematical statistics by introducing, for the first time, the method of differential equations in characteristic function theory. He used this method to solve problems of characterisation of distributions. The method allowed him to invesitgate the independence of the sample mean and sample variance in certain cases. After moving to Cincinnati, Lukacs worked with Szász who had held a post there from 1936. The two worked on probability and wrote a number of joint papers. He continued at Our Lady of Cincinnati College until 1953, although he spent some leave working as a mathematical statistician at the National Bureau of Standards in Washington, D.C. In 1953 Lukacs Eugene joined the Office of Naval Research and he later became the Head of its Statistics. Lukacs moved again, this time in 1955 to the Catholic University of America where, in 1959, he set up the Statistical Laboratory becoming its director. It became an important research establishment, visited by many mathematicians including Harald Cramer, Jerzy Neyman, Alfred Renyi, Paul Lévy, R A Fisher, Mark Kac, Yu Linnik, Paul Erdös, Jacob Wolfowitz, William Cochran and William Feller. Lukacs retired in 1972, and went first to Bowling Green where he remained for four years. After holding visiting posts back in Europe, in particular his home university of Vienna and at Erlangen, he returned to his home in Washington D.C. in 1978. One of his most important books Developments in Characteristic Function Theory was published in 1983, eleven years after he retired. Perhaps his most famous work was Characteristic Functions (1960) which studied the properties of characteristic functions and their applications. Other topics to which Lukacs made major contributions include characterisations of distributions, stability of characterisation results and functional equations.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lukacs.html (2 of 4) [2/16/2002 11:21:04 PM]

Lukacs

In [3] Lukacs's many visiting appointments are listed:Eugene liked to travel. This led him to spend time teaching or lecturing as visiting professor at the Sorbonne (1961-62, 1965-66), the Swiss Federal Institute, Zurich (1961-62), the Institute of Technology, Vienna (1965-66, 1970, 1975-77), the University of Hull (1971), the University of Sheffield (1974-75), the University of Erlangen (1977-78), the University of Brussels (1961-62) and the University of Athens (1961-62). Lukacs's hobbies and interests included stamp and coin collecting, hiking, bird watching, photography and, as stated above, travelling. One of his favourite places was the Mathematics Research Institute at Oberwolfach in southern Germany. He spent part of his holiday at his cottage in the Viennese Woods almost every summer. In [3] his attitude is described:Eugene was a constant source of encouragement to his colleagues and students. He showed a great deal of interest in their work and was always available for consultation. He promptly responded to countless queries that he received from all over the world, mostly on questions concerning characteristic functions. We shall miss Eugene greatly, not only for his contributions to probability and statistics but also as a colleague, a friend and as a human being of integrity. Lukacs undertook many editorial duties. He was an associate editor of the Journal of the American Statistical Association (1951-55, 1961-63), the Annals of Mathematical Statistics (1958-64,1968-70), and the Journal of Multivariate Analysis (1970-83). Jointly with Z W Birnbaum, he was the founding editor of the Academic Press Series in Probability and Mathematical Statistics (1962-85). He received many honors such as being elected a fellow of the Institute of Mathematical Statistics in 1957, a fellow of the American Statistical Association in 1969, and a fellow of the American Association for Advancement of Science in 1958. He was elected to the Austrian Academy of Science in 1973. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lukacs.html (3 of 4) [2/16/2002 11:21:04 PM]

Lukacs

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lukacs.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lukacs.html (4 of 4) [2/16/2002 11:21:04 PM]

Lukasiewicz

Jan Lukasiewicz Born: 21 Dec 1878 in Lvov, Austrian Galicia (now Ukraine) Died: 13 Feb 1956 in Dublin, Ireland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jan Lukasiewicz's father, Luke Lukasiewicz, was a captain in the Austrian army. Perhaps before proceeding we should explain why the father in this Polish speaking family, living in Lvov which is now in the Ukraine, should be in the Austrian army. Galicia, in which Lvov was situated, was attached to Austria in the 1772 partition of Poland. However, by the time Lukasiewicz was born in Lvov, Austria had named the region the Kingdom of Galicia and Lodomeria and given it a large degree of administrative autonomy. Jan's mother, Leopoldine Holtzer, was the daughter of an Austrian civil servant and both Leopoldine and Luke were Roman Catholics. Lukasiewicz was interested in mathematics at school and he entered the University of Lvov where he studied mathematics and philosophy. Following his undergraduate studies, he continued to work for his doctorate which was awarded in 1902 with the highest distinction possible. Wishing to lecture in universities, Lukasiewicz continued to study for his habilitation, submitting his thesis to the University of Lvov in 1906. Once he had been awarded his habilitation, Lukasiewicz began to lecture at a Privatdozent. Then in 1911 he was promoted to an extraordinary professor at Lvov. Soon large changes in Poland would present new opportunities to Lukasiewicz. Since the partition of Poland, Russia had controlled that part of the country called Congress Poland, which included Warsaw. The University of Warsaw had been closed and only a Russian language university operated there. At the outbreak of World War I, the Central Powers (Germany and AustriaHungary) attacked Congress Poland. In August 1915 the Russian forces withdrew from Warsaw.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lukasiewicz.html (1 of 3) [2/16/2002 11:21:06 PM]

Lukasiewicz

Germany and Austria-Hungary took control of most of the country and a German governor general was installed in Warsaw. One of the first moves after the Russian withdrawal was the refounding of the University of Warsaw and it began operating as a Polish university in November 1915. Lukasiewicz was invited to the new University of Warsaw when it reopened in 1915. It was an exciting time in Poland and a new Kingdom of Poland was declared on 5 November 1916. Lukasiewicz was Polish Minister of Education in 1919 and a professor at Warsaw University from 1920 to 1939. During this period between the wars Lukasiewicz was twice rector of Warsaw University. During this time Lukasiewicz and Leshniewski founded the Warsaw School of Logic. Tarski, who was a student of Leshniewski, would make this school internationally famous. Lukasiewicz published his famous text Elements of mathematical logic in Warsaw in 1928 (the English translation appeared in 1963): [1]:... viewing mathematical logic as an instrument of enquiry into the foundations of mathematics and the methodology of empirical science, Lukasiewicz succeeded in making it a required subject for mathematics and science students in Polish universities. His lucid lectures attracted students of the humanities as well. Lukasiewicz had married Regina Barwinska and they suffered great hardships during World War II. Lukasiewicz wrote an autobiography in 1953 and, after his death it was published by his widow Regina Lukasiewicz [6]. The suffering of Lukasiewicz is graphically illustrated in this autobiography. Lukasiewicz and his wife fled from Poland and in 1946 they were in exile in Belgium when he was offered a chair by the University of Dublin in Ireland. He worked on mathematical logic, wrote essays on the principle of non-contradiction and the excluded middle around 1910, developed a three value propositional calculus (1917) and worked on many valued logics. Lukasiewicz introduced the 'Polish notation' which allowed expressions to be written unambiguously without the use of brackets and his studies were to form the basis for Tarski's work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lukasiewicz.html (2 of 3) [2/16/2002 11:21:06 PM]

Lukasiewicz

Glossary index

Quotations index

Mathematicians of the day JOC/EFR May 2000

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lukasiewicz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lukasiewicz.html (3 of 3) [2/16/2002 11:21:06 PM]

Luke

Yudell Leo Luke Born: 26 June 1918 in Kansas City, Missouri, USA Died: 6 May 1983 in Moscow, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yudell Luke was born into a Jewish family, his father, David Luke, being the sexton in a synagogue. He attended Kansas City Missouri Junior College, graduating in 1937. Following that he attended the University of Illinois, graduating with a B.S. in 1939 and a Master's degree from the same university in 1940. After the award of the Master's degree, Luke taught for two years at the University of Illinois but because of World War II he left to do military service. From 1942 until 1946 Luke served in the U.S. navy, being stationed in Hawaii. After his war service ended in 1946, he returned to Kansas City, Missouri with his wife LaVerne Podoll, who was from Chicago, and the two children which they had at that time. Yudell and Laverne Luke had two more children making a total of four girls. Luke was appointed to the Midwest Research Institute soon after he returned to Kansas City in 1946. His first appointment was as Head of the Mathematical Analysis Section, a position he held until he was made Senior Advisor for Mathematics in 1961. Promotion to Senior Advisor in Mathematics in 1967 was only to last until 1971 for at that time the mathematics group at the Midwest Research Institute was disbanded. At the Institute [3]:... in addition to his own research activities and the supervision of the research members of the Applied Mathematics Group, Professor Luke had a variety of responsibilities including that of procuring research projects for the Institute from government and industrial organisations. Several of his research students from this period matured into well-known research mathematicians who often collaborated with him. Professors J Wimp, W Fair and J L Fields are three of his best known former research students. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Luke.html (1 of 3) [2/16/2002 11:21:08 PM]

Luke

However, his posts at the Midwest Research Institute were not the only ones he held. In 1955 Luke had been appointed a lecturer at the University of Missouri in Kansas City. He also taught at the University of Kansas and, after the mathematics group at the Midwest Research Institute was disbanded in 1971, Luke was appointed as professor at the University of Missouri in Kansas City. In 1975 he was honoured by the University of Missouri with the award of the N T Veatch award for Distinguished Research and Creative Activity. Then, in 1978, he was honoured with the appointment as Curator's Professor at the University of Missouri, a post he held until his death. Luke published nearly 100 papers and eight books during his highly distinguished career. This work falls into a number of different areas but it began with applied mathematics and research into aeronautics. In this area he published on the forces on aircraft wings, in particular studying stress and sonic flutter. His work on these topics led him to require much information on special functions and he was led to develop tables of special functions and to use numerical techniques to solve equations. His early work on Bessel functions and hypergeometric functions appeared in his first major text Integrals of Bessel functions which was published in 1962. In order to compute tables of special functions, Luke needed to acquire expertise in approximation theory and in this way he was led to the main area of research on which he was to become a leading world expert. This is explained in [3] as follows:He was one of the first mathematicians to realise the potential of the Tau Method for the analysis and praxis of numerical approximation problems. This approach, due to Cornelius Lanczos, is based on the ideas of best uniform approximation by polynomials and rational functions. Luke used this method at a time when most of the interest in numerical analysis was still centred around finite difference techniques. Not only did he use rational approximation, but Luke also developed series expansions as an approximation method. For example he expanded hypergeometric functions in series of Laguerre and Hermite polynomials. Many of these methods involved great computational problems and Luke was led to another important area of his research, namely the design of algorithms to implement his numerical approximations. Some of his books record his great research achievements. For example The special functions and their approximations (1969) and two further volumes Mathematical functions and their approximation (1975) and Algorithms for the computation of mathematical functions (1977) contain a beautiful survey of the areas on which he worked. These texts are described in [1] as follows:These works contain an amazing wealth of information, theoretical as well as practical, pertaining to special functions, summarising and systemising to a large extent Yudell's own research and that of his collaborators, without neglecting, however, relevant work of others. It was not only through his research, however, that Luke contributed to mathematics. He was an industrious reviewer, reviewing by his own estimation over 1800 papers and books throughout his career. He received awards from Applied Mechanics Reviews in both 1972 and 1981 for his outstanding service. Another of his interests was in classifying information and he made substantial contributions to this in his work in preparing a cumulative index for the first 23 volumes of the Mathematics of Computation. A keen supporter of various mathematical societies, we should mention in particular his efforts in setting up the Visiting Lecturer Program for the Society of Industrial and Applied Mathematics. He, himself, was a visiting lecturer for the Society in 1960-61, 1964-65 and 1975-76.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Luke.html (2 of 3) [2/16/2002 11:21:08 PM]

Luke

In 1982 an exchange programme between the University of Missouri in Kansas City and the University of Moscow was set up. In 1983 Luke travelled to Moscow to lecture there as part of this exchange programme. He gave a wonderful series of lectures on special functions, asymptotic analysis, and approximation theory. Tragically, however, he died while still in Moscow. His interests outside mathematics are described in [2]:Playing bridge and cribbage and participating in baseball and basketball were his favorites. In fact Luke wrote two books on the probabilities of winning at the card game of cribbage. His interests are also described in [1]:He loved opera, philosophy, baseball, among other things. While at MRI he gave an extensive series of lectures on the history of philosophy, focusing especially on Spinoza, whose work he believed, contains the most meaningful elements of those ethical and intellectual ideals which alone can provide a personal bedrock in an uncertain, frenetically changing world. One of his four daughters wrote [2]:He was a very generous person and his religion meant a great deal to him. ... He was very special to me and I would like for everybody to know what a wonderful man he was. His wife, LaVerne, is still living in Kansas City. He has two daughters in Kansas City, one daughter in California, and one daughter in Florida. There are eight grandchildren all over the country who were very close to him. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Luke.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Luke.html (3 of 3) [2/16/2002 11:21:08 PM]

Luzin

Nikolai Nikolaevich Luzin Born: 9 Dec 1883 in Irkutsk, Russia Died: 25 Feb 1950 in Moscow, USSR

Click the picture above to see five larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nikolai Nikolaevich Luzin was born in Irkutsk, and his birthplace was not, as is incorrectly stated in a number of sources, Tomsk. Nikolai's father was a businessman, half Russian and half Buryat. Nikolai was the only son of his parents and the family moved to Tomsk when he was about eleven years old so that he could attend the Gymnasium there. One might expect that Nikolai would have shown a special talent for mathematics at the Gymnasium, but this was far from the case ([13] and [14]):This was because the system of instruction ... was based on mechanical memory: it was required to learn the theorems by heart and to reproduce their proofs exactly. For Luzin this was torture. His progress in mathematics at the Gymnasium became worse and worse, so that his father was obliged to engage a tutor ... Fortunately the tutor was a talented young man who quickly discovered that, despite Luzin's poor performance in mathematics, he could solve hard problems but often using a novel method that the tutor had never seen before. Soon the tutor had shown Luzin that mathematics was not a subject where one had to learn long lists of facts, but a topic where creativity and imagination played a major role. In 1901 Luzin left the Gymnasium and at this time his father sold his business and the family moved to Moscow. There Luzin entered the Faculty of Physics and Mathematics at Moscow University intending to train to become an engineer. At first Luzin lived in the new family home in Moscow, but Luzin's father began to gamble on the stock exchange with the money he had made from the sale of his business. The family soon hit hard times as Luzin's father lost all their savings and the family had to leave their home. Luzin, together with a friend, moved into a room owned by the widow of a doctor. His friend soon became involved with the Revolution and was forced into hiding. Luzin stayed on by himself in the room

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Luzin.html (1 of 5) [2/16/2002 11:21:10 PM]

Luzin

but he clearly got on well with the owners since he later, in 1908, married the widow's daughter. At Moscow University Luzin studied under Bugaev, learning from him the theory of functions which was to influence greatly the direction his research would eventually take. However he was only an average student who seemed to show little flair for mathematics. However, although Luzin appeared to lack talent in mathematics, one of his teachers Egorov spotted his great talent, invited him to his home, and began to set him hard problems. There was a mathematics student at the university, Pavel Florensky, who experienced a crisis after graduating and turned to religion and the study of theology. This had a major effect on Luzin, who was a close friend of Florensky, as we shall describe below. After graduating in the autumn of 1905 Luzin seemed unsure whether to devote himself to mathematics. In fact Luzin's crisis had hit him in the spring of 1905 and, on 1 May 1906, Luzin wrote to Florensky from Paris where Egorov had sent him five months earlier in an attempt to get him through the crisis (see [8]):You found me a mere child at the University, knowing nothing. I don't know how it happened, but I cannot be satisfied any more with analytic functions and Taylor series ... it happened about a year ago. ... To see the misery of people, to see the torment of life, to wend my way home from a mathematical meeting ... where, shivering in the cold, some women stand waiting in vain for dinner purchased with horror - this is an unbearable sight. It is unbearable, having seen this, to calmly study (in fact to enjoy) science. After that I could not study only mathematics, and I wanted to transfer to the medical school. ... I have been here about five months, but have only recently begun to study. Luzin was not only upset by seeing the prostitutes, he also says in the letter how he had been affected by the 'terrible days' of the 1905 Revolution. There are letters from Egorov at this time pleading with Luzin not to give up mathematics. After returning to Russia, Luzin studied medicine and theology as well as mathematics. However in April 1908 he wrote of the joy he was finding in number theory (see [8]):It is a mysterious area that envelops me deeper and deeper. In the same letter he says that he has just married and:... my wife is also very interested and shares my commitment to the search for the profound truths of life. Largely Luzin's crisis seems to have been solved by Florensky to whom Luzin wrote in July 1908:Two times I was very close to suicide - then I came ... looking to talk with you, and both times I felt as if I had leaned on a pillar and with this feeling of support I returned home ... I owe my interest in life to you... His interest in mathematics slowly returned but it was not until 1909 that Luzin seems to have finally committed himself completely to mathematics. Under Egorov's supervision he worked on his master's thesis. In 1910 he was appointed as assistant lecturer in Pure Mathematics at Moscow University. He worked for a year with Egorov and they went on to publish joint papers on function theory which mark the beginnings of the Moscow school of function theory. In 1910 Luzin travelled abroad visiting Göttingen where he was influenced by Landau. He returned to Moscow in 1914 and he completed his thesis The integral and trigonometric series which he submitted http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Luzin.html (2 of 5) [2/16/2002 11:21:10 PM]

Luzin

in 1915. After his oral examination he was awarded a doctorate, despite having submitted his thesis for the Master's Degree. Egorov was extraordinarily impressed by the work and had pressed for the award of the doctorate, but it was written in a style quite different from the accepted Russian style of the time. Some of the results were not rigorously proved but were justified using phrases such as 'it seems to me' and 'I am convinced'. Other mathematicians were not so impressed at the time, for example Steklov wrote comments in the margin such as 'it seems to him, but it doesn't seem to me' and 'Göttingen chatter'. However, the work was of fundamental importance as is stated in [13] and [14]:The influence of Luzin's dissertation on the future development of the theory of functions cannot be overestimated. Its fundamental results, deep methods of investigation and fundamental statements of problems put it into the ranks of works with which it is difficult to compare any dissertation or monograph of the time. In 1914 Luzin and his wife separated for a short time and again Florensky seems to have helped them through the difficult time. He wrote to Luzin's wife (see [8]):Nikolai Nikolaevich is a very sweet and fine person; but in personal relationships he is not at all mature, especially in intuitively perceiving the hidden currents of life. ... You will have to take the relationship in hand and create a family tone, simplicity. Instead, as I perceive it ... you have established the tone of an acquaintanceship rather than a family. Florensky seems to have given good advice since Luzin and his wife returned to a successful marriage. In 1917 Luzin was appointed as Professor of Pure Mathematics at Moscow University just before the Revolution. The Revolution caused Luzin to rethink some of the same thoughts as he had done at the time of his crisis and again he exchanged letters with Florensky. By this stage, however, his mathematical career was extremely successful and the second crisis did not materialise. Over the next ten years Luzin and Egorov built up an impressive research group at the University of Moscow which the students called 'Luzitania'. The first students included P S Aleksandrov, M Ya Suslin, D E Menshov and A Ya Khinchin. The next students included P S Urysohn, A N Kolmogorov, N K Bari, L A Lyusternik and N G Shnirelman. In 1923 P S Novikov and L V Keldysh joined the group. Another of the members of the Luzitania research group at this time was Lavrentev. In fact Lavrentev draws the following picture of the group:Whereas Egorov was reserved and formal, Luzin was extroverted and theatrical, inspiring real devotion among these students and young colleagues. ... There was intense camaraderie ... inspired by Luzin. Luzin's main contributions are in the area of foundations of mathematics and measure theory. He also made significant contributions to descriptive set topology. In the theory of boundary properties of analytic functions he proved an important result in 1919 on the invariance of sets of boundary points under conformal mappings. He also studied, together with Privalov, boundary uniqueness properties of analytic functions. From 1917 onwards, Luzin studied descriptive set theory. He stated the fundamental problem ([13] and [14]):The aim of set theory is a question of great importance: can we regard a line atomistically

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Luzin.html (3 of 5) [2/16/2002 11:21:10 PM]

Luzin

as a set of points: incidentally this question is not new, but goes back to the Greeks. Much of Luzin's work on set theory involved the study of effective sets, that is sets which can be constructed without the axiom of choice. Keldysh describes this work in [10] and [11]:... Luzin proceeded from the point of view of the French school (Borel, Lebesgue), which greatly influenced him. But whereas the French had analysed set-theoretical constructions carried out with the help of the Axiom of Choice, Luzin went considerably further and considered difficulties arising within the theory of effective sets. The study of effective sets that he embarked upon was pursued intensively for more than two decades and led to the solution of many important problems of set theory ... Luzin's school was at its peak during the years 1922 to 1926, but then Luzin concentrated on writing his second monograph on the theory of functions and spent less time with the young mathematicians in the school. Many of these mathematicians turned to other topics such as topology, differential equations, and functions of a complex variable. In 1927 Luzin was elected as a member of the Academy of Sciences of the USSR. Two years later he became a full member of first the Department of Philosophy, then to the Department of Pure Mathematics. He worked from this time until his death in the Academy of Sciences of the USSR. From 1935 he headed the Department of the Theory of Functions of Real Variables at the Steklov Institute. In 1931 Luzin himself turned to a new area when he began to study differential equations and their application to geometry and to control theory. His work in this area led him to study the bending of surfaces which is described in [13] and [14]:The bending of a surface on a principal base is a continuous bending of a surface under which the conjugacy of the net of certain curves on the surface is preserved. ... Finikov had derived differential equations that determine all principal on a given surface, and Byushgens had obtained differential equations that determine surfaces which have a given linear element and admit a bending on a principal base. However, the question of solubility of these equations, in general, remained unclear. ... no example was found in which the equations ... were insoluble ... up to 1938, when Luzin, by means of a subtle analysis of these equations, established that the existence of a principal base is rather rare. Luzin always had an interest in the history of mathematics and late in his career he wrote important articles on Newton and on Euler. As a teacher his remarkable talents are described by Kuznetsov ([13] or [14]):His presentation was always very elegant and at first sight apparently unnecessarily simple - the result of his great pedagogic talent. The solution of the large problems that he undertook is distinguished by their subtlety, elegance, and simplicity of presentation. Keldysh and Novikov wrote in [12]:Thanks to his exceptional intuition and his ability to see deeply into the heart of a question, Luzin frequently predicted mathematical facts whose proof turned out to be possible only after many years and required the creation of completely new mathematical methods. He was one of the outstanding mathematicians and thinkers of our time ... Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Luzin.html (4 of 5) [2/16/2002 11:21:10 PM]

Luzin

Click on this link to see a list of the Glossary entries for this page List of References (19 books/articles) Mathematicians born in the same country Honours awarded to Nikolai Luzin (Click a link below for the full list of mathematicians honoured in this way) Planetary features

Crater Luzin on Mars

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Luzin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Luzin.html (5 of 5) [2/16/2002 11:21:10 PM]

Lyapunov

Aleksandr Mikhailovich Lyapunov Born: 6 June 1857 in Yaroslavl, Russia Died: 3 Nov 1918 in Odessa, Russia

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aleksandr Lyapunov was a school friend of Markov and later a student of Chebyshev. He did important work on differential equations, potential theory, stability of systems and probability theory. His work concentrated on the stability of equilibrium and motion of a mechanical system and the stability of a uniformly rotating fluid. He devised important methods of approximation. Lyapunov's methods, introduced by him in 1899, provide ways of determining the stability of sets of ordinary differential equations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (20 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1890 to 1900

Honours awarded to Aleksandr M Lyapunov (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Lyapunov

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lyapunov.html (1 of 2) [2/16/2002 11:21:12 PM]

Lyapunov

Other Web sites

Pass Magazine (Lyapunov fractals)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lyapunov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lyapunov.html (2 of 2) [2/16/2002 11:21:12 PM]

Lyndon

Roger C Lyndon Born: 18 Dec 1917 in Calais, Maine, USA Died: 8 June 1988

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Roger Lyndon's mother died when he was two years old and his family moved to various towns so, as a consequence, his education took place at a number of different schools. He graduated from Derby School in 1935 and entered Harvard University with the aim of studying literature so that he might become a writer. However, he discovered that, for him, mathematics was easy and required little effort while he had to spend long hours learning literature. The move to mathematics was made and he graduated from Harvard in 1935. Having worked for a year in a bank in Albuquerque, Lyndon returned to Harvard, being awarded a Master's Degree in 1941. He taught at Georgia Tech in session 1941/42, then he returned to Harvard for the third time in 1942 and there taught navigation to pilots while he studied for his doctorate. He was awarded a Ph.D. in 1946 for a thesis on homological algebra, the work being an outstanding early step in the study of spectral sequences. His supervisor was S MacLane and his thesis was entitled The Cohomology Theory of Group Extensions. After attending a course by Tarski, Lyndon and Tarski became good friends and Lyndon was later to work on model theory as a result of attending these lectures. Accepting a position at Princeton, he attended a course on knot theory by R Fox and from this his interest was aroused in combinatorial group theory. Reidemeister was at Princeton for a year in 1948 and again this was a major influence on Lyndon to work on group presentations. Lyndon's first work which came out of these discussions with Reidemeister was published in 1950. In it

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lyndon.html (1 of 2) [2/16/2002 11:21:14 PM]

Lyndon

Lyndon investigated one-relator groups. In particular he computed their cohomology groups. In 1953 Lyndon left Princeton to take up a professorship at the University of Michigan where he remained throughout his career except for a number of posts as visiting professor at Berkeley, Queen Mary College, London, Montpellier, France and Picardie, France. K I Appel, writes in [1]:Lyndon produces elegant mathematics and thinks in terms of broad and deep ideas.... I once asked him whether there was a common thread to the diverse work in so many different fields of mathematics, he replied that he felt the problems on which he had worked had all been combinatorial in nature... on would certainly have to put him in the very first rank of those who have used combinatorial techniques in the last forty years. Lyndon made numerous major contributions to combinatorial group theory. These include the development of 'small cancellation theory', work on Fuchsian groups and the Riemann-Hurwitz formula, his introduction of 'aspherical' presentations of groups and his work on length functions in free products of groups. Lyndon was the coauthor of one of the most important works on combinatorial group theory. Together with Paul Schupp, he wrote Combinatorial group theory (1976). I [EFR] remember how eagerly the book was awaited by those interested in research in this area, and the excitement of seeing the book when it first appeared and was passed round a lecture theatre at a conference I was attending. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Lyndon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Lyndon.html (2 of 2) [2/16/2002 11:21:14 PM]

MacCullagh

James MacCullagh Born: 1809 in Landahaussy (near Strabane), Ireland Died: 24 Oct 1847 in Dublin, Ireland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James MacCullagh's father moved to Strabane when James was young so that he could receive a good education. His mathematical talents were soon evident, see [2]:In Strabane he was, while very young, placed at the only respectable school at that time in the town. Here his genius soon displayed itself. After school hours he was almost constantly employed in solving mathematical problems... To take up a study of the classics he was sent to another school, this time in Lifford. At the age of fifteen MacCullagh entered Trinity College, Dublin. His undergraduate career was one in which he received the highest grade in almost every examination he took. He graduated in 1829 and, after graduating, he entered the Fellowship examinations. In [4] his attempt in this examination is related:MacCullagh entered for the highly competitive fellowship examination, conducted orally in Latin. He was unsuccessful, a hardly surprising result when it is realised how much cramming was required. However, given his youth and inexperience, his performance was nor discreditable, his mark in mathematics being equal to that awarded to the two successful candidates... Not long after failing the Fellowship examination MacCullagh submitted his first papers for publication. One was a paper on giving geometrical results on conic sections, the other was on light, in particular on double refraction in which he gave a clear construction of the Fresnel wave surface. Hamilton wrote a review of these papers in 1830. Two years later MacCullagh tried again for a Fellowship but this time, after being told that his first answer to the first mathematics question was wrong he refused to answer any further questions. Not http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/MacCullagh.html (1 of 3) [2/16/2002 11:21:15 PM]

MacCullagh

surprisingly he failed but immediately he learnt this he sent a letter containing geometrical theorems on the theory of rotations to the examiners. The results were original but, unfortunately for MacCullagh, had been obtained independently by Poinsot who published them in 1834. MacCullagh did succeed in obtaining a fellowship in 1832 and he was appointed junior assistant to the mathematics professor in Dublin. In 1833 Hamilton announced his discovery of conical refraction of light. This was a major discovery and, following Hamilton's announcement, MacCullagh published a note on conical refraction in which he claimed at least partial priority for the discovery of conical refraction. He wrote:The indeterminate cases of my own theorems, which, optically interpreted, mean conical refraction, of course occurred to me at the time... MacCullagh was particularly cross since he knew Hamilton had studied his 1830 papers containing these theorems since Hamilton had reviewed them. Hamilton was equally cross that his priority for the discovery of conical refraction was being challenged. MacCullagh was forced to admit, what was clearly the truth, that although conical refraction could be deduced from his theorems he had only made that deduction after Hamilton had announced the discovery. MacCullagh had been close to a great discovery but had just failed to make the final step. In 1835 MacCullagh published on crystalline refraction and reflection. Later he discovered that Franz Neumann had read a paper on the subject to the Berlin Academy in December 1835 which was published in 1837. The two theories were identical except that Franz Neumann developed the theory much further. MacCullagh wrote:Franz Neumann's paper is very elaborate, and supersedes, in a great measure, the design which I had formed of treating the subject more fully at my leisure... Although MacCullagh had priority, Franz Neumann had stolen the glory and again MacCullagh had missed out. He produced several other papers on light, the most important being in 1839 when he applied methods used by Green to study reflection and refraction of waves at a surface. Also in 1839 MacCullagh was made an honorary member of the St Andrews Literary and Philosophical Society which had been founded by David Brewster. In 1842 at a meeting of the British Association in Manchester, there was a discussion on the wave or particle nature of light in which Hamilton, MacCullagh, Bessel, Jacobi, Peacock, David Brewster and others took part. Later in 1842 MacCullagh was awarded the Copley medal of the Royal Society, a particularly great achievement since Bessel was among those considered for the award. In February 1843 MacCullagh was elected a fellow of the Royal Society. Also in 1843 MacCullagh published his most important work on geometry, namely On surfaces of the second order which described how surfaces such as the ellipsoid could be generated. Although he produced much less work on geometry than on light, it is his work on geometry which has survived and proved in the end the more important. His work on light was of course of less importance after Maxwell published his electro-magnetic theory of light in 1865. MacCullagh corresponded with many scientists, in particular with John Herschel and with Babbage. He also met with many scientists, some of which have been mentioned above, but during a visit to Turin in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/MacCullagh.html (2 of 3) [2/16/2002 11:21:15 PM]

MacCullagh

1840 he was invited to Babbage's apartments and there met, among others, Plana and Menabrea. On Sunday 24 October 1847 MacCullagh committed suicide in his rooms in Dublin. The reasons for his suicide are hard to determine but the best clue may be in a letter which he wrote to Babbage five years before in which he wrote (see [4]):... I have grown very stupid of late, and regularly fail at everything I attempt. What the reason may be I cannot tell. But I begin to be of Newton's opinion, that after a certain age, a man may as well give up mathematics. ... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to James MacCullagh (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1843

Royal Society Copley Medal

Awarded 1842

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/MacCullagh.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/MacCullagh.html (3 of 3) [2/16/2002 11:21:15 PM]

MacLane

Saunders MacLane Born: 4 Aug 1909 in Taftville, Connecticut, USA

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Saunders MacLane graduated in 1926 from high school and, in that year, he entered the University of Yale. MacLane's school education had been interrupted when he was 15 years old for, at that time, his father had died. His father had been a Congregational Minister and, after his death, MacLane moved to Leominster to live with his grandfather who was also a Congregational Minister. MacLane graduated from Yale in 1930 and took up a fellowship at Chicago. At the University of Chicago he was influenced by Eliakim Moore. By this time E H Moore was nearly seventy years old but his advice to MacLane to study for a doctorate at Göttingen in Germany certainly persuaded MacLane to work at the foremost mathematical research centre in the world at that time. Of course Moore had himself studied in Germany as a young man and had created in Chicago an eminent research school of mathematics based on his experiences of German mathematics at that time. Although MacLane went to the greatest mathematics research centre in the world, political events would soon disrupt Göttingen. MacLane began to work for his doctorate under Bernays' supervision but in 1933 the Nazis came to power. They began to remove the top mathematicians from Göttingen, and other universities, who had Jewish connections. MacLane had seen that he had to work quickly for his doctorate and leave Germany as soon as possible before things deteriorated further. He defended his thesis Abbreviated Proofs in the Logical Calculus, with Weyl as examiner, on 19 July 1933 and quickly returned to the United States. The article [6] by MacLane give an interesting account of the events at Göttingen in 1933. On returning to the United States, MacLane spent the session 1933/34 at Yale and then the following two years at Harvard. He left Harvard to take up a post of instructor at Cornell for session 1936/37, he spent

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/MacLane.html (1 of 3) [2/16/2002 11:21:17 PM]

MacLane

the following session back at Chicago before accepting an appointment as an assistant professor at Harvard. It was during these years that he wrote his famous text A survey of modern algebra with G Birkhoff which was published in 1941. Kaplansky writes in [3] about this text:A Survey of Modern Algebra opened to American undergraduates what had until then been largely reserved for mathematicians in van der Waerden's Moderne Algebra, published a decade earlier. The impact of Birkhoff and MacLane on the content and teaching of algebra in colleges and universities was immediate and long sustained. What we recognise in undergraduate courses in algebra today took much of its start with the abstract algebra which they made both accessible and attractive. During World War II MacLane worked in the Applied Mathematics Groups at Columbia. Then in 1947 he was appointed professor of mathematics at Chicago. The research centre there had Stone, Abraham Albert, Kaplansky, Otto Schilling and Weil on the staff and was led by Stone. In 1952, five years after being appointed, MacLane took over the chairmanship of the department from Stone. MacLane's work covered a wide range of mathematics. He worked on and off throughout his career on mathematical logic, no surprise for a student of Bernays, and he did some early work on planar graphs. He studied valuations and their extensions to polynomial rings. In the 1940s he worked on cohomology and introduced the basic notions of category theory. Kelly, in [4], writes:No man could so stimulate others unless, alongside an incisive intellect, he was possessed of enthusiasm and warmth, a deep interest in his fellow man, and a sympathy the more real for being unsentimental. Those who proudly call themselves his friends know these things: others will infer them in reading [his works]. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1940 to 1950

Honours awarded to Saunders MacLane (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh American Maths Society President

1973 - 1974

AMS Colloquium Lecturer

1963

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/MacLane.html (2 of 3) [2/16/2002 11:21:17 PM]

MacLane

Other Web sites

AMS (The AMS article by MacLane about Mathematics at Göttingen under the Nazis) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/MacLane.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/MacLane.html (3 of 3) [2/16/2002 11:21:17 PM]

MacMahon

Percy Alexander MacMahon Born: 26 Sept 1854 in Malta Died: 25 Dec 1929 in Bognor-Regis, England

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Percy MacMahon was born into a military family. His father was Brigadier-General P W MacMahon and, it is interesting to note that as a young child he had a fascination with the way that the ammunition was stacked. This early mathematical interest, before he even knew what mathematics was, is typical of many who go on to became leading mathematicians. MacMahon was educated at school in Cheltenham and was always destined for a military career. In February 1871 he entered Woolwich as a gentleman cadet, and in the following year he became a Lieutenant. After his education at Woolwich, he served in India and Malta for five years before returning to England and teaching in military academies. Promoted to Captain in 1881, he was appointed as Instructor in Mathematics at the Royal Military Academy the following year. He held this post until 1888 when he became Assistant Inspector at the Arsenal. After three years he was appointed as Instructor in Physics at the Artillery College, being promoted to Professor of Physics there before he retired from the services in 1898. Even before he retired from the Artillery College, MacMahon had been elected a Fellow of the Royal Society in 1890. He served as President of the London Mathematical Society from 1894-96 and then took on a number of different roles. In 1901 he served as President of Section A of the British Association for its meeting in Glasgow then, from 1902 to 1914, he was one of the Secretaries to the British Association. From 1906 to 1920 he served as Deputy Warden of the Standards of the Board of Trade. His roles were not only many in number but also varied in nature. He was President of the Royal Astronomical Society in 1917 and served on the Council of the Royal Society of Arts. He was also appointed as a Governor of Winchester College.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/MacMahon.html (1 of 3) [2/16/2002 11:21:19 PM]

MacMahon

MacMahon worked on invariants, following Cayley and Sylvester. Then he studied partitions and Latin squares, giving a Presidential Address to the London Mathematical Society on combinatorial analysis in 1894. MacMahon wrote a two volume treatise on this last mentioned topic (volume one in 1915 and the second volume in the following year) which has become a classic. He wrote An introduction to combinatory analysis in 1920. In 1921 he wrote New Mathematical Pastimes, a book on mathematical recreations. This book shows another of the topics which fascinated MacMahon, namely the construction of patterns which can be repeated to fill the plane. In [2] 89 of his publications are listed. Baker, in [2], describes his personality:His absorption in the mathematical problems he was considering, which was noticeable in his Woolwich days, became more pronounced in later life; but another trait, also noticeable at Woolwich, was manifest to the end, namely, his formal kind courtesy towards all with whom he had dealings; and he had always a desire to know the names of the more distinguished younger mathematicians, and to get some idea of the work they were doing. Many of his friends will always remember his personal cordiality, in which he was so ably assisted by his wife. MacMahon received many honours. In particular he was awarded honorary degrees by Dublin (1897), Cambridge (1904), Aberdeen (1911) and St Andrews (1911). He also won numerous medals: the Royal Medal of the Royal Society (1900), the Sylvester Medal of the Royal Society (1919), and the De Morgan Medal from the London Mathematical Society (1923).

Article by: J J O'Connor and E F Robertson List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

Mathematical games and recreations

Honours awarded to Percy MacMahon (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1890

Royal Society Royal Medal

Awarded 1900

Royal Society Sylvester Medal

Awarded 1919

London Maths Society President

1894 - 1896

LMS De Morgan Medal

Awarded 1923

Other Web sites

MacMahon triangles (in French)

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/MacMahon.html (2 of 3) [2/16/2002 11:21:19 PM]

MacMahon

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/MacMahon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/MacMahon.html (3 of 3) [2/16/2002 11:21:19 PM]

Macaulay

Francis Sowerby Macaulay Born: 11 Feb 1862 in Witney, England Died: 9 Feb 1937 in Cambridge, Cambridgeshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Francis Macaulay's father was a Minister in the Methodist Church. He sent his son Francis to a school for the sons of Methodist Minister's in Bath, namely Kingswood School. Graduating from Kingswood School, Macaulay entered St John's College, Cambridge from which he graduated with distinction. After graduating from Cambridge, Macaulay returned to Kingswood School in Bath, where he had himself studied, and taught mathematics there for two years. He then went to London, becoming a teacher at St Paul's School. In this school he taught many outstanding pupils and he encouraged them into a research career in mathematics. two such students were Watson and Littlewood and we know of the teaching methods employed by Macaulay through the writing of Littlewood [1]:... there was little formal instruction; students were directed to read widely but thoroughly, encouraged to be self-reliant, and inspired to look forward to pursuing research in mathematics. Macaulay wrote 14 papers on algebraic geometry and polynomial ideals. The papers look at algebraic curves, the Riemann-Roch theorem and algebraic polynomials. It is important pioneering work in the development of algebraic geometry. Macaulay discovered the primary decomposition of an ideal in a polynomial ring which is the analogue of the decomposition of a number into a product of prime powers in 1915. This work was independent of that done by Lasker in 1905. The passage of time since Macaulay published The algebraic theory of modular systems in 1916 has

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Macaulay.html (1 of 3) [2/16/2002 11:21:21 PM]

Macaulay

shown us what a remarkable work this is in the development of modern mathematics. The book has recently been reissued, eighty years after it was first published, not merely as an historical document but also because Macaulay's ideas are still highly relevant to present day research. What ideas were there then in this work? The main theme underlying the book is the problem of solving equations of systems of polynomials in several variables. Such problems have no complete solution, but Macaulay looks for structural properties of the set of solutions. In other words, in today's terminology, he is examining ideals in polynomial rings. This leads Macaulay to study Lasker's decomposition of ideals into primary ideals (the analogue of the decomposition of an integer into prime powers) and he also looks at properties which today surround the theory of Grobner bases. Where have Macaulay's ideas led in today's mathematics? Well the ideas in this book have led to ideal theory studied by Krull (see W Krull, Idealtheorie, Berlin, 1935), to Cohen-Macaulay rings, so named by Zariski and Samuel (see O Zariski and P Samuel, Commutative Algebra, Princeton, NJ, 1958), the notion of perfectness studied in (see W Grobner, Moderne algebraische Geometrie, Vienna, 1949), and to the notion of Gorenstein rings. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1910 to 1920

Honours awarded to Francis Macaulay (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1928

Other Web sites

Macaulay 2 (A computer package for calculating with rings) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Macaulay.html (2 of 3) [2/16/2002 11:21:21 PM]

Macaulay

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Macaulay.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Macaulay.html (3 of 3) [2/16/2002 11:21:21 PM]

Macdonald

Hector Munro Macdonald Born: 19 Jan 1865 in Edinburgh, Scotland Died: 16 May 1935 in Aberdeen, Scotland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hector Macdonald was educated at Aberdeen and Cambridge. Elected to the Royal Society in 1901, he became professor of mathematics at Aberdeen in 1905, winning the Society's Royal Medal in 1916. Macdonald worked on electric waves and solved difficult problems regarding diffraction of these waves by summing series of Bessel functions. He corrected his 1903 solution to the problem of a perfectly conducting sphere embedded in an infinite homogeneous dielectric in 1904 after a subtle error was pointed out by Poincaré. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Honours awarded to Hector Macdonald (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1901

Royal Society Royal Medal

Awarded 1916

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Macdonald.html (1 of 2) [2/16/2002 11:21:23 PM]

Macdonald

London Maths Society President

1916 - 1918

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Macdonald.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Macdonald.html (2 of 2) [2/16/2002 11:21:23 PM]

Macintyre

Sheila Scott Macintyre Born: 23 April 1910 in Edinburgh, Scotland Died: 21 March 1960 in Cincinnati, Ohio, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sheila Scott's father was rector of Trinity Academy Edinburgh and she attended the school. She attended Edinburgh Ladies' College from 1926 until 1928 when she graduated as Dux in Mathematics. In the same year, 1928, she entered the University of Edinburgh where she won several scholarships. She graduated M.A. with first class honours in mathematics and natural philosophy in 1932. It was usual for the top students in those days to study at Oxford or Cambridge University after taking their first degree at a Scottish University. Scott spent three years at Girton College at Cambridge. Her final year at Cambridge was one in which she undertook research under Cartwright's supervision. She published her first paper on the work she had done under Cartwright's supervision On the asymptotic periods of integral functions in 1935. Scott returned to Scotland to train as a teacher and she taught for five years in a number of schools from 1934. She taught at St Leonard's, a girls' school in St Andrews, as well as James Allen's School for Girls and Stowe School. During this period Whittaker introduced Sheila Scott to A J Macintyre who was a mathematics lecturer at Aberdeen University. They married in 1940 and the following year Sheila Macintyre was appointed as an assistant lecturer in the same department as her husband in Aberdeen. At Aberdeen Sheila Macintyre completed her doctorate under E M Wright's supervision. Wright wrote:... good as her research was there would have been more of it had she not had a family to look after. Between 1947 and 1958 she published another ten papers during a period when she brought up her two children (a third child died at age two in 1949). Sheila Macintyre was an active member of the Edinburgh Mathematical Society and of the Mathematical association. In 1958 she was elected a Fellow of the Royal Society of Edinburgh. In 1958 Macintyre and her husband accepted visiting research professorships at the University of Cincinnati. There she taught until her early death from cancer. In [1] Cartwright quotes R C Buck who wrote about Macintyre:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Macintyre.html (1 of 2) [2/16/2002 11:21:24 PM]

Macintyre

In her chosen area of analysis, she introduced powerful refinements of techniques, and what is much harder, new and original problems for investigation. Wright, writing in [4], says:Brilliant original mathematician though Sheila was, it is even more as the superb teacher and the gay helpful colleague that we remember her. Her clarity of mind made her a quite exceptionally able lecturer, but it was her warm-hearted interest in each of her students (conbined with a shrewd assessment of their abilities) that made her so successful a tutor and regent. They repaid her by a very real affection and respect. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Sheila S Macintyre (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Macintyre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Macintyre.html (2 of 2) [2/16/2002 11:21:24 PM]

Maclaurin

Colin Maclaurin Born: Feb 1698 in Kilmodan (12 km N of Tighnabruaich), Cowal, Argyllshire, Scotland Died: 14 June 1746 in Edinburgh, Scotland

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Colin Maclaurin was born in Kilmodan where his father, John Maclaurin, was the minister of the parish. The village (population 387 in 1904) is on the river Ruel and the church is at Glendaruel. You can see a picture of Kilmodan Church. John Maclaurin was more of a scholar than one would expect of a parish minister, for he had translated the Psalms into Gaelic. Colin, however, never knew his father, for he died when Colin was six weeks old. Colin was the youngest of three sons, the oldest being John, while the second was Daniel who died at a young age. Colin Maclaurin's mother inherited a small estate in Argyllshire and it was on the estate that Colin spent the early years of his life. His mother wanted a good education for Colin and his brother John, so the family moved to Dumbarton where the boys attended school. In 1707, when Colin was nine years old, his mother died so the task of bringing up Colin and his brother John fell to their uncle Daniel Maclaurin who was the minister at Kilfinnan on Loch Fyne. Colin became a student at the University of Glasgow in 1709 at the age of eleven years. This may seem an unbelievable age for someone to begin their university education, but it was not so amazing at this time as it would be today. Basically Scottish schools and universities competed for the best pupils at that time, rather than a university education being seen as following a school education as is the norm today. Certainly Maclaurin's abilities soon began to show at Glasgow University. His first encounter with advanced mathematics came one year after he entered university, when he found a copy of Euclid's Elements in one of his friend's rooms. This was the standard text for mathematical study at this time, but http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maclaurin.html (1 of 6) [2/16/2002 11:21:26 PM]

Maclaurin

Maclaurin studied it on his own, quickly mastering the first six of the thirteen books of the Elements. At Glasgow Maclaurin came into contact with Robert Simson who was the Professor of Mathematics there. Simson was particularly interested in the geometry of ancient Greece and his enthusiasm for the topic was to influence the young student Maclaurin. Tweddle in [23] looks at the correspondence between Simson and Maclaurin on conic sections 25 years after Maclaurin's student days at Glasgow. At the age of 14 Maclaurin was awarded the degree of M.A. Although a master's degree in name, this was a first degree equivalent to a B.A. but the ancient Scottish universities (including St Andrews, my [EFR] own university) still retain the degree of M.A. as the first degree in Arts. However, Maclaurin had to defend a thesis in a public examination for the award of this degree (which is not the case today), and he chose On the power of gravity as his topic. The thesis, which developed Newton's theories, was written by a 14 year old boy at a time when such advanced ideas would only be familiar to a small number of the leading mathematicians. After graduating with the degree of M.A., Maclaurin remained at the University of Glasgow for a further year to study divinity. It had been his intention to enter the Presbyterian Church but [7]:... becoming disgusted at the dissensions that had at that time crept into the church ... he decided against that career. After leaving Glasgow in 1714, Maclaurin returned to live with his uncle in the manse at Kilfinnan. These were happy years for Maclaurin who studied hard and walked in the nearby hills and mountains for recreation. He clearly attained a very high standard in mathematics for, in August 1717, he was appointed professor of mathematics at Marischal College in the University of Aberdeen. The appointment followed ten days of examinations to find the best candidate and it is clear that, despite there being another outstanding candidate, Maclaurin had the most knowledge of advanced topics. Maclaurin was to make two journeys to London, and the first of these he made in 1719. Maclaurin had already shown himself a very strong advocate of the mathematical and physical ideas of Newton, so it was natural that they should meet during Maclaurin's visit to London. It is surprising that some of Newton's biographers, for example A Rupert Hall in his 1992 biography, should declare that Maclaurin and Newton never met. Maclaurin writing of this visit to London in one of his letters (see for example letter 117 in [3]) states:I received the greatest civility from [members of the Royal Society] and particularly from the great Sir Isaac Newton with whom I was very often. Maclaurin received more than civility from the Royal Society, for he was elected a Fellow of the Royal Society during this visit to London. A rather strange event in Maclaurin's career took place during the time he held the chair of mathematics at Aberdeen. Lord Polwarth was a diplomatic agent of King George II. At this time it was customary for the sons of the top people to make a grand tour of Europe as part of finishing their education. Polwarth invited Maclaurin to accompany his son on such a grand tour and, it is not too surprising that Maclaurin accepted this chance to travel and meet with French mathematicians. What is surprising is that he does not appear to have sought the necessary permission of the university authorities in Aberdeen, although he does appear to have found someone to do his teaching. Turnbull writes however in [5]:... one wonders what was happening to his unshepherded classes in Marischal College. Had

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maclaurin.html (2 of 6) [2/16/2002 11:21:26 PM]

Maclaurin

he forgotten all about them; did he turn a deaf ear to all calls to return; was there something in him, akin to the impenetrable aloofness of Newton, which shut him off from his fellows and his duties at times of mental creativity. It was not a short tour, for Maclaurin spent two years travelling with Polwarth's son. It was an episode which was to end tragically, for while they were visiting Montpellier, Polwarth's son became ill and died. Maclaurin returned to Aberdeen to discover that the University was most certainly highly displeased that he had not been undertaking his duties for two years. It was certainly not the case that Maclaurin had been idle during his time away, for, while in France, he had been awarded a Grand Prize by the Académie des Sciences for his work on the impact of bodies. Despite being reinstated to his chair by the University of Aberdeen, Maclaurin sought a position in the University of Edinburgh. James Gregory, not the famous mathematician of that name but rather the lesser known James Gregory who was a brother of David Gregory, held the chair of mathematics at Edinburgh but had become too ill to carry out the work. The University of Edinburgh sought to appoint someone to a joint professorship with James Gregory and, on 21 August 1725, Newton wrote to Maclaurin offering his support in recommending him for appointment to the post (see [1], or [7] or letter 122 of [3]):I am very glad to hear that you have the prospect of being joined with Mr James Gregory in the Professorship of Mathematics at Edinburgh, not only because you are my friend, but principally because of you abilities, you being acquainted as well with the improvements of Mathematics as with the former state of those sciences. I heartily wish you good success, and shall be very glad to hear of your being elected. In November 1725 Newton wrote to John Campbell, the lord provost of Edinburgh, supporting Maclaurin's appointment (see [1], or [7]):I am glad to understand that Mr Maclaurin is in good repute amongst you, for I think he deserves it very well: And to satisfy you that I do not flatter him, and also to encourage him to accept the place of assisting Mr Gregory, in order to succeed him, I am ready (if you will please give me leave) to contribute twenty pounds per annum towards a provision for him till Mr Gregory's place becomes void, if I live so long. There is no evidence to suggest that Edinburgh took Newton up on his offer to contribute to Maclaurin's salary. Maclaurin began his appointment to the University of Edinburgh on 3 November 1725. Maclaurin was to spend the rest of his career in Edinburgh. In 1733 he married Anne Stewart, who was the daughter of the Solicitor General for Scotland. They were to have seven children but, as was common at that time, not all reached adulthood. Of the seven children, two boys and three girls survived him. Not long after his marriage, Maclaurin worked to expand the Medical Society of Edinburgh into a wider society to include other branches of learning. Maclaurin himself acted as one of the two secretaries of this expanded Society and at the monthly meetings he often read a paper of his own or a letter from a foreign scientist on the latest developments in some topic of current interest. This Society would, after Maclaurin's death, become the Royal Society of Edinburgh. Maclaurin did notable work in geometry, particularly studying higher plane curves. In fact his first important work was Geometrica Organica... published in 1720 while he was at the University of Aberdeen. In 1740 he was awarded a second prize from the Académie des Sciences in Paris, this time for a study of the tides. This prize was jointly awarded to Maclaurin, Euler and Daniel Bernoulli, bracketing Maclaurin with the top two mathematicians of his day. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maclaurin.html (3 of 6) [2/16/2002 11:21:26 PM]

Maclaurin

In 1742 Maclaurin published his 2 volume Treatise of fluxions, the first systematic exposition of Newton's methods written as a reply to Berkeley's attack on the calculus for its lack of rigorous foundations. Maclaurin wrote in the introduction (see for example [1]):[Berkeley] represented the method of fluxions as founded on false reasoning, and full of mysteries. His objections seemed to have been occasioned by the concise manner in which the elements of this method have been usually described, and their having been so much misunderstood by a person of his abilities appeared to me to be sufficient proof that a fuller account of the grounds of this was required. The Treatise of fluxions is a major work of 763 pages, much praised by those who read it but usually described as having little influence. The article [10], however, argues convincingly that Maclaurin's influence on the Continentals has been underrated. Grabiner gives five areas of influence of Maclaurin's treatise: his treatment of the fundamental theorem of the calculus; his work on maxima and minima; the attraction of ellipsoids; elliptic integrals; and the Euler-Maclaurin summation formula. Maclaurin appealed to the geometrical methods of the ancient Greeks and to Archimedes' method of exhaustion in attempting to put Newton's calculus on a rigorous footing. It is in the Treatise of fluxions that Maclaurin uses the special case of Taylor's series now named after him and for which he is undoubtedly best remembered today. The Maclaurin series was not an idea discovered independently of the more general result of Taylor for Maclaurin acknowledges Taylor's contribution. Another important result given by Maclaurin, which has not been named after him or any other mathematician, is the important integral test for the convergence of an infinite series. The Treatise of fluxions is not simply a work designed to put the calculus on a rigorous basis, for Maclaurin gave many applications of calculus in the work. For example he investigates the mutual attraction of two ellipsoids of revolution as an application of the methods he gives. Other topics which Maclaurin wrote on were the annular eclipse of the sun in 1737 and the structure of bees' honeycombs. He also contributed to actuarial studies as one of the founders of the topic and [5]:He laid sound actuarial foundations for the insurance society that has ever since helped the widows and children of Scottish ministers and professors. Maclaurin did become involved in controversy with other mathematicians over a number of results. Two are well documented, one being with William Braikenridge (see our biography of Braikenridge in this archive, and also [16]) on the argument over the result:... if the sides of a polygon pass through fixed points and all but one of the vertices lie on fixed lines, then the remaining vertices describe a conic section or a straight line. In [17] the controversy between Maclaurin and George Campbell over complex roots is described. Again the argument, which Maclaurin calls "a disagreeable dispute", was about priority. We should not only comment on Maclaurin's mathematical research, however, but also on his qualities as a teacher. His teaching at the University of Edinburgh came in for considerable praise [7]:... such was his anxiety for the improvement of his scholars that if at any time they seemed not fully to comprehend his meaning, or if, upon examining them, he found they could not readily demonstrate the propositions from which he had provided, he was apt rather to suspect his own explanation to have been obscure, than their want of genius or attention, and therefore would resume the demonstration in some other method, to try if, by exposing it http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maclaurin.html (4 of 6) [2/16/2002 11:21:26 PM]

Maclaurin

in a different light, he would give then a better view of it. Maclaurin played an active role in the defence of Edinburgh during the Jacobite rebellion of 1745. As the Jacobite army marched towards Edinburgh in September 1745, Maclaurin worked endlessly in attempting to prepare the defences of the city. He described the events (see for example letter 100 of [3]):The care of the walls was recommended to me, in which I laboured night and day under infinite discouragements from superior powers. When I was promised hundreds of workers I could hardly get dozens. This was daily complained of, redress was promised but till the last two days no redress was made, and then it was too late. When the city fell to the attacking Jacobites, Maclaurin fled to England and while in Newcastle he received an invitation from the Archbishop of York to be his guest in York. There he:... lived for some time as happy as was possible for a man who had left his country in such a situation and his family in it behind him. When the Jacobite army marched south from Edinburgh, Maclaurin returned to the city in November 1745. However, he was weakened by his exertions in preparing the defences of Edinburgh, by the difficult journeys to and from York, by the cold winter weather, and by a fall from his horse. On 26 December 1745 he wrote:I have not been [out] since December 3. My illness seemed dangerous, the physicians call it an obstruction in the reins from the severe cold in travelling November 14, 15 and 16. I had a swelling about my stomach. He died the following year in Edinburgh and was buried in Greyfriars Church where his grave can still be seen at the south-west corner. Many wrote of Maclaurin's outstanding kindness. He was described as [7]:... kindly and approachable ... and it was said that the help he gave to his students:... was never wanting; nor was admittance refused to any except in his teaching hours. His friendship was exceedingly highly valued:His acquaintance and friendship were ... centred by the ingenious of all ranks; who by their fondness for his company, took up a great deal of his time, and left him not master of it, even in his country retreat. Maclaurin's Treatise on algebra was published in 1748, two years after his death. Another work Account of Sir Isaac Newton's discoveries was left incomplete on his death but was published in 1750. One of his papers remained unpublished until 1996 when Grabiner published [11]. In this work Maclaurin considers the geometric problem of finding the difference between the volume of the frustum of a solid of revolution which is generated by a conic section and the volume of the cylinder of the same height as the frustum having diameter equal to that of the frustum at the midpoint of its height. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (26 books/articles)

Some Quotations (3)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maclaurin.html (5 of 6) [2/16/2002 11:21:26 PM]

Maclaurin

A Poster of Colin Maclaurin

Mathematicians born in the same country

Some pages from publications

Preface from Treatise on Fluxions (1742). A page from Treatise of Algebra (1742)

Cross-references to History Topics

1. The rise of calculus 2. A visit to James Clerk Maxwell's house 3. Matrices and determinants

Cross-references to Famous Curves

1. Cayley's Sextic 2. Lituus 3. Trisectrix of Maclaurin

Other references in MacTutor

1. 2. 3. 4.

Maclaurin series Maclaurin series for cosine Maclaurin series for sine Chronology: 1740 to 1760

Honours awarded to Colin Maclaurin (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1719

Lunar features

Crater Maclaurin

Other Web sites

1. Rouse Ball 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Maclaurin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maclaurin.html (6 of 6) [2/16/2002 11:21:26 PM]

Madhava

Madhava of Sangamagramma Born: 1350 in Sangamagramma (near Cochin), Kerala, India Died: 1425 in India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Madhava of Sangamagramma was born near Cochin on the coast in the Kerala state in southwestern India. It is only due to research into Keralese mathematics over the last twenty-five years that the remarkable contributions of Madhava have come to light. In [10] Rajagopal and Rangachari put his achievement into context when they write:[Madhava] took the decisive step onwards from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis. Now all the mathematical writings of Madhava have been lost although some of his texts on astronomy have survived. However his brilliant work in mathematics has been largely discovered by the reports of other Keralese mathematicians such as Nilakantha who lived about 100 years later. Madhava discovered the series equivalent to the Maclaurin expansions of sin x, cos x, and arctan x around 1400, which is over two hundred years before they were rediscovered in Europe. Details appear in a number of works written by his followers such as Mahajyanayana prakara which means Method of computing the great sines. In fact this work had been claimed by some historians such as Sarma (see for example [2]) to be by Madhava himself but this seems highly unlikely and it is now accepted by most historians to be a 16th century work by a follower of Madhava. This is discussed in detail in [4]. Jyesthadeva wrote Yukti-Bhasa in Malayalam, the regional language of Kerala, around 1550. In [9] Gupta gives a translation of the text and this is also given in [2] and a number of other sources. Jyesthadeva describes Madhava's series as follows:The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude. This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Madhava.html (1 of 3) [2/16/2002 11:21:27 PM]

Madhava

Perhaps we should write down in modern symbols exactly what the series is that Madhava has found. The first thing to note is that the Indian meaning for sine of q would be written in our notation as r sin q and the Indian cosine of would be r cos q in our notation, where r is the radius. Thus the series is r q = r(r sin q)/1(r cos q) - r(r sin q)3/3r(r cos q)3 + r(r sin q)5/5r(r cos q)5 - r(r sin q)7/7r(r cos q)7 + ... putting tan = sin/cos and cancelling r gives q = tan q - (tan3q)/3 + (tan5q)/5 - ... which is equivalent to Gregory's series tan-1q = q - q3/3 + q5/5 - ... Now Madhava put q =

/4 into his series to obtain

/4 = 1 - 1/3 + 1/5 - ... and he also put q =

/6 into his series to obtain

= 12(1 - 1/(3 3) + 1/(5 32) - 1/(7 33) + ... We know that Madhava obtained an approximation for

correct to 11 decimal places when he gave

= 3.14159265359 which can be obtained from the last of Madhava's series above by taking 21 terms. In [5] Gupta gives a translation of the Sanskrit text giving Madhava's approximation of correct to 11 places. Perhaps even more impressive is the fact that Madhava gave a remainder term for his series which improved the approximation. He improved the approximation of the series for /4 by adding a correction term Rn to obtain /4 = 1 - 1/3 + 1/5 - ... 1/(2n-1) Rn Madhava gave three forms of Rn which improved the approximation, namely Rn = 1/(4n) or Rn = n/(4n2+1) or Rn = (n2+1)/(4n3+5n). There has been a lot of work done in trying to reconstruct how Madhava might have found his correction terms. The most convincing is that they come as the first three convergents of a continued fraction which can itself be derived from the standard Indian approximation to namely 62832/20000. Madhava also gave a table of almost accurate values of half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle. It is thought that the way that he found these highly accurate tables was to use the equivalent of the series expansions sin q = q - q3/3! + q5/5! - ... cos q = 1 - q2/2! + q4/4! - ... Jyesthadeva in Yukti-Bhasa gave an explanation of how Madhava found his series expansions around 1400 which are equivalent to these modern versions rediscovered by Newton around 1676. Historians have claimed that the method used by Madhava amounts to term by term integration. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Madhava.html (2 of 3) [2/16/2002 11:21:27 PM]

Madhava

Rajagopal's claim that Madhava took the decisive step towards modern classical analysis seems very fair given his remarkable achievements. In the same vein Joseph writes in [1]:We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition, making him almost the equal of the more recent intuitive genius Srinivasa Ramanujan, who spent his childhood and youth at Kumbakonam, not far from Madhava's birthplace. Article by: J J O'Connor and E F Robertson List of References (12 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. A chronology of pi 2. An overview of Indian mathematics

Other references in MacTutor

Chronology: 1300 to 1500

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Madhava.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Madhava.html (3 of 3) [2/16/2002 11:21:27 PM]

Magnitsky

Leonty Filippovich Magnitsky Born: 19 June 1669 in Ostashkov, Russia Died: 30 Oct 1739 in Moscow, Russia Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leonty Magnitsky was sent to the Iosifo-Volokolamsky Monastery when he was 15 years old. Before this he had little education. He went from there to the Simonov Monastery in Moscow where he trained to be a priest. He studied at an academy in Moscow until 1694. At this stage he began to earn a living teaching the children of the important families in Moscow. He continued to do this until 1701 when Peter the Great gave him an allowance so that he might work writing his mathematics books. Peter the Great founded the Navigation School in Moscow in 1701 and the following year he appointed Magnitsky a teacher there. Magnitsky remained there for the rest of his life. From 1715 until his death he was director of the Navigation School. Magnitsky wrote Arithmetic in 1703, the first guide to mathematics published in Russia. It remained for 50 years the basic Russian mathematics text. He also produced a Russian edition of Vlacq's log tables again in 1703. Peter the Great was clearly well pleased with Magnitsky's work and books since he had a house built for Magnitsky's family in Moscow in 1704. Peter the Great also used skill which Magnitsky must have had in fortifications, since when Sweden invaded in 1707, Peter had Magnitsky work on fortifications of the city of Tver. Being at the Navigation School, Magnitsky began to work on navigation. He edited Tables for Navigation (1722). Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Magnitsky.html (1 of 2) [2/16/2002 11:21:29 PM]

Magnitsky

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Magnitsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Magnitsky.html (2 of 2) [2/16/2002 11:21:29 PM]

Magnus

Wilhelm Magnus Born: 5 Feb 1907 in Berlin, Germany Died: 15 Oct 1990

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Magnus attended the University of Frankfurt receiving his doctorate from that university in 1931. His doctorate was supervised by Dehn who asked Magnus various questions about one-relator groups in 1928. Magnus was able to answer these questions and published his results on one-relator groups in 1930. In 1932 he published a major result in combinatorial group theory when he proved that the word problem for one-relator groups is soluble. Magnus was appointed to the staff in Frankfurt serving from 1933 until 1938. During this time he spent the session 1934/35 at Princeton University in the United States. During this period Magnus introduced Lie ring methods to study the lower central series of free groups. He studied the automorphism groups of free groups in 1934. In 1935 Magnus gave examples of finitely presented groups which were isomorphic to proper factor groups of themselves. Hopf had originally asked whether such groups exist and, although Jakob Nielsen had shown that free groups of finite rank have this property ten years before Hopf asked the question, nobody - including Jakob Nielsen himself - noticed the question had already been answered. However Magnus's career was to hit problems when he refused to join the Nazi Party and, as a consequence of this, was not allowed to hold an academic post during World War II. Instead he had to work in industry. In 1947 he was offered a professorship at the University of Göttingen but he was not to remain there for long. In 1946 Bateman died and Whittaker was asked to recommend someone who could undertake the project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Magnus.html (1 of 3) [2/16/2002 11:21:30 PM]

Magnus

of organising and publishing Bateman's manuscripts. Whittaker's advice was that Erdélyi should lead the project and, in 1947, Erdélyi went to the California Institute of Technology. The project was a major one and other collaborators were needed. Magnus left Göttingen to join the Bateman project in 1948. He collaborated on the production of three volumes of Higher Transcendental Functions and two volumes of Tables of Integral Transforms. In 1950 Magnus went to the Courant Institute of Mathematical Sciences. He spent 23 years there before moving to a chair at the Polytechnic Institute of New York in 1973. He held this post for five years before retiring. In [2] Magnus's research is described in these terms:Magnus's mathematical expertise was exceptionally wide ranging. In addition to research in group theory and special functions, he worked on problems in mathematical physics, including electromagnetic theory and applications of the wave equation. It was not only in the breadth and depth of research that Magnus excelled. He was also one of the best supervisors of doctoral students, supervising 61 doctoral students during his career. His teaching is described in [2] as 'outstanding' and this is confirmed by his receipt of the Great Teacher Award of New York University in 1969. His nine books cover a wide range of mathematical topics such as elliptic functions, tessellations (Noneuclidean tessellations and their groups (1974) ), combinatorial group theory (a major work Combinatorial group theory (1966) written jointly with A Karrass and D Solitar) and mathematical physics. He was awarded a number of honours including a Rockefeller Fellowship in 1934, a Guggenheim Fellowship in 1969 and Fulbright-Hays Senior Research Scholarship in 1973/74. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1930 to 1940

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Magnus.html (2 of 3) [2/16/2002 11:21:30 PM]

Magnus

JOC/EFR June 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Magnus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Magnus.html (3 of 3) [2/16/2002 11:21:30 PM]

Mahavira

Mahavira Born: about 800 in possibly Mysore, India Died: about 870 in India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mahavira (or Mahaviracharya meaning Mahavira the Teacher) was of the Jaina religion and was familiar with Jaina mathematics. He worked in Mysore in southern Indian where he was a member of a school of mathematics. If he was not born in Mysore then it is very likely that he was born close to this town in the same region of India. We have essentially no other biographical details although we can gain just a little of his personality from the acknowledgement he gives in the introduction to his only known work, see below. However Jain in [10] mentions six other works which he credits to Mahavira and he emphasises the need for further research into identifying the complete list of his works. The only known book by Mahavira is Ganita Sara Samgraha , dated 850 AD, which was designed as an updating of Brahmagupta's book. Filliozat writes [6]:This book deals with the teaching of Brahmagupta but contains both simplifications and additional information. ... Although like all Indian versified texts, it is extremely condensed, this work, from a pedagogical point of view, has a significant advantage over earlier texts. It consisted of nine chapters and included all mathematical knowledge of mid-ninth century India. It provides us with the bulk of knowledge which we have of Jaina mathematics and it can be seen as in some sense providing an account of the work of those who developed this mathematics. Now there were many Indian mathematicians before the time of Mahavira but, perhaps surprisingly, their work on mathematics is always contained in texts which discuss other topics such as astronomy. The Ganita Sara Samgraha by Mahavira is the earliest Indian text which we possess which is devoted entirely to mathematics. In the introduction to the work Mahavira paid tribute to the mathematicians whose work formed the basis of his book. These mathematicians included Aryabhata I, Bhaskara I, and Brahmagupta. Mahavira writes:With the help of the accomplished holy sages, who are worthy to be worshipped by the lords of the world ... I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are picked from the sea, gold from the stony rock and the pearl from the oyster shell; and I give out according to the power of my intelligence, the Sara Samgraha, a small work on arithmetic, which is however not small in importance. The nine chapters of the Ganita Sara Samgraha are:

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mahavira.html (1 of 3) [2/16/2002 11:21:32 PM]

Mahavira

1. Terminology 2. Arithmetical operations 3. Operations involving fractions 4. Miscellaneous operations 5. Operations involving the rule of three 6. Mixed operations 7. Operations relating to the calculations of areas 8. Operations relating to excavations 9. Operations relating to shadows Throughout the work a place-value system with nine numerals is used or sometimes Sanskrit numeral symbols are used. Of interest in Chapter 1 regarding the development of a place-value number system is Mahavira's description of the number 12345654321 which he obtains after a calculation. He describes the number as:... beginning with one which then grows until it reaches six, then decreases in reverse order. Notice that this wording makes sense to us using a place-value system but would not make sense in other systems. It is a clear indication that Mahavira is at home with the place-value number system. Among topics Mahavira discussed in his treatise was operations with fractions including methods to decompose integers and fractions into unit fractions. For example 2/17 = 1/12 + 1/51 + 1/68. He examined methods of squaring numbers which, although a special case of multiplying two numbers, can be computed using special methods. He also discussed integer solutions of first degree indeterminate equation by a method called kuttaka. The kuttaka (or the "pulveriser") method is based on the use of the Euclidean algorithm but the method of solution also resembles the continued fraction process of Euler given in 1764. The work kuttaka, which occurs in many of the treatises of Indian mathematicians of the classical period, has taken on the more general meaning of "algebra". An example of a problem given in the Ganita Sara Samgraha which leads to indeterminate linear equations is the following: Three merchants find a purse lying in the road. One merchant says "If I keep the purse, I shall have twice as much money as the two of you together". "Give me the purse and I shall have three times as much" said the second merchant. The third merchant said "I shall be much better off than either of you if I keep the purse, I shall have five times as much as the two of you together". How much money is in the purse? How much money does each merchant have? If the first merchant has x, the second y, the third z and p is the amount in the purse then p + x = 2(y + z), p + y = 3(x + z), p + z = 5(x + y). There is no unique solution but the smallest solution in positive integers is p = 15, x = 1, y = 3, z = 5. Any solution in positive integers is a multiple of this solution as Mahavira claims. Mahavira gave special rules for the use of permutations and combinations which was a topic of special interest in Jaina mathematics. He also described a process for calculating the volume of a sphere and one for calculating the cube root of a number. He looked at some geometrical results including right-angled triangles with rational sides, see for example [4]. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mahavira.html (2 of 3) [2/16/2002 11:21:32 PM]

Mahavira

Mahavira also attempts to solve certain mathematical problems which had not been studied by other Indian mathematicians. For example, he gave an approximate formula for the area and the perimeter of an ellipse. In [8] Hayashi writes:The formulas for a conch-like figure have so far been found only in the works of Mahavira and Narayana. Now it is reasonable to ask what a "conch-like figure" is. It is two unequal semicircles (with diameters AB and BC) stuck together along their diameters. Although it might be reasonable to suppose that the perimeter might be obtained by considering the semicircles, Hayashi claims that the formulas obtained:... were most probably obtained not from the two semicircles AB and BC. Article by: J J O'Connor and E F Robertson List of References (11 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. An overview of Indian mathematics 2. A history of Zero

Other references in MacTutor

Chronology: 500 to 900

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mahavira.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mahavira.html (3 of 3) [2/16/2002 11:21:32 PM]

Mahendra_Suri

Mahendra Suri Born: about 1340 in Western India Died: about 1410 in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mahendra Suri was a Jain. Jainism began around the sixth century BC and the religion had a strong influence on mathematics particularly in the last couple of centuries BC. By the time of Mahendra Suri, however, Jainism had lost support as a national religion and was much less vigorous. It had been influenced by Islam and in particular Islamic astronomy came to form a part of the background. However, Pingree in [4] writes that this filtering of Islamic astronomy into Indian culture was:... not allowed to affect in any way the structure of the traditional science. Mahendra Suri was a pupil of Madana Suri. He is famed as the first person to write a Sanskrit treatise on the astrolabe. Ohashi writes in [3] of the early history of the astrolabe in the Delhi Sultanate in India:The astrolabe was introduced into India at the time of Firuz Shah Tughluq (reign AD 1351 88), and Mahendra Suri wrote the first Sanskrit treatise on the astrolabe entitled Yantraraja (AD 1370). The Delhi Sultanate was established around 1200 and from that time on Muslim culture flourished in India. The ideas of Islamic astronomy began to appear in works in the Sanskrit language and it is the Islamic ideas on the astrolabe which Mahendra Suri wrote on in his famous text. It is clear from the various references in the text and also from the particular values that Mahendra Suri uses for the angle of the ecliptic etc. that his work is based on Islamic rather than traditional Indian astronomy works. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mahendra_Suri.html (1 of 2) [2/16/2002 11:21:33 PM]

Mahendra_Suri

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mahendra_Suri.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mahendra_Suri.html (2 of 2) [2/16/2002 11:21:33 PM]

Mahler

Kurt Mahler Born: 26 July 1903 in Krefeld, Prussian Rhineland Died: 25 Feb 1988 in Canberra, Australia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kurt Mahler contracted tuberculosis at the age of five years which affected his right knee. Because of these health problems he attended school for only four years leaving in 1917 at the age of 13. In 1918 he took a job in a factory but attended technical schools to learn to become an instrument maker. He was self-taught in mathematics teaching himself while working in the factory, reading works by Landau, Knopp, Klein and Hilbert among others. His father sent small articles his son had written to Klein who passed them to Siegel. Siegel arranged for him to attend Frankfurt University which he entered in 1923. There he attended lectures by Dehn on topology, Hellinger on elliptic functions, Siegel on calculus and Szász. In 1925 Kurt moved to Göttingen where he attended lectures by Emmy Noether, Courant, Landau, Born, Heisenberg, Hilbert and Ostrowski and acted as unpaid assistant to Norbert Wiener. It was through Emmy Noether that he learnt about p-adic numbers which were to be one of the major topics of his research throughout his life. In 1927 he submitted his doctoral dissertation on zeros of the gamma function to Frankfurt. In 1933 Mahler was appointed to his first post at the University of Königsberg but, before he could take up the post, Hitler came to power. Mahler realised at once that, as he was Jewish, he had to leave Germany. He accepted an invitation from Mordell to go to Manchester where he spent 1933-34. He spent 1934-36 in Groningen in the Netherlands. However in 1936 he was involved in a bicycle accident while in Groningen and his knee troubles returned. He underwent operations on his knee back home in Krefeld

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mahler.html (1 of 3) [2/16/2002 11:21:35 PM]

Mahler

and also spent some time in Switzerland where he was finally cured. Mahler returned to Manchester in 1937 but during 1940 he was interned as "an enemy alien" for 3 months and spent some time in the same camp on the Isle of Man as Kurt Hirsch. Returning to Manchester he remained there until 1962 when he went to Canberra for the last 6 years of his career. He worked on transcendence of numbers, showing in 1946 that 0.123456789101112131415161718192021... was transcendental. In [12] it is noted that Mahler regretted that, apart from his own work, little interest had been shown by 20th century mathematicians in the study of arithmetical properties of decimal expansions. He also classified real and complex numbers into classes which are algebraically independent. Other major themes of his work were p-adic numbers, p-adic Diophantine approximation, geometry of numbers (a term coined by Minkowski to describe the mathematics of packings and coverings) and measure on polynomials. Mahler received many awards. He was elected a Fellow of the Royal Society in 1948. The London Mathematical Society awarded him its Senior Berwick Prize in 1950 and its De Morgan Medal in 1971. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) Mathematicians born in the same country Honours awarded to Kurt Mahler (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1948

LMS De Morgan Medal

Awarded 1971

LMS Berwick Prize winner

1950

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mahler.html (2 of 3) [2/16/2002 11:21:35 PM]

Mahler

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mahler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mahler.html (3 of 3) [2/16/2002 11:21:35 PM]

Maior

John Maior Born: 1469 in Gleghornie (near Haddington), Scotland Died: 1550 in St Andrews, Fife, Scotland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Maior worked in Paris and St Andrews where he taught logic and theology. He was interested in mathematics and logic and applied these to physics writing an important text on the infinite Propositum de infinito in 1506. Maior was also important for spreading the work of Bradwardine and Swineshead in his teaching. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Maior.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maior.html [2/16/2002 11:21:36 PM]

Malcev

Anatoly Ivanovich Malcev Born: 27 Nov 1909 in Misheronsky (near Moscow), Russia Died: 7 July 1967 in Novosibirsk, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Like most Russian mathematicians there are different ways to transliterate Malcev's name into the Roman alphabet. The most common way, other than Malcev, is to write it as MAl-tsev. Anatoly Ivanovich Malcev's father was a glass-blower so his background was certainly from that of a poor family. It did not take long for his mathematical abilities to shine, however, and his teachers at secondary school quickly became convinced that he was destined to become an outstanding mathematician. Malcev graduated from school in 1927 and, in the same year, he went to Moscow State University to study mathematics. He graduated in 1931 but before that he had already begun to teach in a secondary school in Moscow in 1930. After graduating he continued with his teaching career then, in 1932, he was appointed as an assistant at the Ivanovo Paedagogical Institute which was in Ivanovo to the north-east of Moscow. The town, on both banks of the Uvod River, had been known as Ivanovo-Voznesensk but was renamed Ivanovo in 1932, the year Malcev started to work there. When Malcev began teaching in Ivanovo he had only a first degree but Ivanovo had a good rail link with Moscow and he was able to make frequent trips there to discuss his research with Kolmogorov. Malcev's first publications were on logic and model theory and resulted from work he had begun entirely on his own. Ideas from these papers were later to reappear in Robinson's work on non-standard analysis. Kolmogorov invited Malcev to join his graduate programme at Moscow University, and he held a studentship there for a year although he continued his teaching post at the Ivanovo Paedagogical Institute

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malcev.html (1 of 4) [2/16/2002 11:21:38 PM]

Malcev

during the year. Malcev always considered himself to have been Kolmogorov's student and certainly during the year he held the studentship at Moscow University, Malcev was directed by Kolmogorov to certain algebra problems. In 1937 Malcev published a paper on the embeddability of a ring in a field, answering a question posed by Kolmogorov. The question had been posed originally by van der Waerden as to whether there existed rings without zero-divisors which could not be embedded in a field. Malcev answered this question by constructing a ring whose multiplicative semigroup was not embeddable in a group. He was led to investigate the existence of rings whose multiplicative semigroup was embeddable in a group yet the ring still was not embeddable in a field. This led, two years later, to another fundamental paper of Malcev where he gave necessary and sufficient conditions for a semigroup to be embeddable in a group. In 1937 Malcev wrote a dissertation on Torsion free abelian groups of finite rank then, between 1939 and 1941, he studied for his doctorate at the Steklov Institute of the Academy of Sciences. During this period in which he undertook doctoral research Malcev continued as a lecturer at the Ivanovo Paedagogical Institute. In 1941 he received the degree of Doctor of Science for a dissertation Structure of isomorphic representable infinite algebras and groups. Malcev became a professor at Ivanovo Paedagogical Institute in 1944. His work in group theory continued and he proved important results on linear groups and, in particular, on linear soluble groups. In [2] and [3] his work on representations of infinite groups by matrices is commented on:Among these results we mention the local theorem for the class of groups representable by matrices of a given order, and also the theorem on residual finiteness of finitely generated linear groups. This last theorem implies, in particular, the proposition that free groups are residually finite. Malcev also studied Lie groups and topological algebras, producing a synthesis of algebra and mathematical logic. For example he wrote on semisimple subgroups of Lie groups in 1944 and Free topological algebras in 1957. In 1946 he was awarded a State Prize for his work on Lie groups. Malcev also created a synthesis of the theory of algebras and of algorithms called constructive algebras. In 1960, Malcev was appointed to a chair in mathematics at the Mathematics Institute at Novosibirsk and to be chairman of the Algebra and Logic Department at Novosibirsk State University. At Novosibirsk ([2] and [3]):Malcev founded the Siberian section of the Mathematics Institute of the Academy of Sciences, a logic-algebraic school with many members, and directed the world famous seminar Algebra i Logika. In 1962 Malcev founded the specialised journal Algebra i Logika and from that time he was editor of this journal. He was also the founder of the Siberian Mathematical Society and its first president, chief editor and active leader of Sibirsk. Mat. Zh.; and a member of one of the oldest journals in our country, Mat. Sbornik. During the early 1960s Malcev work on problems of decidability of elementary theories of various algebraic structures. He showed the undecidability of the elementary theory of finite groups, of free nilpotent groups, of free soluble groups and many others. He investigated the undecidability of the elementary theory of classical linear groups and proved that the class of locally free algebras had a decidable theory.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malcev.html (2 of 4) [2/16/2002 11:21:38 PM]

Malcev

In 1948 Malcev wrote the undergraduate text Foundation of linear algebra which appeared in English translation in 1962. If I [EFR] may add a personal note, when I began to teach linear algebra in around 1970, I discovered that the publisher Freeman gave away free copies of texts that lecturers might consider adopting for use by a class. I received Malcev's book Foundation of linear algebra from them and I was astonished to find this remarkable approach to linear algebra in a book superbly written to give an understanding of the subject rather than the approach adopted by a large number of linear algebra books which seem on the whole written to provide good examination questions. At the regrettably early age of 57 Malcev died while taking part in the Novosibirsk Topology Conference which he had helped to organise ([2] and [3]):It is impossible to forget the sorrow and deep concern that Malcev's death caused to all the mathematicians, those who lived in Novosibirsk as well as the guests. All felt they were saying farewell to a great scientist, to whom many were indebted... Just before his death Malcev had delivered his final lecture at this Novosibirsk Topology Conference. It was on algebras, now called Malcev algebras, which are natural generalisations of Lie algebras. Malcev had introduced these algebras in 1955 and in this lecture he gave a survey of his work on this topic over the twelve preceding years. Malcev received many honours, and world-wide recognition for his innovative work. He was elected a member of the USSR Academy of Sciences in 1958. We mentioned some of the prizes he received above, such as the State Prize in 1946, but another important honour which he received in 1964 was a Lenin Prize for his series of papers on the applications of mathematical logic to algebra. In [2] and [3] Malcev's contribution is summed up by pointing out that his:... mathematical work is distinguished by the abundance of new ideas and the creation of new mathematical trends on the one hand, and the solution of a number of classical problems on the other hand. He is described as:... a person of great charm, an interesting companion and a wise counsellor on scientific and worldly affairs. In [4] some of Malcev's interests are described:Malcev liked very long walks and long distance swimming. He played the violin from a very early age until he was 30, when he switched to the piano ... He liked history, and there are a number of volumes on this subject in his library, including the history of mathematics. Other interests that he had included poetry and his particular tastes in music included Bach and Russian folk songs. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malcev.html (3 of 4) [2/16/2002 11:21:38 PM]

Malcev

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Malcev.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malcev.html (4 of 4) [2/16/2002 11:21:38 PM]

Malebranche

Nicolas Malebranche Born: 6 Aug 1638 in Paris, France Died: 13 Oct 1715 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nicolas Malebranche's father (also called Nicolas Malebranche) was a secretary to the king, while his mother, Catherine de Lauzon, was a very gifted lady. It is probable that Malebranche's great literary style came from his mother's tuition. Malebranche was the youngest of a large number of children, but his life was much influenced by illness. He was crippled all his life with a deformed spine and this meant that he did not attend school in the usual way but was educated at home to the age of sixteen. Malebranche studied philosophy and theology at the Collège de la Marche from 1654 to 1656 and graduated Master of Arts. P André writes in [1] that he found theology:... neither great nor true, full of vain subtleties, perpetual equivocation, lacking in taste and Christian spirit. Malebranche went to the Sorbonne in Paris until 1659, again intending to make theology this life's work but he found it no more to his liking than he had before. He considered it [1]:... only a confused mass of human opinions, frivolous discussions and hair-splitting subtleties, without any order or principle or rational interconnection. Refusing to accept a canonry at Notre Dame, he joined the Congregation of the Oratory in 1660. The Congregation of the Oratory of Jesus and Mary Immaculate, also called the Bérullians, was founded by Pierre de Bérulle in 1611. Its chief aim was, and still is, training candidates for the priesthood. De Bérulle was a friend of Descartes and by the time Malebranche studied at the Oratory its teaching were strongly based on Descartes' philosophy. In 1664 Malebranche was ordained a priest having studied ecclesiastical history, Hebrew and Biblical criticism.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malebranche.html (1 of 4) [2/16/2002 11:21:40 PM]

Malebranche

Malebranche read Descartes' Traité de l'homme and this turned him towards a study of mathematics and physics. In [1] his reaction to Descartes' book is recounted:The joy of becoming acquainted with so large a number of discoveries caused him such palpitations of the heart that he was obliged to stop reading in order to recover his breath. Malebranche himself said that Descartes had:... in thirty years discovered more truths than all the other philosophers put together. Malebranche was also influenced by Leibniz who visited Paris in 1672. The two had many meetings when they discussed ideas both of philosophy and of mathematics and, in particular, Leibniz conveyed many of his ideas about his new calculus to Malebranche. Malebranche became professor of mathematics at the Congregation of the Oratory from 1674. He had a large influence on the development of mathematics and science, principally through the group which he built up in Paris which was seen as the leading one in France. Mathematicians such as Varignon, de L'Hôpital, Guisnée and Reyneau all became part of this circle at the Oratory. Although Malebranche made no outstanding mathematical discoveries, he is of major importance in the development of mathematics since through him the work of Leibniz and Descartes in mathematics was spread and developed. One of Malebranche's direct contributions to mathematics was his editorial role in the publication of de L'Hôpital's Analyse des infiniment petits pour l'intelligence des lignes courbes. Malebranche also had a strong influence through his teaching, in particular he taught mathematics and physics to Privat de Molières and Reyneau. Others were not so much his disciples as his opponents, for example he was in dispute with Arnauld for many years. Malebranche is a major philosopher and follower of Descartes. His metaphysics is his belief that we see all things in God. Human knowledge of the world is only possible through a relation between man and God. He developed Descartes' ideas to bring them more in line with standard Roman Catholic orthodox belief. At first Malebranche's ideas of the physical world followed closely those of Descartes and were based on a belief in a rational geometrical world. He based his laws of motion on the abstract laws of collisions between idealised solid objects. However Leibniz tried, with some success, to persuade Malebranche that the laws of motion were not entirely mathematical laws but were the consequence of God's creation. When it came to an understanding of force, Malebranche found great difficulties with the ideas of his fellow scientists. He wrote:It seems to me that people make very great errors and even very dangerous ones regarding force which gives movement and which transports bodies. How did Malebranche explain force? He did not believe in Descartes' idea of a "clockwork universe' which God set in motion and then it ran itself, only determined by completely general laws of mathematics. Malebranche needed to have a more active role for God in his universe and he did this through his concept of force. He basically believed that if two spheres collided then there was no force which changed the direction of their motion. Rather he saw the collision as an occasion for God to act and since a perfect God would act in the simplest way then the result would always result in the same change in motion. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malebranche.html (2 of 4) [2/16/2002 11:21:40 PM]

Malebranche

Malebranche's most important work is the three volumes of De la recherche de la vérité (1674-75). The work received much acclaim from many and was translated into many languages. Criticism of his work, in particular by Arnauld, led to Malebranche's publication of Traité de la nature et de la grâce (1680), which was banned by the Roman Catholic Church ten years later. Another important work is Entretiens sur la métaphysique et sur la religion (1688) in which Malebranche sets out in the clearest way his metaphysics and philosophy. Malebranche's other work includes research into the nature of light and colour, studies in the infinitesimal calculus and work on vision. This mathematical and scientific work was published in Réflexions sur la lumière, les couleurs et la génération du feu in 1699. He was elected to the Académie des Sciences in the same year of 1699 mainly as a result of his work Traité des lois de la communication du mouvement. What was the opinion of contemporary and later writers on Malebranche and his ideas? Fontenelle considered him a great mathematician and physicist and also a great writer:His diction is pure and chaste, and has all the dignity which the subject requires and all the grace of which it admits. D'Alembert also praises his writing but not his philosophy:I think that he is in all respects very inferior to Bayle and Gassendi as a philosopher; it even seems to me that he was less a great philosopher than an excellent writer on philosophy ... I see him as a good demolisher but a bad architect. All d'Alembert can find in the way of praise of Malebranche's De la recherche de la vérité is to say it contained:... a few useful truths concealed as it were stifled beneath a heap of systems which have long since been forgotten. However d'Alembert finds that Malebranche writes in:... the most suitable language for philosophy, the only one worthy of it, methodical without dryness, thoroughly developed but without verbiage, interesting and sensible without false warmth, great without effort, and noble without turgidity. Voltaire echoed the same thoughts when he compared Locke and Malebranche saying:A single page of Locke contains more truths than all the volumes of Malebranche; but a single line of Malebranche reveals more subtlety, imagination, finesse, and genius, perhaps, than all of Locke's enormous book. Malebranche was to have a strong influence on many who visited Paris while he and his disciples exerted a strong influence there. One who was strongly influenced was Berkeley who visited Paris in 1713 and met with Malebranche. Malebranche was taken ill in 1715 while staying at the house of a friend at Villeneuve- Saint- Georges. He was taken back to the Oratory in Paris and died four months later after great suffering. Fontenelle writes that his illness:... adapted itself to his philosophy. The body, which he so much despised, was reduced to nothing; but like the mind, accustomed to supremacy, continued sane and sound. He http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malebranche.html (3 of 4) [2/16/2002 11:21:40 PM]

Malebranche

remained throughout a calm spectator of his own long death, the last moment of which was such that it was believed he was merely resting. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) A Poster of Nicolas Malebranche

Mathematicians born in the same country

Honours awarded to Nicolas Malebranche (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Malebranche (5th Arrondissement)

Other Web sites

1. Internet Encyclopedia of Philosophy 2. The Galileo Project 3. The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Malebranche.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malebranche.html (4 of 4) [2/16/2002 11:21:40 PM]

Malfatti

Gian Francesco Malfatti Born: 1731 in Ala, Trento, Italy Died: 9 Oct 1807 in Ferrara, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gianfrancesco Malfatti studied under Vincenzo Riccati, F M Zanotti, and G Manfredi at the College of San Francesco Saverio in Bologna. Then, in 1754, Malfatti went to Ferrara where he taught mathematics and physics in a school that he started up there. University of Ferrara was erected by Alberto V who reigned over Ferrara from 1388 to 93. The University was obtained by Pope Boniface IX as a concession in 1391 and this is taken as its founding date. Copernicus studied there in 1503 but the buildings which house the University today date from the end of the 16th-century. The University of Ferrara was re-established in its 16th-century buildings in 1771 and Malfatti was appointed to the chair of mathematics there in that year. Struik, describing the papers in [2], writes that Malfatti:... was a quiet, scholarly man who spent most of his life as a librarian and professor in Ferrara. He was one of the founders of the "Societa Italiana delle Scienze" (1782) and was active in academic reform, especially in the Napoleonic period. Malfatti wrote an important work on equations of the fifth degree. In 1802 he gave the first solution to the problem of describing in a triangle three circumferences that are mutually tangent, each of which touches two sides of the triangle, the so-called Malfatti problem. His solution was published in a paper of 1803 on un problema stereotomica. Jacob Bernoulli had solved this for an isosceles triangle while, after Malfatti, the problem was also solved by Steiner and Clebsch, the latter solving it using elliptic functions.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malfatti.html (1 of 2) [2/16/2002 11:21:42 PM]

Malfatti

Malfatti's interests extended beyond this geometrical problem, however:His papers dealt with many subjects from probability to mechanics and he participated in the debate around Ruffini's attempt to prove the impossibility of solving (in the meaning of that period) equations of higher degree than four. There are several papers in [2] which describe Malfatti's work. These include: Problems and methods of mathematical analysis in the work of Gianfrancesco Malfatti, Contributions of Gianfrancesco Malfatti to combinatorial analysis and to the theory of finite difference equations, The work of Malfatti in the realm of mechanics, The geometrical research of Gianfrancesco Malfatti, Gianfrancesco Malfatti and the theory of algebraic equations, and Gianfrancesco Malfatti and the support problem. We should note some interesting results which he obtained. In 1781 he showed that the lemniscate had the property that [1]:... a mass point moving on it under gravity goes along any arc of the curve in the same time as it traverses the subtending arc. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Cross-references to Famous Curves

Lemniscate of Bernoulli

Other Web sites

Clark Kimberling

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Malfatti.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malfatti.html (2 of 2) [2/16/2002 11:21:42 PM]

Malus

Etienne Louis Malus Born: 23 July 1775 in Paris, France Died: 24 Feb 1812 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Etienne Louis Malus's father was Louis Malus de Mitry and was Treasurer of France. Etienne Louis was first educated at home where he was instructed in literature and mathematics. He then attended the engineering school, Ecole Royale de Genie, at Mézières. There he was taught by Monge who realised Malus had special mathematical talents. In 1793 Malus left the school, having been dismissed for political reasons. On leaving Mézières, Malus joined the army and was posted to Dunkerque. There his abilities were noticed and he was sent to the Ecole Polytechnique as a pupil. Here he was taught by Fourier, and he was perhaps the most able of all of Fourier's pupils. He was to remain associated with Ecole Polytechnique as an examiner all his life. While studying at the Ecole Polytechnique Malus began to undertake original research, writing papers on the path of light through materials of differing refractive indices. After graduating from the Ecole Polytechnique Malus rejoined the army, this time taking part in campaigns on the Rhine in 1797. As an army engineer Malus was ordered to accompany Napoleon's invasion of Egypt in 1798. This did not greatly please him since at the time that he received the order he was stationed in the town of Giessen, as part of the occupying force, and he was about to marry the daughter of the Chancellor of the University of Giessen. While in Cairo, Napoleon's fleet was destroyed in Aboukir Bay and Malus wrote, see [3], From then on we realised that all our communications with Europe were broken. We began to lose hope of ever seeing our native land again. At Napoleon's instigation, while they were in Cairo, the Cairo Institute was set up having 12 mathematical members. As well as Malus these included Monge, Fourier and Napoleon Bonaparte himself. After returning in 1801 Malus held posts in Antwerp, Strasbourg, and Paris. His mathematical work was almost entirely concerned with the study of light. This involved him in studying geometrical systems called ray systems, closely connected to Plücker's line complexes. He conducted experiments to verify Huygens' theories of light and rewrote the theory in analytical form. His discovery of the polarisation of light by reflection was published in 1809 and his theory of double

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malus.html (1 of 2) [2/16/2002 11:21:43 PM]

Malus

refraction of light in crystals in 1810. In 1811 Malus served, along with Lagrange, Legendre, Laplace and Haüy, on the committee to decide on who to award the prize to for the best work on the propagation of heat in solid bodies. They awarded the prize to Fourier. Malus received many honours for his work, in particular he was awarded a prize from the Académie des Sciences in 1810 for his memoir on double refraction. In the same year he was elected to the Académie des Sciences and the following year, despite the war between England and France, Malus was awarded the Rumford medal of the Royal Society of London. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Honours awarded to Etienne Louis Malus (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Malus (5th Arrondissement)

Commemorated on the Eiffel Tower Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Malus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Malus.html (2 of 2) [2/16/2002 11:21:43 PM]

Manava

Manava Born: about 750 BC in India Died: about 750 BC in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Manava was the author of one of the Sulbasutras. The Manava Sulbasutra is not the oldest (the one by Baudhayana is older) nor is it one of the most important, there being at least three Sulbasutras which are considered more important. We do not know Manava's dates accurately enough to even guess at a life span for him, which is why we have given the same approximate birth year as death year. Historians disagree on 750 BC, and some would put this Sulbasutra later by one hundred or more years. Manava would have not have been a mathematician in the sense that we would understand it today. Nor was he a scribe who simply copied manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in mathematics for its own sake, merely interested in using it for religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for religious rites and it would appear an almost certainty that Manava himself would be a Vedic priest. The mathematics given in the Sulbasutras is there to enable accurate construction of altars needed for sacrifices. It is clear from the writing that Manava, as well as being a priest, must have been a skilled craftsman. Manava's Sulbasutra, like all the Sulbasutras, contained approximate constructions of circles from rectangles, and squares from circles, which can be thought of as giving approximate values of . There appear therefore different values of throughout the Sulbasutra, essentially every construction involving circles leads to a different such approximation. The paper [1] is concerned with an interpretation of verses 11.14 and 11.15 of Manava's work which give = 25/8 = 3.125. See the article Indian Sulbasutras for more information on the Sulbasutras in general and the mathematical results which they contain. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Manava.html (1 of 2) [2/16/2002 11:21:44 PM]

Manava

Cross-references to History Topics

1. An overview of Indian mathematics 2. The Indian Sulbasutras

Other references in MacTutor

Chronology: 30000BC to 500BC

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Manava.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Manava.html (2 of 2) [2/16/2002 11:21:44 PM]

Mandelbrot

Benoit Mandelbrot Born: 20 Nov 1924 in Warsaw, Poland

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Benoit Mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature. Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles. Mandelbrot's family emigrated to France in 1936 and his uncle Szolem Mandelbrojt, who was Professor of Mathematics at the Collège de France and the successor of Hadamard in this post, took responsibility for his education. In fact the influence of Szolem Mandelbrojt was both positive and negative since he was a great admirer of Hardy and Hardy's philosophy of mathematics. This brought a reaction from Mandelbrot against pure mathematics, although as Mandelbrot himself says, he now understands how Hardy's deep felt pacifism made him fear that applied mathematics, in the wrong hands, might be used for evil in time of war. Mandelbrot attended the Lycée Rolin in Paris up to the start of World War II, when his family moved to Tulle in central France. This was a time of extraordinary difficulty for Mandelbrot who feared for his life on many occasions. In [3] the effect of these years on his education was emphasised:The war, the constant threat of poverty and the need to survive kept him away from school and college and despite what he recognises as "marvellous" secondary school teachers he was largely self taught. Mandelbrot now attributed much of his success to this unconventional education. It allowed him to think in ways that might be hard for someone who, through a conventional education, is strongly encouraged to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html (1 of 5) [2/16/2002 11:21:46 PM]

Mandelbrot

think in standard ways. It also allowed him to develop a highly geometrical approach to mathematics, and his remarkable geometric intuition and vision began to give him unique insights into mathematical problems. After studying at Lyon, Mandelbrot entered the Ecole Normale in Paris. It was one of the shortest lengths of time that anyone would study there, for he left after just one day. After a very successful performance in the entrance examinations of the Ecole Polytechnique, Mandelbrot began his studies there in 1944. There he studied under the direction of Paul Lévy who was another to strongly influence Mandelbrot. After completing his studies at the Ecole Polytechnique, Mandelbrot went to the United States where he visited the California Institute of Technology. From there he went to the Institute for Advanced Study in Princeton where he was sponsored by John von Neumann. Mandelbrot returned to France in 1955 and worked at the Centre National de la Recherche Scientific. He married Ailette Kagan during this period back in France, but he did not stay there too long before returning to the United States. Clark gave the reasons for his unhappiness with the style of mathematics in France at this time [3]:Still deeply concerned with the more exotic forms of statistical mechanics and mathematical linguistics and full of non standard creative ideas he found the huge dominance of the French foundational school of Bourbaki not to his scientific tastes and in 1958 he left for the United States permanently and began his long standing and most fruitful collaboration with IBM as a Research Fellow and Research Professor at their world renowned laboratories in Yorktown Heights in New York State. IBM presented Mandelbrot with an environment which allowed him to explore a wide variety of different ideas. He has spoken of how this freedom at IBM to choose the directions that he wanted to take in his research presented him with an opportunity which no university post could have given him. In 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 paper claiming that it was a masterpiece and a potential source of interesting problems, but Mandelbrot did not like it. Indeed he reacted rather badly against suggestions posed by his uncle sice he felt that his whole attitude to mathematics was so different from that of his uncle. Instead Mandelbrot chose his own very different course which, however, brought him back to Julia's paper in the 1970s after a path through many different sciences which some characterise as highly individualistic or nomadic. In fact the decision by Mandelbrot to make contributions to many different branches of science was a very deliberate one taken at a young age. It is remarkable how he was able to fulfil this ambition with such remarkable success in so many areas. With the aid of computer graphics, Mandelbrot who then worked at IBM's Watson Research Center, was able to show how Julia's work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics. The Mandelbrot set is a connected set of points in the complex plane. Pick a point Z0 in the complex plane. Calculate: Z1 = Z02 + Z0 http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html (2 of 5) [2/16/2002 11:21:46 PM]

Mandelbrot

Z2 = Z12 + Z0 Z3 = Z22 + Z0 ... If the sequence Z0, Z1, Z2, Z3, ... remains within a distance of 2 of the origin forever, then the point Z0 is said to be in the Mandelbrot set. If the sequence diverges from the origin, then the point is not in the set. His work was first put elaborated in his book Les objets fractals, forn, hasard et dimension (1975) and more fully in The fractal geometry of nature in 1982. On 23 June 1999 Mandelbrot received the Honorary Degree of Doctor of Science from the University of St Andrews. At the ceremony Peter Clark gave an address [3] in which he put Mandelbrot's achievements into perspective. We quote from that address:... at the close of a century where the notion of human progress intellectual, political and moral is seen perhaps to be at best ambiguous and equivocal there is one area of human activity at least where the idea of, and achievement of, real progress is unambiguous and pellucidly clear. That is mathematics. In 1900 in a famous address to the International Congress of mathematicians in Paris David Hilbert listed some 25 open problems of outstanding significance. Many of those problems have been definitively solved, or shown to be insoluble, culminating as we all know most recently in the mid-nineties with the discovery of the proof of Fermat's Last Theorem. The first of Hilbert's problems concerned a thicket of issues about the nature of the continuum or the real line, a major concern of 19th and indeed of 20th century analysis. The problem was both one of geometry concerning the nature of the line thought of as built up of points and of arithmetic thought of as the theory of the real numbers. The integration of those two fields was one of the great achievements of Richard Dedekind and George Cantor, the latter of whom we [St Andrews University] were intelligent enough to honour in 1911. Now lurking about so to speak in the undergrowth of that achievement lay certain very extraordinary geometric objects indeed. To all at the time, they seemed strange, indeed rather pathological monsters. Odd indeed they were, there were curves - one dimensional lines in effect - which filled two dimensional spaces, there were curves which were well behaved, that is nice and continuous but which had no slope at any point (Not just some points, ANY points) and they went by strange names, the Peano Space filling curve, the Sierpinski gasket, the Koch curve, the Cantor Ternary set. Despite their pathological qualities, their extraordinary complexity, especially when viewed in greater and greater detail, they were often very simple to describe in the sense that the rules which generated them were absurdly simple to state. So odd were these objects that mathematicians set about barring these monsters and they were set aside as too strange to be of interest. That is until our honorary graduand created out of them an entirely new science, the theory of fractal geometry: it was his insight and vision which saw in those objects and the many new ones he discovered, some of which now bear his name, not mathematical curiosities, but signposts to a new mathematical universe, a new geometry with as much system and generality as that of Euclid and a new physical science. As well as IBM Fellow at the Watson Research Center Mandelbrot was Professor of the Practice of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html (3 of 5) [2/16/2002 11:21:46 PM]

Mandelbrot

Mathematics at Harvard University. He also held appointments as Professor of Engineering at Yale, of Professor of Mathematics at the Ecole Polytechnique, of Professor of Economics at Harvard, and of Professor of Physiology at the Einstein College of Medicine. Mandelbrot's excursions into so many different branches of science was, as we mention above, no accident but a very deliberate decision on his part. It was, however, the fact that fractals were so widely found which in many cases provided the route into other areas [3]:I should not ... give the impression that we have here before us a mathematician alone. Let me explain why. The first of his great insights was the discovery that the extraordinarily complex almost pathological structures, which had been long ignored, exhibited certain universal characteristics requiring a new theory of dimension to treat them adequately which he had generalised from earlier work of Hausdorff and Besicovitch but the second great insight was that the fractal property so discovered, the general theory of which he had provided, was present almost universally in Nature. What he saw was that the overwhelming smoothness paradigm with which mathematical physics had attempted to describe Nature was radically flawed and incomplete. Fractals and pre-fractals once noticed were everywhere. They occur in physics in the description of the extraordinarily complex behaviour of some simple physical systems like the forced pendulum and in the hugely complex behaviour of turbulence and phase transition. They occur as the foundations of what is now known as chaotic systems. They occur in economics with the behaviour of prices and as Poincaré had suspected but never proved in the behaviour of the Bourse or our own Stock exchange in London. They occur in physiology in the growth of mammalian cells. Believe it or not ... they occur in gardens. Note closely and you will see a difference between the flower heads of broccoli and cauliflower, a difference which can be exactly characterised in fractal theory. Mandelbrot has received numerous honours and prizes in recognition of his remarkable achievements. For example, in 1985 Mandelbrot was awarded the 'Barnard Medal for Meritorious Service to Science'. The following year he received the Franklin Medal. In 1987 he was honoured with the Alexander von Humboldt Prize, receiving the Steinmetz Medal in 1988 and many more awards including the Nevada Medal in 1991 and the Wolf prize for physics in 1993 Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

Some Quotations (3)

A Poster of Benoit Mandelbrot

Mathematicians born in the same country

Other references in MacTutor

1. The Mandelbrot set 2. Chronology: 1970 to 1980 3. Chronology: 1980 to 1990

Other Web sites

1. Mandelbrot's home page 2. Andy Burbanks (Some Mandelbrot set images)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html (4 of 5) [2/16/2002 11:21:46 PM]

Mandelbrot

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mandelbrot.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mandelbrot.html (5 of 5) [2/16/2002 11:21:46 PM]

Mannheim

Victor Mayer Amédée Mannheim Born: 17 July 1831 in Paris, France Died: 11 Dec 1906 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Amédée Mannheim entered the Ecole Polytechnique in Paris in 1848 at the age of 17. Two years later he went to Metz where he attended the Ecole d'Application. Although slide rules existed before Mannheim's time, invented by Oughtred and Gunter and others, it was Mannheim who standardised the modern version of the slide rule which was in common use until pocket calculators took over a few years ago. It was while he was a student at Metz that the ideas for this slide rule came to Mannheim. I [EFR] purchased a slide rule of the Mannheim type when I was at school and my parents paid 5 pounds for it. That would be the equivalent of perhaps 100 pounds today so the calculator has not only given us a better calculating tool but also a much cheaper one. However, I still have the slide rule and treasure it, while I have thrown out all my early calculators. After graduating from the Ecole d'Application in Metz, Mannheim became an officer of the French artillery. After several years in the military, Mannheim was appointed to the Ecole Polytechnique in Paris, while continuing his army career. His first appointment at the Ecole Polytechnique was as a répétiteur in 1859, then in 1863 he was appointed as an examiner. In the following year Mannheim was appointed as Professor of Descriptive Geometry at the Ecole Polytechnique. Koppelman writes in [1]:He was a dedicated and popular teacher, strongly devoted to the Ecole Polytechnique, and was one of the founders of the Société Amicale des Anciens Elèves de l'Ecole. Mannheim retired from his army post in 1890, having attained the rank of colonel in the engineering corps. He continued teaching at the Ecole Polytechnique until he retired in 1901 at the age of 70. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mannheim.html (1 of 2) [2/16/2002 11:21:48 PM]

Mannheim

He made numerous contributions to geometry and for his outstanding contributions to the subject he was awarded the Poncelet Prize of the Académie des Science in 1872. He studied the polar reciprocal transformation introduced by Chasles and applied his results to kinetic geometry. Mannheim's own definition of kinetic geometry considered it to be the study of motion of a figure without reference to any forces, time or other properties external to the figure. Mannheim's work on the exact synthesis of mechanisms is studied in [6]. He also studied surfaces, in particular Fresnel's wave surfaces. The paper [5] studies this topic of his work in detail. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1850 to 1860

Other Web sites

G Pastori (Slide rule history)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mannheim.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mannheim.html (2 of 2) [2/16/2002 11:21:48 PM]

Mansion

Paul Mansion Born: 3 June 1844 in Marchin (near Huy), Belgium Died: 16 April 1919 in Ghent, Belgium

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Mansion studied at the Ecole Normale des Sciences at Ghent. He began his studies in 1862 when he was eighteen years old. By 1867 he was teaching advanced mathematics courses there. He was later appointed as Professor of Mathematics at the University of Ghent. Of course living in Ghent made Mansion particularly aware of the famous mathematician Quetelet who was born in Ghent some 50 years before Mansion was born and who had studied at the university there. Quetelet and Garnier had edited the Belgium publication Correspondance mathématique et physique. and in 1874 Mansion, together with Catalan and Neuberg, founded a journal Nouvelle correspondance mathématique named to honour the earlier Correspondance mathématique et physique. This journal which Mansion, Catalan and Neuberg had founded was published between 1874 and 1880. After this Catalan encouraged Mansion and Neuberg to collaborate in publishing a new journal and, indeed, they did precisely this, publishing Mathesis from 1881 onwards. Mansion became director of Mathesis and continued with this project until he retired in 1910. According to Pelseneer [1], Mansion:... held an eminent position in the scientific world of Belgium despite his extreme narrow-mindedness. Mansion translated into French many mathematical works by Riemann, Plücker and Clebsch. He did not restrict himself to translating mathematical texts into French, however, for he translated works by other famous authors such as Dante. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mansion.html (1 of 2) [2/16/2002 11:21:50 PM]

Mansion

He wrote on the history of Greek mathematics and on Copernicus, Galileo and Kepler. He also wrote on the history of physics and on Greek astronomy. Being a narrow-minded man with highly orthodox Roman Catholic views, his evaluation of the work of authors such as Copernicus and Galileo was somewhat biased and indeed many would consider his writings on these authors as intended to justify the views of the Roman Catholic Church. Despite the rather critical tone of these comments we should point out that Mansion was highly productive. When Demoulin wrote an obituary of Mansion (published in 1929) he included a list of 349 of his works. Nor were these works merely of local interest, for many were considered important enough to be translated into German or to be republished in foreign publications. Among the honours which Mansion received was election to the Royal Academy of Belgium. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mansion.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mansion.html (2 of 2) [2/16/2002 11:21:50 PM]

Mansur

Abu Nasr Mansur ibn Ali ibn Iraq Born: 970 in (possibly) Khwarazm (now Kara-Kalpakskaya, Uzbekistan) Died: 1036 in (probably) Ghazna (now Ghazni, Afganistan) Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Abu Nasr Mansur was a native of Gilan which is mentioned in The Regions of the World, a Persian geography book of 982. His family, the Banu Iraq, were rulers of Khwarazm the region adjoining the Aral Sea, and it was in this region that Abu Nasr Mansur studied and became a disciple of Abu'l-Wafa. Abu Nasr Mansur was teaching in this area when he first began his association with al-Biruni whom he taught from about 990. This began an important collaboration which was to go on for many years. The end of the 10th century and beginning of the 11th century was a period of great unrest in the Islamic world and there were civil wars in the region in which Abu Nasr Mansur was living. Khwarazm was at this time part of the Samanid Empire which ruled from Bukhara. Other states in the region were the Ziyarid state with its capital at Gurgan on the Caspian sea. Further west the Buwayhid dynasty ruled over the area between the Caspian sea and the Persian Gulf, and over Mesopotamia. Another kingdom which was rapidly rising in influence was the Ghaznavids whose capital was at Ghazna in Afganistan. In 995 the Banu Iraq, of which Abu Nasr Mansur was a prince, was overthrown in a coup. It is not clear what happened to Abu Nasr Mansur at this stage but certainly his pupil al-Biruni fled at the outbreak of the civil war. It seems that Abu Nasr Mansur was, some little time later, employed at the court of Ali ibn Ma'mun and remained at the court when his brother Abu'l Abbas Ma'mun succeeded him. Both these brother married sisters of the ruler Mahmud from the powerful state at Ghazna which would eventually take control of Ma'mun's kingdom. Both Ali ibn Ma'mun and Abu'l Abbas Ma'mun were patrons of the sciences and supported a number of top scientists at their court. Not only did Abu Nasr Mansur work there but from about 1004 al-Biruni also worked there, renewing the collaboration between him and his teacher. The wars in the region, however, were to disrupt the scientific work of Abu Nasr Mansur and eventually he and al-Biruni left in about 1017. Mahmud was extending his influence over the region from his base in Ghazna and made a demand of Abu'l Abbas Ma'mun in 1014 to have his name inserted into the Friday prayers. This was a signal that he wanted an end to Ma'mun's rule and for the region to come under his control. After Ma'mun had at least partially agreeded to Mahmud's demands, he was killed by his own army for what they considered to be an act of treachery. Following this Mahmud marched his army into the region and gained control. Both Abu Nasr Mansur and al-Biruni seem to have left with the victorious Mahmud. It seems that Abu Nasr Mansur spent most of the rest of his life at Mahmud's court in Ghazna.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mansur.html (1 of 3) [2/16/2002 11:21:51 PM]

Mansur

Abu Nasr Mansur is perhaps most famous for his collaboration with al-Biruni. Certainly Abu Nasr Mansur worked on many topics as a result of requests from al-Biruni and a total of twenty-five works are known to have been written by him. It is possible that in this list of twenty-five two names of works that have come down to us are different titles for the same work while another title may refer to a fragment of a larger work (in which case there are only twenty-three). Seventeen works have survived and they show that Abu Nasr Mansur was an extremely able astronomer and mathematician. Of Abu Nasr Mansur's works seven are on mathematics, the rest are on astronomy. All the surviving works have been published, most have been translated into at least one European language, and this gives some indication of the importance attached to his work. Many of Abu Nasr Mansur's works were dedicated to his student al-Biruni. In fact al-Biruni himself lists twelve works which he says Abu Nasr Mansur dedicated to him (although some historians read al-Biruni's words as meaning that he wrote the works himself, but this interpretation seems highly unlikely). The first such work which Abu Nasr Mansur dedicated to al-Biruni was written around 997, soon after the civil war had disrupted their work. In his own writings al-Biruni sometimes quotes results due to Abu Nasr Mansur which he says he worked on at al-Biruni's request. Certainly both men seem keen to give full credit to the other's contributions. Abu Nasr Mansur's main achievements are his commentry on the Spherics of Menelaus, his role in the development of trigonometry from Ptolemy's calculation with chords towards the trigonometric functions used today, and his development of a set of tables which give easy numerical solutions to typical problems of spherical astronomy. Abu Nasr Mansur's reworking of the Spherics of Menelaus is particularly important since the Greek original of Menelaus work has been lost, although there are several Arabic versions. Menelaus's work formed the basis for Ptolemy's numerical solutions of spherical astronomy problems in the Almagest. The work is in three books: the first book studies properties of spherical triangles, the second book investigates properties of systems of parallel circles on a sphere as they intersect great circles, while the third book gives a proof of Menelaus's theorem. In his work on trigonometry Abu Nasr Mansur discovered the sine law a/sin A = b/sin B = c/sin C. Abu'l-Wafa may have discovered this law first and Abu Nasr Mansur may have learnt it from him. Certainly which of the two has priority is hard to determine and will almost certainly never be known with certainty. A third person who is sometimes credited with this same discovery is al-Khujandi but it seems lees likely that he was the discoverer since, as Samso writes in [1]:... he was essentially a practical astronomer, unconcerned with theoretical problems. The claims of al-Khujandi to be the discoverer of the sine law are further discussed in his biography. The article [4] is a description and study of The table of minutes of Abu Nasr Mansur. The author of [4] traces the origins of the work to the 10th century Damascene tables by Habash. Abu Nasr Mansur's treatise discusses the five trigonometric functions which are used to solve problems in spherical astronomy. The article shows the improvement achieved by Abu Nasr Mansur in using 1 as the value of the radius, instead of 60 as was done by most Arabic astronomers.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mansur.html (2 of 3) [2/16/2002 11:21:51 PM]

Mansur

Other work by Abu Nasr Mansur on astronomical topics included four works on the construction and application of the astrolabe. His proof of the sine law appears a number of times in his works, for example in Almagest of the Shah, Book of the azimuth, Treatise on the determination of shperical arcs, and Treatise in which some geometrical questions addressed to him are answered. The questions referred to in the title of this last work were addressed to him by al-Biruni. Others works include Treatise in which a difficulty in the thirteenth book of the Elements is solved and A chapter from a book on the sphericity of the heavens. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Longitude and the Académie Royale 2. Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mansur.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mansur.html (3 of 3) [2/16/2002 11:21:51 PM]

Marchenko

Vladimir Aleksandrovich Marchenko Born: 7 July 1922 in Kharkov, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vladimir Marchenko attended Kharkov University, graduating in 1945. He was appointed onto the staff after graduating and worked at Kharkov University until 1961. Half way through this period, in 1953, he was promoted to professor. In 1961 Marchenko was appointed to the Physical-Technical Institute of Low Temperature in Kharkov. In the 1950s Marchenko obtained important results in approximation theory. His results concern the approximation theory of almost periodic functions. Also in the 1950s he studied the asymptotic behaviour of the spectral measure and of the spectral function for the Sturm-Liouville equation. In [7] this work is described:He is well known for his original results in the spectral theory of differential equations, including the discovery of new methods for the study of the asymptotic behaviour of spectral functions and the convergence expansions in terms of eigenfunctions. He also obtained fundamental results in the theory of inverse problems in spectral analysis for the Sturm-Liouville and more general equations. In fact Marchenko later applied his methods to the Schrödinger equation. The periodic case of the Korteweg-de Vries equation was solved by Marchenko in 1972. He used the method of the inverse problem in the theory of dissipation. Petryshyn in [7] describes other work of Marchenko including his work on self adjoint differential operators:Marchenko made significant contributions to the theory of self-adjoint differential operators http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Marchenko.html (1 of 2) [2/16/2002 11:21:53 PM]

Marchenko

with infinitely many independent variables and also to the theory of spaces of functions of infinitely many variables as inductive limits of locally convex function spaces. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Marchenko.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Marchenko.html (2 of 2) [2/16/2002 11:21:53 PM]

Marcinkiewicz

Józef Marcinkiewicz Born: 4 March 1910 in Cimoszka, Bialystok, Poland Died: 1940

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Józef Marcinkiewicz grew up with some health problems, in particular he had lung trouble, but this did not prevent him taking an active part in sports. Swimming and skiing were two sports at which he became particularly proficient. He had a schooling which resulted in a love of both mathematics and of Polish literature. He found making the decision as to which of these subjects to study at university a difficult one, and of course studying both was impossible. It was not that his training had lacked a science base, for he had a wide knowledge of physics and astronomy, but rather he just had broad interests. He decided to take a university course in mathematics but Polish literature would remain a hobby. In the autumn of 1930 Marcinkiewicz entered the Division of Mathematical and Natural Sciences of the University of Stefan Batory in Wilno. The town was then in Poland but it had been known by its Russian name of Vilnius when it was the capital of Lithuania. The university there had been named after Stefan Batory who was king of Poland from 1575 to 1586. The university had three professors of mathematics, the most famous of them being Antoni Zygmund who was appointed in the year that Marcinkiewicz began his undergraduate course. Zygmund was undertaking research on trigonometric series and in 1931-32 he gave a course on this topic at Wilno for the first time. It was, by his own description, an ambitious course, with an course on Lebesgue integration preceding it. Zygmund writes that the course was [5]:... too difficult for the average student of the second year ... Although it did not form part of his degree course, Marcinkiewicz asked Zygmund if he could take the course:... to my great pleasure he approached me without any prompting on my part. This was the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Marcinkiewicz.html (1 of 3) [2/16/2002 11:21:55 PM]

Marcinkiewicz

beginning of our mathematical collaboration. It is surprising that a professor would start a mathematical collaboration with an undergraduate in their second year of study but that is exactly what happened. Moreover, Zygmund explains that it was a rather natural thing to take place at the university:In the quiet and provincial pre-war Wilno a close contact between the student and professor was natural and easy, and we spent much time on mathematical discussions. The mathematical development of Marcinkiewicz was progressing at a very fast rate and the richness and originality of new ideas in him was always a source of amazement to me. In 1933, after three years of study, Marcinkiewicz was awarded his Master's degree. He had already obtained important new mathematical results and the thesis that he wrote for his Master's degree contained them. In particular he had found a continuous periodic function whose trigonometric interpolating polynomials, corresponding to equally spaces mesh points, diverge almost everywhere. Two years later Marcinkiewicz was awarded his doctorate, something which would be impressive enough if he had spent two years working for it, but he did not have this luxury since he had to undertake a years military service immediately after the award of his Master's degree. Once he had completed military training he was appointed as a junior assistant at the university in Wilno. His doctoral thesis Trigonometric interpolation of absolutely continuous functions was an extended form of the work which he had submitted for his Master's degree. He received the doctorate in 1935. After being awarded his doctorate, Marcinkiewicz received a Fellowship from the Fund for National Culture which enabled him to spend a year at the University of Lwów. There he collaborated with Julius Schauder who had returned to Lwów a year earlier having spending time in Paris working with Hadamard and Leray. Zygmund writes [5]:The influence of Schauder was particularly beneficial and would probably have led to important developments had time permitted. For in the field of real variable Marcinkiewicz had exceptionally strong intuition and technique, and the results he obtained in the theory of conjugate functions, had they been extended to functions of several variables might have given (as we see clearly now) a strong push to the theory of partial differential equations. The only visible trace of Schauder's influence is a very interesting paper of Marcinkiewicz on the multipliers of Fourier series, a paper which originated in connection with a problem proposed by Schauder ... During his year in Lwów, Marcinkiewicz also collaborated with Kaczmarz. He suggested problems on general orthogonal systems to Marcinkiewicz and this resulted in a series of papers from him on this topic. Returning to Wilno in the autumn of 1936, Marcinkiewicz became a senior assistant there, and in the following year he became a Dozent. Perhaps through his contacts with Schauder, who had greatly benefited from his time in Paris, Marcinkiewicz applied to the Fund for National Culture for another Fellowship, this time to study in Paris. He was successful and in the spring of 1939 he went to Paris. While he was there he received an offer of a Chair of Mathematics at the University of Poznan, and was awaiting approval of the post by the Ministry of Education so that he could take it up at the beginning of the 1939-40 academic year. From Paris Marcinkiewicz went to England and he was there in August 1939 when the deteriorating http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Marcinkiewicz.html (2 of 3) [2/16/2002 11:21:55 PM]

Marcinkiewicz

political situation made him decide to return to Poland. His colleagues in England tried to persuade him to stay, rather than return to Poland, but through his years military training he was an officer in the army reserve and felt that it was his duty to his country to return. War broke out a few days after Marcinkiewicz returned to Wilno. Zygmund writes [5]:On September 2, the second day of the war, I came across him accidentally in the street in Wilno, already in military uniform ... We agreed to meet the same day in the evening but apparently circumstances prevented him from coming since he did not show up at the appointed place. A few months later came the news that he was a prisoner of war and was asking for mathematical books. It seems that this was the last news about Marcinkiewicz. During his time in Paris and England, Marcinkiewicz had produced some mathematical work which he had written down in manuscript form. After returning to Poland he gave these manuscripts to his parents for safe keeping. Sadly Marcinkiewicz's parents suffered the same fate as he did and died during the war. No trace of the manuscripts was ever found. Zygmund tells us in [5] a little about Marcinkiewicz as a person:[He was] a tall and hansome boy, lively, sensitive, warm and ambitious, with a great sense of duty and honour. He did not shun amusement, and in particular was quite fond of dancing and the game of bridge. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Marcinkiewicz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Marcinkiewicz.html (3 of 3) [2/16/2002 11:21:55 PM]

Marczewski

Edward Marczewski Born: 15 Nov 1907 in Warsaw, Poland Died: 17 Oct 1976 in Wroclaw, Poland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edward Marczewski attended the Batory Secondary School in Warsaw, a school named after Stefan Batory who was king of Poland from 1575 to 1586. He graduated from the school in 1925 and entered the Department of Mathematics and Physics of the University of Warsaw. At the University of Warsaw he was taught by Kuratowski who was teaching the first year calculus course that Marczewski attended. Kuratowski writes in [1]:... during the first tutorials in that subject [calculus] he attracted my attention by his extraordinary ingenuity. The audience then was exceptionally large (about 300 persons), the largest in the pre-war period; Marczewski was the best. Kuratowski directed Marczewski's studies and gave him much personal attention. However it was two other mathematicians at the University of Warsaw who influenced Marczewski more than Kuratowski. These were Mazurkiewicz and Sierpinski who interested Marczewski in measure theory and related topics. Remaining at the University of Warsaw to study for his doctorate under Sierpinski's supervision, he submitted his thesis in 1932. He had remained interested in collaborating with Kuratowski and, in the same year as his doctorate was awarded, a joint paper by Marczewski and Kuratowski was published in Fundamenta Mathematicae. This was the golden age of Polish mathematics. From very unpromising times up to World War I, with the recreation of the Polish nation at the end of that war, Polish mathematics entered a golden age. In 1936, at the height of this remarkable explosion of mathematical talent, Marczewski wrote (see for example [1]):http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Marczewski.html (1 of 3) [2/16/2002 11:21:57 PM]

Marczewski

Poland always possessed great individuals, who worked, often with success, for the many, and not infrequently for whole institutions and sometimes for whole generations. But now it has among its mathematicians not only outstanding individuals but also a numerous, organised group of people whole-heartedly devoted to creative scientific work; it has its own school of mathematics. Marczewski was right, but sadly time were about to change. With the German invasion of Poland in 1939 the intellectual life of the country was destroyed (or at least there was a concerted effort by the invaders to destroy Polish intellectual life). Marczewski was visiting Lvov when it was occupied by the German army and here, as throughout Poland, intellectuals perished. For example A Lomnicki, S Ruziewicz, and W Stozek were three mathematicians from the Technical University of Lvov who were shot by a German firing squad in July 1941. Now Marczewski returned to Warsaw, but in the capital things were equally difficult. Mathematics lecturers at the University of Warsaw who died around this time included S Kwietniewski and A Lindenbaum in 1941 and S Saks murdered in November 1942. For much of the war Marczewski survived in Warsaw suffering great hardships. However, near the end of the war he was captured and sent to a labour camp in Breslau, as the Germans called the town, but Wroclaw to give it its Polish name. In Wroclaw, Marczewski was a prisoner of the Germans while the Russian forces besieged the city. The German defenders of Wroclaw continued to resist even after the fall of Berlin. On 6 May 1945 the Russian forces captured Wroclaw and Marczewski was set free. Perhaps the rather surprising turn of events was that he decided to remain in Wroclaw and throw himself vigorously into the rebuilding of the university and educational system there. The most important outcome which Marczewski worked for was the setting up of a Polish university in Wroclaw. This he achieved with remarkable speed. On 24 August 1945, only three weeks after the Potsdam Conference, the Polish government issued a decree to:... convert the University and Technical University of Wroclaw into Polish state academic institutions. Lectures began at the new Polish University of Wroclaw on 15 November 1945. In addition to Marczewski, who was appointed as a professor, Steinhaus was appointed as Dean of the Faculty of Mathematics, Physics and Chemistry. Marczewski later became rector of the University and served for four years in this role. The Polish Mathematical Society was equally quick to set up a new Wroclaw section. At its first meeting following the end of the war in October 1945 it passed a resolution:... to welcome delegates from the Society's new Section in Wroclaw. The meeting considers the vigorous start of the Wroclaw mathematical centre's activities, and the presence of its delegates not only as a manifestation of the return of the Western Territories to Poland, but also as a proof of the rebirth of Polish culture in those territories. The General Meeting considers it to be a matter of great importance that the Wroclaw Section, under the direction of Professors B Knaster, E Marczewski, H Steinhaus and W Slebodzinski and in cooperation with the University and Technical University of Wroclaw, should constitute one of the most active scientific centres radiating across the Western Territories. Marczewski founded the journal Colloquium Mathematicum in 1946 in Wroclaw, and he was its

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Marczewski.html (2 of 3) [2/16/2002 11:21:57 PM]

Marczewski

editor-in-chief for 30 years. In 1948 the Polish Mathematical Institute was set up, based mainly in Warsaw but having divisions in other cities. Marczewski was proposed as Deputy General Secretary of the Mathematical Institute. He was a cofounder of the Wroclaw Centre of Excellence in Mathematics. We should end by making comments on Marczewski's mathematics. His main work was in set theory, general topology, and measure theory. In [6] there is a bibliography of Marczewski's publications including 94 mathematical and 47 other research publications. In [1] Kuratowski discusses some of the mathematical contributions made by Marczewski:He obtained particularly interesting and frequently applied results on the duality between the notions of a set of the first category and a set of measure zero; and similarly - between a set with Baire's property and a measurable set. He also devoted much attention to analytic sets (Suslin sets), operation (A) and uniformization ... . The great universality, elegance and simplicity of his proofs is a characteristic feature of his papers. Kuratowski also paid this fine tribute to Marczewski:... he was a man of exceptional wisdom and exceptional kindness, which won him adherents and numerous friends. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Marczewski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Marczewski.html (3 of 3) [2/16/2002 11:21:57 PM]

Margulis

Gregori Aleksandrovic Margulis Born: 24 Feb 1946 in Moscow, Russia

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gregori Margulis was educated at Moscow High School, graduating in 1962. In that year he began his undergraduate studies at Moscow University and he was awarded his first degree in 1967. Margulis remained at Moscow University for his postgraduate studies. He showed great potential as a mathematician and the first important award which he won was during his time as a postgraduate student when he received the young mathematicians prize from the Moscow Mathematical Society in 1968. Margulis completed his graduate studies in 1970 and he was awarded the degree of Candidate of Science for a thesis On some problems in the theory of U-systems. After being awarded the Candidate of Science degree (the equivalent of a British or American Ph.D.), Margulis began to work in the Institute for Problems in Information Transmission. He was a Junior scientific worker there from 1970 to 1974 when he was promoted to Senior scientific worker. He held this post until 1986 when he was promoted again, this time to Leading scientific worker. International honour was given to Margulis in 1978 when he was awarded a Fields Medal at the International Congress at Helsinki. However it was not a happy occasion for Margulis who was not permitted by the Soviet authorities to travel to Helsinki to receive the Medal. Tits, delivering the address [7] spoke of his sadness that Margulis could not be present:... I cannot but express my deep disappointment - no doubt shared by many people here - in the absence of Margulis from this ceremony. In view of the symbolic meaning of this city of Helsinki, I had indeed grounds to hope that I would have a chance at last to meet a mathematician whom I know only through his work and for whom I have the greatest respect and admiration.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Margulis.html (1 of 3) [2/16/2002 11:21:58 PM]

Margulis

Perhaps Tits' comment about 'symbolic meaning' should be explained. He delivered the address in the Finlandia Hall in Helsinki where Margulis should have received the Medal and where the Helsinki Accords had been signed on 1 August 1975. This major agreement was signed at the end of the first Conference on Security and Cooperation in Europe. The Helsinki Accords, signed by all the countries of Europe (excluding Albania) and by the United States and Canada, were designed to reduce the cold war tension by accepting the European boundaries as they then were. Tits talks in [7] about the range of Margulis's work in combinatorics, differential geometry, ergodic theory, dynamical systems and discrete subgroups of Lie groups. The award of the Fields Medal was mainly for his work on this latter topic:Already Poincaré wondered about the possibility of describing all discrete subgroups of finite covolume in a Lie group G. The profusion of such subgroups in G = PSL2(R) makes one at first doubt of any such possibility. However, PSL2(R) was for a long time the only simple Lie group which was known to contain non-arithmetic discrete subgroups of finite covolume, and further examples discovered in 1965 by Makarov and Vinberg involved only few other Lie groups, thus adding credit to conjectures of Selberg and Pyatetski-Shapiro to the effect that "for most semisimple Lie groups" discrete subgroups of finite covolume are necessarily arithmetic. Margulis's most spectacular achievement has been the complete solution of that problem and, in particular, the proof of the conjecture in question. Margulis was soon able to leave the Soviet bloc and, in 1979, he was able to spend three months at the University of Bonn. Between 1988 and 1991 Margulis made a number of visits to the Max Planck Institute in Bonn, to the Institut des Hautes Etudes and to the Collège de France, to Harvard and to the Institute for Advanced study in Princeton. From 1991 he has held a chair at Yale University. The Oppenheim conjecture was made in 1929 and concerns values of indefinite irrational quadratic forms at integer points. Early work was based on results of Jarnik and Walfisz. In the 1940's Davenport and Heilbronn contributed by proving special cases and in 1946 Watson extended their results showing the conjecture to be true for further special cases. Margulis proved the full conjecture in 1986 and gives a beautiful survey of the work leading to this solution in [3]. There Margulis explains that:The different approaches to this and related conjectures (and theorems) involve analytic number theory, the theory of Lie groups and algebraic groups, ergodic theory, representation theory, reduction theory, geometry of numbers and some other topics. Margulis has received many honours for his work. In addition to the Fields Medal he has been awarded the Medal of the Collège de France (1991) and in the same year he was elected an honorary member of the American Academy of Arts and Sciences. In 1995 he received the Humboldt Prize and in 1996 he was honoured by election as a member of the Tata Institute of fundamental research. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Margulis.html (2 of 3) [2/16/2002 11:21:58 PM]

Margulis

Mathematicians born in the same country Other references in MacTutor

Chronology: 1980 to 1990

Honours awarded to Gregori Margulis (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1978

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Margulis.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Margulis.html (3 of 3) [2/16/2002 11:21:58 PM]

Marinus

Marinus of Neapolis Born: about 450 in Neapolis, Palestine (called Shechem in Bible, now Nablus, Israel) Died: about 500 in possibly Athens, Greece Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Marinus of Neapolis was probably a Samaritan, but just possibly a Jew. He became a convert to the Greek way of life and joined the Academy in Athens where he was a pupil of Proclus who was head of the Academy. In fact when Proclus wrote a commentary on the Myth of Er, he dedicated it to Marinus. Marinus succeeded Proclus as head of the Academy at Athens in 485. We are told in [5] that:... he lectured on Pappus' commentary to Book V of the Almagest (in particular his discussion of parallax); and there are still extant lecture notes on the Data of Euclid. Marinus [2]:... wrote a commentary, or rather introduction to the Data of Euclid. It is mainly taken up with a discussion of the question - what is meant by given? In fact Marinus was a great believer in mathematics, something which he shared with fellow late Neoplatonists. He said (see for example [1]):I wish everything were mathematics. Although Marinus followed closely the views of his teacher, Proclus he did show originality which is much to his credit. The level of his regard for Proclus is evident in the biography that he wrote Life of Proclus in which he [1]:... praised and eulogised [Proclus]. Yet when Marinus felt that Proclus was in error he was quite prepared to give his own views. For example Proclus had claimed that Plato's Parmenides was concerned with gods. Marinus, quite correctly, pointed out that Plato's work in rather concerned with 'forms' The article [7] by Tihon is an interesting account of two previously unknown commentaries on astronomical topics by Marinus. One concerns the Milky Way and Marinus discusses whether it is affected by precession. As early as the 5th century BC Democritus had correctly understood the Milky Way to be due to a multitude of faint stars. However this was still not the accepted view in the time of Marinus, who argues against this hypothesis in this commentary. Marinus did claim, however, that the Milky Way was part of the sphere of fixed stars and so underwent precession in the same way as the fixed stars. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Marinus.html (1 of 2) [2/16/2002 11:22:00 PM]

Marinus

In the second commentary by Marinus described in [7], he corrects the rules for the direction of parallax in longitude given by Theon of Alexandria in his small commentary on Ptolemy's Handy Tables. Marinus makes use of ideas by Pappus in making his corrections. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Marinus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Marinus.html (2 of 2) [2/16/2002 11:22:00 PM]

Markov

Andrei Andreyevich Markov Born: 14 June 1856 in Ryazan, Russia Died: 20 July 1922 in Petrograd (now St Petersburg), Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Andrei A Markov was a graduate of Saint Petersburg University (1878), where he began a professor in 1886. Markov's early work was mainly in number theory and analysis, continued fractions, limits of integrals, approximation theory and the convergence of series. After 1900 Markov applied the method of continued fractions, pioneered by his teacher Pafnuty Chebyshev, to probability theory. He also studied sequences of mutually dependent variables, hoping to establish the limiting laws of probability in their most general form. He proved the central limit theorem under fairly general assumptions. Markov is particularly remembered for his study of Markov chains, sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. This work launched the theory of stochastic processes. In 1923 Norbert Wiener became the first to treat rigorously a continuous Markov process. The foundation of a general theory was provided during the 1930s by Andrei Kolmogorov. Markov was also interested in poetry and he made studies of poetic style, interestingly Kolmogorov had similar interests. Markov had a son (of the same name) who was born on September 9, 1903 and followed his father in also becoming a renowned mathematician. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Markov.html (1 of 2) [2/16/2002 11:22:02 PM]

Markov

Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Andrei A Markov

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1900 to 1910

Honours awarded to Andrei A Markov (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Markov

Other Web sites

1. University of Minnesota 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Markov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Markov.html (2 of 2) [2/16/2002 11:22:02 PM]

Mascheroni

Lorenzo Mascheroni Born: 13 May 1750 in Bergamo, Lombardo-Veneto (now Italy) Died: 14 July 1800 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lorenzo Mascheroni was ordained as a priest at the age of 17. At first he taught rhetoric then, from 1778, he taught physics and mathematics at the seminary at Bergamo. In 1786 Mascheroni became professor of algebra and geometry at the University of Pavia. He later became rector of the university. In Adnotationes ad calculum integrale Euleri (1790) Mascheroni calculated Euler's constant to 32 decimal places. In fact only the first 19 places were correct and the rest was corrected by Johann von Soldner in 1809. Mascheroni's work shows a deep understanding of the Euler's calculus. Mascheroni is also known as a poet and he dedicated one of his books Geometria del compasso (1797) to Napoleon in verse. In this work Mascheroni proved that all Euclidean constructions can be made with compasses alone, so a straight edge in not needed. In fact this was (unknown to Mascheroni) proved in 1672 by a little known Danish mathematician Georg Mohr. Mascheroni also wrote a well-composed book Nuove ricerchi su l'equilibrio delle vòlte (1785) on statics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mascheroni.html (1 of 2) [2/16/2002 11:22:03 PM]

Mascheroni

A Poster of Lorenzo Mascheroni

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1780 to 1800

Other Web sites

1. Cut the knot (Compass only constructons) 2. Euler-Mascheroni constant

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Mascheroni.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mascheroni.html (2 of 2) [2/16/2002 11:22:03 PM]

Maschke

Heinrich Maschke Born: 24 Oct 1853 in Breslau, Germany (now Wroclaw, Poland) Died: 1 March 1908 in Chicago, Illinois, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Heinrich Maschke's father was an important medical man. Heinrich attended the Gymnasium in Breslau where he showed great ability. He entered the University of Heidelberg in 1872, studying there under Königsberger. Military service was required at that time so Maschke spent a year in the army before he continued his studies at the University of Berlin. At Berlin he was taught by some outstanding mathematical teachers including Weierstrass, Kummer and Kronecker. In common with the standard practice of the time he moved around different German universities, next going to Göttingen from where he received his doctorate in 1880. His first teaching post was in the Luisenstädtische Gymnasium in Berlin. He then returned to Göttingen for the years 1886-87 working with Klein. He returned to the Gymnasium in Berlin, then began part-time study of electrotechnics at the Polytechnicum in Charlottenburg. In 1890 he resigned his teaching post and took up full-time technical training. In 1891 Maschke emigrated to the USA and worked for a year with the Western Electrical Instrument Company, Newark, New Jersey. However he returned to mathematics and he was appointed professor of mathematics at the University of Chicago when the university first opened in 1892. Under Klein's inspiration while at Göttingen, Maschke had worked in group theory, in particular working on finite groups of linear transformations. He is best known today for Maschke's theorem which states that if the order of the finite group G is not divisible by the characteristic of the field K, then the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maschke.html (1 of 2) [2/16/2002 11:22:05 PM]

Maschke

(finite-dimensional) K-representations of G are completely reducible. Maschke's second area of work was on differential geometry in particular the theory of quadratic differential quantics. In this area he led the symbolic treatment of the subject. At Chicago, together with Eliakim Moore, Maschke was responsible for the rapid rise to eminence of the University in mathematics research. David Eugene Smith writing in [4] says:He devoted the remainder of his life to the training of mathematicians and to assisting in building up and maintaining a strong department in that university. He was a teacher of great ability and his courses were made more valuable by his all-round culture, by his originality of thought, and by his personal interest in the large numbers of young mathematicians who attended his lectures. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Maschke.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maschke.html (2 of 2) [2/16/2002 11:22:05 PM]

Maseres

Francis Maseres Born: 15 Dec 1731 in London, England Died: 19 May 1824 in Reigate, Surrey, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Francis Maseres graduated from Clare College Cambridge in 1752 with a degree in classics and mathematics. He practiced law for a few years without much success, then served as Attorny General for Quebec until 1769. After returning to England he was appointed Cursitor Baron of the Exchequer and held this office until his death at the age of 93. Maseres wrote many mathematical works which show a complete lack of creative ability. He rejected negative numbers and that part of algebra which is not arithmetic, despite writing 150 years after Viète and Harriot. It is probable that Maseres rejected all mathematics which he could not understand. He writes:If any single quantity is marked either with the sign + or the sign - without affecting some other quantity, the mark will have no meaning or significance, thus if it be said that the square of -5, or the product of -5 into -5, is equal to +25, such an assertion must either signify no more than 5 times 5 is equal to 25 without any regard for the signs, or it must be mere nonsense or unintelligible jargon. Maseres had an unfortunate influence on the teaching of algebra in Britain for several decades. An example of this is seen in the algebra text of Nicolas Vilant, Regius Professor at St Andrews from 1765 to 1807. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maseres.html (1 of 2) [2/16/2002 11:22:07 PM]

Maseres

List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Francis Maseres (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1771

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Maseres.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maseres.html (2 of 2) [2/16/2002 11:22:07 PM]

Maskelyne

Nevil Maskelyne Born: 5 Oct 1732 in London, England Died: 9 Feb 1811 in Greenwich, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nevil Maskelyne's father died when Nevil was 12 years old leaving the family rather poor. He attended Westminster School and was still being educated there when his mother died in 1748. Just before the death of his mother his interest in astronomy had begun after seeing the eclipse of 25 July 1748. Maskelyne writing about his school education said:Great mathematicians have become astronomers from the facility mathematics gave them in the attainment of astronomy; but here the love of astronomy was the motive of application to mathematics without which our astronomer soon found he could not make the progress he wished in his favourite science; in a few months, without any assistance he made himself master of the elements of geometry and algebra. With these helps he soon read the principal books in astronomy and optics and also in ... mechanics, pneumatics and hydrostatics. The considerable progress he had made in these sciences led him naturally to the University of Cambridge... Maskelyne entered Cambridge in 1749 where he studied mathematics and graduated seventh wrangler in mathematics in 1754. He was ordained a minister in 1755. Then he became a fellow of Trinity College Cambridge in 1756. In 1758 he was admitted to the Royal Society, his recommendation describing him as ... a person well versed in mathematical learning and natural philosophy ... In 1761 the Royal Society sent Maskelyne to the island of St Helena to observe a transit of Venus. This was important since accurate measurements would allow the distance from the Earth to the Sun to be accurately measured and the scale of the solar system determined. During the voyage he experimented http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maskelyne.html (1 of 3) [2/16/2002 11:22:09 PM]

Maskelyne

with the lunar position method of determining longitude. Maskelyne returned to Chipping Barnet in 1761, where he was a curate, and worked on publishing a book. He published the lunar distance method for determining longitude in The British Mariner's Guide (1763). In 1764 he went on a voyage to Barbados to carry out trials of Harrison's timepiece. Soon after his return, in 1765, he was appointed Astronomer Royal. He published the first volume of the Nautical Almanac in 1766 and continued to work on this project up to the time of his death. Maskelyne proposed to the Royal Society in 1772, an experiment for determining the Earth's density with the use of a plumb line. He was not the first to suggest such an experiment. Bouguer and La Condamine had tried such an experiment over 30 years before. Maskelyne carried out the experiment in 1774 on Schiehallion, a mountain in Perthshire, Scotland. Schiehallion was chosen because it was surprisingly regular and conical in shape so its volume could be determined accurately. From his observations Maskelyne computed that the Earth's density is approximately 4.5 times that of water. He was awarded the Copley medal of the Royal Society in 1775 for this work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

English attack on the Longitude Problem

Honours awarded to Nevil Maskelyne (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1758

Royal Society Copley Medal

Awarded 1775

Lunar features

Crater Maskelyne

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maskelyne.html (2 of 3) [2/16/2002 11:22:09 PM]

Maskelyne

JOC/EFR February 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Maskelyne.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maskelyne.html (3 of 3) [2/16/2002 11:22:09 PM]

Mason

Charles Max Mason Born: 26 Oct 1877 in Madison, Wisconsin, USA Died: 23 March 1961 in Claremont, California, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Max Mason studied at the University of Wisconsin where he received a B.Litt. in 1898 then went to Germany where studied for his doctorate at the University of Göttingen. His received his doctorate in 1903 for a dissertation entitled Randwertaufgaben bei gewöhnlichen Differentialgleichungen which had been supervised by Hilbert. Mason then returned to the United States, accepting a post of instructor in mathematics at Massachusetts Institute of Technology. After spending 1903-4 at MIT, Mason spent the next four years as assistant professor of mathematics at Yale. In 1908 he was appointed professor of mathematical physics at the University of Wisconsin. Mason remained at Wisconsin until 1925 but during World War I he worked as a member of the submarine committee of the NRC. During this period of war work he invented submarine detection devices. Mason left Wisconsin in 1925 to become president of the University of Chicago. Then from 1928 he was the natural sciences director of the Rockefeller Foundation in New York. Between 1929 and 1936 he was president of the Rockefeller Foundation. He then became chairman of the team directing the construction of the Palomar Observatory which was completed in 1945. Mason's research interests lay in differential equations, the calculus of variations and electromagnetic theory. He developed the relation between the algebra of matrices and integral equations as well as studying boundary value problems. Other topics in the large range of applied mathematics topics which he studied were existence theorems and asymptotic expansions. He also invented acoustical compensators. He wrote a number of books, in particular The New Haven Mathematical Colloquium (1910) and he co-authored The Electromagnetic Field. A strong supporter of the American Mathematical Society, he was an associate editor of the Transactions of the American Mathematical Society between 1911 and 1917. He was elected a member of the National Academy of Sciences. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mason.html (1 of 2) [2/16/2002 11:22:10 PM]

Mason

List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Max Mason (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1906

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mason.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mason.html (2 of 2) [2/16/2002 11:22:10 PM]

Mathews

George Ballard Mathews Born: 23 Feb 1861 in London, England Died: 19 March 1922 in Liverpool, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Mathews attended Ludlow Grammar School. From there he went to University College, London where he studied mathematics. He then studied at St John's College, Cambridge from where he graduated as First Wrangler in 1883. The following year Mathews was elected a Fellow of St John's College, and later in 1884 he was appointed to the chair of mathematics at the University College of North Wales at Bangor in the year the university opened. He taught at the University College of North Wales in Bangor and in Cambridge for periods through his life. He resigned his chair at Bangor in 1896 to return to Cambridge, but in 1911 he resigned from Cambridge and was appointed to a special lectureship at Bangor. Most of Mathews' research was on number theory but he also wrote texts on Bessel functions and on projective geometry. In his two volume work Theory of numbers (1892) topics covered included Gauss's theory of quadratic forms and their development by mathematicians such as Dirichlet, Eisenstein and Smith. The book also discusses prime numbers and Riemann's memoir on primes but, since it was written two or three years before the prime number theorem was proved, this part of the work became dated rather quickly. The book A treatise on Bessel functions and their applications to physics (1895) was written by Mathews in collaboration with Andrew Gray who was the professor of physics at Bangor. It was the first major treatise on Bessel functions in English and covered topics such as applications of Bessel functions to electricity, hydrodynamics and diffraction. Here Mathews was luckier than with the number theory work, since even when Watson's treatise on Bessel functions was published in 1922, it did not cover the applications of Mathews's book which continued to be useful and well used. Mathews also wrote Algebraic equations (1907) which is a clear exposition of Galois theory, and Projective geometry (1914). This latter book develops the subject of projective geometry without using the concept of distance and it bases projective geometry on a minimal set of axioms. The book also treats von Staudt's theory of complex elements as defined by real involutions. The book contains a wealth of information concerning the projective geometry of conics and quadrics. In addition to his treatises and many papers on the classical theory of numbers, Mathews also wrote some

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mathews.html (1 of 2) [2/16/2002 11:22:11 PM]

Mathews

articles for Encyclopaedia Britannica, in particular writing the article on universal algebra and the one on number. Another role Mathews played in mathematics teaching and administration was as an examiner to the Universities of Ireland and to the University of Manchester. He is described by T A Broadbent in [1] as follows:Mathews was an accomplished classical scholar; and besides Latin and Greek he was proficient in Hebrew, Sanskrit and Arabic. He also possessed great musical knowledge and skill. His versatility led a colleague at Bangor to assert that Mathews could equally well fill four or more chairs at the college. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Honours awarded to George Mathews (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1897

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mathews.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mathews.html (2 of 2) [2/16/2002 11:22:11 PM]

Mathieu_Claude

Claude Louis Mathieu Born: 25 Nov 1783 in Mâcon, France Died: 5 March 1875 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Claude-Louis Mathieu began his career as an engineer but soon became a mathematician at the Bureau des Longitudes in 1817 and later professor of astronomy at Collège de France in Paris. From 1829 he was professor of analysis at the Ecole Polytechnique in Paris. For many years Claude Mathieu edited the work on population statistics L'Annuaire du Bureau des Longitudes produced by the Bureau des Longitudes. He also worked on determining the distances to stars. He published L'Histoire de l'astronomie au XVIII siècle in 1827. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Honours awarded to Claude-Louis Mathieu (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Impasse Mathieu (15th Arrondissement)

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mathieu_Claude.html (1 of 2) [2/16/2002 11:22:12 PM]

Mathieu_Claude

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mathieu_Claude.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mathieu_Claude.html (2 of 2) [2/16/2002 11:22:12 PM]

Mathieu_Emile

Emile Léonard Mathieu Born: 15 May 1835 in Metz, France Died: 19 Oct 1890 in Nancy, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Emile Mathieu is remembered especially for his discovery (in 1860 and 1873) of five sporadic simple groups named after him. These were studied in his thesis on transitive functions. Mathieu was brought up in Metz, and he attended school in that town. He excelled at school, first in classical studies showing remarkable abilities in Latin and Greek. However, once he had met mathematics when he was in his teenage years, it became the only subject which he wanted to pursue. Entering the Ecole Polytechnique in Paris his progress was almost unbelievable, even given the remarkable achievements of the brilliant mathematicians in this archive who also attended this institution. It took Mathieu only eighteen months to complete the whole course and he continued to study for a doctorate. By 1859 he had been awarded his Docteur ès Sciences for a thesis on transitive functions, the work which led to his initial discovery of sporadic simple groups. Progress as remarkable as that achieved by Mathieu would seem to put him in the ideal position to obtain a university appointment but this was not forthcoming. He took on work as a private tutor of mathematics and he continued in this role for ten years. He did suffer a rather serious illness in 1866 and this seems to have stopped him taking over Lamé's courses at the Sorbonne in that year. He was appointed professor of mathematics at Besançon in 1869 and, after five years teaching at Besançon, Mathieu moved to Nancy to take up the chair of mathematics there. Mathieu's main work, after his initial interest in pure mathematics, was in mathematical physics although he did do some important work on the hypergeometric function. As Grattan-Guinness writes in [1]:Although Mathieu showed great promise in his early years, he never received such normal signs of approbation as a Paris chair or election to the Académie des Sciences. From his late twenties his main efforts were devoted to the then unfashionable continuation of the great French tradition of mathematical physics, and he extended in sophistication the formation and solution of partial differential equations for a wide range of physical problems. Perhaps had Mathieu continued to follow up his remarkable discoveries in group theory, he might have achieved more fame and better posts in his lifetime. Some of his earliest work in mathematical physics was related to his study of light and he looked at the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mathieu_Emile.html (1 of 2) [2/16/2002 11:22:13 PM]

Mathieu_Emile

surfaces of vibrations arising from Fresnel waves. He also worked on the polarisation of light where he highlighted some weaknesses in Cauchy's results on the topic. He worked on potential theory applied to elasticity, heat diffusion, and the vibration of bells, a very hard problem. Mathieu studied fluids, in particular examining capillary forces. He also studied magnetic induction and the three body problem where he applied his work to the perturbations of Jupiter and Saturn. In addition to being remembered for the Mathieu groups, he is also remembered for the Mathieu functions. He discovered these functions, which are special cases of hypergeometric functions, while solving the wave equation for an elliptical membrane moving through a fluid. The Mathieu functions are solutions of the Mathieu equation which is d2u/dz2 = (a + 16b cos2z)u = 0. In [1] Grattan-Guinness describes Mathieu's nature as:... shy and retiring [which] may have accounted to some extent for the lack of worldly success in his life and career; but among his colleagues he won only friendship and respect. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other Web sites

1. Eric's World of Mathematics (Mathieu functions) 2. Eric's World of Mathematics (Mathieu groups)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mathieu_Emile.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mathieu_Emile.html (2 of 2) [2/16/2002 11:22:13 PM]

Matsushima

Yozo Matsushima Born: 11 Feb 1921 in Sakai City, Osaka Prefecture, Japan Died: 9 April 1983 in Osaka, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yozo Matsushima attended Naniwa High School. After graduating he entered Osaka Imperial University (later named Osaka University) where he was taught by Kenjiro Shoda, among others. These were difficult years for anyone to be studying in Japan and the next few years, as World War II drew to an end, would be even more difficult. He graduated with the degree of Bachelor of Science in September 1942. Matsushima was appointed as an assistant in the Mathematical Institute of Nagoya Imperial University (like Osaka and other Japanese universities it would soon drop the name "Imperial" from its title) immediately he had received his B.Sc. degree, and he was in post in time for the 1942-43 academic year. There were major difficulties in carrying out research in these war years since, quite apart from military reasons and problems caused by bombing, international mathematical journals were not reaching Japan. Equally, it was very difficult for Japanese mathematicians to publish the results that they were discovering. The first paper which Matsushima published contained a proof that a conjecture of Zassenhaus was false. Zassenhaus had conjectured that every semisimple Lie algebra L over a field of prime characteristic, with [L, L] = L, is the direct sum of simple ideal and Matsushima was able to construct a counterexample. He then embarked on research which enabled him to prove that Cartan subalgebras of a Lie algebra are conjugate, but due to being out of touch with current research, he was to publish this result while unaware that Chevalley had already published a proof. When he was able to obtain details of another of Chevalley's papers through a review in Mathematical Reviews he was able to construct the proofs for himself. The 1947 volume of the Proceedings of the Japan Academy, containing two papers by Matsushima, did http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Matsushima.html (1 of 4) [2/16/2002 11:22:15 PM]

Matsushima

not appear until 1950 while the first volume of Journal of the Mathematical Society of Japan contained three of his papers. This shows the quantity of mathematics he had managed to produce in the final years of the war and the next couple of years which were almost as difficult ones in which to carry out research in Japan. In 1952-53 he organised a seminar on Lie pseudogroups and differential systems at Nagoya, a topic on which he was now working. One of the students who attended this seminar, Kuranishi, went on to prove a famous result on this topic. Reminiscing on these years later in his life, Matsushima wrote that:... although he himself got relatively little out of differential systems for the time and effort spent, he felt much gratified with Kuranishi's great success. Having earlier being promoted to associate professor, in 1953 Matsushima became a full professor at Nagoya University. Chevalley visited him in Nagoya in the autumn of 1953, spending three months there. It was a visit which Matsushima greatly enjoyed, and which was most useful to him. Equally Chevalley enjoyed the visit and invited Matsushima to spend the following year in France. He left for France in the autumn of 1954, spending time at the University of Strasbourg, and then in Paris as a member of C.N.R.S. at the invitation of Chevalley and Henri Cartan. Matsushima presented some of his results to Ehresmann's seminar in Strasbourg, extending Cartan's classification of complex irreducible Lie algebras to the case of real Lie algebras. Having arrived in Paris by the spring of 1955, he lectured on Lie pseudogroups at the Bourbaki seminar. Kobayashi ([2] or [3]) writes:Matsushima returned to Nagoya in December 1955. His sojourn in France seems to have determined the course of his research for the next several years. When Shoda retired from the Chair of Algebra at Osaka University, Matsushima was appointed to fill the vacant chair in early 1960. His research in Osaka took a somewhat different direction and he wrote a series of papers on cohomology of locally symmetric spaces. In particular he collaborated with Murakami on this topic. In September 1962 he went to the Institute for Advanced Study in Princeton where he spent a year. Back in Osaka, Matsushima jointly began to organise the United States-Japan Seminar in Differential Geometry which was held in Kyoto in June 1965. Shortly after the end of this seminar, he set off for another visit to France to spend the academic year 1965-66 as visiting professor at the University of Grenoble. In September 1966 Matsushima accepted a chair at the University of Notre Dame in Indiana in the United States. While there he did not end the collaboration which he had been carrying out with Murakami and they continued to undertake joint research. He became an editor of the new Journal of Differential Geometry in 1967, remaining on the editorial board for the rest of his life. Matsushima spent 14 years as a professor at Notre Dame before returning to Japan in 1980. A conference was organised in his honour in May 1980 before he left Notre Dame. He reached the age of 60 in February 1981 and a volume of papers by colleagues and former students was published: Manifolds and Lie groups, Papers in honour of Yozo Matsushima. The volume contains some papers presented to the conference held in Notre Dame in the previous May. Sadly, after his return to Osaka he had less than three years in his chair before he died from pneumonia at the young age of 62. Among the honours that Matsushima received, perhaps the most prestigious was the Asahi Prize which was presented to him for his research on continuous groups in 1962. We have mentioned several topics on which Matsushima worked in the course of this biography. For http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Matsushima.html (2 of 4) [2/16/2002 11:22:15 PM]

Matsushima

completeness we list the topics on which Matsushima's contribution is described by in detail by Kobayashi in [2]:(1) Lie theory - Lie algebras and Lie groups; Lie pseudogroups and differential systems; (2) homogeneous complex manifolds - homogeneous Kahler manifolds; homogeneous Stein manifolds; Schubert varieties; (3) vector bundle valued harmonic forms - Betti numbers of locally symmetric spaces; cohomology of vector bundles over locally symmetric spaces; second fundamental forms and curvature forms; (4) automorphisms of complex manifolds - automorphisms of Siegel domains; first Chern class and automorphisms of compact Kahler manifolds; (5) complex tori - vector bundles over complex tori; ample bundles and subvarieties of complex tori; affine structures. Murakami writes this tribute to Matsushima in [5] (or see [6]):Matsushima's love for mathematics is evidenced by the discipline and diligence he consistently applied to his research. He was a man whose concern lay not only in mathematics, but he had a keen interest in human nature and a constant curiosity about the world that resulted in his being an avid reader of a variety of books and a most interesting conversationalist. His friends and colleagues will miss not only his mathematical talent, but the warmth and cynical humour which lay behind his outwardly serious countenance. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Matsushima.html (3 of 4) [2/16/2002 11:22:15 PM]

Matsushima

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Matsushima.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Matsushima.html (4 of 4) [2/16/2002 11:22:15 PM]

Mauchly

John William Mauchly Born: 30 Aug 1907 in Cincinnati, Ohio, USA Died: 8 Jan 1980 in Ambler, Pennsylvania, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Mauchly went to school in Washington DC and in 1925 he was awarded a scholarship by the State of Maryland to allow him to attend the Johns Hopkins University in Baltimore, Maryland. He began studying engineering at Johns Hopkins University but his interests changed in the course of his studies towards pure science and his first degree was in physics. Mauchly continued studying physics after taking his first degree and he was awarded his doctorate in 1932. He then taught physics at a number of different colleges and spent some time at the Carnegie Institution in Washington DC undertaking research. By 1940 Mauchly was teaching physics at Ursinus College near Philadelphia. While there he became interested in developing electronic computers which combined his interests in physics and engineering. It may seem strange today that someone with an interest in engineering would be drawn towards building computers but at this time a computer was a huge mechanical construction. Also Mauchly's interests were in electrical engineering and he looked for ways to develop electrical circuits for computation. Work was going on in the area of producing electrical circuits to do arithmetic and Mauchly, together with some of his students from Ursinus College, visited establishments where such development was being undertaken. Mauchly began to experiment in constructing electrical circuits for counting at Ursinus College aimed at trying out new ideas which he brought to the subject. Now 1941 was the time when World War II was strongly affecting the directions of academic research as the USA geared up for research specifically directed towards the war effort. Mauchly took a training course in electronics, designed for defence purposes, at the Moore School of Electrical Engineering at the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mauchly.html (1 of 4) [2/16/2002 11:22:17 PM]

Mauchly

University of Pennsylvania. Having completed the course Mauchly was offered a position as an instructor on the course. Research at the Moore School of Electrical Engineering was being carried out using early forms of computers. In particular the School used a Bush analyser, designed by Vannevar Bush specifically to integrate systems of ordinary differential equations. The machine consisted of replaceable shafts, gears, wheels, handles, electric motors, and disks and required much manual work to set it up. Mauchly found that [2]:When large and related problems that could be set on the machine for weeks were to be solved it was in constant use, but when small problems were being dealt with, engineers found it more expedient to solve then mathematically, without taking days to set up the machine. Mauchly had already developed his own ideas on how to construct a better computer and he tried to interest other members of staff at the Moore School but with little success. One person who was interested in his ideas, however, was John Eckert who had been one of his instructors at Moore College when Mauchly was a student on the training course. The two became close friends and discussed their ideas about electronic computers at almost every possible opportunity. Mauchly wrote a report on the design of an electronic computer which would, in his opinion, be far easier to use and allow results to be obtained much more quickly than the Bush analyser. The Ballistic Research Laboratory had been set up at Aberdeen, in Harford county, northeastern Maryland as part of the Aberdeen Proving Ground, a military weapons testing site which had been established in 1917 during World War I. The Ballistic Research Laboratory consisted of staff from the Moore School and staff from the Aberdeen Proving Ground. When a new director was put in charge of the Ballistic Research Laboratory in 1942 he worked both at Aberdeen and at the Moore School. He read Mauchly's report in March 1943, eighteen months after it was written, and was very impressed. Various committees then considered the proposal before money could be assigned to the project of building Mauchly's computer, and in April 1943 Veblen approved the scheme. Mauchly and John Eckert then collaborated in the construction of the Electronic Integrator and Computer (ENIAC). The machine was intended to be a general purpose one, but it was also designed for a very specific task, namely compiling tables for the trajectories of bombs and shells. ENIAC is described in [1]:Completed by February 1946, the ENIAC was the first general-purpose electronic digital computer. It contained roughly 18000 vacuum tubes and measured about 2.5 metres in height and 24 metres in length. The machine was more than 1000 times faster than its electromechanical predecessors and could execute up to 5000 additions per second. Its operation was controlled by a program that was set up externally by wires on plugboards. The ENIAC was the most complex and influential electronic computer of its time ... Of course by 1946 World War II was over but the machine was used intensively, particularly on top secret problems associated with the development of nuclear weapons. Von Neumann was working on this project and became involved with the ENIAC computer and used it to solve systems of partial differential equations which were crucial in the work on atomic weapons at Los Almos. Both Mauchly and John Eckert left the Moore School at the University of Pennsylvania in October 1946.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mauchly.html (2 of 4) [2/16/2002 11:22:17 PM]

Mauchly

They started up the Electronic Control Company which they received an order from Northrop Aircraft Company to build the Binary Automatic Computer (BINAC). One of the major advances of this machine, which was used from August 1950, was that data was stored on magnetic tape rather than on punched cards. The Electronic Control Company become the Eckert-Mauchly Computer Corporation and it received an order from the National Bureau of Standards to build the Universal Automatic Computer (UNIVAC). This was the first computer to be produced commercially in the United States with 46 UNIVACs being built. One of the major advances which the UNIVAC introduced was an ability to handle both numerical and alphabetical information with equal success. John Eckert and Mauchly were better at computer design than they were at the economics of running a company. The problem really lay in the fact that this was such a new area that costs of production were extremely hard to estimate. As a consequence the Eckert-Mauchly Computer Corporation soon hit financial difficulties. In 1950 the Remington Rand Corporation acquired the Eckert-Mauchly Computer Corporation and changed its name to the Univac Division of Remington Rand. Mauchly left the company and formed Mauchly Associates of which he was president from 1959 to 1965 when he became chairman of the board. In 1966 Mauchly received the Harry M Goode Memorial Award, a medal and $2,000 awarded by the Computer Society:For his pioneering contributions to automatic computing by participating in the design and construction of the ENIAC, the world's first all-electronic computer, and of the BINAC and the UNIVAC, and for his pioneering efforts in the application of electronic computers to the solution of scientific and business problems. Mauchly served as president of Dynatrend Inc. from 1968 to his death in 1980 and also president of Marketrend Inc. from 1970 again until his death in 1980. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor Other Web sites

Chronology: 1940 to 1950 1. Kalamazoo 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mauchly.html (3 of 4) [2/16/2002 11:22:17 PM]

Mauchly

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mauchly.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mauchly.html (4 of 4) [2/16/2002 11:22:17 PM]

Maupertuis

Pierre Louis Moreau de Maupertuis Born: 28 Sept 1698 in Saint Malo, France Died: 27 July 1759 in Basel, Switzerland

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre de Maupertuis became a member of the Académie des Sciences in Paris in 1731. In 1732 he introduced Newton's theory of gravitation to France. He was a member of an expedition to Lapland in 1736 which set out to measure the length of a degree along the meridian. Maupertuis' measurements verified Newton's predictions that the Earth would be an oblate speroid. They corrected earlier results of Cassini. Maupertuis gained fame from this expedition and he was invited to Germany by Frederick the Great. After going to Berlin he accompanied the Prussian army in the field and was taken prisoner in 1741. He became a member of the Berlin Academy of Sciences in 1741 and, four years later, became its president. He held the post of president for eight years. Maupertuis published on many topics including mathematics, geography, astronomy and cosmology. In 1744 he first enunciated the Principle of Least Action and he published it in Essai de cosmologie in 1750. Maupertuis hoped that the principle might unify the laws of the universe and combined it with an attempted proof of the existence of God. Maupertuis was accused by Samuel König of plagiarizing Leibniz's work but he was defended by Euler. Voltaire was so critical of Maupertuis' work that eventually he left Berlin in 1753. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maupertuis.html (1 of 2) [2/16/2002 11:22:19 PM]

Maupertuis

List of References (7 books/articles) A Poster of Pierre de Maupertuis

Mathematicians born in the same country

Cross-references to History Topics

Orbits and gravitation

Honours awarded to Pierre de Maupertuis (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1728

Lunar features

Crater Maupertuis and Rimae Maupertuis

Other Web sites

1. Helsinki, Finland (Maupertuis' expedition to Lapland) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Maupertuis.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maupertuis.html (2 of 2) [2/16/2002 11:22:19 PM]

Maurolico

Francisco Maurolico Born: 16 Sept 1494 in Messina, Italy Died: 22 July 1575 in Messina, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Francisco Maurolico was ordained a priest in 1521. He later became a Benedictine and lived his whole life in Sicily except for short periods in Rome and Naples. He served as head of the mint in Sicily, he was in charge of the fortifications of Messina and was appoined to write a history of Sicily. Maurolico wrote important books on Greek mathematics, restored many ancient works from scant information and translated many ancient texts such as those by Theodosius, Menelaus, Autolycus, Euclid, Apollonius and Archimedes. He gave a table of secants and, although Delambre credited him with the first use of this function, it had appeared earlier in the work of Copernicus. Maurolico also worked on geometry, the theory of numbers (L E Dickson notes some of his results), optics, conics and mechanics, writing important books on these topics. Maurolico gave methods for measuring the Earth in Cosmographia which were later used by Jean Picard in measuring the meridian in 1670. He made astronomical observations, in particular he observed the supernova which appeared in Cassiopeia in 1572 now known as 'Tycho's supernova'. Tycho Brahe published details of his observations in 1574. Some details of Maurolico's observations were published by Clavius but full details of Maurolico's observations were never published and only rediscovered comparatively recently. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1500 to 1600

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maurolico.html (1 of 2) [2/16/2002 11:22:20 PM]

Maurolico

Honours awarded to Francisco Maurolico (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Maurolycus

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Maurolico.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maurolico.html (2 of 2) [2/16/2002 11:22:20 PM]

Maxwell

James Clerk Maxwell Born: 13 June 1831 in Edinburgh, Scotland Died: 5 Nov 1879 in Cambridge, Cambridgeshire, England

Click the picture above to see nine larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James Clerk Maxwell was born at 14 India Street in Edinburgh, a house built by his parents in the 1820s, but shortly afterwards his family moved to their home at Glenlair in Kirkcudbrightshire about 20 km from Dumfries. There he enjoyed a country upbringing and his natural curiosity displayed itself at an early age. In a letter written on 25 April 1834 when 'The Boy' was not yet three years old he is described as follows, see [4]:He is a very happy man, and has improved much since the weather got moderate; he has great work with doors, locks, keys etc., and 'Show me how it doos' is never out of his mouth. He also investigates the hidden course of streams and bell-wires, the way the water gets from the pond through the wall and a pend or small bridge and down a drain ... When James was eight years old his mother died. His parents plan that they would educate him at home until he was 13 years old, and that he would then be able to go the Edinburgh University, fell through. A 16 year old boy was hired to act as tutor but the arrangement was not a successful one and it was decided that James should attend the Edinburgh Academy. James, together with his family, arrived at 31 Heriot Row, the house of Isabella Wedderburn his father's sister, on 18 November 1841. He attended Edinburgh Academy where he had the nickname 'Dafty'. P G Tait, although almost the same age, was one class below James. Tait, who would become a close school friend and friend for life, described Maxwell's school days [39]:At school he was at first regarded as shy and rather dull. he made no friendships and spent his occasional holidays in reading old ballads, drawing curious diagrams and making rude mechanical models. This absorption in such pursuits, totally unintelligible to his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maxwell.html (1 of 7) [2/16/2002 11:22:22 PM]

Maxwell

schoolfellows, who were then totally ignorant of mathematics, procured him a not very complimentary nickname. About the middle of his school career however he surprised his companions by suddenly becoming one of the most brilliant among them, gaining prizes and sometimes the highest prizes for scholarship, mathematics, and English verse. In early 1846 at the age of 14, Maxwell wrote a paper on ovals. In this work he generalised the definition of an ellipse by defining the locus of a point where the sum of m times the distance from one fixed point plus n times the distance from a second fixed point is constant. If m = n = 1 then the curve is an ellipse. Maxwell also defined curves where there were more than two foci. This became his first paper On the description of oval curves, and those having a plurality of foci which was read to the Royal Society of Edinburgh on 6 April 1846. These ideas were not entirely new as Descartes had defined such curves before but the work was remarkable for a 14 year old. Maxwell was not dux of the Edinburgh Academy, this honour going to Lewis Campbell who later became the professor of Greek at the University of St Andrews. Lewis Campbell was a close friend of Maxwell's and he wrote the biography [3] and its second edition [4]. These biographies make fascinating reading filled with personal memories. At the age of 16, in November 1847, Maxwell entered the second Mathematics class taught by Kelland, the natural philosophy (physics) class taught by Forbes and the logic class taught by William Hamilton. Tait, also at the University of Edinburgh, later wrote in the Proceedings of the Royal Society of Edinburgh (1879-80) [4]:The winter of 1847 found us together in the classes of Forbes and Kelland, where he highly distinguished himself. With the former he was a particular favourite, being admitted to the free use of the class apparatus for original experiments. ... During this period he wrote two valuable papers which are published in our Transactions, on The Theory of Rolling Curves and The Equilibrium of Elastic Solids. The University of Edinburgh still has a record of books that Maxwell borrowed to take home while an undergraduate. These include Cauchy, Calcul Differentiel Fourier, Theorie de la Chaleur Monge, Géometrie Descriptive Newton, Optics Poisson, Mechanics Taylor, Scientific Memoirs Willis, Principles of Mechanism Maxwell went to Peterhouse Cambridge in October 1850 but moved to Trinity where he believed that it was easier to obtain a fellowship. Again we quote Tait's article in the Proceedings of the Royal Society of Edinburgh (1879-80):... he brought to Cambridge in the autumn of 1850, a mass of knowledge which was really immense for so young a man, but in a state of disorder appalling to his methodical private tutor. Though the tutor was William Hopkins, the pupil to a great extent took his own way, and it may safely be said that no high wrangler of recent years ever entered the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maxwell.html (2 of 7) [2/16/2002 11:22:22 PM]

Maxwell

Senate-house more imperfectly trained to produce 'paying' work than did Clerk Maxwell. But by sheer strength of intellect, though with the very minimum of knowledge how to use it to advantage under the conditions of the Examination, he obtained the position of Second Wrangler, and was bracketed equal with the Senior Wrangler, in the higher ordeal of the Smith's Prizes. Thomson [39] describes Maxwell's undergraduate days:... Scholars dined together at one table. This bought Maxwell into daily contact with the most intellectual set in the College, among whom were many who attained distinction in later life. These in spite of his shyness and some eccentricities recognised his exceptional powers. ... The impression of power which Maxwell produced on all he met was remarkable; it was often much more due to his personality than to what he said, for many found it difficult to follow him in his quick changes from one subject to another, his lively imagination started so many hares that before he had run one down he was off on another. Maxwell obtained his fellowship and graduated with a degree in mathematics from Trinity College in 1854. The First Wrangler in that year was Edward Routh, who as well as being an excellent mathematician was a genius at mastering the cramming methods required to succeed in the Cambridge Tripos of that time. Maxwell remained at Cambridge where he took pupils, then was awarded a Fellowship by Trinity to continue work. One of Maxwell's most important achievements was his extension and mathematical formulation of Michael Faraday's theories of electricity and magnetic lines of force. His paper On Faraday's lines of force was read to the Cambridge Philosophical Society in two parts, 1855 and 1856. Maxwell showed that a few relatively simple mathematical equations could express the behaviour of electric and magnetic fields and their interrelation. However, in early 1856, Maxwell's father became ill and Maxwell wanted to be able to spend more time with him. He therefore tried to obtain an appointment in Scotland, applying for the post of Professor of Natural Philosophy at Marischal College in Aberdeen when Forbes told him it was vacant. Maxwell travelled to Edinburgh for the Easter vacation of 1856 to be with his father and the two went together to Glenlair. On 3 April his father died and, shortly after, Maxwell returned to Cambridge as he had planned. Before the end of April he learnt that he had been appointed to the chair at Marischal College. In November 1856 Maxwell took up the appointment in Aberdeen. When the subject announced by St John's College Cambridge for the Adams Prize of 1857 was The Motion of Saturn's Rings Maxwell immediately interested. Maxwell and Tait had thought about the problem of Saturn's rings in 1847 while still pupils at the Edinburgh Academy. Maxwell decided to compete for the prize and his research at Aberdeen in his first two years was taken up with this topic. He showed that stability could be achieved only if the rings consisted of numerous small solid particles, an explanation now confirmed by the Voyager spacecraft. In a letter to Lewis Campbell, written on 28 August 1857, while he was at Glenlair, Maxwell wrote:I have effected several breaches in the solid ring, and now am splash into the fluid one, amid a clash of symbols truly astounding. When I reappear it will be in the dusky ring, which is something like the siege of Sebastopol conducted from a forest of guns 100 miles one way, and 30,000 miles the other, and the shot never to stop, but go spinning away round a circle, radius 170,000 miles...

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maxwell.html (3 of 7) [2/16/2002 11:22:22 PM]

Maxwell

Maxwell's essay won him the Adams Prize and Airy wrote:It is one of the most remarkable applications of mathematics to physics that I have ever seen. Maxwell became engaged to marry Katherine Mary Dewar in February 1858 and they married in June 1859. Despite the fact that he was now married to the daughter of the Principal of Marischal College, in 1860, when Marischal College and King's College combined, Maxwell, as the junior of the department, had to seek another post. His scientific work, however, had been proceeding with great success. Stokes had written to him on 7 November 1857:I have just received your papers on the dynamical top, etc., and the account of experiments on the perception of colour. The latter, which I missed seeing at the time when it was published, I have just read with great interest. The results afford most remarkable and important evidence in favour of the theory of three primary colour-perceptions, a theory which you, and you alone, as far as I know, have established on an exact numerical basis. When the Chair of Natural Philosophy at Edinburgh became vacant in 1859, Forbes having moved to St Andrews, it seemed that fate had smiled on Maxwell to bring him back to his home town. He asked Faraday to act as a referee for him, in a letter written on 30 November 1859. Many of Maxwell's friends were also applicants for this post including Tait and Routh. Maxwell lost out to Tait despite his outstanding scientific achievements. When the Edinburgh paper, the Courant, reported the result it noted that:Professor Maxwell is already acknowledged to be one of the most remarkable men known to the scientific world. The reason he was not appointed must have been those given by the paper when they wrote:... there is another quality which is desirable in a Professor in a University like ours and that is the power of oral exposition proceeding on the supposition of imperfect knowledge or even total ignorance on the part of pupils. The claim that he was not the best person to teach poorly qualified pupils may have been a fair one but it is certainly not the case that he was a poor lecturer. Stokes wrote in 1854 that he had:... once been present when [Maxwell] was giving an account of his geometrical researches to the Cambridge Philosophical Society, on which occasion I was struck with the singularly lucid manner of his exposition. Again Fleming, who had attended Maxwell's lectures, expressed similar thoughts [19]:Maxwell in short had too much learning and too much originality to be at his best in elementary teaching. For those however who could follow him his teaching was a delight. In 1860 Maxwell was appointed to the vacant chair of Natural Philosophy at King's College in London. The six years that Maxwell spent in this post were the years when he did his most important experimental work. The duties of the post were more demanding than those at Aberdeen. Campbell writes in [3]:There were nine months of lecturing in the year, and evening lectures to artisans, etc., were recognised as a part of the Professor's duties. In London, around 1862, Maxwell calculated that the speed of propagation of an electromagnetic field is approximately that of the speed of light. He proposed that the phenomenon of light is therefore an electromagnetic phenomenon. Maxwell wrote the truly remarkable words:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maxwell.html (4 of 7) [2/16/2002 11:22:22 PM]

Maxwell

We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena. Maxwell also continued work he had begun at Aberdeen, considering the kinetic theory of gases. By treating gases statistically in 1866 he formulated, independently of Ludwig Boltzmann, the Maxwell-Boltzmann kinetic theory of gases. This theory showed that temperatures and heat involved only molecular movement. This theory meant a change from a concept of certainty, heat viewed as flowing from hot to cold, to one of statistics, molecules at high temperature have only a high probability of moving toward those at low temperature. Maxwell's approach did not reject the earlier studies of thermodynamics but used a better theory of the basis to explain the observations and experiments. Maxwell left King's College, London in the spring of 1865 and returned to his Scottish estate Glenlair. He made periodic trips to Cambridge and, rather reluctantly, accepted an offer from Cambridge to be the first Cavendish Professor of Physics in 1871. He designed the Cavendish laboratory and helped set it up. The Laboratory was formally opened on 16 June 1874. The four partial differential equations, now known as Maxwell's equations, first appeared in fully developed form in Electricity and Magnetism (1873). Most of this work was done by Maxwell at Glenlair during the period between holding his London post and his taking up the Cavendish chair. They are one of the great achievements of 19th-century mathematics. One of the tasks which occupied much of Maxwell's time between 1874 and 1879 was his work editing Henry Cavendish's papers. Cavendish, see [13]:... published only two papers [and] left twenty packages of manuscript on mathematical and experimental electricity. ... Maxwell entered upon this work with the utmost enthusiasm: he saturated his mind with the scientific literature of Cavendish's period; he repeated many of his experiments, and copied out the manuscript with his own hand. ... The volume entitled The Electrical Researches of the Honourable Henry Cavendish was published in 1879, and is unequalled as a chapter in the history of electricity. Fleming attended Maxwell's last lecture course at Cambridge. He writes [19]:During the last term in May 1879 Maxwell's health evidently began to fail, but he continued to give his lectures up to the end of the term. ... To have enjoyed even a brief personal acquaintance with Professor Maxwell and the privilege of his oral instruction was in itself a liberal education, nay more, it was an inspiration, because everything he said or did carried the unmistakable mark of a genius which compelled not only the highest admiration but the greatest reverence as well. Maxwell returned with his wife, who was also ill, to Glenlair for the summer. His health continued to deteriorate and he suffered much pain although remained remarkably cheerful. On 8 October 1879 he returned with his wife to Cambridge but, by this time he could scarcely walk. One of the greatest scientists the world has known passed away on 5 November. His doctor, Dr Paget, said:No man ever met death more consciously or more calmly. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maxwell.html (5 of 7) [2/16/2002 11:22:22 PM]

Maxwell

Click on this link to see a list of the Glossary entries for this page List of References (40 books/articles)

Some Quotations (6)

A Poster of James Clerk Maxwell

Mathematicians born in the same country

Cross-references to History Topics

1. A comment from Thomas Hirst's diary 2. A visit to Maxwell's house in Edinburgh. 3. General relativity 4. An overview of the history of mathematics 5. Special relativity 6. Topology and Scottish mathematical physics

Other references in MacTutor

1. Chronology: 1840 to 1850 2. Chronology: 1850 to 1860 3. Chronology: 1860 to 1870 4. Chronology: 1870 to 1880

Honours awarded to James Clerk Maxwell (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1861

Royal Society Bakerian lecturer

1866

Fellow of the Royal Society of Edinburgh Lunar features

Crater Maxwell

Other Web sites

1. I Hutchison 2. Ham Radio showcase (The Life of James Clerk Maxwell by Campbell and Garnett (1882)) 3. James Maxwell Foundation 4. Edingburghers 5. Waterloo University 6. IEEE (Exhibition on Maxwell and Faraday) 7. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maxwell.html (6 of 7) [2/16/2002 11:22:22 PM]

Maxwell

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR November 1997 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Maxwell.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Maxwell.html (7 of 7) [2/16/2002 11:22:22 PM]

Mayer_Adolph

Christian Gustav Adolph Mayer Born: 15 Feb 1839 in Leipzig, Germany Died: 11 April 1907 in Gries bei Bozen, Austria (now Bolzano, Italy)

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adolph Mayer's father was a merchant from Leipzig with a prosperous business so that the family were well off. Mayer, as was the custom of German students at this time, studied at a number of different universities during his education. As well at studying at his home university in Leipzig, Mayer also studied at Göttingen, Heidelberg and Königsberg where he worked under Franz Neumann. He received his doctorate from Heidelberg in 1861. Continuing his studies at Heidelberg, Mayer submitted his habilitation thesis to that university and gained the right to teach at universities in 1866. He taught at Heidelberg for the rest of his life, becoming an extraordinary professor in 1871 and marrying Margerete Weigel in the following year. He was promoted to an ordinary professor in 1890. Wussing writes in [1] that:As a professor, Mayer enjoyed great respect from his colleagues and students. His activities as a researcher ... earned him membership in numerous learned societies ... Mayer worked on differential equations, the calculus of variations and mechanics. He emphasised the principle of least action in all his work which followed the path of Lagrange and Jacobi. His work on the integration of partial differential equations and a search to determine maxima and minima using variational methods brought him close to the investigations which Lie was carrying out around the same time. Engel received his doctorate from Leipzig in 1883 after studying under Mayer and writing a thesis on

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mayer_Adolph.html (1 of 2) [2/16/2002 11:22:24 PM]

Mayer_Adolph

contact transformations. Engel became a valuable assistant to Lie for several years but towards the end of the 1880s the relationship between Engel and Lie broke down. In 1892 the lifelong friendship between Lie and Klein broke down and the following year Lie publicly attacked Klein. Mayer was connected with the whole episode through his friendship with Klein, being an editor of Mathematische Annalen, and perhaps most significantly since his work was closely related to that of Lie. The book [2] contains a collection of 186 letters exchanged between Klein and Mayer over the years from 1871 to 1907. The letters provide insights into the scientific and personal relations among Klein, Mayer and Lie over the period. Wussing writes in [1] that:through the subsequent works of Mayer, Lie's achievements became famous relatively quickly. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Adolph Mayer (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater C Mayer

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mayer_Adolph.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mayer_Adolph.html (2 of 2) [2/16/2002 11:22:24 PM]

Mayer_Tobias

Johann Tobias Mayer Born: 17 Feb 1723 in Marbach, Württemberg, Germany Died: 20 Feb 1762 in Göttingen, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Tobias Mayer was a self taught mathematician who worked as a cartographer in Nürnberg. He discovered the libration of the Moon and this gained him fame which led to his appointment as professor of economics and mathematics at Göttingen in 1751. Mayer began calculating lunar and solar tables in 1753 and in 1755 he sent them to the British government. These tables were good enough to determine longitude at sea with an accuracy of half a degree. Mayer's method of determining longitude by lunar distances and a formula for correcting errors in longitude due to atmospheric refraction were published in 1770 after his death. In a preface written to his tables written in 1760 Mayer says I am the more unwilling my tables should lie any longer concealed; especially as the most celebrated astronomers of almost every age have ardently wished for a perfect theory of the Moon ... on account of its singular use in navigation. I have constructed theses tables ... with respect to the inequalities of motions, from that famous theory of the great Newton, which that eminent mathematician Eulerus first elegantly reduced to general analytic equations. In the first issue of the Nautical Almanac there was a description by Maskelyne of Mayer's tables The Tables of the Moon had been brought by the late Professor Mayer of Göttingen to a sufficient exactness to determine the Longitude at Sea to within a Degree, as appeared by the Trials of several Persons who made use of them. The Difficulty and Length of the necessary Calculations seemed the only Obstacles to hinder them from becoming of general Use. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mayer_Tobias.html (1 of 2) [2/16/2002 11:22:26 PM]

Mayer_Tobias

The Board of Longitude sent Mayer's widow 3000 as an award for the tables. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Tobias Mayer

Mathematicians born in the same country

Cross-references to History Topics

English attack on the Longitude Problem

Honours awarded to Tobias Mayer (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater T Mayer

Other Web sites

1. Linda Hall Library (Drawings of the Moon) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mayer_Tobias.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mayer_Tobias.html (2 of 2) [2/16/2002 11:22:26 PM]

Mazur

Stanislaw Mazur Born: 1 Jan 1905 in Poland Died: 5 Nov 1981

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stanislaw Mazur went to the Polytechnic Institute in Lvov to work with Orlicz, Nikliborc and Kaczmarz. In [1] an incident from this time is recalled by Andrzej Turowicz. Mazur wrote his first paper while still an undergraduate and he submitted it to Steinhaus. The paper was to be read by Mazur at a meeting of the Lvov Scientific Society but only hours before the meeting Steinhaus summoned Mazur to tell him that he had handed him four blank sheets of paper. Students could only afford cheap quality yellow paper and Mazur had diluted his ink with water to make it last longer. Steinhaus said to Mazur:Well, Mr Mazur, perhaps there is something written here after all. But if you intend to devote your life to scientific pursuits, why don't you first supply yourself with white paper and black ink. There is no record of whether Mazur took Steinhaus's advice, but he certainly devoted himself to mathematical pursuits. He became a student of Banach's who taught at the university in Lvov. His doctorate, under Banach's supervision, was awarded in 1935. Mazur was a close collaborator with Banach at Lvov and became a member of the Lvov School of Mathematics, a group of about a dozen mathematicians working in functional analysis, real functions and probability theory. He wrote several papers in collaboration with Banach during the 1930s and, having a better knowledge than Banach of German, he polished the language used in the joint papers which they wrote in German. The collaboration between Mazur and Banach in Lvov was important for both men. There is no doubt that of of all Banach's colleagues in Lvov, Mazur was the one closest to him. The way that the mathematicians worked in Lvov has become famous. They spent many hours thinking about mathematical problems in the Scottish Café. These sessions are described in [1]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mazur.html (1 of 4) [2/16/2002 11:22:27 PM]

Mazur

Usually they began arriving between 5 and 7 pm - always occupying the same tables - and for the next several hours they worked with total concentration, covering the marble table tops with mathematical farmulas. But saying that they worked with total concentration is not completely accurate, as there was no meeting without jokes, heated discourse, shouting and drinking. Ulam, in [2], describes the particular way that the collaboration between Mazur and Banach in the Scottish Café worked:We discussed problems proposed right there, often with no solution evident even after several hours of thinking. The next day Banach was likely to appear with several small sheets of paper containing outlines of proofs he had completed. If they were not polished or even not quite correct, Mazur would frequently put them into a more satisfactory form. It was in the Scottish Café that the famous Scottish Book consisting of open questions posed by the mathematicians working there came into being. Mazur contributed 24 problems to the book with himself as the sole author, and a further 19 problems jointly contributed with others such as Banach. It was not only Banach with who Mazur collaborated but others including Ulam. In [2] Ulam describes their collaborations on function spaces:We found a solution to a problem involving infinite dimensional vector spaces. The theorem we proved - that a transformation preserving distances is linear - is now part of the standard treatment of the geometry of function spaces. We wrote a paper which was published in Compte Rendus ... It was Mazur ... who introduced me to certain large phases of mathematical thinking and approaches. From him I learned much about the attitudes and psychology of research. During the 1930s Mazur was an active member of the Polish Communist Party. This would stand him in good stead when the Communists came to power after the war. Mazur's habilitation thesis was submitted in 1936. After the award of his habilitation, Mazur taught at Lvov until 1946. In [2] Ulam recalls a conversation he had with Mazur in the summer of 1939 about whether he thought there would be a war:People in general expected another crisis in the style of Munich and were not prepared for the coming World War. Mazur said "The possibility of a World War is real. What are we going to do with the Scottish Book and our joint unpublished papers? You are going to the United States, and most surely you are going to be safe. In case the city is bombed, I will pack the manuscripts and the Book in a chest and bury it". We even decided on an exact place - next to the goal post at the soccer pitch on the city outskirts. I do not know if that's what happened. Although it is not known for certain how the Scottish Book survived the war, we do know that it was brought to Wroclaw by Banach's wife and it ended up in the possession of his son. There is a charming story about one of the most famous of the problems in the Scottish Book which was posed by Mazur. This was problem number 153, which Mazur inserted into the Book on 6 November 1936. The problem asked (although not in these words) about the existence of Schauder bases in separable Banach spaces. As with many of the problems in the Scottish Book the proposer would offer a prize for their solution. Prizes offered included wine, spirits, or a meal in Cambridge but Mazur offered a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mazur.html (2 of 4) [2/16/2002 11:22:27 PM]

Mazur

live goose as the prize for this particular problem. Per Enflo showed in 1972 that the problem had a negative solution and, while in Warsaw lecturing on his solution, Mazur presented him with his prize, the live goose! From 1948 Mazur worked at the University of Warsaw. Then, given his earlier involvement with the Polish Communist Party, Mazur became a high official in the science establishment. Mazur made important contributions to geometrical methods in linear and non-linear functional analysis and to the study of Banach algebras. The mean ergodic theorem in Banach spaces was announced by Mazur in 1932 but a proof does not appear in print until 1938 when Yosida and by Kakutani published the result. He was also interested in summability theory, infinite games and recursive functions. Ulam recounts in [2] how Mazur gave the first examples of infinite games in a café in Lvov. You can see a picture of the Scottish Café. The author of [4] points out how many of Mazur's original contributions are not explicitly identified as such but appear in print only as remarks in Banach's Theorie des operations lineaires. For example, the weak-basis theorem, due to Mazur, is given by Banach in his book but no proof appears. In 1978 Mazur was honoured by receiving honorary life membership in the Polish Mathematical Society. The article [6] presents the depth of Mazur's functional-analytic contribution which led to this honour. In 1980 the University of Warsaw awarded Mazur an honorary doctorate. The award was made in recognition that he was a leading Polish mathematician and a cofounder with Banach of the Polish School of Functional Analysis. In the article [3] Mazur's reply at the awarding ceremony is given. In his speach Mazur concentrated on his activities at the University of Lvov during World War II. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mazur.html (3 of 4) [2/16/2002 11:22:27 PM]

Mazur

JOC/EFR February 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mazur.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mazur.html (4 of 4) [2/16/2002 11:22:27 PM]

Mazurkiewicz

Stefan Mazurkiewicz Born: 25 Sept 1888 in Warsaw, Poland Died: 19 June 1945 in Grodzisk, Mazowiecki (near Warsaw), Poland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stefan Mazurkiewicz's father was a barrister in Warsaw. Stefan attended secondary school in Warsaw, graduating in 1907. Mazurkiewicz decided not to remain in Warsaw for his university education, taking what was a typical route for Poles at this time. We shall give some background to the political situation in Poland to explain why this was so. The first thing to note is that when Mazurkiewicz graduated from secondary school in 1907, Poland did not formally exist. Poland had been partitioned in 1772 and the south was called Galicia and under Austrian control. Russia controlled much of the rest of the country and in the years prior to Mazurkiewicz's birth there had been strong moves by Russia to make "Vistula Land", as it was called, be dominated by Russian culture. In a policy implemented between 1869 and 1874, all secondary schooling was in the Russian language. Warsaw only had a Russian language university after the University of Warsaw was closed by the Russian administration in 1869. Galicia, although under Austrian control, retained Polish culture and was often where Poles from "Vistula Land" went for their education. Indeed Mazurkiewicz first went to Kraków, in Galicia but, again following the pattern of students of the time to spend sessions at a number of different universities, he went next to Munich, then to the renowned mathematical research centre at Göttingen. Mazurkiewicz returned to Galicia for his doctorate, which was supervised by Sierpinski on space filling curves at the University of Lvov. His doctorate was awarded in 1913, but World War I began the following year and it was to bring major changes in Poland and to Mazurkiewicz's life. In August 1915 the Russian forces which had held Poland for many years withdrew from Warsaw. Germany and Austria-Hungary took control of most of the country and a German governor general was installed in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mazurkiewicz.html (1 of 4) [2/16/2002 11:22:29 PM]

Mazurkiewicz

Warsaw. One of the first moves after the Russian withdrawal was the refounding of the University of Warsaw and it began operating as a Polish university in November 1915. At this point Mazurkiewicz became a professor at this reborn University of Warsaw and he would remain on the staff of the university for the rest of his life. Kuratowski attended seminars given by Mazurkiewicz in Warsaw before the end of the war. He writes in [2]:As early as 1917 [Janiszewski and Mazurkiewicz] were conducting a topology seminar, presumably the first in that new, exuberantly developing field. The meeting of that seminar, taken up to a large extent with sometimes quite vehement discussions between Janiszewski and Mazurkiewicz, were a real intellectual treat for the participants. The role that Mazurkiewicz played in the creation of the Polish School of Mathematics was an important one. Kuratowski writes in [2]:... Stefan Mazurkiewicz was the central figure among professors of mathematics, especially in the early years of the university's existence. A brilliant lecturer, a very active research worker, he had a great influence on young people and encouraged them to do research of their own in modern fields of mathematics ... His main work was in topology and the theory of probability. His notion of dimension of a compact set preceded that of Menger and Urysohn by seven years. Mazurkiewicz applied topological methods to the theory of functions, obtaining powerful results. His theory gave particularly strong results when applied to the Euclidean plane, giving deep knowledge of its topological structure. Many of the ideas introduced by Mazurkiewicz were studied independently by Hahn. They independently proved that [1]:... every continuous function that transforms a compact linear set into a plane set with interior points takes the same value in at least three points. Other results by Mazurkiewicz gave information about the topological structure of curves. He proved strong results on continuous functions containing Sierpinski's curve and wrote several papers on functional spaces. The style employed by Mazurkiewicz, both in research and teaching, is described by Kuratowski in [2]:Mazurkiewicz's passion was solving problems and raising new and often very profound ones. This unusually creative scholar's almost sportsmanlike attitude towards mathematics was in some sense manifested in the way he lectured and prepared his results for publication: Mazurkiewicz used no notes while lecturing, and his lectures were not always completely elaborated but they were greatly admired by his audience for their ingenuity and deep intelligence. Very often, however, his publications were not sufficiently polished and presented only a draft of an argument and therefore were not easily understandable; but as a rule they contained new ideas and fascinated reader by their author's inventive powers and the wealth of his methods. Mazurkiewicz held many important positions in the University of Warsaw as it flourished between the two wars. He was elected vice-rector of the University and he was Dean of the Faculty of Mathematical and Natural Sciences for nine years. He was also president of the Polish Mathematical Society in 1933-35 and he continued the tradition established in Lvov of meetings in coffee houses. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mazurkiewicz.html (2 of 4) [2/16/2002 11:22:29 PM]

Mazurkiewicz

After Janiszewski died in 1920, Mazurkiewicz and Sierpinski took over the editorship of the journal Fundamenta Mathematicae which Janiszewski had set up. In [6] Tamarkin wrote:Under masterful guidance of its editors, S Mazurkiewicz and W Sierpinski, Fundamenta Mathematicae immediately developed into a unique periodical which attracted international recognition and cooperation, and whose history became the history of development of the modern theory of functions and point sets. Throughout his career Mazurkiewicz was interested in the theory of probability. He proved the strong law of large numbers in 1922, a result which was proved independently by Cantelli. He also considered axiom systems for probability theory, publishing different versions in the years 1933 and 1934. After the German invasion of Poland in 1939 life there became extremely difficult. There was a strategy by the Nazi invaders to put an end to the intellectual life of Poland and to achieve this they sent many academics to concentration camps and murdered others. It was during this period of occupation that Mazurkiewicz wrote a treatise on probability including his own results on the topic. Near the end of the war, Warsaw became the centre for Polish resistance. In 1944 the people, knowing that Soviet forces were nearing the city, rose up against the German garrison which by this time was severely depleted. However, German reinforcements were sent and the Polish resistance was defeated. Hitler ordered the destruction of the city in retaliation for the uprising. The Germans systematically destroyed the buildings using explosives on some and setting fire to others. Around 175,000 of the population of Warsaw died as a result of the uprising. Mazurkiewicz escaped with his life, but the manuscript of his treatise on probability was destroyed as the buildings burned. At this time Mazurkiewicz was greatly weakened as a result of the difficult life he had led in Warsaw and also through an illness from which he now suffered. However, he attempted to rewrite his treatise on probability which had been burnt in the destruction. Although he only managed to recreate part of the work it was completed eleven years after his death and published as number 32 in the Mathematical Monographs series. Despite being gravely ill, Mazurkiewicz thought only of the recreation of Polish mathematics at as the war drew to a close, being filled with the same enthusiasm which he had displayed at the end of World War I. On 25 February 1945 he submitted a report to the Ministry of Education on the recovery route that mathematics should take. He argued strongly for the creation of a Mathematical Institute along the lines of Kuratowski's report made before the start of the war. Already moves were under way to re-establish the University of Warsaw and the Technical University. Mazurkiewicz took part in these meetings despite his failing health. Desperately ill he was taken to a hospital at Grodzisk Mazowiecki on the outskirts of Warsaw where he died during an operation for a gastric ulcer. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mazurkiewicz.html (3 of 4) [2/16/2002 11:22:29 PM]

Mazurkiewicz

Mathematicians born in the same country Other references in MacTutor

Chronology: 1920 to 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mazurkiewicz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mazurkiewicz.html (4 of 4) [2/16/2002 11:22:29 PM]

McClintock

John Emory McClintock Born: 19 Sept 1840 in Carlisle, Pennsylvania , USA Died: 10 July 1916 in Bay Head, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John McClintock studied at Dickinson College, Carlisle from 1854 until he entered Yale University in 1856. He then studied at Columbia College in New York (later Columbia University) from 1857 receiving his A.B. in 1859. McClintock spent a year in Europe studying chemistry at the University of Paris and then at Göttingen. He represented the US Consul in England from 1863 to 1866 when he became associated with a banking firm in Paris. Returning to the USA he was an actuary in New York, then in Milwaukee before becoming an actuary in the Mutual Life Insurance Company in New York in 1889 where he remained until he retired in 1911. In fact McClintock was for many years the leading actuary in America. He published 30 papers between 1868 and 1877 on actuarial questions. His publications were not confined to questions relating to life insurance policies however. He published about 22 papers on mathematical topics. One paper treats difference equations as differential equations of infinite order and others look at quintic equations which are soluble algebraically. Another work is on quadratic residues. Archibald writes [1]:McClintock is known to have expressed regret that he had not followed an academic career, which would have permitted him to give a large share of his time to research... In such a direction he would probably have gone far. In 1889 when McClintock took up his actuarial post with the Mutual Life Insurance Company in New York, the New York Mathematical Society was just coming into existence. McClintock joined the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/McClintock.html (1 of 2) [2/16/2002 11:22:31 PM]

McClintock

Society in December 1889 and was elected vice-president of the Society. In the following year he was elected president and he has the distinction of being the only president of the Society to serve for four years. During McClintock's presidential term, Klein visited the Society and talked on non-euclidean spherical trigonometry. Study also addressed the Society during McClintock's term as president and talked on his work with Engel. When McClintock's term ended he gave the first presidential address to the Society on The past and future of the Society. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to John McClintock (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1891 - 1894

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/McClintock.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/McClintock.html (2 of 2) [2/16/2002 11:22:31 PM]

McDuff

Margaret Dusa Waddington McDuff Born: 18 Oct 1945 in London, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dusa McDuff was christened Margaret Dusa Waddington. Her father, Conrad Hal Waddington, was appointed Professor of Genetics at the University of Edinburgh in Scotland while her mother, Margaret Justin Blanco White, was an architect who had a Civil Service post in Edinburgh. Dusa was educated at a girls school in Edinburgh and, although the standard was less good than at the boys school, nevertheless she had an exceptionally good mathematics teacher. She wrote:I always wanted to be a mathematician (apart from a time when I was eleven when I wanted to be a farmer's wife), and assumed that I would have a career, but I had no idea how to go about it: I didn't realise that the choices which one made about education were important and I had no idea that I might experience real difficulties and conflicts in reconciling the demands of a career with life as a woman. By the time that Dusa completed her secondary schooling in Edinburgh she had a boyfriend. This led to her choosing the University of Edinburgh for her undergraduate studies, turning down a scholarship which she had won to go to Cambridge University. During her undergraduate years at Edinburgh Dusa married her boyfriend and took his name becoming Dusa McDuff. Awarded a B.Sc. from Edinburgh in 1967, Dusa went to Girton College, Cambridge for her doctoral studies. At Cambridge McDuff was supervised by G A Reid and she worked on problems in functional analysis. This time her husband followed her to Cambridge. She solved a difficult problem on von Neumann algebras, constructing infinitely many different factors of type II1, and published the work in the Annals of Mathematics. After completing her doctorate in 1971 McDuff was appointed to a two-year Science Research Council Postdoctoral Fellowship at Cambridge. Now McDuff followed her husband again, this time with a six http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/McDuff.html (1 of 4) [2/16/2002 11:22:33 PM]

McDuff

month visit to Moscow. He was studying the Russian Symbolist poet Innokenty Annensky and Dusa had no specific plans, yet it would turn out a very profitable visit for her mathematically. She met Israil Gelfand in Moscow and he gave her a deeper appreciation of mathematics. McDuff wrote:Gelfand amazed me by talking of mathematics as though it were poetry. He once said about a long paper bristling with formulas that it contained the vague beginnings of an idea which he could only hint at and which he had never managed to bring out more clearly. I had always thought of mathematics as being much more straightforward: a formula is a formula, and an algebra is an algebra, but Gelfand found hedgehogs lurking in the rows of his spectral sequences! After the Moscow visit, where she studied Gelfand-Fuchs cohomology, McDuff returned to Cambridge. There she attended Frank Adams' topology lectures and around this time her first child was born. However her research at this time was not well focused and she began to lose her way a little. Appointed to a position as a lecturer at the University of York in 1973 she began to work with Graeme Segal on classifying spaces of categories. To a certain extent she considered this as a second doctorate to regain direction to her research. Perhaps 1974 was a turning point for McDuff. She was invited to spend a year at Massachusetts Institute of Technology. She wrote:This was a turning point. While there I realised how far away I was from being the mathematician I felt that I could be, but also realised that I could do something about it. For the first time, I met some other women whom I could relate to and who also were trying to become mathematicians. I became much less passive: I applied to the Institute for Advanced Study and got in, and even had a mathematical idea again, which grew into a joint paper with Segal on the group-completion theorem. Back in England, McDuff separated from her husband and, soon afterwards, she was appointed to a post at the University of Warwick in 1976. McDuff had a friend, the mathematician Jack Milnor who worked in Princeton. After two years at Warwick, McDuff resigned her tenured post there and accepted an untenured post at the State University of New York at Stony Brook so that she could be close to Jack Milnor. McDuff wrote:I still worked very much in isolation and there are only a few people who are interested in what I did, but it was a necessary apprenticeship. I had some ideas, and gained confidence in my technical abilities. Of course, I was influenced by the clarity of Jack Milnor's ideas and approach to mathematics, and was helped by his encouragement. I kept my job in Stony Brook, even though it meant a long commute to Princeton and a weekend relationship, since it was very important to me not to compromise on my job as my mother had done. After several years, I married Jack and had a second child. From the early 1980s McDuff worked on symplectic topology. During a sabbatical term at the Institut des Hautes Etudes Scientifique in Paris in 1985 she studied Gromov's work on elliptic methods which became the basis for much of her later work. In 1984 she was promoted to full professor at Stony Brook, being Chair of the Mathematics Department there from 1991 to 93. McDuff has received many honours for her remarkable mathematical achievements. In 1991 she was awarded the Ruth Lyttle Satter Prize of the American Mathematical Society. The quotations we have given in this article are taken from the acceptance speech she gave on the occasion of the presentation of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/McDuff.html (2 of 4) [2/16/2002 11:22:33 PM]

McDuff

the Prize. Many other honours have come her way, perhaps the most prestigious of which is her election as a Fellow of the Royal Society of London in 1994. The citation for her election to the Fellowship read:McDuff is best known for her work in the geometry of multi-dimensional structures. Her work in symplectic geometry, functional analysis and diffeomorphism groups has provided understanding and unexpected results in a whole range of areas of great importance. Her work is based on a deep and wide mathematical understanding, and has opened an extraordinarily fertile new branch of mathematics. In 1995 she was elected a Fellow of the American Academy of Arts and Sciences. In addition to these honours McDuff has been invited to give many prestigious lectures. She gave the Invited Address at the American Mathematical Society Winter meeting in Atlanta in 1988, the first Progress in Mathematics lecture at the American Mathematical Society Summer meeting in Boulder in 1989, an Invited Address at the International Congress of Mathematicians in Kyoto in 1990, and a Plenary Address at the Second European Congress in Budapest in 1996. Although her research contributions to mathematics have been truly outstanding, McDuff has given service to mathematics in many other ways. She has been involved in reform of undergraduate teaching at Stony Brook, is on the editorial board of Notices of the American Mathematical Society, and has been an active member of Women in Science and Engineering. We give one further quote from her acceptance speech of the Ruth Lyttle Satter Prize concerning women in mathematics:I think that there is quite an element of luck in the fact that I have survived as a mathematician. I also got real help from the feminist movement, both emotionally and practically. I think things are somewhat easier now: there is at least a little more institutional support of the needs of women and families, and there are more women in mathematics so that one need not be so isolated. But I don't think that all the problems are solved. Outside mathematics McDuff says that her interests are reading, chamber music, playing the cello, gardening, walking, and talking to friends. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article)

A Quotation

Mathematicians born in the same country Honours awarded to Dusa McDuff (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society Other Web sites

1994 1. Autobiographical notes 2. Agnes Scott College 3. Home page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/McDuff.html (3 of 4) [2/16/2002 11:22:33 PM]

McDuff

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/McDuff.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/McDuff.html (4 of 4) [2/16/2002 11:22:33 PM]

McShane

Edward James McShane Born: 10 May 1904 in New Orleans, Louisiana, USA Died: 1 June 1989 in Charlottesville, Virginia, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edward McShane was the son of Augustus McShane, a medical doctor. His mother was Harriet Kenner Butler, who had been a school teacher. After attending school in New Orleans, McShane entered Tulane University. There he studied mathematics and engineering, graduating in 1925 with both Science and Engineering degrees. After completing his first degrees (for he had two!), McShane continued to study at Tulane for his Master's degree in mathematics while teaching mathematics as an instructor. It is noted in [2] that he:... once remarked that he had never regretted his decision to become a mathematician rather than an engineer. With the award of a Master's degree from Tulane in 1927, McShane went to the University of Chicago to undertake studies for his doctorate. However, financial difficulties caused him to interrupt his studies and take out the session 1928-29 when he taught at the University of Wichita to earn some money. Returning to Chicago, where his studies were supervised by Bliss, he completed work on his thesis and he was awarded his PhD in 1930. McShane spent part of the following two years at Princeton, part at Ohio State, part at Harvard and part at Chicago. During this period he married Virginia Haun on 10 September 1931. They would have two daughters, Jennifer and Ginger, and one son, Neill, all of whom were highly talented in both music and mathematics. Now 1932 was not a good year to be trying to find a university post in the United States due to the effects of the Depression. McShane and his wife spent the session 1932-33 at Göttingen, but if 1932 was a bad http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/McShane.html (1 of 3) [2/16/2002 11:22:35 PM]

McShane

year in the United States, 1933 was certainly a bad year in Germany with Hitler coming to power and bringing in his first anti-Jewish legislation. Having seen the frightening start of the Nazi rise to power, the McShane family returned to the United States where McShane spent the next two years at Princeton. In 1935 McShane was appointed to a professorship at the University of Virginia at Charlottesville, where, except for temporary appointments elsewhere, he was to remain for the rest of his life. One of these periods away was when he went to head the Ballistics Research Laboratory at the Aberdeen Proving Ground, Maryland in 1942-45 during the Second World War. His joint treatise Exterior Ballistics written during this period, and published in 1953, was considered the leading work on ballistics at the time. In [2] a nice story is told regarding McShane during the McCarthy era. In the early 1950s United States senator Joseph R McCarthy whipped up strong feelings against communism. McShane had been asked to complete a questionnaire. One question asked:... whether he had ever been involved with organisations that had at any time advocated the violent overthrow of the U.S. government. It was quite a brave move for McShane to reply "yes", because he was an employee of the State of Virginia! At the University of Virginia this sense of humour added to his popularity with both staff and graduate students who liked [2]:... his clear lectures, his amusing anecdotes, and his willingness to think about their problems. McShane is famous for his work in the calculus of variations, Moore-Smith theory of limits, the theory of the integral, stochastic differential equations, and ballistics. In fact he wrote three important books on integration, the first being Integration written in 1944 to provide a readable and clear introduction to Lebesgue integration for students. In 1953 he wrote Order preserving maps and integration processes which was [2]:... an outgrowth of his search for a mathematically correct setting in which to treat the divergent integrals in quantum physics. In 1974, the year he retired and was made Professor Emeritus at Virginia, McShane published Stochastic calculus and stochastic models which again reflected his work on the mathematical setting for quantum mechanics. Then in 1983 he published his third book on integration, Unified integration which provides [2]:... a theory of integrals with applications to physics. McShane was much involved with both the American Mathematical Society and with the Mathematical Association of America. He was American Mathematical Society Colloquium Lecturer in 1943, and later served the Society as President in 1959-60. He also served as President of the Mathematical Association of America in 1953-54. Among the honours which he received was the Award for Distinguished Service to Mathematics from the Mathematical Association of America in 1964. Another honour is the fact that the paper [1] belongs to a special volume of the SIAM Journal on Control and Optimization dedicated to him. He is described in [2] as:... a deeply cultured man, with a flair for languages and a great variety of other interests. He was widely known for his warmth, generosity, and modesty, and for his fund of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/McShane.html (2 of 3) [2/16/2002 11:22:35 PM]

McShane

humorous and interesting anecdotes. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Edward McShane (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1959 - 1960

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/McShane.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/McShane.html (3 of 3) [2/16/2002 11:22:35 PM]

Meissel

Daniel Friedrich Ernst Meissel Born: 31 July 1826 in Germany Died: 11 March 1895

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernst Meissel studied at the University of Berlin working under Jacobi. He also had contacts with Dirichlet. His doctorate is from Halle. He taught in a number of places, teaching in Kiel from 1871 until the end of his life. Meissel's mathematical work covers a number of areas. He worked on prime numbers giving the result that there are 50847478 primes less than 109. Lehmer showed, 70 years later, that this is 56 too few. In addition to other number theory work on Möbius inversion and the theory of partitions, Meissel wrote on Bessel functions, asymptotic analysis, refraction of light and the three body problem. His main skill was in numerical calculations and manipulation of complicated expressions. His work was based on the mathematics he learnt as a student and he appears not to have kept up with new developments. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Ernst Meissel

Mathematicians born in the same country

Other Web sites

Steve Finch (Hadamard-de la Vallée Poussin or Meissel-Mertens constants)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Meissel.html (1 of 2) [2/16/2002 11:22:37 PM]

Meissel

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Meissel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Meissel.html (2 of 2) [2/16/2002 11:22:37 PM]

Mellin

Robert Hjalmar Mellin Born: 1854 in Liminka, Northern Ostrobothnia, Finland Died: 1933

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hjalmar Mellin, the son of a clergyman, was born in Liminka, northern Ostrobothnia, in Finland in 1854. He grew up and received his schooling in Hämeenlinna (about 100 km north of Helsinki) and undertook his university studies in Helsinki, where his teacher was the Swedish mathematician G. Mittag-Leffler. In the autumn of 1881 Mellin defended his doctoral dissertation on algebraic functions of a single complex variable. He made two sojourns in Berlin in 1881 and 1882 to study under K. Weierstrass and in 1883-84 he returned to continue his studies with Mittag-Leffler in Stockholm. Mellin was appointed as a docent at the University of Stockholm from 1884-91 but never actually gave any lectures. Also in 1884 he was appointed a senior lecturer in mathematics at the recently founded Polytechnic Institute which was later (in 1908) to become the Technical University of Finland. In 1901 Mellin withdrew his application for the vacant chair of mathematics at the University of Helsinki in favour of his illustrious (and younger) fellow countryman E. Lindelöf (1870-1946). During the period 1904-07 Mellin was Director of the Polytechnic Institute and in 1908 he became the first professor of mathematics at the new university. He remained at the university for a total of 42 years, retiring in 1926 at the age of 72. With regard to the ever-burning language question, Mellin was a fervent fennoman with an apparently fiery temperament. It must be recalled, at this juncture, that Finland had for a long time been part of the kingdom of Sweden and had consequently been subjected to its language and culture. (After the Napoleonic wars Finland became an autonomous Grand Duchy under Russia, to finally emerge as an independent republic in the aftermath of the First World War.) Mellin was one of the founders of the Finnish Academy of Sciences in 1908 as a purely Finnish alternative to the predominantly Swedish-speaking Society of Sciences. From 1908 until his death in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mellin.html (1 of 2) [2/16/2002 11:22:39 PM]

Mellin

1933, at the age of 79, he represented his country on the editorial board of Acta Mathematica. Mellin's research work was principally in the area of the theory of functions which resulted from the influence of his teachers Mittag-Leffler and Weierstrass. He studied the transform which now bears his name and established its reciprocal properties. He applied this technique systematically in a long series of papers to the study of the gamma function, hypergeometric functions, Dirichlet series, the Riemann zeta function and related number-theoretic functions. He also extended his transform to several variables and applied it to the solution of partial differential equations. The use of the inverse form of the transform, expressed as an integral parallel to the imaginary axis of the variable of integration, was developed by Mellin as a powerful tool for the generation of asymptotic expansions. In this theory, he included the possibility of high-order poles (thereby leading to the inclusion of logarithmic terms in the expansion) and to several sequences of poles yielding sums of asymptotic expansions of very general form. During the last decade of his life Mellin was, rather curiously for an analyst, preoccupied by Einstein's theory of relativity and he wrote no less than ten papers on this topic. In these papers, where he was largely concerned with general philosophical problems of time and space, he adopted a quixotic standpoint in his attempt to refute the theory as being logically untenable. Article by: R Paris, University of Abertay, Scotland Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mellin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mellin.html (2 of 2) [2/16/2002 11:22:39 PM]

Menabrea

Luigi Federico Menabrea Born: 4 Sept 1809 in Chambéry, Savoy, France Died: 24 May 1896 in St Cassin (near Chambéry), France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Luigi Menabrea studied engineering and mathematics at the University of Turin, then became an engineer in the army. He became professor of mechanics at Turin and, in 1842, published a paper extending ideas relating to Babbage's mechanical calculator. Menabrea began a political career which saw him become Italian Premier and Foreign Minister in 1867. During this period of politics he still did excellent scientific work, giving the first precise formulation of methods of structual analysis based on the principle of virtual work. He also studied elasticity and the principle of least work. He published, jointly with J L F Bertrand, the first correct proof of this principle in 1870. Castigliano, with whom Menabrea was in dispute regarding this principle, became better known for the concepts of work and energy in analytical mechanics. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Some pages from publications

Title page of A sketch of the analytical engine invented by Charles Babbage (1843)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menabrea.html (1 of 2) [2/16/2002 11:22:41 PM]

Menabrea

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Menabrea.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menabrea.html (2 of 2) [2/16/2002 11:22:41 PM]

Menaechmus

Menaechmus Born: about 380 BC in Alopeconnesus, Asia Minor (now Turkey) Died: about 320 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Menaechmus is mentioned by Proclus who tells us that he was a pupil of Eudoxus in the following quote (see for example [3]):Amyclas of Heraclea, one of the associates of Plato, and Menaechmus, a pupil of Eudoxus who had studied with Plato, and his brother Dinostratus made the whole of geometry still more perfect. There is another reference in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Menaechmus was (see for example [1]):... a Platonic philosopher of Alopeconnesus, or according to some of Proconnesus, who wrote works of philosophy and three books on Plato's Republic... Alopeconnesus and Proconnesus are quite close, the first in Thrace and the second in the sea of Marmara, and both are not far from Cyzicus where Menaechmus's teacher Eudoxus worked. The dates for Menaechmus are consistent with his being a pupil of Eudoxus but also they are consistent with an anecdote told by Stobaeus writing in the 5th century AD. Stobaeus tells the rather familiar story which has also been told of other mathematicians such as Euclid, saying that Alexander the Great asked Menaechmus to show him an easy way to learn geometry to which Menaechmus replied (see for example [1]):O king, for travelling through the country there are private roads and royal roads, but in geometry there is one road for all. Some have inferred from this (see for example [4]) that Menaechmus acted as a tutor to Alexander the Great, and indeed this is not impossible to imagine since as Allman suggests Aristotle may have provided the link between the two. There is also an implication in the writings of Proclus that Menaechmus was the head of a School and this is argued convincingly by Allman in [4]. If indeed this is the case Allman argues that the School in question was the one on Cyzicus where Eudoxus had taught before him. Menaechmus is famed for his discovery of the conic sections and he was the first to show that ellipses, parabolas, and hyperbolas are obtained by cutting a cone in a plane not parallel to the base. It has generally been thought that Menaechmus did not invent the words 'parabola' and 'hyperbola', but that these were invented by Apollonius later. However recent evidence in Diocles' On burning mirrors

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menaechmus.html (1 of 3) [2/16/2002 11:22:42 PM]

Menaechmus

discovered in Arabic translation in the 1970s, led G J Toomer to claim that both the names 'parabola' and 'hyperbola' are older than Apollonius. Menaechmus made his discoveries on conic sections while he was attempting to solve the problem of duplicating the cube. In fact the specific problem which he set out to solve was to find two mean proportionals between two straight lines. This he achieved and therefore solved the problem of the duplicating the cube using these conic sections. Menaechmus's solution is described by Eutocius in his commentary to Archimedes' On the sphere and cylinder. Suppose that we are given a, b and we want to find two mean proportionals x, y between them. Then a : x = x : y = y : b so, doing a piece of modern mathematics, a/x = x/y so x2 = ay, and a/x = y/b so xy = ab. We now see that the values of x and y are found from the intersection of the parabola x2 = ay and the rectangular hyperbola xy = ab. Of course we must emphasis that this in no way indicates the way that Menaechmus solved the problem but it does show in modern terms how the parabola and hyperbola enter into the solution to the problem. Immediately following this solution, Eutocius gives a second solution. Again a piece of modern mathematics illustrates it: a/x = x/y so x2 = ay, and x/y = y/b so y2 = bx. We now see that the values of x and y are found from the intersection of the two parabolas x2 = ay and y2 = bx. [1], [3] and [4] all consider a problem associated with these solutions. Plutarch says that Plato disapproved of Menaechmus's solution using mechanical devices which, he believed, debased the study of geometry which he regarded as the highest achievement of the human mind. However, the solution described above which follows Eutocius does not seem to involve mechanical devices. Experts have discussed whether Menaechmus might have used a mechanical device to draw his curves. Allman [4] suggests that Menaechmus might have drawn the curves by finding many points on them and that this might be considered as a mechanical device. The solution proposed to this question in [1], however, seems particularly attractive. What has come to be known as Plato's solution to the problem of duplicating the cube is widely recognised as not due to Plato since it involves a mechanical instrument. Heath [3] writes:... it seems probable that someone who had Menaechmus's second solution before him worked to show how the same representation of the four straight lines could be got by a mechanical construction as an alternative to the use of conics. The suggestion made in [1] is that the 'someone' of this quote was Menaechmus himself. Other references to Menaechmus include one by Theon of Smyrna who suggests that he was a supporter of Eudoxus's theory of the heavenly bodies based on concentric spheres. In fact Theon of Smyrna claims that Menaechmus developed the theory further by adding further spheres. There have been conjectures made as to where this information was written down by Menaechmus so that it was available to Theon of Smyrna. One conjecture is that it appeared in Menaechmus's commentaries on Plato's Republic referred

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menaechmus.html (2 of 3) [2/16/2002 11:22:42 PM]

Menaechmus

to in the quote above from the Suda Lexicon. Proclus writes about Menaechmus saying that he studied the structure of mathematics [4]:... he discussed for instance the difference between the broader meaning of the word element (in which any proposition leading to another may be said to be an element of it) and the stricter meaning of something simple and fundamental standing to consequences drawn from it in the relation of a principle, which is capable of being universally applied and enters into the proof of all manner of propositions. Another matter relating to the structure of mathematics which Menaechmus discussed was the distinction between theorems and problems. Although many had claimed that the two were different, Menaechmus on the other hand claimed that there was no fundamental distinction. Both are problems, he claimed, but in the usage of the terms they are directed towards different objects. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

Doubling the cube

Other references in MacTutor

1. Conic sections 2. Menaechmus's construction for the duplication of the cube

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Menaechmus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menaechmus.html (3 of 3) [2/16/2002 11:22:42 PM]

Menelaus

Menelaus of Alexandria Born: about 70 in (possibly) Alexandria, Egypt Died: about 130 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Although we know little of Menelaus of Alexandria's life Ptolemy records astronomical observations made by Menelaus in Rome on the 14th January in the year 98. These observation included that of the occultation of the star Beta Scorpii by the moon. He also makes an appearance in a work by Plutarch who describes a conversation between Menelaus and Lucius in which Lucius apologises to Menelaus for doubting the fact that light, when reflected, obeys the law that the angle of incidence equals the angle of reflection. Lucius says (see for example [1]):In your presence, my dear Menelaus, I am ashamed to confute a mathematical proposition, the foundation, as it were, on which rests the subject of catoptrics. Yet it must be said that the proposition, "All reflection occurs at equal angles" is neither self evident nor an admitted fact. This conversation is supposed to have taken place in Rome probably quite a long time after 75 AD, and indeed if our guess that Menelaus was born in 70 AD is close to being correct then it must have been many years after 75 AD. Very little else is known of Menelaus's life, except that he is called Menelaus of Alexandria by both Pappus and Proclus. All we can deduce from this is that he spent some time in both Rome and Alexandria but the most likely scenario is that he lived in Alexandria as a young man, possibly being born there, and later moved to Rome. An Arab register of mathematicians composed in the 10th century records Menelaus as follows (see [1]):He lived before Ptolemy, since the latter makes mention of him. He composed: "The Book of Spherical Propositions", "On the Knowledge of the Weights and Distribution of Different Bodies" ... Three books on the "Elements of Geometry", edited by Thabit ibn Qurra, and "The Book on the Triangle". Some of these have been translated into Arabic. Of Menelaus's many books only Sphaerica has survived. It deals with spherical triangles and their application to astronomy. He was the first to write down the definition of a spherical triangle giving the definition at the beginning of Book I:A spherical triangle is the space included by arcs of great circles on the surface of a sphere ... these arcs are always less than a semicircle. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menelaus.html (1 of 4) [2/16/2002 11:22:43 PM]

Menelaus

In Book I of Sphaerica he set up the basis for treating spherical triangles as Euclid treated plane triangles. He used arcs of great circles instead of arcs of parallel circles on the sphere. This marks a turning point in the development of spherical trigonometry. However, Menelaus seems unhappy with the method of proof by reductio ad absurdum which Euclid frequently uses. Menelaus avoids this way of proving theorems and, as a consequence, he gives proofs of some of the theorems where Euclid's proof could be easily adapted to the case of spherical triangles by quite different methods. It is also worth commenting that [3]:In some respects his treatment is more complete than Euclid's treatment of the analogous plane case. Book 2 applies spherical geometry to astronomy. It largely follows the propositions given by Theodosius in his Sphaerica but Menelaus give considerably better proofs. Book 3 deals with spherical trigonometry and includes Menelaus's theorem. For plane triangles the theorem was known before Menelaus:... if a straight line crosses the three sides of a triangle (one of the sides is extended beyond the vertices of the triangle), then the product of three of the nonadjacent line segments thus formed is equal to the product of the three remaining line segments of the triangle. Menelaus produced a spherical triangle version of this theorem which is today also called Menelaus's Theorem, and it appears as the first proposition in Book III. The statement is given in terms of intersecting great circles on a sphere. Many translations and commentaries of Menelaus Sphaerica were made by the Arabs. Some of these survive but differ considerably and make an accurate reconstruction of the original quite difficult. On the other hand we do know that some of the works are commentaries on earlier commentaries so it is easy to see how the original becomes obscured. There are detailed discussions of these Arabic translations in [6], [9], and [10]. There are other works by Menelaus which are mentioned by Arab authors but which have been lost both in the Greek and in their Arabic translations. We gave a quotation above from the 10th century Arab register which records a book called Elements of Geometry which was in three volumes and was translated into Arabic by Thabit ibn Qurra. It also records another work by Menelaus was entitled Book on Triangles and although this has not survived fragments of an Arabic translation have been found. Proclus referred to a geometrical result of Menelaus which does not appear in the work which has survived and it is thought that it must come from one of the texts just mentioned. This was a direct proof of a theorem in Euclid's Elements and given Menelaus's dislike for reductio ad absurdum in his surviving works this seems a natural line for him to follow. The new proof which Proclus attributes to Menelaus is of the theorem (in Heath's translation of Euclid):If two triangles have the two sides equal to two sides respectively, but have the base of one greater than the base of the other, it will also have the angle contained by the equal straight lines of the first greater than that of the other. Another Arab reference to Menelaus suggests that his Elements of Geometry contained Archytas's solution of the problem of duplicating the cube. Paul Tannery in [8] argues that this make it likely that a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menelaus.html (2 of 4) [2/16/2002 11:22:43 PM]

Menelaus

curve which it is claimed by Pappus that Menelaus discussed at length was the Viviani's curve of double curvature. Bulmer-Thomas in [1] comments that:It is an attractive conjecture but incapable of proof on present evidence. Menelaus is believed by a number of Arab writers to have written a text on mechanics. It is claimed that the text studied balances studied by Archimedes and those devised by Menelaus himself. In particular Menelaus was interested in specific gravities and analysing alloys.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Trisecting an angle 2. Arabic mathematics : forgotten brilliance? 3. The trigonometric functions 4. An overview of Indian mathematics

Other references in MacTutor

1. Menelaus's theorem 2. Chronology: 1AD to 500

Honours awarded to Menelaus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Menelaus and Rimae Menelaus

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menelaus.html (3 of 4) [2/16/2002 11:22:43 PM]

Menelaus

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Menelaus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menelaus.html (4 of 4) [2/16/2002 11:22:43 PM]

Menger

Karl Menger Born: 13 Jan 1902 in Vienna, Austria Died: 5 Oct 1985 in Chicago, Illinois, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karl Menger attended the Doblinger Gymnasium in Vienna (1913-1920) where one of his fellow students was Pauli. Karl entered the University of Vienna in 1920 to study physics. However Hahn became a lecturer in Vienna in 1921 and Menger attended a course he gave on What's new concerning the concept of a curve. Menger became interested in the topic and was encouraged by Hahn to work on the topic. Menger's work led him to a definition of dimension independently of Urysohn, however Urysohn had died in a drowning accident before he could publish his work and Menger was not aware of it. After a severe lung disease which forced Menger to spend more than a year in a sanatorium, he returned with important papers he had written on dimension while in the sanatorium and completed his doctorate in 1924. In 1925 Menger was invited by Brouwer to take up a post in the University of Amsterdam where he spent two years working with Brouwer. In 1927 Menger was invited by Hahn to accept the chair of geometry at the University of Vienna when Reidemeister left for Königsberg. Menger was not sorry to leave Amsterdam since he had become involved in a priority dispute with Brouwer and they were not on the best of terms. In 1938, as a result of the political situation in Austria, he resigned his chair and accepted a post in the USA at the University of Notre Dame. Karl Signund writes:After the war, the reconstruction of the bombed-out State Opera was accorded highest priority by democratic new Austria. Men like ... Menger, however, were politely told that the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menger.html (1 of 2) [2/16/2002 11:22:45 PM]

Menger

University of Vienna had no place for them. At Notre Dame Menger organised a Mathematical Colloquium based on that at Vienna. He arranged a visit by Gödel to Notre Dame but failed to persuade him to accept a post there. However after the war began to affect the USA in 1941, academic life was disrupted and Menger's Mathematical Colloquium failed to become influential as the Vienna Circle had been. Around this time Menger's interests in mathematics broadened and he began to work on hyperbolic geometry, probabilistic geometry and the algebra of functions. Menger's work on geometry failed to have the impact that his work on dimension theory had. This is almost certainly because geometry, at this time, was a rather neglected area of mathematics, particularly in the USA. In 1948 Menger went to the Illinois Institute of Technology and he was to remain in Chicago for the rest of his life. In [1] his interests are described as follows:He had a great love of music. ... he built up a notable collection of decorative tiles from all over the world. ... he ate meat sparingly, particularly in his last years. But he was always glad to sample cuisines, from Cuban to Ethiopian, that were new to him. He liked baked apples. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Menger.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menger.html (2 of 2) [2/16/2002 11:22:45 PM]

Mengoli

Pietro Mengoli Born: 1626 in Bologna, Italy Died: 1686 in Bologna, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pietro Mengoli was taught mathematics by Cavalieri at Bologna the studied for a doctorate in philosophy which was awarded by Bologna in 1650. Three years later he obtained a second doctorate in civil and canon law, also from Bologna. When Cavalieri died, Mengoli was appointed to his chair at the University of Bologna. Mengoli was professor of arithmetic (1648-1649), professor of mechanics (1649-1668) and professor of mathematics (1668-1686). In addition to these posts he was also a parish priest in Bologna from 1660. Mengoli used infinite series to good effect. He showed that the harmonic series does not converge and that the harmonic series with alternating signs converges to log(2). This series was also investigated by Nicolaus Mercator. Mengoli wrote Novae quadraturae arithmeticae (1650) on infinite series, Geometriae speciosae elementa (1659) on limits, and Circolo (1672) where he found an infinite product expansion for /2. Other work by Mengoli included work on astronomy, work on refraction in the atmosphere and a book Speculazioni di musica (1670) on the theory of music. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) Mathematicians born in the same country Cross-references to History Topics

The rise of the calculus

Other references in MacTutor

Chronology: 1650 to 1675

Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mengoli.html (1 of 2) [2/16/2002 11:22:49 PM]

Mengoli

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mengoli.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mengoli.html (2 of 2) [2/16/2002 11:22:49 PM]

Menshov

Dmitrii Evgenevich Menshov Born: 18 April 1892 in Moscow, Russia Died: 25 Nov 1988

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dmitrii Evgenevich Menshov's father, Evgenii Titovich Menshov, was a medical doctor who worked in the New Ekaterininskii Hospital and the Lazarevskii Institute of Oriental Languages. His mother, Alexandra Nikolaevna Tatishcheva, was an important early influence in Menshov's education. She was a highly educated woman who, as well as educating her son, also sometimes gave French lessons to other children. In 1904, at the age of 12 years, Menshov began his secondary schooling. He attended the gymnasium section of the Lazarevskii Institute of Oriental Languages where his father acted as school doctor. Influenced by his mother's tuition in foreign languages, Menshov's first love at school was indeed for languages. He went on to study French, German, English, Latin, and Armenian at school. However, Menshov had an outstanding mathematics teacher and, like many children who are influenced by an outstanding teacher, Menshov began to show a strong interest in mathematics from about the age of 13. He also was strongly attracted to physics so, with wide interests across many subjects, he graduated from the school in 1911 with the gold medal for outstanding achievement. After leaving school, Menshov sat the entrance examination for the Moscow Engineering College and began his studies there in the autumn of 1911. However he only studied there for six months before deciding to leave and work on his own on learning advanced mathematics. Then in the autumn of 1912 Menshov entered the Department of Physics and Mathematics at Moscow University. There he attended lectures by Egorov, Lakhtin, Andreev and he took his first course on functions of a real variable given by Byushgens. Perhaps the most significant event for Menshov, however, was that Luzin returned from Göttingen to Moscow in the autumn of 1914 and began to lecture on functions of a real variable. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menshov.html (1 of 4) [2/16/2002 11:22:51 PM]

Menshov

Menshov attended Luzin's lecture course, and when Luzin posed the open problem of whether the Denjoy integral and the Borel integral were equivalent, he was able to solve the problem. The Denjoy integral is the more general of the two and Menshov showed that this was the case. He showed Luzin his solution to the problem that Luzin had just posed and before the end of 1914 the two had begun a firm mathematical friendship. Menshov's discovery, made while still an undergraduate, became his first publication. It appeared as the paper The relationship between the definitions of the Denjoy and Borel integrals in 1916. Luzin quickly established a School of Mathematics at Moscow University and Menshov became one of his fist research students along with P S Aleksandrov, M Ya Suslin, and A Ya Khinchin. Menshov's first degree was awarded in 1916 for the thesis which he wrote on The Riemann theory of trigonometric series which was examined by Egorov and Luzin. However, only three weeks after he graduated, Menshov discovered one of his most fundamental results on the uniqueness problem for trigonometric series. Let us describe this result. Consider the trigonometric series a0/2 +

(ancos nx + bnsin nx).

Cantor had proved that if this series converges to 0 for all x in [0, 2 ] - E, for a countable set E, then an = bn = 0 for all n. Vallée Poussin had proved that if the above series converged to a finite Lebesgue integrable function f(x) then the given series is the Fourier series of f(x). It was expected that Vallée Poussin's result would still hold if the countable set E was replaced by a set E of measure zero. The remarkable, and unexpected, result that Menshov discovered in 1916 was that this was not so, for he constructed a trigonometric series which converges to 0 for all x in [0, 2 ] - E, for a set E of measure zero, yet not all the coefficients of the trigonometric series are zero. By the end of 1918 Menshov had been awarded his Master's degree and he went to Ivanovo north-east of Moscow, which at that time was the temporary capital of the revolutionary government, but he soon moved to Nizhnii-Novgorod where he was appointed as a professor at the University. He taught at Nizhnii-Novgorod during 1919 and early 1920 but he returned to Ivanovo in May 1920 where he was appointed as a professor at the Ivanovo Pedagogic Institute. In addition to this appointment he also taught at the Polytechnic Institute at Ivanovo from January 1921. At this time Luzin and other members of his research school were in Ivanovo so Menshov was certainly in the mainstream of the exciting mathematics that was being developed. In the autumn of 1922 Menshov returned to Moscow and began teaching at the University. He also taught for a few years at the Moscow Institute of Forest Technology. It may have been noticed by an attentive reader that we have still not noted that Menshov being awarded a doctorate (equivalent to the habilitation or D.Sc.). In fact he never submitted a thesis for a doctorate but, despite this, he was awarded the doctorate in 1935 since ([1] or [2]):... he had already been acknowledged as one of the world's most outstanding specialists on the theory of functions of a real and a complex variable. Together with the award of the doctorate came Menshov's appointment to a professorship at Moscow University.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menshov.html (2 of 4) [2/16/2002 11:22:51 PM]

Menshov

In 1933 a new chair of Analysis and Theory of Functions was created at Moscow University and Lavrentev appointed. In 1938 the Faculty of Mechanics and Mathematics at Moscow University founded two chairs, the chair of the Theory of Functions and the chair of Functional Analysis. Privalov held this first chair up to 1941 but then, on Privalov's early death in that year, Menshov was appointed to the chair of the Theory of Functions. Lusternik held the chair of Functional Analysis from 1938. In 1943 these two chairs were combined and the Department of Theory of Functions and Functional Analysis was created with Menshov as its head. Menshov also worked a the Steklov Mathematical Institute of the Academy of Sciences of the USSR from 1934 to 1941 and then again from 1947. Menshov's mathematical interests and the style of his mathematics is described in ([3] and [4]):His scientific interests relate principally to the theory of trigonometric series, the theory of orthogonal series and the problem of monogenity of functions of a complex variable. He published more than eighty papers on these subjects, which have had an exceptionally great effect on the development of the whole theory of functions. Menshov does not belong among the ranks of those mathematicians who undertake the solution of comparatively easy problems, or who continue the research of other authors on a course that has already been indicated. A characteristic feature of scientific activity is that in his work on the theory of functions he solved a number of extremely difficult key problems which had baffled many eminent mathematicians. For his work on the representation of functions by trigonometric series, Menshov was awarded a State Prize in 1951. He was then elected a Corresponding Member of the Academy of Sciences of the USSR in 1953. In 1958 Menshov attended the International Congress of Mathematicians in Edinburgh and he was invited to address the Congress with his paper On the convergence of trigonometric series. The first of the two pictures of Menshov which we have given was taken while he was at the Congress in Edinburgh, Scotland in 1958. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menshov.html (3 of 4) [2/16/2002 11:22:51 PM]

Menshov

JOC/EFR January 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Menshov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Menshov.html (4 of 4) [2/16/2002 11:22:51 PM]

Meray

Hugues Charles Robert Méray Born: 12 Nov 1835 in Chalon-sur-Saône, France Died: 2 Feb 1911 in Dijon, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Méray studied at the Ecole Normale Supérieure in Paris. He began his studies in 1854 when he was eighteen years old, and graduated in 1857. After graduating, Méray taught at the Lycée of St Quentin for two years but then left teaching for seven years during which time he lived in a small village near Chalon-sur-Saône. Following these seven years when he chose not to work, Méray took up a teaching position again in 1866, this time lecturing at the University of Lyon for a year before being appointed as Professor of Mathematics at the University of Dijon. He would continue to work in Dijon for the rest of his career. Robinson writes in [1]:In his time he was a respected but not a leading mathematician. Méray is remembered for having anticipated, clearly and with only minor differences of style, Cantor's theory of irrational numbers, one of the main steps in the arithmetisation of analysis. So here we have a case of a mathematician who produced work which might have made him one of the leading mathematicians in the world. However, as happened many times throughout history, Méray was unlucky for the genius of his work was not recognised at the time. Others (we give details below) published the same ideas and it would be their work rather than that of Méray which influenced the direction of mathematics. All we can do now is to give Méray the credit he deserves for his remarkable work, even if fate did not allow Méray a role of importance in the development of the subject. In 1869 Méray was the first to publish an arithmetical theory of irrational numbers in his paper Remarques sur la nature des quantités définies par la condition de servir de limites à des variables données. Others such as Ohm (1829), Bolzano (1835) and Hamilton (1833) had published work on irrational numbers but none of these earlier authors gave a rigorous account. Méray's is the earliest coherent and rigorous theory of the irrational numbers to appear in print. His work not being influenced by Weierstrass (whose work was unpublished) or Dedekind who only published his theories after Cantor's important paper appeared in 1872. Méray followed Lagrange's earlier work but gave rigorous proofs of what Lagrange had only conjectured. Méray published a second important work in 1872. This work is a book Nouveau précis d'analyse infinitésimale which aims to present the theory of functions of a complex variable using power series. It http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Meray.html (1 of 2) [2/16/2002 11:22:52 PM]

Meray

is another rigorous work and in fact between 1872 and 1894 Méray produced a series of papers which remove geometric considerations from analytic proofs. Méray's work consistently follows Lagrange in basing the whole of analysis on the concept of functions written as Taylor series. We have noted above that Méray's work had no real influence on the development of mathematics despite being almost exactly the same as the work which would transform the direction of mathematics. It was not that Méray's work went unnoticed. His 1872 book Nouveau précis d'analyse infinitésimale was reviewed by Hermann Laurent in 1873. Hermann Laurent, in his review, ignored Méray's irrational numbers [1]:... while gently chiding the author for using too narrow a notion of a function and for being too rigorous in a supposed textbook. At that time there was not in France - as there was in Germany - a sufficient appreciation of the kind of problem considered by Méray, and not until much later was it realised that he had produced a theory of a kind that had added lustre to the names of some of the greatest mathematicians of the period. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1870 to 1880

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Meray.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Meray.html (2 of 2) [2/16/2002 11:22:52 PM]

Mercator_Gerardus

Gerardus Mercator Born: 5 March 1512 in Rupelmonde, Flanders (now Belgium) Died: 2 Dec 1594 in Duisburg, Duchy of Cleve (now Germany)

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gerardus Mercator's original name was Gerard de Cremere. He was educated at 'sHertogenbosch in the Netherlands, then, in 1530, he entered the University of Louvain where he studied the humanities and philosophy. He graduated from Louvain with an M.A. in 1532. After graduating, Mercator began to have worries on how to reconcile the account of the origin of the universe given in the Bible with that given by Aristotle. He travelled to a number of places while going through this personal crisis including Antwerp and Mechelen. His travels did little for his religious worries but gave him a deep interest in geography. Mercator returned to Louvain where he now studied mathematics under Gemma Frisius. He also learnt about applications of mathematics to geography and astronomy. He learnt to be an engraver and instrument maker at this time from Gaspar à Myrica. In 1535-1536 Mercator working in Louvain with Myrica and with Frisius constructed a terrestrial globe. In 1537 they constructed a globe of the stars. Mercator produced a map of Palestine (1537), a map of the world with a new projection (1538) and a map of Flanders (1540). Mercator was charged with heresy in 1544. This was partly due to his Protestant beliefs, partly due to the fact that he travelled so widely to acquire data for his maps that suspicions were aroused. He spent seven months in prison. He was released, mainly due to strong support from the University of Louvain, and in 1552 he moved to Duisburg where he opened a cartographic workshop. In Duisburg Mercator completed a project to produce a new map of Europe (1554) and he taught

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mercator_Gerardus.html (1 of 2) [2/16/2002 11:22:54 PM]

Mercator_Gerardus

mathematics there from 1559 to 1562. Further maps followed, one of Lorraine (1564) and one of the British Isles (1564). He was appointed Court Cosmographer to Duke Wilhelm of Cleve, also in 1564. During this period he began to perfect a new map projection for which he is best remembered. The map projection that bears his name he first used in 1569. He is also the first to use the term 'atlas' for a collection of maps. He published corrected and updated versions of Ptolemy's maps in 1578 as the first part of his 'atlas'. His 'atlas' continued with a further series of maps of France, Germany and the Netherlands in 1585. Although the project was never completed Mercator did publish a further series in 1589 including maps to the Balkans (then called Sclavonia) and Greece. Some maps which were incomplete at his death were completed and published by his son in 1595. Mercator's break from the methods of Ptolemy was as important for geography as was Copernicus for astronomy. Article by: J J O'Connor and E F Robertson List of References (8 books/articles) A Poster of Gerardus Mercator

Mathematicians born in the same country

Some pages from publications

The frontispiece of Atlas, sive cosmographica meditationes (1630)

Honours awarded to Gerardus Mercator (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Mercator and Rupes Mercator

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mercator_Gerardus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mercator_Gerardus.html (2 of 2) [2/16/2002 11:22:54 PM]

Mercator_Nicolaus

Nicolaus Mercator Born: 1620 in Eutin, Schleswig-Holstein, Denmark (now Germany) Died: 14 Jan 1687 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nicolaus Mercator entered the University of Rostock in 1632. He received a degree in 1641, then went to Leiden for a short period. After his return to Rostock in 1642 he was appointed to a post in the university. In 1648 Nicolaus moved to the University of Copenhagen but, after working there for six years, he had to leave when the university was closed due to the plague. From this time on things were not too good for him. He went to England in 1660 attempting to bring in some money and did some private tutoring. He was recognised in England however since he was elected a Fellow of the Royal Society in 1666. This was on the strength of a pendulum clock which Mercator designed to work at sea and thus be used to determine longitude. He also made measurements of air pressure for the Royal Society. In 1682 he moved to France, this time with a specific position, namely to design the waterworks at Versailles. While he was working at Copenhagen, Mercator published a number of textbooks on spherical trigonometry, geography and astronomy. These were Trigonometria sphaericorum logarithmica (1651), Cosmographia (1651), and Astronomica sphaerica (1651). He published further works in astronomy while in England, for example Hypothesis astronomia nova (1664) and Institutiones astronomicae (1676). Mercator discovered the well known series, sometimes called Mercator's series, ln(1+x) = x - x2/2 + x3/3 - x4/4 + ... He published this in Logarithmotechnia 1668. This series was also investigated by Mengoli. There is some reason to confuse Nicolaus Mercator with Gerardus Mercator since Nicolaus also worked on Gerardus Mercator's map projection. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mercator_Nicolaus.html (1 of 2) [2/16/2002 11:22:55 PM]

Mercator_Nicolaus

Mathematicians born in the same country Other references in MacTutor

Chronology: 1650 to 1675

Honours awarded to Nicolaus Mercator (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1666

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mercator_Nicolaus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mercator_Nicolaus.html (2 of 2) [2/16/2002 11:22:55 PM]

Mercer

James Mercer Born: 15 Jan 1883 in Bootle, Liverpool, England Died: 21 Feb 1932 in England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James Mercer was educated in Liverpool, attending University College there before entering Trinity College Cambridge to study the mathematical Tripos. Among his contemporaries there were G N Watson and J E Littlewood while he was taught by Whitehead, Whittaker and Hardy. Now Hardy had just come onto the staff at Trinity and he acted as a private tutor to Mercer. In [2] Littlewood describes the Mathematical Tripos examinations which he and Mercer sat. Describing Part I of the Tripos, Littlewood writes:[The examination] consisted of 7 papers ('first four days') on comparatively elementary subjects, the riders, however, being quite stiff, followed a week later by another 7 ('second four days'). A pass on the first four days qualified for a degree, but the second four days carried double the marks, and since it was impossible to revise everything the leading candidates concentrated on the second four days ... On the problem paper [for the first four days] Mercer got 270 out of 760 for 18 questions (I got only 180). ... In the second four days (ignoring the problem paper) Mercer and I each got about 2050 out of 4500 (each about 330 out of 1340 in the 18 question problem paper). Mercer graduated in 1907 bracketed Senior Wrangler (first equal) with Littlewood. He was awarded a Smith's prize and, in 1909, he was elected a Fellow of Trinity. He returned to his home town of Liverpool for a while, being appointed as an Assistant Lecturer there, but soon after, when the opportunity came to return to Cambridge, he accepted a Fellowship and Lectureship in Christ's College. During the First World War Mercer served as a Naval Instructor. He saw active service at the Battle of Jutland, the only major encounter between the British and German fleets during the First World War. Fought off the coast of Denmark, it began on 31 May 1916. The battle was inconclusive but the British fleet suffered heavy losses. Mercer survived the battle and at the end of the war returned to his duties at Cambridge. Back in Cambridge he resumed his mathematical research, continuing his work on function theory. He was elected a Fellow of the Royal Society in 1922 but after this time his health, which had been poor for some time, began to fail. He took ill health retirement in 1926. Hobson [1] gives this overview of Mercer's mathematical achievements:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mercer.html (1 of 2) [2/16/2002 11:22:56 PM]

Mercer

Mercer was a mathematical analyst of originality and skill; he made noteworthy advances in the theory of integral equations, and especially in the theory of the expansion of arbitrary functions in series of orthogonal functions. His output of original work would no doubt have been much larger had he not been continually hampered by bad health. Mercer's theorem about the uniform convergence of eigenfunction expansions for kernels of operators appears in his 1909 paper Functions of positive and negative types and their connection with the theory of integral equations published in the Philosophical Transactions. There have been many papers written since then generalising Mercer's theorem to various other settings. For example Mathematical Reviews contains at least 65 papers studying such generalisations which have been published since 1980. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Mercer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mercer.html (2 of 2) [2/16/2002 11:22:56 PM]

Merrifield

Charles Watkins Merrifield Born: 1827 in England Died: 1884

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Merrifield trained as a lawyer but never practised. He became an Examiner at the Education Department in 1863 was elected to the Royal Society. He went to the Royal School of Naval Architecture in 1867. As an expert on tables and interpolation he served on the committee which was set up to evaluate Babbage's Analytic Machine. He was also active in promoting improvements in the teaching of geometry. Article by: J J O'Connor and E F Robertson

Mathematicians born in the same country Honours awarded to Charles Merrifield (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1863

London Maths Society President

1878 - 1880

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Merrifield.html (1 of 2) [2/16/2002 11:22:58 PM]

Merrifield

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Merrifield.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Merrifield.html (2 of 2) [2/16/2002 11:22:58 PM]

Merrill

Winifred Edgerton Merrill Born: 24 Sept 1862 in Ripon, Wisconsin, USA Died: 6 Sep 1951 in Fairfield, Connecticut, USA

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Winifred Edgerton Merrill made a vast impact on the male orientated world of mathematics. She left behind the Victorian ideal that a wellborn woman should stay at home, and went about continuing her education in mathematics to Ph.D. level. This was a fantastic achievement and Merrill became the first American woman to obtain a Ph.D. in mathematics. Her determination to obtain graduate education is an example that many have followed since. Merrill was born on 24 September 1862 in Ripon, Wisconsin. Nothing is known of her parents, but it can be assumed they were financially stable, as private tutors provided Merrill's early education. In 1883 she graduated with a B.A. degree from Wellesley College, Wellesley, Massachusetts. After working at Harvard, in 1884 Merrill applied, to Columbia University, in New York, to be allowed to study mathematics and astronomy. Up until this time Columbia University was an institute for men only and due to this was the case Merrill's initial request was refused. However, with the support and advice of Frederick A P Barnard, the 10th president of Columbia University who was a campaigner for women's education, Merrill visited each trustee individually to plead her case. She argued the points that to be able to study astronomy Merrill needed a telescope and only Columbia University had one and also that the Professor of Astronomy required an assistant at the time. At the next meeting of the trustees, they voted to allow Merrill to pursue her studies individually. Her work in the field of mathematical astronomy included in 1883 the computation of the orbit of a comet. On completion of the required credits, and having written an original thesis that:...dealt with geometric interpretations of multiple integrals and translations and relations of various systems of co-ordinates, Merrill applied to the trustees to be awarded a Ph.D. In 1886 Merrill became the first American woman to be awarded such a distinction in mathematics, which she earned cum laude. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Merrill.html (1 of 3) [2/16/2002 11:23:00 PM]

Merrill

She was offered a position as a professor of mathematics at Smith College in 1887, which she declined, as that year Merrill married Frederick James Hamilton Merrill, an 1885 graduate of Columbia's School of Mines who received his Ph.D. from Columbia in 1890. Frederick Merrill served as the New York State geologist from 1899 to 1904. Before his death in 1916, he had also been director of the New York State Museum. Marriage took Merrill from her pursuit of a scholarly career. However, in 1889 she became one of the committee of five people who drafted the proposals that resulted in the foundation of Barnard College at Columbia in the same year. Barnard College was New York's first secular institution to award women the liberal arts degree. Merrill withdrew from the campaign when her husband objected, to what he saw as the impropriety of committee meetings being held in men's offices in the city. Merrill taught mathematics at several institutions for a few years after her graduation from Columbia. In 1906 she founded the Oaksmere School for Girls, now located at Mamaroneck, New York. It was a school recognised for its high scholastic standards, a branch of this school was opened in Paris in 1912. After her retirement from the school, in 1926, Merrill moved to New York City. Whilst living there she wrote articles on education for publication in journals and became a popular speaker on educational topics. For some years Merrill served as an alumna trustee of Wellesley College. Frederick and Winifred Merrill had three children. One was a daughter who married Robert Russell Bennett of New York. Their younger son was Col. Edgerton Merrill. It was with her elder son, Hamilton Merrill, that Merrill lived during the last two years of her life. She died on the 6 September 1951 in Fairfield, Connecticut. On the 50th anniversary of Merrill's graduation from Wellesley, a portrait of her was presented to Columbia. It now hangs in Columbia's Philosophy Hall. The inscription beneath it symbolises her achievement in mathematics and her contribution to the further education of women, it reads:She opened the door. Article by: J J O'Connor and E F Robertson based on a project by Suzanne Davidson. A Reference (One book/article) Mathematicians born in the same country Other Web sites

Agnes Scott College

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Merrill.html (2 of 3) [2/16/2002 11:23:00 PM]

Merrill

JOC/EFR May 2001

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Merrill.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Merrill.html (3 of 3) [2/16/2002 11:23:00 PM]

Mersenne

Marin Mersenne Born: 8 Sept 1588 in Oize in Maine, France Died: 1 Sept 1648 in Paris, France

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Marin Mersenne attended school at the College of Mans, then, from 1604 spent five years in the Jesuit College at La Fleche. From 1609 to 1611 he studied theology at the Sorbonne. Mersenne joined the religious order of the Minims in 1611. The name of the order comes since the Minims regard themselves as the least (minimi) of all the religious; they devote themselves to prayer, study and scholarship. Mersenne continued his education within the order at Nigeon and then at Meaux. He returned to Paris where in 1612 he became a priest at the Place Royale. He taught philosophy at the Minim convent at Nevers from 1614 to 1618. In 1619 he returned again to Paris to the Minims de l'Annociade near Place Royale. His cell in Paris became a meeting place for Fermat, Pascal, Gassendi, Roberval, Beaugrand and others who later became the core of the French Academy. Mersenne corresponded with other eminent mathematicians and he played a major role in communicating mathematical knowledge throughout Europe at a time when there were no scientific journals. Mersenne investigated prime numbers and he tried to find a formula that would represent all primes. Although he failed in this, his work on numbers of the form 2p - 1, p prime has been of continuing interest in the investigation of large primes. It is easy to prove that if the number n = 2p-1 is prime then p must be a prime. In 1644 Mersenne claimed that n is prime if p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 but composite for the other 44 primes

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mersenne.html (1 of 3) [2/16/2002 11:23:02 PM]

Mersenne

smaller than 257. Over the years it has been found that Mersenne was wrong about 5 of the primes less than or equal to 257 (he claimed two that did not lead to a prime (67 and 257) and missed 3 that did: 61, 89, 107). Mersenne defended Descartes and Galileo against theological criticism and struggled to expose the pseudo sciences of alchemy and astrology. He continued some of Galileo's work in acoustics and stimulated some of Galileo's own later discoveries. He proposed the use of the pendulum as a timing device to Huygens, thus inspiring the first pendulum clock. In 1633 Mersenne published Traité des mouvements, and in 1634 he published Les Méchanique de Galilée which was a version of Galileo's lectures on mechanics. He translated parts of Galileo's Dialogo into French and in 1639 he published a transation of Galileo's Discorsi. It is through Mersenne that Galileo's work became known outside Italy. Two important publications in mathematical physics were L'Harmonie Universelle (1636) and Cogitata Physico-Mathematica (1644). Mersenne also wrote Traité d'harmonie universelle (1627), a work on music, musical instruments and acoustics. After his death letters in his cell were found from 78 different correspondents including Fermat, Huygens, Pell, Galileo and Torricelli. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (25 books/articles) A Poster of Marin Mersenne Cross-references to History Topics

Mathematicians born in the same country 1. Perfect numbers 2. Prime numbers

Other references in MacTutor

Chronology: 1600 to 1625

Honours awarded to Marin Mersenne (Click a link below for the full list of mathematicians honoured in this way) Lunar features Other Web sites

Crater Mersenius 1. The Prime Pages (Mersenne numbers) 2. Luke Welsh (A "home page" for Mersenne) 3. The Catholic Encyclopedia 4. The Galileo Project 5. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mersenne.html (2 of 3) [2/16/2002 11:23:02 PM]

Mersenne

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mersenne.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mersenne.html (3 of 3) [2/16/2002 11:23:02 PM]

Mertens

Franz Mertens Born: 20 March 1840 in Schroda, Posen, Prussia (now Sroda, Poland) Died: 5 March 1927 in Vienna, Austria

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Franz Mertens studied at Berlin with Kronecker and Kummer. In 1865 he obtained a doctorate with a dissertation on potential theory. He worked first in Krakóv, then in Graz, then from 1894 in Vienna. Mertens was a number theorist who is best remembered for his elementary proof of the Dirichlet theorem which appears in most modern textbooks. He worked on analytic number theory and algebra. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other Web sites

Steve Finch (Hadamard-de la Vallée Poussin or Meissel-Mertens constants)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mertens.html (1 of 2) [2/16/2002 11:23:03 PM]

Mertens

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mertens.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mertens.html (2 of 2) [2/16/2002 11:23:03 PM]

Meshchersky

Ivan Vsevolodovich Meshchersky Born: 10 Aug 1859 in Arkhangelsk, Russia Died: 7 Jan 1935 in Leningrad, USSR (now St Petersburg, Russia) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ivan Meshchersky taught at St Petersburg for 58 years. He studied mechanics, in particular the motion of bodies of variable mass. He applied this to comets being the first to study the inverse problem of determining the loss of mass from a knowledge of the orbit and the farces acting. His work formed the basis for the rocket technology developed after 1945. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Ivan V Meshchersky (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Meshcherskiy

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Meshchersky.html (1 of 2) [2/16/2002 11:23:04 PM]

Meshchersky

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Meshchersky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Meshchersky.html (2 of 2) [2/16/2002 11:23:04 PM]

Meyer

Freidrich Wilhelm Franz Meyer Born: 2 Sept 1856 in Magdeburg, Germany Died: 11 June 1934 in Königsberg, Germany (now Kaliningrad, Russia)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Meyer studied at Berlin taking courses by Kummer, Weierstrass and Kronecker, then he taught at Königsberg for 27 years. He studied algebraic geometry, algebraic curves and invariant theory. Together with Heinrich Weber and Klein, he founded the Encyclopädie der Mathematisches Wissenschaften which was a massive project producing 20 volumes between 1900 and 1930. Meyer wrote many articles and was an exceptional editor since he had written 2000 reviews for Fortschritte der Mathematik. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Wilhelm Meyer

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Meyer.html (1 of 2) [2/16/2002 11:23:06 PM]

Meyer

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Meyer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Meyer.html (2 of 2) [2/16/2002 11:23:06 PM]

Miller

George Abram Miller Born: 31 July 1863 in Lynnville, Pennsylvania, USA Died: 10 Feb 1951 in Urbana, Illinois, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Miller was awarded his doctorate from Cumberland University in 1892. He spent the years from 1895 to 1897 in Europe attending lectures by Lie and Jordan. He wrote more than 800 articles over a period of 40 years about half at research level, the others aimed at school teachers. Miller worked mostly on group theory but he was also interested in the history of mathematics. Although interesting because it was done at an early stage, his work fails to show much depth. A bequest of one million dollars to the University of Illinois (where he taught for 25 years) on his death showed the skill he had in making financial investments. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to George A Miller (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Miller.html (1 of 2) [2/16/2002 11:23:07 PM]

Miller

Lunar features

Crater Miller

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Miller.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Miller.html (2 of 2) [2/16/2002 11:23:07 PM]

Milne

Edward Arthur Milne Born: 14 Feb 1896 in Hull, Yorkshire, England Died: 21 Sept 1950 in Dublin, Ireland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Arthur Milne was born in Hull where his father, Sidney Milne, was the headmaster of a Church of England school. His mother, Edith Cockcroft, was also a teacher. Arthur was the oldest of his parents' three children, all boys, all of whom became scientists. The family moved to Hessle, near Hull, when Arthur was nine years old. Milne attended Hymers College in Hull for his secondary schooling and from there he won an open scholarship in mathematics and natural science to study at Trinity College, Cambridge. It was a remarkable achievement for not only did he win this scholarship in 1914, he gained the largest number of marks which had ever been awarded in the examination. At Cambridge, in the eighteen months during which he studied there, he was inspired by Chapman and Hardy. Milne's defective eyesight prevented him from active service in World War I. However, in 1916 he joined a team working on ballistics at the Anti-Aircraft Section. Meg Weston Smith writes in [10]:In 1916 he abandoned his studies to join an innovative research team examining the behaviour of shells, fuses etc. Beyond producing the relevant mathematics, he flew in early aircraft, hanging over the wing to take readings of temperature and pressure, and he supervised the eminent statistician Karl Pearson in drawing up new firing tables, based on the team's findings, which were issued throughout the armed services. Milne also increased the understanding of wind and sound, in the course of refining the huge binaural listening trumpets which detected aircraft at night. Under the team's leader A V Hill, and R H Fowler (with whom Milne later collaborated) Milne received a marvellous training in how to do research. Thus his three years with the team probably ranked equally in importance in his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Milne.html (1 of 4) [2/16/2002 11:23:09 PM]

Milne

education with his one and half years at Cambridge. In 1919 Milne returned to Cambridge but not with the intention of either completing his undergraduate degree or of studying for a doctorate. Elected a Fellow of Trinity College shortly after he came back to Cambridge, Milne was appointed assistant director of the Solar Physics Observatory in Cambridge in 1920. He worked there under its director H F Newell who suggested that he turn his attention to work on stellar atmospheres. Milne's work on the atmospheres of stars extended work done earlier by Schuster in 1905 and by Schwarzschild in 1906. Schuster had studied the transfer of radiation where it was assumed that no absorption was taking place in the atmosphere, while Schwarzschild studied equilibrium states for radiation in an absorbing atmosphere. Milne combined the two approaches and came up with an integral equation of great mathematical interest which is now known as Milne's integral equation. In 1922 Milne won a Smith's Prize at Cambridge for an essay on the darkening of the limb of a stellar disk. He calculated the amount of darkening of the limb resulting from a given energy distribution in the star's spectrum, and compared his theoretical results with the known values for the sun. He then studied the inverse problem of deducing the energy distribution in a star's spectrum from the limb darkening. We mentioned above that Milne collaborated with R H Fowler they cooperated in 1923 in studying absorption lines in stellar spectra. Sydney Chapman resigned as Beyer professor of applied mathematics at Manchester in 1924 in order to take up the chair of mathematics at Imperial College London. Milne was appointed to succeed Chapman as Beyer professor of applied mathematics at Manchester, taking up the post in 1925. While at Manchester he continued his research into radiative equilibrium and the structure of stellar atmospheres, and his work on these topics led to his election to a Fellowship of the Royal Society. In 1928 he accepted the Rouse Ball Chair at Oxford, becoming the first holder when he took up the appointment in January 1929. His Royal Society Bakerian lecture in 1929 on The Structure and Opacity of a Stellar Atmosphere marks the end of his research into this topic. From around the time that Milne took up the Rouse Ball Chair at Oxford, he moved on to another research topic, this time studying the structure of stars. The main thrust of his work on this topic was to give different ideas to those of Eddington. Whitrow writes in [1]:Although much of Milne's criticism of Eddington's work has not been generally accepted, his methods led to important developments ... After about three years concentrating on a mathematical theory of stellar structure, Milne turned his attention to cosmology. He developed a new form of relativity called kinematic relativity, an alternative to Einstein's general theory of relativity, which also met with considerable opposition. However, his work made people rethink old ideas and led to new approaches to the fundamental concepts of space and time. Milne's books include Thermodynamics of the Stars (1930) which contains material relating to his Smith's Prize essay discussed above, Relativity, Gravitation and World-Structure (1935), and Kinematic Relativity (1948). These were all scholarly texts written for his fellow academics and, unlike most of the other famous astronomers of his time, he wrote no texts intended for the general public. Milne received many honours during his career. He was elected to the Royal Society in 1926 and invited http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Milne.html (2 of 4) [2/16/2002 11:23:09 PM]

Milne

to give its Bakerian lecture in 1929. In 1941 Milne was awarded the Royal Medal of the Royal Society:... for his researches on the atmosphere of the Earth and the sun, on the internal constitution of the stars, and on the theory of relativity. He was President of the Royal Astronomical Society from 1943 to 1945 having been awarded the Society's gold medal in 1935. As to Milne's character Whitrow writes in [1]:Small in stature, Milne had outstanding qualities of mind and was a continual fount of inspiration to others as well as himself. ... Milne had the humility and simplicity of character that often goes with scientific genius, and he bore personal misfortunes with courage, dignity, and religious conviction. As a young man Milne had suffered from encephalitis. He had a type of inflammation known as encephalitis lethargica which swept across Europe in an epidemic after World War I. Milne contracted the disease in 1924, and had made a good recovery by 1925. Usually there are Parkinsonian symptoms later in life, and this was so with Milne, in whom the after effects appeared around 1945. He died from a heart attack in Dublin while attending a conference. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country Honours awarded to Arthur Milne (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1926

Royal Society Royal Medal

Awarded 1941

Royal Society Bakerian lecturer

1929

London Maths Society President

1937 - 1939

ASP Bruce Medallist

1945

Lunar features

Crater Milne

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Milne.html (3 of 4) [2/16/2002 11:23:09 PM]

Milne

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Milne.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Milne.html (4 of 4) [2/16/2002 11:23:09 PM]

Milnor

John Willard Milnor Born: 20 Feb 1931 in Orange, New Jersey, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Milnor was educated at the University of Princeton, receiving his A.B. in 1951. He began research at Princeton after graduating and, in 1953 before completing his doctoral studies, he was appointed to the faculty in Princeton. In 1954 Milnor received his doctorate for his thesis Isotopy of Links written under Ralph Fox's supervision. Milnor remained on the staff at Princeton where he was an Alfred P Sloan fellow from 1955 until 1959. He was promoted to professor in 1960 then, in 1962, Milnor was appointed to the Henry Putman chair. Milnor was awarded a Fields Medal at the 1962 International Congress of Mathematicians in Stockholm. His most remarkable achievement, which played a major role in the award of the Fields Medal, was his proof that a 7-dimensional sphere can have several differential structures. This work opened up the new field of differential topology. Milnor showed that 28 different differentiable structures exist on the seven-dimensional sphere. He distinguished between these structures using numerical invariants based on the Todd polynomials. The Todd polynomials were first studied in algebraic geometry and it is surprising that they play this fundamental role in classification of manifolds. The reason that Milnor could use them to distinguish the differential properties of manifolds is because they have arithmetic properties, involving the Bernoulli numbers, which reflect in a deep and not fully understood way these differential properties. The references [2] to [6] give a good indication of the wide influence of Milnor's work up to 1992 (when these articles were written). The article [2] is a survey of Milnor's work in algebra, particularly in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Milnor.html (1 of 3) [2/16/2002 11:23:11 PM]

Milnor

algebraic K-theory, where his work continues to have important influences. The article [5] looks at nine papers which Milnor had written on differential geometry. It discusses Milnor's theorem, which shows that the total curvature of a knot is at least 4 . Among other results discussed are Milnor's result showing that we cannot necessarily "hear the shape" of a 16-dimensional torus, and another result giving upper and lower bounds on the number of distinct words of a given length in a finitely generated subgroup of the fundamental group. In the 1950s Milnor did a substantial amount of work on algebraic topology which is discussed in [6]. He constructed the classifying space of a topological group and gave a geometric realisation of a semi-simplicial complex. He also studied the Steenrod algebra and its dual, investigated the structure of Hopf algebras, and studied characteristic classes and their relation to mathematical physics. Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarised by Peter Makienko in his review of [3]:It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid-1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems. Milnor has received many awards for work. He received the National Medal of Science in 1967 and was elected a member of the National Academy of Science, the American Academy of Arts and Science. He is a member of the American Philosophy Society and has played a major role in the American Mathematical Society. In August 1982 Milnor received the Leroy P Steele Prize:...for a paper of fundamental and lasting importance, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64 (1956), 399-405. Among the many services he has rendered to mathematics is editorial work, being editor of the Annals of Mathematics from 1962. Since 1988 he has been at the State University of New York at Stony Brook. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1950 to 1960

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Milnor.html (2 of 3) [2/16/2002 11:23:11 PM]

Milnor

Honours awarded to John Milnor (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1962

AMS Colloquium Lecturer

1968

Other Web sites

1. John Milnor's home page 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Milnor.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Milnor.html (3 of 3) [2/16/2002 11:23:11 PM]

Minding

Ernst Ferdinand Adolf Minding Born: 23 Jan 1806 in Kalisz, Poland Died: 3 May 1885 in Dorpat, Russia (now Tartu, Estonia)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ferdinand Minding's family moved from Kalisz to Hirschberg when he was one year old. Hirschberg is now Jelenia Góra in Poland. He attended Hirschberg Gymnasium graduating in 1824. From there he went to the universities of Halle and Berlin where he studied philology, philosophy and physics. He graduated from Berlin in 1827, then taught mathematics in schools for several years. Now Minding had not studied mathematics at university so how did he become a mathematics teacher? He was self taught in mathematics having studied the subject on his own while pursuing other topics at university. While he was a school teacher, he studied for his doctorate which was awarded by Halle for a thesis on approximating the values of double integrals. In 1831 Minding became a mathematics lecturer at the University of Berlin then, in 1843 he was to become professor at Dorpat, a post he held for 40 years. At Dorpat Minding taught algebra, analysis, geometry, probability, mechanics and physics. From 1851 until 1855 he was Dean of the Faculty at Dorpat. In 1864 Minding became a Russian citizen and, in the same year, was elected to the St Petersburg Academy. His work, which continued Gauss's study of 1828 on the differential geometry of surfaces, greatly influenced Peterson. In 1830 Minding published on the problem of the shortest closed curve on a given surface enclosing a given area. He introduced the geodesic curvature although he did not use the term which was due to Bonnet who discovered it independently in 1848. In fact Gauss had proved these

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Minding.html (1 of 2) [2/16/2002 11:23:13 PM]

Minding

results, before either Minding of Bonnet, in 1825 but he had not published them. Minding also studied the bending of surfaces proving what is today called Minding's theorem in 1839. The following year he published in Crelle's Journal a paper giving results about trigonometric formulas on surfaces of constant curvature. Lobachevsky had published, also in Crelle's Journal, related results three years earlier and these results by Lobachevsky and Minding formed the basis of Beltrami's interpretation of hyperbolic geometry in 1868. Minding also worked on differential equations, algebraic functions, continued fractions and analytic mechanics. In differential equations he used integrating factor methods. This work won Minding the Demidov prize of the St Petersburg Academy in 1861. It was further developed by A N Korkin. Darboux and Emile Picard pushed these results still further in 1878. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Minding.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Minding.html (2 of 2) [2/16/2002 11:23:13 PM]

Mineur

Henri Mineur Born: 7 March 1899 in Lille, France Died: 7 May 1954 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henri Mineur applied for admission to Ecole Normale Supérieur in Paris in 1917. He was placed first but because of the war he decided to join the army. This did not hold him up for long for he was awarded his degree in 1921. He had already started research in mathematics by 1920 and after graduating he taught mathematics in a school in Dusseldorf while he completed his thesis. He was awarded his doctorate in 1924 for a thesis on functional equations in which he established an addition theorem for Fuchsian functions. The work of his thesis was published in 1925 and, three years later, he published a paper on the differential calculus. Mineur had always been interested in astronomy and, in 1925, he left teaching to take up a post in the Paris Observatory. He contributed to many areas of astronomy and mathematics including celestial mechanics, analytic mechanics, statistics and numerical analysis. In 1938 Mineur wrote an important work on the least square method, Technique de la méthods des moindres carrés. His most important work on numerical methods was Technique de calcul numérique (1952). In astronomy Mineur made many significant discoveries. He discovered that the speeds of the stars varied according to their distance from the plane of the galaxy. He also discovered the retrograde rotation of the system of globular clusters around the galaxy. Another significant discovery was to realise that there was a error in the accepted period-luminosity law for Cepheid variables. Since the whole scale of the Universe is based on distances determined using this relationship, this discovery doubled the size of the Universe in the sense that all objects in the Universe where now shown to be twice as far away as previously thought. This had a huge impact for cosmology. Mineur was the main force behind the setting up of the Institute d'Astrophysique in Paris in 1936 and he was its first director, a post he held for the rest of his life. During the years 1940-44 Mineur was an active member of the French Resistance risking his life on many occasions. Mineur had five years of bad health with heart and liver problems before his death at the early age of 55.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mineur.html (1 of 2) [2/16/2002 11:23:14 PM]

Mineur

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Henri Mineur (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Mineur

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mineur.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mineur.html (2 of 2) [2/16/2002 11:23:14 PM]

Minkowski

Hermann Minkowski Born: 22 June 1864 in Alexotas, Russian Empire (now Kaunas, Lithuania) Died: 12 Jan 1909 in Göttingen, Germany

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hermann Minkowski studied at the Universities of Berlin and Königsberg. He received his doctorate in 1885 from Königsberg. He taught at several universities, Bonn, Königsberg and Zurich. In Zurich, Einstein was a student in several of the courses he gave. Minkowski accepted a chair in 1902 at the University of Göttingen, where he stayed for the rest of his life. At Göttingen he learnt mathematical physics from Hilbert and his associates. He participated in a seminar on electron theory in 1905 and he learnt the latest results and theories in electrodynamics. By 1907 Minkowski realised that the work of Lorentz and Einstein could be best understood in a non-educlidean spave. He considered space and time, which were formerly thought to be independent, to be coupled together in a four-dimensional 'space-time continuum'. Minkowski worked out a four-dimensional treatment of electrodynamics. His major works in this area are Raum und Zeit (1907) and Zwei Abhandlungen über die Grundgleichungen der Elektrodynamik (1909). This space-time continuum provided a framework for all later mathematical work in relativity. These ideas were used by Albert Einstein in developing the general theory of relativity. Minkowski was mainly interested in pure mathematics and spent much of his time investigating quadratic forms and continued fractions. His most original achievement, however, was his 'geometry of numbers'. This study led on to work on convex bodies and to questions about packing problems, the ways in which figures of a given shape can be placed within another given figure.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Minkowski.html (1 of 2) [2/16/2002 11:23:16 PM]

Minkowski

At the young age of 44, Minkowski died suddenly from a ruptured appendix. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles)

Some Quotations (2)

A Poster of Hermann Minkowski

Mathematicians born in the same country

Cross-references to History Topics

Special relativity

Honours awarded to Hermann Minkowski (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Minkowski

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Minkowski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Minkowski.html (2 of 2) [2/16/2002 11:23:16 PM]

Mirsky

Leon Mirsky Born: 19 Dec 1918 in Russia Died: 1 Dec 1983 in Sheffield, England

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leon Mirsky's family left Russia for Germany when he was 8 years old, but in 1933 they were forced to leave Germany and settled in Bradford, England. Leon entered King's College, London in 1936. At first he studied for the Intermediate examination after which he was awarded a scholarship to study for a degree in mathematics. During this time he took a passionate interest in the theory of numbers and became a great admirer of Edmund Landau. Leon graduated from London in 1940 and began work on an M.Sc. However the war meant that students from King's College were sent to Bristol University and it was there that he completed his Master's degree. In 1942 he was appointed to Sheffield University and, except for short spell at Manchester and Bristol (working with Heilbronn), he was to spend the rest of his career at Sheffield (where he was to work unsupervised for a Ph.D. which was awarded in 1949). Mirsky had three main areas of research. (i) The theory of numbers, where he studied r-free numbers, i.e. numbers not divisible by the r th power of any integer. He obtained analogues of Vinogradov's result on the representation of an odd integer as the sum of three primes, the Goldbach conjecture on the representation of an even integer as the sum of two primes, and the twin prime conjecture. (ii) Linear algebra, where he wrote his famous text An introduction to linear algebra (1955) and went on to publish 35 papers on the topic. In particular he proved results on the existence of matrices with given eigenvalues and given diagonal elements. (iii) Combinatorics, where he developed ideas coming from Hall's theorem:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mirsky.html (1 of 2) [2/16/2002 11:23:17 PM]

Mirsky

A finite family of sets has a system of distinct representatives if and only if the union of every k sets of the family contains at least k elements. Mirsky's talent for teaching is described in [1] as follows:Leon was a born teacher. He welcomed the challenge of presenting a whole theory or just one proof in a logical, efficient, clear and elegant manner. ... His lectures ... were highly individual performances. There was never any hint of familiarity with his audience and Leon always wore a gown to emphasize the formality of the occasion. On the other hand, the alert student could spot a succession of jokes all made with an entirely straight face and no change of tone. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of Leon Mirsky

Mathematicians born in the same country

Cross-references to History Topics

Matrices and determinants

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mirsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mirsky.html (2 of 2) [2/16/2002 11:23:17 PM]

Mises

Richard von Mises Born: 19 April 1883 in Lemberg, Austria (now Lvov, Ukraine) Died: 14 July 1953 in Boston, Massachusetts, USA

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Richard von Mises was born in Lvov when it was under Austrian control and known as Lemberg. His father, Arthur Edler von Mises, worked for the Austrian State Railways as a technical expert and his mother was Adele von Landau. Richard was the second son of the family, the elder son being Ludwig von Mises who went on to become as famous as Richard. Ludwig, who was about eighteen months older than Richard, went on to become an economist who contributed to liberalism in economic theory and made his belief in consumer power an important part of that theory. Richard also had a younger brother, who died as an infant. It was on the technical course that von Mises embarked, studying mathematics, physics and engineering at the Technische Hochschule in Vienna. After graduating he was appointed as Georg Hamel's assistant in Brünn. The city of Brünn is today called Brno in the south-eastern Czech Republic. Up to World War II the inhabitants were predominantly German, although today they are now mainly Czech. Von Mises was awarded a doctorate from Vienna in 1907 and the following year he was awarded his habilitation from Brünn, becoming qualified to lecture on engineering and machine construction. He was professor of applied mathematics at Strasburg from 1909 until 1918, although this was a period which was interrupted by World War I. Even before the outbreak of the war von Mises had qualified as a pilot and he gave the first university course on powered flight in 1913. When war broke out von Mises joined the Austro-Hungarian army and piloted aircraft. He had lectured on the design of aircraft before the war and he now put this into practice leading a team which constructed a 600-horsepower plane for the Austrian army in 1915. After the end of the war von Mises was appointed to a new chair of hydrodynamics and aerodynamics at http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mises.html (1 of 5) [2/16/2002 11:23:19 PM]

Mises

the Technische Hochschule in Dresden. Appointed in 1919 he soon moved again, this time to the University of Berlin to become the director of the new Institute of Applied Mathematics which had been set up there. Schmidt had argued for the setting up of the Institute in 1918:The pervasion of practical life by mathematical methods, as a result of the development of technology before the war and, above all, the unexpected need for ... mathematicians during the war make it an undeniable necessity to install applied mathematics at the largest Prussian university... Among university students, however, one frequently finds the opinion that applied mathematics is a subject of inferior importance, which does not require one's full attention. to create a new tradition, it needs an important personality of approved name. Such a personality can only be attracted by a full professorship. Theses words by Schmidt show great wisdom, and the authorities did indeed create the full professorship and made an inspired choice for the first holder in von Mises. Ostrowski wrote in a 1965 lecture (see for example [16]):Only with the appointment of Richard von Mises to the University of Berlin did the first serious German school of applied mathematics with a broad sphere of influence come into existence. Von Mises was an incredibly dynamic person and at the same time amazingly versatile like Runge. He was especially well versed in the realm of technology. The Institute of Applied Mathematics flourished under his control. In 1921 he founded the journal Zeitschrift für Angewandte Mathematik und Mechanik and he became the editor of the journal. In the first edition he wrote an introduction stressing the wide range of applied mathematics and also the fact that the line between pure and applied mathematics is not a fixed one, but one which changes over time as different areas of "pure mathematics" find applications in practical situations. He set up a new curriculum for applied mathematics at the university which spread over six semesters and included applications of mathematics to astronomy, geodesy and technology. It was not a "soft option" and von Mises went out of his way to stress that applied mathematics was every bit as rigorous as pure mathematics requiring [13]:... a mathematical model of the widest possible generality, where the argument could be made with clarity, elegance, and rigour. His Institute rapidly became a centre for research into areas such as probability, statistics, numerical solutions of differential equations, elasticity and aerodynamics. Von Mises was also an excellent lecturer. Collatz, one of his students, wrote:I was enrolled in Berlin in 1930. ... Professor Dr Richard von Mises [gave] excellent, very clear and stimulating lectures on applied analysis .... The paper [8], written by Collatz, discusses von Mises' work on numerical mathematics, discusses his founding of the journal Zeitschrift für Angewandte Mathematik und Mechanik and looks at the difficulties faced by von Mises in bringing up the status of applied mathematics during the 1920s and early 1930s. On 30 January 1933, however, Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Von Mises in one sense was not Jewish for he was a Roman Catholic by religion. He still fell under the non-Aryan definition of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mises.html (2 of 5) [2/16/2002 11:23:19 PM]

Mises

the act but there was an exemption clause which exempted non-Aryans who had fought in World War I. Von Mises certainly qualified under this clause and it would have allowed him to keep his chair in Berlin in 1933. He realised, quite correctly, that the exemption clause would not save him for long. On the 10 June 1933 he wrote to von Kármán about a young German, Walter Tollmien, who was looking for a position:I have to advise you that the irrevocable prerequisite for any kind of employment or scholarship or suchlike is to make a statement on his word of honour that his four grandparents are Aryan and in particular are of non-Jewish descent. ... I believe that in a favourable case the prospects are not quite so bad as indeed a large part of all the previous candidates can be omitted under the present law. Von Kármán forwarded the letter to Tollmien, writing "Indeed a document of our time" on the back! Von Mises saw an offer of a chair in Turkey as a way out of his predicament in Germany but he tried to ensure that his pension rights were preserved. On 12 October 1933 he wrote to the ministry explaining that it would benefit Germany if he accepted a post in Turkey and that he should be allowed to retain his pension rights for his 24 years of service. He received the reply that he would have to relinquish all rights of a salary, a pension or support for his dependants. He protested in a further letter to the Ministry that he was legally entitled to rights that he was not prepared to relinquish. The Nazi Theodor Vahlen wanted to take over as director of the Institute despite poor academic qualities. He promised von Mises that if he would support him to succeed as Director of the Institute then he would ensure that von Mises would not lose his pension rights. In October 1933 von Mises wrote his letter to support Vahlen as his successor. Collatz, von Mises' student, wrote:I took my Staatsexamen in November 1933, and Professor von Mises examined me on the day before his departure. The same day, he talked to me for about one hour, giving advice for my further research ... Vahlen was appointed Director of the Institute in December 1933. Having taken up the new chair in Istanbul, von Mises received a letter in January 1934 denying him any rights at all. It was something that von Mises continued to feel extremely aggrieved about, writing to the Ministry in 1953, shortly before his death, still trying to restore his rights. In 1938 Kemal Atatürk died and those in Turkey who had fled from the Nazis feared that their safe haven would become unsafe. In 1939 von Mises left Turkey for the United States. He became professor at Harvard University and in 1944 he was appointed Gordon-McKay Professor of Aerodynamics and Applied Mathematics there. Von Mises worked on fluid mechanics, aerodynamics, aeronautics, statistics and probability theory. He classified his own work, not long before his death, into eight areas: practical analysis, integral and differential equations, mechanics, hydrodynamics and aerodynamics, constructive geometry, probability calculus, statistics and philosophy. He introduced a stress tensor which was used in the study of the strength of materials. His studies of wing theory for aircraft led him to investigate turbulence. Much of his work involved numerical methods and this led him to develop new techniques in numerical analysis. His most famous, and at the same time most controversial, work was in probability theory. He made considerable progress in the area of frequency analysis which was started by Venn. He http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mises.html (3 of 5) [2/16/2002 11:23:19 PM]

Mises

combined the idea of a Venn limit and a random sequence of events. Ostrowski in the same lecture which we quoted from above wrote (see for example [16]):Because of his dynamic personality his occasional major blunders were somehow tolerated. One has even forgiven him his theory of probability. This judgement by Ostrowski is rather harsh, however, and many consider von Mises' theories to be important in the development of the subject. After the measure theory approach by Kolmogorov had become favoured by almost all statisticians over von Mises' limiting frequency theory approach, there was a return to von Mises ideas and there was an attempt to incorporate them into the measure theoretic approach of Kolmogorov who wrote himself in 1963:... that the basis for the applicability of the results of the mathematical theory of probability to real 'random phenomena' must depend on some form of the frequency concept of probability, the unavoidable nature of which has been established by von Mises in a spirited manner. In paper [18] discusses:... von Mises' notion of a random sequence in the context of his approach to probability theory. [The author claims] that the acceptance of Kolmogorov's rival axiomatisation was due to a different intuition about probability getting the upper hand, as illustrated by the notion of a martingale. Phillip Frank, writing in [4] says:... in looking over the work of von Mises ... we cannot fail to recognise a whole spectrum of research, extending from the philosophical meaning of science to practical methods of numerical computation. ... von Mises has always been a truly broad-minded man ... notwithstanding the wide range of his topics, his work shows a great intrinsic unity: starting from a definite center, it branches out in systematic investigations of a great diversity of problems. ... von Mises chose the topics according to a very definite view-point, determined by his ideas about the essence and method of every thoroughly scientific research. In von Mises' book Positivism: A Study in Human Understanding (1951) he expressed his views on science and life. He subscribes to a doctrine of positivism in the book saying:Positivism does not claim that all questions can be answered rationally, just as medicine is not based on the premise that all diseases are curable, or physics does not start out with the postulate that all phenomena are explicable. But the mere possibility that there may be no answers to some questions is no sufficient reason for not looking for answers, or for not using those which are attainable. This interest in philosophy was only one of von Mises' interests outside the realm of mathematics. Another was the fact that he was an international authority on the Austrian poet Rainer Maria Rilke (1875-1926). In 1950 von Mises was offered honorary membership of the East German Academy of Science. This was difficult for von Mises, particularly in McCarthy era America where any link with communism would have been viewed as a serious offence. He sadly declined in a letter written on 15 September 1950:I would very gladly accept the nomination in remembrance of my teaching activities in Berlin and thus re-establish the bond which connected me for a long time to the German

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mises.html (4 of 5) [2/16/2002 11:23:19 PM]

Mises

scientific life. Unfortunately the present circumstances in Germany as well as those in this country are such that the acceptance of such a distinction could be interpreted as a political demonstration on my part. ... I only relinquish acceptance of this nomination under the pressure of outward circumstances, a nomination which I regard as a great honour in every respect. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (18 books/articles) A Poster of Richard von Mises

Mathematicians born in the same country

Other references in MacTutor

1. Chronology: 1920 to 1930 2. Chronology: 1930 to 1940

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mises.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mises.html (5 of 5) [2/16/2002 11:23:19 PM]

Mittag-Leffler

Magnus Gösta Mittag-Leffler Born: 16 March 1846 in Stockholm, Sweden Died: 7 July 1927 in Stockholm, Sweden

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gösta Mittag-Leffler's father was Johan Olof Leffler while his mother was Gustava Wilhelmina Mittag. Now the reader will have noticed that Gösta Mittag-Leffler had a surname which combined both his mother's name Mittag and his father's name Leffler so might suppose that when his parents married they combined their names; however this was not so. Gösta's mother took the name Gustava Wilhelmina Leffler on her marriage and when their first child Gösta was born, he took the name Gösta Leffler. It was not until Gösta Leffler was a twenty year old student that he decided to add "Mittag" to his name. This was a clear indication of his feelings for his mother and for his grandparents on his mother's side. Gösta's father was a school teacher who became a headmaster of a high school in Stockholm. He also served a spell as a member of parliament. As we mentioned, Gösta was the eldest of the family and he was born in the schoolhouse at the school where his father taught. After a while his parents bought a house of their own and in this new home Gösta's sister and two brothers were born; none of them would follow Gösta in adding Mittag to their names. Both sides of the family were of German origin but had lived for several generations in Sweden. Gösta's grandfather on his father's side was a sail maker who, like his son would do, served for a time as a member of parliament. Gösta's grandfather on his mother's side was a dean in the church, living in a country area of Sweden. As a young boy Gösta spent each summer holiday with his mother's parents and he had a great affection for his mother's family. It was clear from his later life that Mittag-Leffler had many talents in addition to his mathematics and it was his upbringing that did much to foster these talents. His parents house was frequently filled with their friends and it provided an environment in which Gösta, together with his brothers and sister, learnt much in addition to their schooling. Gösta also showed his many talents as he progressed through school,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mittag-Leffler.html (1 of 6) [2/16/2002 11:23:22 PM]

Mittag-Leffler

and his teachers in elementary school and later those at the Gymnasium in Stockholm realised that he had an outstanding ability for mathematics. Garding writes [7]:From an early age Gösta Leffler kept an irregular diary. His early entries show an interest in literature and later, when he was around twenty, the diary gives the picture of a well-behaved and well-educated young man with general interests. Mittag-Leffler trained as an actuary but later took up mathematics. He studied at Uppsala, entering in 1865. During his studies he supported himself by taking private pupils. He submitted and:... defended a less than remarkable thesis in 1872 on applications of the argument principle. As a result Mittag-Leffler was appointed as a Docent at the University of Uppsala in 1872. Perhaps the event which would have the greatest lasting effect on his life was being awarded a salary which came through an endowment with a condition attached which said that the holder had to spend three years abroad. In October 1873 Mittag-Leffler set off for Paris. Although Mittag-Leffler met many mathematicians in Paris, such as Bouquet, Briot, Chasles, Darboux, and Liouville, the main aim of the visit was to learn from Hermite. Mittag-Leffler attended Hermite's lectures on elliptic functions but found them hard going. His stay in Paris is described in detail in a diary he kept. His time consisted of:... a visit to the theatre (Sarah Bernhardt), discussions on political and religious matters, lessons in French and English, a visit to the workers' quarter, reflections on people met and so on. Sometimes there is a sigh over Hermite's lectures, so difficult to understand. Certainly Hermite spoke in glowing terms about Weierstrass and the contributions he and other German mathematicians were making, so Mittag-Leffler made a decision to go to Berlin in the Spring of 1875. There he attended Weierstrass's lectures which proved to be extremely influential in setting the direction of Mittag-Leffler's subsequent mathematical work. Much later in life described his three years abroad:I got a vivid impression of the sharp tension between academic circles in Paris and Berlin during my visits to the two capitals. I was therefore struck by my experience that Hermite and Weierstrass were absolutely free of nationalistic feelings or leanings. Both were born Catholics and Hermite, as Cauchy before him, was a warm believer. Weierstrass was interested in, or rather amused by, talking to learned prelates about the finest points of the church dogmas. While Mittag-Leffler was in Berlin he learnt that the professor of mathematics at Helsingfors (today called Helsinki), Lorenz Lindelöf, the father of Ernst Lindelöf, was leaving the chair to take up an administrative post. Mittag-Leffler wrote about these events:In 1875 during my stay in Berlin, I was informed in a roundabout way through Uppsala that the professorship in mathematics in Helsingfors was open. It was L Lindelöf, an important mathematician and rector of the university, who was going to the National School Board... and I decided to apply for his post. When I payed a visit to my teacher, the famous mathematician Weierstrass, and told him about my plans, he exclaimed: No, please, do not do that! I have written to the minister of culture and asked him to institute an extraordinary professorship for you here in Berlin and I just got the message that my application has been granted! I was not blind to the great advantages, mathematically speaking, of such a position http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mittag-Leffler.html (2 of 6) [2/16/2002 11:23:22 PM]

Mittag-Leffler

compared to the one in Helsingfors. Hardly ever was there such a brilliant collection of distinguished mathematicians: Weierstrass, Kummer, Kronecker, Helmholtz, Kirchhoff, Borchardt etc. But the conditions were not endurable for a foreigner. It was not so long after Germany's victorious war against France, and German arrogance was at a high point. Foreigners were treated with haughty condescension; der grosse Kaiser, Bismarck, Moltke etc. were words heard everywhere. It was taken as a matter of course that Holland, Sweden etc. would be members of a German Bund. For those who were not born Germans, it was impossible to live under such conditions. At least this is what I thought. Now things are different, the brilliant victory has not born the fruits that the Germans imagined and they have taken a more realistic view. However, my application to Helsingfors was not sent immediately ... We have quoted extensively from Mittag-Leffler's writing about this overseas trip and his reaction to national rivalries. We have done so because it is important in determining the international role which Mittag-Leffler went on to play, for his passion for international cooperation in mathematics was a direct consequence of what he saw on his three year trip abroad. Mittag-Leffler was appointed to a chair at the University of Helsinki in 1876 and, five years later, he returned to his home town of Stockholm to take up a chair at the University. He was the first holder of the mathematics chair in the new university of Stockholm (called Stockholms Högskola at this time). Soon after taking up the appointment, he began to organise the setting up of the new international journal Acta Mathematica. We will discuss this important aspect of his contribution below. In 1882 he married Signe af Lindfors who came from a wealthy family. They had met while Mittag-Leffler was living in Helsinki and, although Signe was from a Swedish family, they too were living in Finland. Mittag-Leffler made numerous contributions to mathematical analysis (concerned with limits and including calculus, analytic geometry and probability theory). He worked on the general theory of functions, concerning relationships between independent and dependent variables. His best known work concerned the analytic representation of a one-valued function, this work culminated in the Mittag-Leffler theorem. This study began as an attempt to generalise results in Weierstrass's lectures where he had described his theorem on the existence of an entire function with prescribed zeros each with a specified multiplicity. Mittag-Leffler tried to generalise this result to meromorphic functions while he was studying in Berlin. He eventually assembled his findings on generalising Weierstrass's theorem to meromorphic functions into a paper which he published (in French) in 1884 in Acta Mathematica . In this paper Mittag-Leffler proposed a series of general topological notions on infinite point sets based on Cantor's new set theory [8]:With this paper ... Mittag-Leffler became the sole proprietor of a theorem that later became widely known and with this he took his place in the circle of internationally known mathematicians. Mittag-Leffler was one of the first mathematicians to support Cantor's theory of sets but, one has to remark, a consequence of this was that Kronecker refused to publish in Acta Mathematica . Between 1900 and 1905 Mittag-Leffler published a series of five papers which he called "Notes" on the summation of divergent series. The aim of these notes was to construct the analytical continuation of a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mittag-Leffler.html (3 of 6) [2/16/2002 11:23:22 PM]

Mittag-Leffler

power series outside its circle of convergence. The region in which he was able to do this is now called Mittag-Leffler's star. In 1882 Mittag-Leffler founded Acta Mathematica and served as chief editor of the journal for 45 years. The original idea for such a journal came from Sophus Lie in 1881, but it was Mittag-Leffler's understanding of the European mathematical scene, together with his political skills, which ensured its success. His wife Signe was, as we have mentioned, extremely wealthy and her money helped support the setting up of the Journal but most of the money needed came from an appeal and from subscriptions. However, periodicals like Acta Mathematica do not succeed just because of money. It required an international base and certainly Mittag-Leffler fully understood this. He needed leading mathematicians to send papers to Acta Mathematica and Cantor and Poincaré contributed many papers to the first few volumes. But there was another important factor in its success as Hardy points our [9]:.. Mittag-Leffler was always a good judge of the quality of the work submitted to him for publication. Even in his later years, when most of the editorial work was delegated to others, he retained that curious sense which enables the great editor to feel the value of the work at which he has hardly glanced... In 1884, the year Mittag-Leffler published his masterpiece, Kovalevskaya arrived in Stockholm at his invitation. On her death in 1891 Mittag-Leffler wrote:She came to us from the centre of modern science full of faith and enthusiasm for the ideas of her great master in Berlin, the venerable old man who has outlived his favourite pupil. Her works, all of which belong to the same order of ideas, have shown by new discoveries the power of Weierstrass's system. By the early 1890s Mittag-Leffler had built for his family a wonderful new home in Djursholm in the suburbs of Stockholm. His life he now divided between time spent at his home in Djursholm and at his country home at Tallberg, around 300 km north of Stockholm. In his home in the garden suburbs of Stockholm he had the finest mathematical library in the world. Hardy spent time with Mittag-Leffler in his library and describes it in [9]:All books and periodical were there, ... and if one got tired one could read the correspondence of all the mathematicians in the world, or enjoy the view of Stockholm from the roof. Of Mittag-Leffler's country home Hardy writes:... there Mittag-Leffler appeared at his best, a most entertaining mixture of the great international mathematician and the rather naive country squire. He was a strong nationalist, in spite of his internationalism, as anyone who lived in so beautiful a country might be; and he loved his house and his garden and his position as the landowner of the countryside. Mittag-Leffler and his wife bequeathed their home in Stockholm to the Academy of Sciences in 1916. World War I caused a loss in the value of money left by the Mittag-Lefflers to fund the mathematical foundation which they proposed. However, eventually the Mittag-Leffler Institute was set up based on the house and today is a major mathematical research centre. Hardy, writing in [9], describes the regard that Mittag-Leffler was held in, particularly in his own country:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mittag-Leffler.html (4 of 6) [2/16/2002 11:23:22 PM]

Mittag-Leffler

I can remember well the occasion when he lectured for the last time to a Scandinavian Congress, at Copenhagen in 1925, and the whole audience rose and stood as he entered the room. It was a reception rather astonishing at first to a visitor from a less ceremonious country; but it was an entirely spontaneous expression of the universal feeling that to him, more than to any other single man, the great advance in the status of Scandinavian mathematics during the last fifty years was due. Mittag-Leffler received many honours. He was an honorary member of almost every mathematical society in the world. He was elected a Fellow of the Royal Society of London in 1896. His contribution is nicely summed up by Hardy [9]:Mittag-Leffler was a remarkable man in many ways. He was a mathematician of the front rank, whose contributions to analysis had become classical, and had played a great part in the inspiration of later research; he was a man of strong personality, fired by an intense devotion to his chosen study; and he had the persistence, the position, and the means to make his enthusiasm count. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles)

A Quotation

A Poster of Gösta Mittag-Leffler

Mathematicians born in the same country

Cross-references to History Topics

The beginnings of set theory

Other references in MacTutor

Chronology: 1880 to 1890

Honours awarded to Gösta Mittag-Leffler (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1896

Other Web sites

1. FAQ's in Mathematics (Why is there no Nobel prize in Mathematics?) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mittag-Leffler.html (5 of 6) [2/16/2002 11:23:22 PM]

Mittag-Leffler

JOC/EFR September 2001

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mittag-Leffler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mittag-Leffler.html (6 of 6) [2/16/2002 11:23:22 PM]

Mobius

August Ferdinand Möbius Born: 17 Nov 1790 in Schulpforta, Saxony (now Germany) Died: 26 Sept 1868 in Leipzig, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

August Möbius was the only child of Johann Heinrich Möbius, a dancing teacher, who died when August was three years old. His mother was a descendant of Martin Luther. Möbius was educated at home until he was 13 years old when, already showing an interest in mathematics, he went to the College in Schulpforta in 1803. In 1809 Möbius graduated from his College and he became a student at the University of Leipzig. His family had wanted him study law and indeed he started to study this topic. However he soon discovered that it was not a subject that gave him satisfaction and in the middle of his first year of study he decided to follow him own preferences rather than those of his family. He therefore took up the study of mathematics, astronomy and physics. The teacher who influenced Möbius most during his time at Leipzig was his astronomy teacher Karl Mollweide. Although an astronomer, Mollweide is well known for a number of mathematical discoveries in particular the Mollweide trigonometric relations he discovered in 1807-09 and the Mollweide map projection which preserves angles and so is a conformal projection. In 1813 Möbius travelled to Göttingen where he studied astronomy under Gauss. Gauss was the director of the Observatory in Göttingen but of course the greatest mathematician of his day, so again Möbius studied under an astronomer whose interests were mathematical. From Göttingen Möbius went to Halle where he studied under Johann Pfaff, Gauss's teacher. Under Pfaff he studied mathematics rather than astronomy so by this stage Möbius was very firmly working in both fields. In 1815 Möbius wrote his doctoral thesis on The occultation of fixed stars and began work on his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mobius.html (1 of 4) [2/16/2002 11:23:23 PM]

Mobius

Habilitation thesis. In fact while he was writing this thesis there was an attempt to draft him into the Prussian army. Möbius wrote This is the most horrible idea I have heard of, and anyone who shall venture, dare, hazard, make bold and have the audacity to propose it will not be safe from my dagger. He avoided the army and completed his Habilitation thesis on Trigonometrical equations. Mollweide's interest in mathematics was such that he had moved from astronomy to the chair of mathematics at Leipzig so Möbius had high hopes that he might be appointed to a professorship in astronomy at Leipzig. Indeed he was appointed to the chair of astronomy and higher mechanics at the University of Leipzig in 1816. His initial appointment was as Extraordinary Professor and it was an appointment which came early in his career. However Möbius did not receive quick promotion to full professor. It would appear that he was not a particularly good lecturer and this made his life difficult since he did not attract fee paying students to his lectures. He was forced to advertise his lecture courses as being free of charge before students thought his courses worth taking. He was offered a post as an astronomer in Greifswald in 1816 and then a post as a mathematician at Dorpat in 1819. He refused both, partly through his belief in the high quality of Leipzig University, partly through his loyalty to Saxony. In 1825 Mollweide died and Möbius hoped to transfer to his chair of mathematics taking the route Mollweide had taken earlier. However it was not to be and another mathematician was preferred for the post. By 1844 Möbius's reputation as a researcher led to an invitation from the University of Jena and at this stage the University of Leipzig gave him the Full Professorship in astronomy which he clearly deserved. From the time of his first appointment at Leipzig Möbius had also held the post of Observer at the Observatory at Leipzig. He was involved the rebuilding of the Observatory and, from 1818 until 1821, he supervised the project. He visited several other observatories in Germany before making his recommendations for the new Observatory. In 1820 he married and he was to have one daughter and two sons. In 1848 he became director of the Observatory. In 1844 Grassmann visited Möbius. He asked Möbius to review his major work Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844) which contained many results similar to Möbius's work. However Möbius did not understand the significance of Grassmann's work and did not review it. He did however persuade Grassmann to submit work for a prize and, after Grassmann won the prize, Möbius did write a review of his winning entry in 1847. Although his most famous work is in mathematics, Möbius did publish important work on astronomy. He wrote De Computandis Occultationibus Fixarum per Planetas (1815) concerning occultations of the planets. He also wrote on the principles of astronomy, Die Hauptsätze der Astronomie (1836) and on celestial mechanics Die Elemente der Mechanik des Himmels (1843). Möbius's mathematical publications, although not always original, were effective and clear presentations. His contributions to mathematics are described by his biographer Richard Baltzer in [3] as follows:The inspirations for his research he found mostly in the rich well of his own original mind. His intuition, the problems he set himself, and the solutions that he found, all exhibit something extraordinarily ingenious, something original in an uncontrived way. He worked http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mobius.html (2 of 4) [2/16/2002 11:23:23 PM]

Mobius

without hurrying, quietly on his own. His work remained almost locked away until everything had been put into its proper place. Without rushing, without pomposity and without arrogance, he waited until the fruits of his mind matured. Only after such a wait did he publish his perfected works... Almost all Möbius's work was published in Crelle's Journal, the first journal devoted exclusively to publishing mathematics. Möbius's 1827 work Der barycentrische Calkul, on analytical geometry, became a classic and includes many of his results on projective and affine geometry. In it he introduced homogeneous coordinates and also discussed geometric transformations, in particular projective transformations. He introduced a configuration now called a Möbius net, which was to play an important role in the development of projective geometry. Möbius's name is attached to many important mathematical objects such as the Möbius function which he introduced in the 1831 paper Über eine besondere Art von Umkehrung der Reihen and the Möbius inversion formula. In 1837 he published Lehrbuch der Statik which gives a geometric treatment of statics. It led to the study of systems of lines in space. Before the question on the four colouring of maps had been asked by Francis Guthrie, Möbius had posed the following, rather easy, problem in 1840. There was once a king with five sons. In his will he stated that on his death his kingdom should be divided by his sons into five regions in such a way that each region should have a common boundary with the other four. Can the terms of the will be satisfied? The answer, of course, is negative and easy to show. However it does illustrate Möbius's interest in topological ideas, an area in which he is most remembered as a pioneer. In a memoir, presented to the Académie des Sciences and only discovered after his death, he discussed the properties of one-sided surfaces including the Möbius strip which he had discovered in 1858. This discovery was made as Möbius worked on a question on the geometric theory of polyhedra posed by the Paris Academy. Although we know this as a Möbius strip today it was not Möbius who first described this object, rather by any criterion, either publication date or date of first discovery, precedence goes to Listing. A Möbius strip is a two-dimensional surface with only one side. It can be constructed in three dimensions as follows. Take a rectangular strip of paper and join the two ends of the strip together so that it has a 180 degree twist. It is now possible to start at a point A on the surface and trace out a path that passes through the point which is apparently on the other side of the surface from A. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) A Poster of August Möbius

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mobius.html (3 of 4) [2/16/2002 11:23:23 PM]

Mobius

Cross-references to History Topics

1. The development of group theory 2. Topology enters mathematics 3. Abstract linear spaces

Other references in MacTutor

1. A picture of a Möbius band 2. Chronology: 1820 to 1830 3. Chronology: 1830 to 1840 4. Chronology: 1850 to 1860

Honours awarded to August Möbius (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Mobius

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mobius.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mobius.html (4 of 4) [2/16/2002 11:23:23 PM]

Mohr

Georg Mohr Born: 1 April 1640 in Copenhagen, Denmark Died: 26 Jan 1697 in Kieslingswalde (near Görlitz), Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Mohr was educated by his parents and learnt enough mathematics from them to want to study further. He went to Holland intending to study mathematics under Huygens. He also studied in France and England. Mohr was little known as a mathematician. His book Euclides danicus published in 1672 was forgotten about until discovered in a bookstore in 1928. Perhaps no copies were ever sold! The book contains the theorem and its proof that all Euclidean constructions can be carried out with compasses alone. Mascheroni, who is credited with this result, did not prove it until 125 years after Mohr's book was published. Mohr spent part of his life in Holland and part in Denmark. He fought in the Dutch-French wars around 1672 and was a prisoner of war. He was back in Denmark around 1681 but, having decided not to accept a post as supervisor of the king's shipbuilding, he returned to Holland in 1687. He corresponded with Tschirnhaus whom he had met on several occasions in Holland, France and England. In 1695 Mohr accepted a job from Tschirnhaus. Mohr also corresponded with Leibniz. Article by: J J O'Connor and E F Robertson List of References (7 books/articles) Mathematicians born in the same country Other references in MacTutor Other Web sites

Chronology: 1650 to 1675 1. Theseus 2. The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mohr.html (1 of 2) [2/16/2002 11:23:25 PM]

Mohr

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mohr.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mohr.html (2 of 2) [2/16/2002 11:23:25 PM]

Molien

Molien This biography is now under Molin. You should be automatically forwarded to the correct page. If your browser leaves you on this page, then press HERE. JOC/EFR May 2000

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Molien.html [2/16/2002 11:23:25 PM]

Molin

Fedor Eduardovich Molin Born: 10 Sept 1861 in Riga, Russia (now Latvia) Died: 25 Dec 1941 in Tomsk, USSR Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Theodor Molien (or Fedor Molin) was the son of Eduard Molien who was a teacher at a Gymnasium in Riga. Theodor attended the Gymnasium where his father taught, and from there he entered the Faculty of Physics and Mathematics at the University of Dorpat. Well it is not strictly accurate to call it the University of Dorpat for, although this is its name today, this was not its name when Molien entered. The town, known today as Tartu in Estonia, was known as Derpt up to 1893 when it became Yuryev although Dorpat was the German name for the town and, for consistency, we shall refer to it as Dorpat. Molien graduated from the University of Dorpat in 1883 after studying astronomy under A Lindstedt as well as mathematics, and continued to work there to become a university professor. He was sent to Leipzig University later in 1883 as part of his studies, and there he attended lectures by Klein and wrote a Master's thesis under Klein's supervision. Molien also attended lectures by Carl Neumann, E Study , W Killing and G Scheffers, before returning to Dorpat where he submitted his Master's thesis and was examined. After this he became a Privatdozent in Dorpat. In his doctoral thesis On higher complex numbers which was examined in 1892, Molien classified the complex semisimple algebras; later Cartan classified the real semisimple algebras and Wedderburn in 1907 gave the result for semisimple algebras over an arbitrary field. Molien introduced the idea of a group ring in his study of group representations. Around the same time Frobenius produced similar results by different techniques. Molien published important papers such as Uber Systeme höherer complexer Zahlen (1893) and Uber die Invarianten der linearen Substitutionsgruppen (1897). This last paper is discussed in detail in [7]. In a letter written to Dedekind on 24 February 1898, Frobenius wrote (see [2] or [5]):You will have noticed that a young mathematician, Theodor Molien in Dorpat, has considered the group determinant independently of me. In volume 41 of the Mathematische Annalen he published a very beautiful, but difficult, work "Uber Systeme höherer complexer Zahlen", in which he has investigated noncommutative multiplication and obtained important general results of which the properties of the group determinant are special cases. Since he was entirely unknown to me, I have made some inquiries regarding his personal circumstances. Details are still lacking. This much I have already learned: that he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Molin.html (1 of 3) [2/16/2002 11:23:27 PM]

Molin

is still a Privatdozent in Dorpat; that his position there is uncertain and that he has not advanced as far as he would have deserved in view of his undoubtedly strong mathematical talent. I would very much like to interest you in this talented young man; here and there you are virtually privy councillor; if an opportunity presents itself, please think of Herr Molien, and if you have time, look at his work. It is not known whether Dedekind made any attempts to help Molien with his career in Dorpat. Certainly he did not advance to a professorship so by 1900 he was looking to find such a post elsewhere. He was appointed professor of mathematics in Tomsk at the Technological Institute in 1900 and he remained in Tomsk for the rest of his career, moving to the chair of mathematics in Tomsk University in 1918. Bashmakova points out in [1] that once Molien moved to Tomsk, in west-central Siberia:... he found himself cut off from centres of scientific activity. The outstanding work he did while at Dorpat never seems to have been so successfully followed up once he went to Tomsk. While still in Dorpat, however, Molien studied Frobenius's work on character theory (as Frobenius had studied the work of Molien) and used it to study polynomial invariants of finite groups. Molien studied how many times a given irreducible representation of a finite group appears in a complete reduction of the representation of the group on the vector space of homogeneous polynomials of degree n over the complex numbers. He gave a generating function to compute the number of times the irreducible character occurs in 1898. Emmy Noether, referring to Molien's paper Uber Systeme höherer complexer Zahlen (1893), wrote (see for example [2]):The most general theorems about algebras go back to Molien. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Molin.html (2 of 3) [2/16/2002 11:23:27 PM]

Molin

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Molin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Molin.html (3 of 3) [2/16/2002 11:23:27 PM]

Monge

Gaspard Monge Born: 9 May 1746 in Beaune, Bourgogne, France Died: 28 July 1818 in Paris, France

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gaspard Monge became the Conte de Péluse later in his life and he is sometimes known by this name. His father was Jacques Monge, a merchant who came originally from Haute-Savoie in southeastern France. Gaspard's mother, whose maiden name was Jeanne Rousseaux, was a native of Burgundy and it was in the town of Beaune in Burgundy that Gaspard was brought up. Around the time that Gaspard was born Beaune, after a period of decline, was becoming prosperous again due to the success of the wine trade. Monge attended the Oratorian College in Beaune. This school was intended for young nobles and was run by priests. The school offered a more liberal education than other religious schools, providing instruction not only in the humanities but also in history, mathematics, and the natural sciences. It was at this school that Monge first showed his brilliance. In 1762, at the age of 16, Monge went to Lyons where he continued his education at the Collège de la Trinité. Despite being only 17 years of age at the time, Monge was put in charge of teaching a course in physics. Completing his education there in 1764, Monge returned to Beaune where he drew up a plan of the city. The plan of Beaune that Monge constructed was to have a major influence in the direction that his career took, for the plan was seen by a member of staff at the Ecole Royale du Génie at Mézières. He was very impressed by Monge's work and, in 1765, Monge was appointed to the Ecole Royale du Génie as a draftsman. Of course, in this post Monge was undertaking tasks that were not entirely to his liking, for he aspired to a position in life which made far more use of his mathematical talents. However the Ecole Royale du Génie brought Monge into contact with Charles Bossut who was the professor of mathematics there. At first Monge's post did not require him to use his mathematical talents, but Monge worked in his

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Monge.html (1 of 7) [2/16/2002 11:23:29 PM]

Monge

own time developing his own ideas of geometry. About a year after becoming a draftsman, Monge was given a task which allowed him to use his mathematical skill to attack the task he was given. Asked to draw up a fortification plan which prevented an enemy from either seeing or firing at a military position no matter what the position of the enemy, Monge devised his own graphical method to construct such a fortification rather than use the complicated methods then available. This method made full use of the geometrical techniques which Monge was developing in his own time. His mathematical abilities were now recognised at the Ecole Royale du Génie and it was realised that Monge was someone with exceptional abilities in both theoretical and practical subjects. Bossut was elected to the Académie des Sciences in 1768 and he left the Ecole in Mézières to became professor of hydrodynamics at the Louvre. On 22 January 1769 Monge wrote to Bossut explaining that he was writing a work on the evolutes of curves of double curvature. He asked Bossut to give an opinion on the originality and usefulness of the work. Bossut must have replied in a very positive fashion for in June a publication in the Journal Encyclopedique by Monge (his first publication) appeared giving a summary of the results which he had obtained. This paper, in which Monge generalised the results obtained by Huygens on space curves (as part of Huygens's investigation of the pendulum) and added many important new discoveries, is described in detail in [19]. The completed work was submitted to the Académie des Sciences in Paris in October 1770 and read before the Académie in August 1771 (although it was not published by the Académie until 1785). When Bossut left the Ecole Royale du Génie at Mézières, Monge was appointed to succeed him in January 1769. In 1770 he received an additional post at the Ecole Royale du Génie when he was appointed as instructor in experimental physics. Although this was a large step forward for Monge's career, he was more interested in making his name as a mathematician in the highest circles. Realising that he had to obtain advice from the leading mathematicians, Monge approached d'Alembert and Condorcet early in 1771. Condorcet must have been impressed with the depth of the mathematics that Monge showed him, for he recommended that he present memoirs to the Académie des Sciences in each of the four areas of mathematics in which he was undertaking research. The four memoirs that Monge submitted to the Académie were on a generalisation of the calculus of variations, infinitesimal geometry, the theory of partial differential equations, and combinatorics. Over the next few years he submitted a series of important papers to the Académie on partial differential equations which he studied from a geometrical point of view. His interest in subjects other than mathematics began to grow and he became interested in problems in both physics and chemistry. In 1777 Monge married Catherine Huart and, since his wife had a forge, he became interested in metallurgy in addition to his wide range of mathematical and scientific interests. Still deeply involved in teaching at the Ecole Royale du Génie at Mézières he organised the setting up of a chemistry laboratory there. From 1780, however, he devoted less time to his work at the Ecole at Mézières since in that year he was elected as adjoint géomètre at the Académie des Sciences in Paris. From that time he spent long periods in Paris, teaching a course in hydrodynamics as a substitute for Bossut as well as participating in projects undertaken by the Académie in mathematics, physics and chemistry. It was not possible to do all this and to teach all his courses at Mézières but he kept his posts there and received his full salary out of which he paid others to teach some courses in his place.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Monge.html (2 of 7) [2/16/2002 11:23:29 PM]

Monge

After three years of dividing his time between Paris and Mézières, Monge was offered yet another post, namely to replace Bézout as examiner of naval cadets. Monge would have liked to keep all these positions, but after attempting to organise an impossible schedule for about a year, he decided that he would have to resign his posts in Mézières, which he did in December 1784. Over the next five years, despite heavy duties as an examiner, Monge undertook research in a wide range of scientific subjects presenting papers to the Académie on [1]:...the composition of nitrous acid, the generation of curved surfaces, finite difference equations, partial differential equations (1785); double refraction and the structure of Iceland spar, the composition of iron, steel, and cast iron, and the action of electricity sparks on carbon dioxide gas (1786); capillary phenomena (1787); and the causes of certain meteorological phenomena (1788); and a study in physiological optics (1789). Now of course 1789 was an eventful year in French history with the storming of the Bastille on 14 July 1789 marking the start of the French Revolution. This was to completely change the course of Monge's life. At the onset of the Revolution he was one of the leading scientists in Paris with an outstanding research record in a wide variety of sciences, experience as an examiner and experience in school reforms which he had undertaken in 1786 as part of his duties as an examiner. Politically Monge was a strong supporter of the Revolution, and his first actions were to show his support by joining various societies supporting the Revolution, but he continued his normal duties as an examiner of naval cadets, and as a major figure in the work of the Académie. By this time he was on the major Académie Commission on Weights and Measures. Louis XVI attempted to flee the country on 20 June 1791, but was stopped at Varennes and brought back to Paris, and this put an end to attempts to share government between the king and an assembly. Relations with Europe deteriorated when the National Assembly declared that a people had the right of self-determination. France declared war on Austria and Prussia on 20 April 1792. French defeats led to unrest in France and, on 10 August 1792, there was further revolutions by the people with nobles and clergy murdered during September. On 21 September the monarchy was abolished in France and a republic was declared. Monge was offered the post of Minister of the Navy in the government by the National Convention. Without disrespect to Monge, it was impossible to satisfy the quite extreme views of many people, and Monge's period as Minister of the Navy cannot be viewed as a success. Although he tried hard in difficult circumstances, he survived only eight months in the post before he gave up the incessant battle with those around him, and he submitted his resignation on 10 April 1793. For a few months Monge returned to his work with the Académie des Sciences but this did not last long for, on 8 August 1793, the Académie des Sciences was abolished by the National Convention. Still a strong republican and supporter of the Revolution, Monge worked on various military projects relating to arms and explosives. He wrote papers on the topics and also gave courses on these military topics. He continued to serve on the Commission on Weights and Measures which survived despite ending the Académie des Sciences. He also proposed educational reforms to the National Convention but, despite being accepted on 15 September 1793, it was rejected on the following day. Such was the volatile nature of decisions at this unstable time. Monge was appointed by the National Convention on 11 March 1794 to the body that was put in place to establish the Ecole Centrale des Travaux Publics (soon to become the Ecole Polytechnique). Not only http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Monge.html (3 of 7) [2/16/2002 11:23:29 PM]

Monge

was he a major influence in setting up the Ecole using his experience at Mézières to good effect, but he was appointed as an instructor in descriptive geometry on 9 November 1794. His first task as instructor was to train future teachers of the school which began to operate from June 1795. Monge's lectures on infinitesimal geometry were to form the basis of his book Application de l'analyse à la géométrie. Another educational establishment, the Ecole Normale, was set up to train secondary school teachers and Monge gave a course on descriptive geometry. He was also a strong believer in the Académie des Sciences and worked hard to see it reinstated as the Institut National. The National Convention approved the new body on 26 October 1795. However from May 1796 to October 1797, Monge was in Italy on a commission to select the best art treasures for the conquerors and bring them to France. Of particular significance was the fact that he became friendly with Napoleon Bonaparte during his time in Italy. Napoleon had defeated Austria and signed the Treaty of Campo Formio on 17 October 1797 which was an exceptionally good treaty for France, preserving most of the French conquests. Monge returned to Paris bringing the text of the Treaty of Campo Formio with him. Back in Paris Monge slotted back into his previous roles and was appointed to the prestigious new one of Director of the Ecole Polytechnique. By February 1798 Monge was back in Rome, involved with the setting up of the Republic of Rome. In [17] the author describes these events using letters which Monge sent to his wife from Rome at that time. In particular Monge proposed a project for advanced schools in the Republic of Rome. Napoleon Bonaparte now asked Monge to join him on his Egyptian expedition and, somewhat reluctantly, Monge agreed. Monge left Italy on 26 May 1798 and joined Napoleon's expeditionary force. The expedition, which included the mathematicians Fourier and Malus as well as Monge, was at first a great success. Malta was occupied on 10 June 1798, Alexandria taken by storm on 1 July, and the delta of the Nile quickly taken. However, on 1 August 1798 the French fleet was completely destroyed by Nelson's fleet in the Battle of the Nile, so that Napoleon found himself confined to the land that he was occupying. Monge was appointed president of the Institut d'Egypte in Cairo on 21 August. The Institut had twelve members of the mathematics division, including Fourier, Monge, Malus and Napoleon Bonaparte. During difficult times with Napoleon in Egypt and Syria, Monge continued to work on perfecting his treatise Application de l'analyse à la géométrie. Napoleon abandoned his army and returned to Paris in 1799, he soon held absolute power in France. Monge was back in Paris on 16 October 1799 and took up his role as director of the Ecole Polytechnique. He discovered that his memoir Géométrie descriptive had been published earlier in 1799. This had been done at his wife's request and had been put together by Hachette from Monge's lectures at the Ecole Normale. On 9 November 1799 Napoleon and two others seized power in a coup and a new government, the Consulate, was set up. Napoleon named Monge a senator on the Consulate for life. Monge accepted with pleasure, although his republican views should have meant that he was opposed to the military dictatorship imposed by Napoleon on France. The truth must be that Monge was [1]:... dazzled by Napoleon ... and accepted all the honours and gifts the emperor bestowed upon him: grand officer of the Legion of Honour in 1804, president of the Senate in 1806, Count of Péluse in 1808, among others. Over the next few years Monge continued a whole range of activities, undertaking his role as a senator while maintaining an interest in research in mathematics but mostly his mathematical work involved

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Monge.html (4 of 7) [2/16/2002 11:23:29 PM]

Monge

teaching and writing texts for the students at the Ecole Polytechnique. Slowly he became less involved in mathematical research, then from 1809 he gave up his teaching at the Ecole Polytechnique as his health began to fail. In June 1812 Napoleon assembled his Grande Armée of about 453,000 men, including men from Prussia and from Austria who were forced to serve, and marched on Russia. The campaign was a disaster but by September Napoleon's army had entered a deserted Moscow. Napoleon withdrew, the Prussians and Austrians deserted the Grande Armée and in there were attempts at a coup against Napoleon in Paris. Monge was dismayed at the situation and his health suddenly collapsed. Slowly his health returned after Napoleon left the remains of his army and returned to Paris to assert his authority. After Napoleon had some military success in 1813, the allied armies against him strengthen. Monge was sent to Liège to organise the defence of the town against an attack. The allied armies began to move against France and Monge fled. When Napoleon abdicated on 6 April 1814, Monge was not in Paris, but soon after he did return and tried to pick up his life again. Napoleon escaped from Elba, where he had been banished, and by 20 March 1815 he was back in Paris. Monge immediately rallied to Napoleon and gave him his full support. After Napoleon was defeated at Waterloo, Monge continued to see him until he was put on board a ship on 15 July. By October Monge feared for his life and fled from France. Monge returned to Paris in March 1816. Two days after his return he was expelled from the Institut de France and from then on his life was desperately difficult as he was harassed politically and his life was continually threatened. On his death the students of the Ecole Polytechnique paid tribute to him despite the insistence of the French Government that no tributes should be paid. In [9] Monge's political career is treated kindly but G Jorland, in a review of that paper, takes a harder view:[Monge's] tenure at the Ministry of the Navy was a complete failure and he presided over the cultural pillage of Italy and Egypt. If Napoleon actually said that Monge loved him like a mistress, it proves that the utmost mathematical clarity can go hand in hand with political blindness. We have commented quite frequently regarding Monge's scientific work above. He is considered the father of differential geometry because of his work Application de l'analyse à la géométrie where he introduced the concept of lines of curvature of a surface in 3-space. He developed a general method of applying geometry to problems of construction. He also introduced two planes of projection at right angles to each other for graphical description of solid objects. These techniques were generalised into a system called géométrie descriptive, which is now known as orthographic projection, the graphical method used in modern mechanical drawing. The basic philosophy behind Monge's approach to mathematics is discussed in [13] where the author states that Monge's aims were the:... geometrisation of mathematics based on: (a) the analogy or correspondence of operations in analysis with geometric transformations; (b) the genetic classification and parametrisation of surfaces through analysis of the movement of generating lines. Monge regarded analysis as being [13]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Monge.html (5 of 7) [2/16/2002 11:23:29 PM]

Monge

... not a self-contained language but merely the 'script' of the 'moving geometrical spectacle' that constitutes reality. His:... new approach addressed itself to the most profound, intimate and universal relations in space and their transformations, putting him in a position to interconnect geometry and analysis in a fertile, previously unheard-of fashion. Practical concerns induced Monge to perceive the object and function of mathematics in a new way, in violation of the formalistic (linguistic) standards set by the approved patrons of mathematics ... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (25 books/articles) A Poster of Gaspard Monge

Mathematicians born in the same country

Cross-references to History Topics

1. An overview of the history of mathematics 2. A history of group theory

Other references in MacTutor

1. Chronology: 1760 to 1780 2. Chronology: 1780 to 1800

Honours awarded to Gaspard Monge (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Monge

Paris street names

Place Monge and Rue Monge (5th Arrondissement)

Commemorated on the Eiffel Tower Other Web sites

1. Rouse Ball 2. Minnesota (One of Monge's geometry theorems and its relationship to Desargues theorem) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Monge.html (6 of 7) [2/16/2002 11:23:29 PM]

Monge

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Monge.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Monge.html (7 of 7) [2/16/2002 11:23:29 PM]

Monte

Guidobaldo Marchese del Monte Born: 11 Jan 1545 in Pesaro, Italy Died: 6 Jan 1607 in Montebaroccio, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Guidobaldo del Monte studied at the University of Padua. After this he studied in Urbino. He had no need to work as he inherited the family estate of Montebaroccio when his father died as well as the title Marchese del Monte. His father had been made Marchese del Monte by Duke Guidobaldo II of Urbino. For a while Guidobaldo served in the army during the war against the Ottoman in Hungary. The Ottomans had occupied and annexed Hungary in 1541 which resulted in almost continuous conflict over many years. After serving in the army, Guidobaldo returned to his estate where he was able to spend his time doing research into mathematics, mechanics, astronomy and optics. Guidobaldo's book Liber mechanicorum (1577) was regarded as the greatest work on statics since Greek times. It was a return to classical Greek rigour deliberately rejecting the approach of Jordanus, Tartaglia and Cardan. His approach was adopted by Galileo who was Guidobaldo's friend for 20 years. Guidobaldo also wrote astronomy books, for example Planisphaeriorum (1579) and Problematum astronomicorum (1609). In 1600 he published a book on perspective Perspectivae libri sex. He wrote on refraction in water but it was unpublished on his death. Also interested in machines of many different types, Guidobaldo wrote on the Archimedean screw to raise water. He also invented a number of instruments including mathematical instruments and compasses. He corresponded with several mathematicians including Barocius and, as mentioned above, Galileo.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Monte.html (1 of 2) [2/16/2002 11:23:30 PM]

Monte

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Guidobaldo del Monte

Mathematicians born in the same country

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Monte.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Monte.html (2 of 2) [2/16/2002 11:23:30 PM]

Montel

Paul Antoine Aristide Montel Born: 29 April 1876 in Nice, France Died: 22 Jan 1975 in Paris, France

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Montel was educated at the Lycée in Nice. In 1894 he entered Ecole Normale Supérieure in Paris. Montel then taught at several lycées but was persuaded by friends to return to Paris and work on a doctorate. This he did and obtained his doctorate in 1907. He was appointed to his first university post in Paris in 1918 at the age of 42. He worked mostly on the theory of analytic functions of a complex variable. He introduced a set of functions called a normal family and used these ideas to simplify classical results in function theory such as the mapping theorem of Riemann and Hadamard's characterisation of entire functions of finite order. Montel also investigated the relation between the coefficients of a polynomial and the location of its zeros in the complex plane. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Montel.html (1 of 2) [2/16/2002 11:23:32 PM]

Montel

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Montel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Montel.html (2 of 2) [2/16/2002 11:23:32 PM]

Montmort

Pierre Rémond de Montmort Born: 27 Oct 1678 in Paris, France Died: 7 Oct 1719 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Rémond (only later to become de Montmort) followed his father's advice and began to study law. However he became bored with this subject and decided to go abroad. He went to England and toured round the country, then going to Germany and again visiting a number of places. By the age of 21 he was back in France where he began to study under Malebranche. Malebranche taught Pierre philosophy and Descartes's physics. Pierre went on to study the latest mathematics, in particular studying algebra and geometry. When Pierre returned to France in 1699 he came into a large inheritance from his father. He used this wealth to purchase an estate at Montmort (and therefore became Pierre Rémond de Montmort). He lived most of his life in Château de Montmort on his estate and often invited top mathematicians to visit him. For instance Nicolaus(I) Bernoulli spent three months at Château de Montmort. Montmort's reputation was made by his book on probability Essay d'analyse sur les jeux de hazard which appeared in 1708. The book, which is a collection of combinatorial problems, is a systematic study of games of chance and shows that there is important mathematics in this area. Montmort collaborated with Nicolaus(I) Bernoulli and he was also a friend of Taylor. At a time of high feelings in the Newton-Leibniz controversy it says a lot for Montmort that he could be friends with followers of both camps. In addition to those mentioned above, Montmort corresponded with Craig, Halley, Hermann and Poleni. Montmort was elected to be a Fellow of the Royal Society in 1715, when he was on a trip to England. The following year he was elected to the Académie Royal des Sciences. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Montmort.html (1 of 2) [2/16/2002 11:23:33 PM]

Montmort

Mathematicians born in the same country Cross-references to History Topics

Mathematical games and recreations

Honours awarded to Pierre Rémond (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1715

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Montmort.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Montmort.html (2 of 2) [2/16/2002 11:23:33 PM]

Montucla

Jean Etienne Montucla Born: 5 Sept 1725 in Lyon, France Died: 18 Dec 1799 in Versailles, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean Montucla was educated in the Jesuit College in his home town of Lyon. Then he went to Paris where he became friends with Diderot and d'Alembert. He had many friends among the top mathematicians of his day, in particular he was much influenced by his two friends d'Alembert and Diderot. Montucla worked at the Gazette de Paris at this time. In 1761 he went to Grenoble where he was appointed to the position of secretary to the Intendant. He went to Cayenne in 1764 as part of an astronomical expedition. After his return to France he received two major appointments, as Head Clerk of Royal Buildings and as Royal Censor. In 1788 Fourier sent an algebra paper to Montucla for his opinion. Fourier did not receive an opinion and wrote in a letter a year later:I begin to take M Montucla at his word when he tells us he has fallen out with learned analysis: I wait calmly for him to be reconciled with it. This may mean that Montucla had stopped doing mathematics at this time but Fourier's comment is not properly understood. Not surprisingly Montucla's royal posts vanished when the French Revolution began. At this time he retired and went to live at Versailles where he worked on history of mathematics texts. Montucla wrote a number of history of mathematics books. The first, in 1754, was a history of the problem of squaring the circle and while the subject has changed greatly since, this is still a classic of its kind. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Montucla.html (1 of 2) [2/16/2002 11:23:35 PM]

Montucla

His other major work was published in two volumes in 1758 and, together with the work by Wallis, is the most esteemed of all the early works on the History of Mathematics. Montluca greatly enlarged and improved a book written by Jacques Ozanam on mathematical puzzles and recreations. His work was influential in popularising geometric dissection problems involve the cutting of geometric figures into pieces that can be arranged to form other geometric figures. Charles Hutton translated Montucla's mathematical puzzles book into English in 1803. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles)

A Quotation

A Poster of Jean Montucla

Mathematicians born in the same country

Cross-references to History Topics

1. Mathematical games and recreations 2. Squaring the circle

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Montucla.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Montucla.html (2 of 2) [2/16/2002 11:23:35 PM]

Moore_Eliakim

Eliakim Hastings Moore Born: 26 Jan 1862 in Marietta, Ohio, USA Died: 30 Dec 1932 in Chicago, Illinois, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eliakim Moore's father was David Hastings Moore while his mother was Julia Sophia Carpenter. David Moore was a Methodist minister and he and his wife gave their son Eliakim an excellent education. As a child Eliakim played with Martha Morris Young and they would later marry. He attended Woodward High School from 1876 to 1879 and there he prepared for his studies at Yale University. It was at this time that he became interested in mathematics and this happened through a summer job which he took. One summer he worked as an assistant to Ormond Stone, who was the director of the Cincinnati Observatory, and from this time on Eliakim knew that he wanted to study mathematics and astronomy at university. In 1879 Moore entered Yale University and took, as he had planned, courses in mathematics and astronomy. His interests became more clearly in the area of mathematics and he received his B.A. in 1883. He remained at Yale to study for his Ph.D. under the supervision of Hubert Anson Newton. Moore's doctoral dissertation was entitled Extensions of Certain Theorems of Clifford and Cayley in the Geometry of n Dimensions and this led to the award of his doctorate in 1885. Newton encouraged Moore to go to Europe for a year and helped to finance the trip. Moore spent the year in Germany, going first to Göttingen where he spent the summer of 1885 studying the German language, but spending most of the academic year 1885-86 attending lectures by Kronecker and Weierstrass at the University of Berlin. Parshall writes in [13]:While direct influences of this German study tour on Moore's subsequent mathematical career are difficult to isolate, it is undeniable that Moore returned to the United States with a sense of the importance and desirability of the active and sustained pursuit of original

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Eliakim.html (1 of 4) [2/16/2002 11:23:37 PM]

Moore_Eliakim

research. It is worth adding that American academics were really forced to train in Europe in Moore's day but when six years later he set up his research school in Chicago it provided for the first time the opportunity for American mathematicians to train in a research-intensive environment in the United States. On his return to the United States, Moore was appointed as an instructor at Northwestern University for the year 1886-87, then he spent two years as a mathematics tutor at Yale before spending the years 1889 to 1892 back at Northwestern University. During this second spell at Northwestern Moore was approached by William Rainey Harper and offered a post at Chautauqua in New York. Moore refused, knowing that this would mean taking up a post which involved a lot of low level teaching, while he wanted to concentrate on a research. Perhaps it was a fortunate episode since, a year later in 1891, Harper was President-designate of the University of Chicago and wanted to staff the mathematics department with research active staff. Now he could offer Moore the post he wanted to satisfy his research ambitions and Moore quickly accepted. Moore was appointed professor and acting head of the mathematics department at Chicago when the university first opened in 1892. Prior to this he had married Martha Morris Young on 21 June 1893 in Columbus, Ohio; they had been friends from their childhood days. The Moores had two sons, only one of whom reached adulthood. In 1896 Moore became head of the mathematics department at Chicago, a post he retained until 1931. When he was appointed at Chicago, Moore persuaded the university authorities to appoint two young German mathematicians Bolza and Maschke to his department. Archibald, in [3], describes this Chicago mathematics team:These three men supplemented one another remarkably. Moore was a fiery enthusiast, brilliant, and keenly interested in the popular mathematical research movements of the day; Bolza, a product of the meticulous German school of analysis led by Weierstrass, was an able, and widely read research scholar; Maschke was more deliberate than the other two, sagacious, brilliant in research, and a most delightful lecturer in geometry. During the period 1892-1908 the University of Chicago was unsurpassed in America as an institution for the study of higher mathematics. Among Moore's Ph.D. students at Chicago were Dickson, Veblen, Anna Pell Wheeler and G D Birkhoff. Although Robert Moore had Veblen as his supervisor in Chicago he worked with, and was strongly influenced by, Eliakim Moore. Moore's first main areas of research, which he studied from about 1892 to 1900, were algebra and groups where he proved in 1893 that every finite field is a Galois field. He also studied infinite series of finite simple groups. In his work on the foundations of geometry begun around 1900 Moore examined the independence of Hilbert's axioms. He reformulated these in terms of points as the only undefined quantities, rather than points, lines and planes as Hilbert had done. His 1902 paper On the projective axioms of geometry showed that Hilbert's axiom system contained redundant axioms. In [1] Moore is described as follows:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Eliakim.html (2 of 4) [2/16/2002 11:23:37 PM]

Moore_Eliakim

Although a gentle man, he sometimes displayed impatience as he strove for excellence in his classes. Parshall in [13] writes in a similar vein:Moore's style of teaching - characterised by a quick-paced presentation of new research ideas and what could be stinging impatience with those who failed to follow - certainly proved effective, at least for the most talented. Moore was a man of high intellectual and academic standards; he expected much from himself, his students, and his colleagues. The third interest that Moore took up after 1906 was on the foundations of analysis [1]:He brought to culmination the study of improper integrals before the appearance of the more effective integration theories of Borel and Lebesgue. He diligently advanced general analysis, which for him meant the development of a theory of classes of functions on a general range. ... Throughout his work in general analysis, Moore stressed fundamentals, as he sought to strengthen the foundations of mathematics. Moore brought precision and rigour to all the fields he studied. Other topics he worked on include algebraic geometry, number theory and integral equations. In 1893 Moore was one of the main organisers of the first international mathematical congress to be held in the United States. He then approached the New York Mathematical Society with a view to publishing the Proceedings of the congress. In the process he persuaded this society to take a more national role and to change their name to the American Mathematical Society. Setting up a Chicago branch of the Society, which he led, Moore helped in the having the Society reach across the United States. He became a strong supporter of the American Mathematical Society being Vice-President from 1898 to 1900, President from 1901 to 1902 and Colloquium Lecturer in 1906. He acted as an editor of the Transactions of the American Mathematical Society from 1899 to 1907. In [3] his contribution is summed up as follows:Moore was an extraordinary genius, vivid, imaginative, sympathetic, foremost leader in freeing American mathematicians from dependence on foreign universities, and in building up a vigorous American School, drawing unto itself workers from all parts of the world. Moore received many honours from around the world for his contributions. He was elected to the National Academy of Sciences, the American Academy of Arts and Sciences, and the American Philosophical Society. He received honorary degrees from Göttingen, Yale, Clark, Toronto, Kansas, and Northwestern.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles)

Some Quotations (2)

Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Eliakim.html (3 of 4) [2/16/2002 11:23:37 PM]

Moore_Eliakim

Honours awarded to Eliakim Moore (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1901 - 1902

AMS Colloquium Lecturer

1906

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Moore_Eliakim.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Eliakim.html (4 of 4) [2/16/2002 11:23:37 PM]

Moore_Jonas

Jonas Moore Born: 8 Feb 1627 in Whitelee, Pendle Forest, Lancashire, England Died: 25 Aug 1679 in Godalming, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jonas Moore became clerk to the Chancellor of Durham. In 1640, at the age of 13, he took up the study of mathematics. Moore was greatly influenced by Oughtred in his mathematical studies. In 1647 he became mathematics tutor to the Duke of York, brother of the future King Charles II, but he did not hold this job for very long since he was removed from the post by political intrigue. Indeed this was a time of political turmoil in England with battles and religious argument. Only two years later Charles I was executed and his son proclaimed Charles II by the Scots in defiance of the English. Charles II was defeated by Cromwell in 1650 and by 1651 was in exile in France. After loosing his post as tutor to the Duke of York (whose father Charles I was still King at this time) Moore went to London where he hoped to make a living as a teacher of mathematics. However, he found it difficult to find sufficiently many pupils so Moore was happy to be appointed as a surveyor in 1649. His task was to work on the draining of the Fens, a natural region of about 40,100 sq km of reclaimed marshland in eastern England between Lincoln and Cambridge. Around the same time as he took up this post as a surveyor, he published a mathematical textbook Arithmetick (1650). Moore made a reputation for himself in this job and soon was appointed to other surveying jobs. In [3] Moore's work at this time is described as follows:He gained reputation by his success in keeping the sea out of Norfolk, surveyed the coasts, and constructed a map of Cambridgeshire... The Convention Parliament of 1660 declared the restoration of the king and lords. They disbanded the army and established an income for the king, Charles II, by keeping the parliamentary innovation of the excise tax. Charles II returned to London and Moore republished his Arithmetick together with A New http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Jonas.html (1 of 3) [2/16/2002 11:23:39 PM]

Moore_Jonas

Contemplation General upon the Ellipsis and Conical Sections taken from Mydorge. The dedication of the republished work shows that Moore was working hard to find favour with the new regime. In 1663 Moore was sent to Tangier to conduct a survey and to report on its fortifications. Here he was involved in the ambitious project to build a massive harbour wall. He received a knighthood in 1669 and was appointed to high office as Surveyor-General of the Ordnance. However Moore is not particularly famous for the mathematics which he did: as a mathematician he is best known as the first to use the notation cot. Rather Moore is famous for his strong support of mathematics and astronomy which made many other mathematical and astronomical advances possible. Perhaps his most important contribution was in his efforts to set up the Royal Observatory at Greenwich and his efforts to support Flamsteed. In 1674 he invited Flamsteed to London, see [3]:... with the design of installing him in a small observatory of his own in Chelsea College, but procured from the king instead the foundation of the Royal Observatory. He furnished him, moreover, at his private expense, with a seven foot sextant employed in Flamsteed's observations until 1688 as well as two clocks... These clocks were used by Flamsteed in his work involving finding the longitude. A recent work by Willmoth [2] goes futher than earlier authors in describing Moore's contribution to the founding of the Royal Observatory claiming, with much supporting evidence, that he was:... the sole driving force behind the scheme. Moore, together with the famous diary writer Samuel Pepys, founded the Royal Mathematical School within Christ's Hospital. This School was set up with the specific aim of training boys in navigation techniques so that they could serve the King at sea. Moore became a governor of the school and together with Perkins, a master at the school, he wrote a major mathematical work intended for use at the Royal Mathematical School. Moore died however before the work could be published. The work, A New system of the Mathematicks appeared in 1681. Moore wrote the sections on arithmetic, geometry, trigonometry and cosmography while the sections on algebra, Euclid and navigation were written by Perkins. Aubrey describes Moore in [1] as:... one of the most accomplished gentlemen of his time: a good mathematician, and a good fellow. ... he was tall and very fat, thin skin, fair, clear grey eyes ... Moore died in 1679 while on a journey from Portsmouth to London. He wrote several books, other than those described above, including Modern Fortification (1673) and A Mathematical Compendium (1674). His Arithmetick was published for a third time in 1698, nearly 20 years after his death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Jonas.html (2 of 3) [2/16/2002 11:23:39 PM]

Moore_Jonas

Cross-references to History Topics

1. English attack on the Longitude Problem 2. The trigonometric functions

Honours awarded to Jonas Moore (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1674

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Moore_Jonas.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Jonas.html (3 of 3) [2/16/2002 11:23:39 PM]

Moore_Robert

Robert Lee Moore Born: 14 Nov 1882 in Dallas, Texas, USA Died: 4 Oct 1974 in Austin, Texas, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Lee Moore's father, Charles Jonathan Moore, owned a hardware store in Dallas. Originally from Connecticut, Charles had moved to the south of the United States during the civil war to fight on the side of the South. Robert Lee Moore's mother was Louisa Ann Moore and she did not need to change her name on marrying Charles since her maiden name was also Moore. Charles and Louisa had six children, with Robert being the second youngest in the family. Robert received a good education at a private high school in Dallas, and before he entered university he had learnt university level calculus by studying the university textbooks. He entered the University of Texas in 1898 and there he took courses by Halsted and Dickson. He graduated with a Sc.B. in 1901 and after a year as a teaching fellow at the University of Texas, Moore spent the academic year 1902-03 as a mathematics instructor at the High School in Marshall, Texas. In fact Moore would have remained at Texas University rather than spend the year teaching in a high school but, for some reason which is not clear, the university regents refused to renew his appointment despite strong protests from Halsted. Now Halsted had suggested a problem in one of his classes which had led Moore to prove that one of Hilbert's geometry axioms was redundant. Eliakim Moore, who was the head of mathematics at Chicago University, heard of this contribution and, since his research interests at the time were precisely on the foundations of geometry, Eliakim Moore organised the award of a scholarship that would allow Robert Moore to study for his doctorate in Chicago. We should note that despite the fact that Eliakim Moore and Robert Moore shared the same surname and the same research interests, they were not related. Veblen

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Robert.html (1 of 5) [2/16/2002 11:23:41 PM]

Moore_Robert

supervised Moore's Ph.D. at University of Chicago and the degree was awarded in 1905 for a dissertation entitled Sets of Metrical Hypotheses for Geometry. It was while Moore was attending lectures in Chicago during this period that he first hit on his original teaching methods [5]:With his quick mind and restless spirit he found the lecture method rather boring - in fact, mind dulling. To liven up a lecture he would run a race with his professor by seeing if he could discover the proof of an announced theorem before the lecturer had finished his presentation. Quite frequently he won the race. But in any case, he felt that he was better off from having made the attempt. Moore spent the year 1905-06 as an assistant professor at the University of Tennessee, then two years as an instructor at Princeton University. In 1908 he was appointed as an instructor at the Northwestern University and then, after three years there, he went to the University of Pennsylvania in 1911. The year before, in 1910, he had married Margaret MacLelland Key of Brenham, Texas; they had no children. After a promotion to assistant professor at the University of Pennsylvania in 1916, he remained there for a further four years. It was at the University of Pennsylvania that Moore first tried out his teaching methods in a Foundations of Geometry course he taught there. He began to have success with what became known as the Moore Method of teaching [5]:Here was a fresh, relatively new area where Moore had himself tested the difficulty of some of the theorems. We shall describe the Moore Method below. Moore was appointed to the staff at the University of Texas in 1920 as an associate professor, being made a full professor three years later. Moore was delighted to return to the University of Texas, his home university. By the time he was appointed in 1920 he had published 17 papers on point-set topology (a term which he coined). For his doctoral thesis Moore had worked on the foundations of topology. In 1915 he published On a set of postulates which suffice to define a number-plane published in the Transactions of the American Mathematical Society. Writing about this paper in 1927, Chittenden wrote:The importance of the regularly and perfectly separable, therefore metric, spaces in the analysis of continua is indicated by the fact that nine years before the publication of the discoveries of Urysohn, R L Moore assumed these properties in the first of a system of axioms for the foundations of plane analysis situs. Moore wrote up his work on point-set topology in the important book Foundations of point set topology published in 1932. This volume, published in the Colloquium Lectures Series of the American Mathematical Society, arose from the colloquium lectures which Moore gave in 1929 and is a self-contained introduction to the topic concentrating on Moore's own contributions to the subject. We should comment on Moore's teaching methods, for their success influenced others to use similar methods. These methods are described by F Burton Jones, who himself was a student of Moore, and himself taught very successfully with a modified version, in [5]:Moore would begin his graduate course in topology by carefully selecting the members of the class. If a student had already studied topology elsewhere or had read too much, he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Robert.html (2 of 5) [2/16/2002 11:23:41 PM]

Moore_Robert

would exclude him (in some cases, he would run a separate class for such students). The idea was to have a class as homogeneously ignorant (topologically) as possible. He would usually caution the group not to read topology but simply to use their own ability. Plainly he wanted the competition to be as fair as possible, for competition was one of the driving forces. ... Having selected the class he would tell them briefly his view of the axiomatic method: there were certain undefined terms (e.g., "point" and "region") which had meaning restricted (or controlled) by the axioms (e.g., a region is a point set). He would then state the axioms that the class was to start with ... After stating the axioms and giving motivating examples to illustrate their meaning he would then state some definitions and theorems. He simply read them from his book as the students copied them down. He would then instruct the class to find proofs of their own and also to construct examples to show that the hypotheses of the theorems could not be weakened,' omitted, or partially omitted. When the class returned for the next meeting he would call on some student to prove Theorem 1. After he became familiar with the abilities of the class members, he would call on them in reverse order and in this way give the more unsuccessful students first chance when they did get a proof. He was not inflexible in this procedure but it was clear that he preferred it. When a student stated that he could prove Theorem x, he was asked to go to the blackboard and present his proof. Then the other students, especially those who had not been able to discover a proof, would make sure that the proof presented was correct and convincing. Moore sternly prevented heckling. This was seldom necessary because the whole atmosphere was one of a serious community effort to understand the argument. When a flaw appeared in a "proof" everyone would patiently wait for the student at the board to "patch it up." If he could not, he would sit down. Moore would then ask the next student to try or if he thought the difficulty encountered was sufficiently interesting, he would save that theorem until next time and go on to the next unproved theorem (starting again at the bottom of the class). Mary Ellen Rudin, who was also a student of Moore's presents a similar picture [3]:His way of teaching was to present you with things that had not yet been proved, and with all kinds of things which might turn out to have a counterexample, and sometimes unsolved problems - that is, unsolved by anyone, not only unsolved by you. So you had some idea of what it meant to be a mathematician - more than the average undergraduate does today. Although the Moore Method proved good for Mary Ellen Rudin, she understood that it was not right for everyone:I wouldn't for anything have let my children go to school with Moore! That is, I think that he was destructive to anyone who didn't fit exactly into his pattern, he did not succeed in giving the people that worked with him an education. It's a mistake to go to school under those circumstances in general. Moore taught at Texas until he was 86 years old, and he wished to carry on teaching but the University

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Robert.html (3 of 5) [2/16/2002 11:23:41 PM]

Moore_Robert

authorities forced him to retire. A number of students strongly supported his bid to remain in post but to no avail. The university authorities were not concerned at his abilities to teach, rather it was the great success of his methods which made his employers fear that bright young mathematicians might not wish to teach there due to his continuing dominating influence. In the picture above he is aged 87 and still in his office in Austin, Texas. The University of Texas did Moore a great honour, however, for in 1973 they named a new physics, mathematics and astronomy building after him. A strong supporter of the American Mathematical Society, Moore was an editor of the Colloquium Publications from 1929 to 1936, being editor-in-chief from 1930 to 1933. He was president of the American Mathematical Society from 1936 to 1938. He was elected to the National Academy of Sciences in 1931. Finally we should make some negative comments about his bigoted attitudes. The quotation below is from a personal communication from Chandler Davis which is based on:... conversations and correspondence with my good friend E E Moise. Chandler Davis writes:R L Moore was firmly anti-black, refusing to teach any black students. He was pretty bigoted against women and Jews too, as many anecdotes attest. Two of his supervisees who went on to brilliant careers and who remained grateful for his teaching were, however, Mary Ellen Rudin and E E Moise. Moore took quite some time, I am told, to adjust to working with a woman and with a Jew, but after he got used to it he treated them well. (Moise was of mixed background, but as he bore the name of his Jewish grandfather he was a Jew in Moore's eyes.) As Chandler Davis suggests, Mary Ellen Rudin was certainly happy with Moore [3]:He encouraged people to believe in themselves as mathematicians because he felt that this was one of the principal tools for doing mathematics - to have confidence. ... I probably would not be a mathematician had I not worked with Moore. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles)

A Quotation

Mathematicians born in the same country Honours awarded to Robert Lee Moore (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1937 - 1938

AMS Colloquium Lecturer

1929

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Robert.html (4 of 5) [2/16/2002 11:23:41 PM]

Moore_Robert

Previous

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Moore_Robert.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moore_Robert.html (5 of 5) [2/16/2002 11:23:41 PM]

Morawetz

Cathleen Synge Morawetz Born: 5 May 1923 in Toronto, Canada

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Cathleen Morawetz was christened Cathleen Synge. Her father was John Lighton Synge, a mathematician who has a biography in this archive, while her mother, Eleanor Mabel Allen Synge, also had some training as a mathematician. Both Cathleen's parents were Irish but she was born in Toronto while her father held the position of assistant professor of mathematics at the University of Toronto. However, when she was two years old the family returned to Ireland when her father was appointed to the chair of Natural Philosophy at Dublin University. When Cathleen was seven years old the family returned to Toronto and it was in Toronto that she attended school. Cathleen won a scholarship and entered the University of Toronto to study mathematics. Her parents both encouraged her interest in mathematics and science but her father jokingly said that if she became a mathematician:... we might fight like the Bernoulli brothers. Of course the years that Cathleen spent as an undergraduate at Toronto was the time of World War II and she undertook war work in 1943-44 as a technical assistant. Returning to the University of Toronto she was awarded her B.A. degree in Mathematics in 1945. Cecilia Krieger, who taught Cathleen mathematics while she was an undergraduate, had been a family friend for many years. She strongly encouraged Cathleen continue her study of mathematics. Cathleen married Herbert Morawetz, who was a chemist, on 28 October 1945. She then went to Massachusetts Institute of Technology to study for her Master's Degree which was awarded in 1946. At this stage she hesitated and considered whether she should pursue her mathematical studies further. There were few job opportunities for women with doctorates in matematics and she considered taking a job in Bell Laboratories in New Jersey. However, strongly encouraged by Cecilia Krieger to study for her http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morawetz.html (1 of 4) [2/16/2002 11:23:43 PM]

Morawetz

doctorate, Morawetz went to New York University to undertake research. She was asked to edit Courant and Friedrichs Supersonic Flow and Shock Waves and, by the time this task was completed, she was fascinated by transonic flow and associated phenomena. She wrote her doctotal thesis, which was supervised by Friedrichs, on the stability of a spherical implosion and was awarded her Ph.D. in 1951. During this period Morawetz applied for US citizenship and she was granted this in 1950. She was a research associate at the Massachusetts Institute of Technology for a short spell but returned to the Courant Institute of Mathematical Sciences of New York University in 1952 as a research associate. Steady promotion there saw her become an assistant professor in 1957, an associate professor in 1960, and the a full professor in 1965. In 1966-67 she held a prestigious Guggenheim Fellowship. In 1978 Morawetz became the associate director of the Courant Institute of Mathematical Sciences a position which she held until 1984 when she was appointed Director of the Courant Institute. With this appointment she became the first woman to hold this position, moreover she was the first woman to hold any comparable directorship within the mathematical sciences in the United States. She was elected as President of the American Mathematical Society, serving in this role in 1995-96. On the announcement that she would become President of the American Mathematical Society the Courant Institute issued a press release saying:Morawetz is an outstanding mathematician, and has lond been one of the leading lights at our prestigious Courant Institute. [We] know that the American Mathematical Society will benefit greatly by her considerable acumen and compelling leadership. The article [2], nominating her for that position, describes her remarkable research achievements:In a series of three significant papers in the late 1950s, Cathleen Morawetz used functional analysis coupled with ingenious new estimates for an equation of mixed type, i.e. with both elliptic and hyperbolic regions, to prove a striking new theorem for boundary value problems for partial differential equations. This theorem was motivated by applications and leads to a startling practical prediction. Namely, if oe starts with a smooth, steady irrotational flow around an aerodynamic profile like a wing, then in general, if one changes the shape of the profile slightly, there cannot be a smooth, steady transonic flow around the purturbed profile. Morawetz's predictions have been confirmed subsequently through both actual experiments and careful numerical simulation which indicate the appearance of shock waves in the flow past the purturbed profile. In the 1960s Morawetz worked on the scattering of sound waves and electromagnetic waves striking objects. Morawetz showed that, for a medium with constant light speed outside a reflecting star-shaped object, high frequency waves are, asymptotically, streams of particles moving along the rays. During the 1970s she extended this work to examine other solutions to the wave equation. She proved many important results relating to the non-linear wave equation. In her later work she continued to study shock waves and transonic flow. In 1998 Morawetz was awarded the National Medal of Science. Established by the United States Congress in 1959 and first awarded to Theodore von Kármán in 1962, it is the highest scientific honour which the United States can give. The citation for the award says that it was given to Morawetz:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morawetz.html (2 of 4) [2/16/2002 11:23:43 PM]

Morawetz

... for pioneering advances in partial differential equations and wave propagation resulting in applications to aerodynamics, acoustics and optics. In her speech accepting the National Medal of Science, Morawetz said:This is an occasion of great moment for me. I am filled with gratitude to all those, and there were a great many, who helped me over many years, and I am proud to be the first woman mathematician to receive the medal. My biggest wish would be that it could help move more women forward in mathematics, be it in grade school or graduate school. The list of honours which Morawetz has received certainly does not stop at those already mentioned above. She has been awarded honorary degrees by Eastern Michigan University, Smith College, and Brown University in 1982; Princeton University in 1986; and Duke University, and New Jersey Institute of Technology in 1988. In 1993 she was named Outstanding Woman Scientist by the Association for Women in Science. In 1997 she received the Krieger-Nelson Award from the Canadian Mathematical Society. She has been elected a Fellow of the American Association for the Advancement of Science and a member of the American Academy of Arts and Sciences. In addition she was the first woman member of the Applied Mathematics Section of the National Academy of Sciences. The list of her remarkable achievements gives her own joke a deeper meaning:Maybe I became a mathematician because I was so crummy at housework. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Cathleen Morawetz (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1995 - 1996

AMS Gibbs Lecturer

1981

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morawetz.html (3 of 4) [2/16/2002 11:23:43 PM]

Morawetz

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Morawetz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morawetz.html (4 of 4) [2/16/2002 11:23:43 PM]

Mordell

Louis Joel Mordell Born: 28 Jan 1888 in Philadelphia, Pennsylvania, USA Died: 12 March 1972 in Cambridge, Cambridgeshire, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Louis Mordell was educated at Cambridge and began research in number theory. He studied solutions of y2 = x3 + k which had been studied by Fermat. Thue showed that this equation had only finitely many solutions but Mordell only learned about Thue's work later. Mordell lectured at Manchester College of Technology from 1920 to 1922. During this time he discovered the result for which he is best known, namely the finite basis theorem, which proved a conjecture of Poincaré. In 1922 he went to Manchester University where he remained until he succeeded Hardy at Cambridge in 1945. Together with Davenport, he initiated great advances in the geometry of numbers. Mordell was elected a Fellow of the Royal Society in 1924, and received its Sylvester Medal in 1949:... for his distinguished researches in pure mathematics, especially for his discoveries in the theory of numbers. He was elected President of the London Mathematical Society in 1943, holding the post until 1945. He had already won the De Morgan Medal of the Society in 1941. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mordell.html (1 of 2) [2/16/2002 11:23:44 PM]

Mordell

List of References (6 books/articles)

Some Quotations (6)

A Poster of Louis Mordell

Mathematicians born in the same country

Honours awarded to Louis Mordell (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1924

Royal Society Sylvester Medal

Awarded 1949

London Maths Society President

1943 - 1945

LMS De Morgan Medal

Awarded 1941

LMS Berwick Prize winner

1946

Other Web sites

AMS (an article by S Lang about a review by Mordell)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mordell.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mordell.html (2 of 2) [2/16/2002 11:23:44 PM]

Mori

Shigefumi Mori Born: 23 Feb 1951 in Nagoya, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Shigefumi Mori attended Kyoto University and he received his B.A. from there in 1973 and his M.A. in 1975. In that year he was appointed as an assistant at Kyoto and studied there for his doctorate under Masayoshi Nagata's supervision. He was awarded his doctorate in 1978 for a thesis The endomorphism rings of some abelian varieties. After the award of his Ph.D., Mori remained as an assistant at Kyoto until 1980 when he was appointed as a lecturer in mathematics at the University of Nagoya. He was promoted to assistant professor in 1982 and, in 1988, to full professor. In 1990 Mori returned to a chair at Kyoto University. During the years from 1977 to 1988 he spent much time in the United States despite the positions he held in Japan. He was visiting professor at Harvard during 1977-1980, the Institute for Advanced Study in 1981-82, Columbia University 1985-87 and the University of Utah for periods during 1987-89 and again during 1991-92. Mori works on algebraic geometry. To put his work in perspective we should note that, as in many areas of mathematics, classification is the ultimate aim. As Hironaka writes in [4]:... to classify algebraic varieties has always been a fundamental problem of algebraic geometry and even an ultimate dream of algebraic geometers. Major progress was made on classifying algebraic surfaces during the first part of the 20th century by the great Italian algebraic geometers led by Castelnuovo, Enriques and Severi. Progress in this line continued with Zariski's contribution during the 1950s, followed by Kodaira's work in the following decade. Mori's work achieved a remarkable continuation of classification efforts in algebraic geometry and in many ways provides a fitting chapter in the progress of algebraic geometry through the 20th century. Mori was awarded a Fields Medal at the 1990 International Congress which was held in the city in which http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mori.html (1 of 3) [2/16/2002 11:23:46 PM]

Mori

he had studied as a student, namely Kyoto in Japan. He received this Medal for some remarkable work over a 12 year period. He worked on algebraic manifolds with ample tangent bundles and was the first to prove the Hartshorne conjecture in 1978. This conjecture, posed in 1970, claimed that projective spaces are the only smooth complete algebraic varieties with ample tangent bundles. In 1981 Mori completed the classification of Fano 3-folds and worked on the minimal model programme. Hironaka, speaking of Mori's work which led to the award of the Fields Medal said [4]:The most profound and exciting development in algebraic geometry during the last decade or so was the Minimal Model Program or Mori's Program in connection with th classification problems of algebraic varieties of dimension three. Shigefumi Mori initiated the program with a decisively new and powerful technique, guided the general research direction with some good collaborators along the way, and finally finished up the program by himself overcoming the last difficulty. Mori has received many other awards for his outstanding work. Before receiving the Fields Medal in 1990 he had already been awarded the Japan Mathematical Society's Yanaga Prize in 1983, the Chunichi Culture Prize in 1984. In 1988, jointly with Y Kawamata, he received a Prize from the Japan Mathematical Society for:...outstanding work in the minimal model theory for algebraic varieties. In 1989 he received the Inoue Prize for:...outstanding work in the theory of higher dimensional algebraic varieties and in particular for the proof of existence of minimal models for 3-dimensional algebraic varieties. The same year as he was awarded the Fields Medal, in 1990, Mori was awarded the Cole Prize in Algebra from the American Mathematical Society. The citation for the award stares [8]:The committee unanimously recommends that the 1990 Cole Prize in Algebra be awarded to Shigefumi Mori for his outstanding work on the classification of algebraic varieties. Mori took the decisive steps over a ten-year period in extending the classical theory of algebraic surfaces to dimension three: prior to Mori's breakthroughs this problem seemed out of reach. Mori's beautiful work also makes majr inroads into the problem in higher dimensions. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1970 to 1980

Honours awarded to Shigefumi Mori (Click a link below for the full list of mathematicians honoured in this way) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mori.html (2 of 3) [2/16/2002 11:23:46 PM]

Mori

Fields' Medal

Awarded 1990

AMS Cole Prize winner

1990

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mori.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mori.html (3 of 3) [2/16/2002 11:23:46 PM]

Moriarty

James Moriarty Born: 1835 in England Died: 1891 in Riechenbach, Switzerland

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James Moriarty was a mathematical genius who had a great influence on many aspects of Victorian society. He is best known as the criminal adversary of Sherlock Holmes. His biographical details are sketchy and the best account of his early life is [1]: His career has been an extraordinary one. He is a man of good birth and excellent education. Endowed by nature with a phenomenal mathematical faculty. At the age of twenty-one he wrote a treatise upon the binomial theorem, which has had a European vogue. On the strength of it he won the mathematical chair at one of our smaller universities, and had, to all appearances, a most brilliant career before him. But the man had hereditary tendencies of the most diabolical kind. A criminal strain ran in his blood, which, instead of being modified, was increased and rendered infinitely more dangerous by his extraordinary mental powers. Dark rumors gathered round him in the university town, and eventually he was compelled to resign his chair and to come down to London, where he set up as an army coach. He is known [2] to have had an interest in the applications of Pure Mathematics also: He is the celebrated author of "The Dynamics of an Asteroid", a book which ascends to such rarefied heights of pure mathematics that it is said that there was no man in the scientific press capable of criticizing it? He was also an early exponent of the subject of Game Theory, well in advance of Nash and Von Neumann. Oskar Morgenstern analysed his contributions in [3]. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moriarty.html (1 of 2) [2/16/2002 11:23:48 PM]

Moriarty

List of References (3 books/articles) Mathematicians born in the same country Other Web sites

1. C Redmond 2. M O'Brien 3. F Eugeni

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 2002

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Moriarty.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moriarty.html (2 of 2) [2/16/2002 11:23:48 PM]

Morin

Arthur Jules Morin Born: 19 Oct 1795 in Paris, France Died: 7 Feb 1880 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Arthur Morin was a student at the Ecole Polytechnique and took up a military career, becoming professor at Metz. Then, in 1839, he became professor at Conservatoire des Arts et Métiers, becoming its director in 1849. Morin was an applied mathematician and he wrote on friction and hydraulics, in particular writing on turbines and water-wheels. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Honours awarded to Arthur J Morin (Click a link below for the full list of mathematicians honoured in this way) Commemorated on the Eiffel Tower

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morin.html (1 of 2) [2/16/2002 11:23:49 PM]

Morin

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Morin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morin.html (2 of 2) [2/16/2002 11:23:49 PM]

Morin_Jean-Baptiste

Jean-Baptiste Morin Born: 23 Feb 1583 in Villefranche, Beaujolais, France Died: 6 Nov 1656 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean-Baptiste Morin studied philosophy at Aix in 1609, then two years later he went to Avignon where he studied medicine, receiving a medical degree in 1613. During the years 1613 to 1621 Morin was employed by the Bishop of Boulogne. He was sent to Germany and Hungary. His main task seems to have been to visit mines and make studies of metals. His skill in astrology seems to have been his main use to the Bishop. His next employer was the Duke of Luxembourg, for whom he worked until 1629. During this period Morin published a defence of Aristotle (1624) and he also worked on optics. However astrology remained his main interest although he worked with Gassendi on observational astronomy. Morin was appointed professor of mathematics at the Collège Royal in 1630 and he was to hold this post until his death. From about the time of his appointment he attacked Galileo and his views. He was to continue his attacks against Galileo after the trial of 1633. Morin remained firmly convinced that the Earth was fixed in space. Morin is best remembered for his attempts to solve the longitude problem. His solution, proposed in 1634, was based on measuring absolute time by the position of the Moon relative to the stars. He was certainly not the first to propose the method but he did add one important new piece of understanding, namely he took lunar parallax into account. Since Morin put forward his method for a longitude prize, a committee was set up by Cardinal Richelieu to evaluate it. Etienne Pascal, Mydorge, Beaugrand, Hérigone, J C Boulenger and L de la Porte served on the committee and they were in dispute with Morin for the five years after he made his proposal. Morin realised that instruments had to be improved, improved methods of solving spherical triangles had to be found and better lunar tables were needed. He made some advances in these areas but his method, although theoretically sound, could not achieve either the computational or observational accuracy to succeed. Morin refused to listen to objections to his proposal. Even while the dispute was going on, in 1638, Morin attacked Descartes saying that he had realised as soon as they met how bad his philosophy was. These disputes alienated Morin from the scientific community. He was to spend the latter part of his life isolated from other scientists although Cardinal http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morin_Jean-Baptiste.html (1 of 2) [2/16/2002 11:23:50 PM]

Morin_Jean-Baptiste

Richelieu's successor Cardinal Mazarin did award him a pension for his work on the longitude in 1645. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

Longitude and the Académie Royale

Honours awarded to Jean-Baptiste Morin (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Square Jean Morin (12th Arrondissement)

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Morin_Jean-Baptiste.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morin_Jean-Baptiste.html (2 of 2) [2/16/2002 11:23:50 PM]

Morley

Frank Morley Born: 9 Sept 1860 in Woodbridge, Suffolk, England Died: 17 Oct 1937 in Baltimore, Maryland, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Frank Morley entered King's College Cambridge in 1879, having won an open scholarship. However ill health disrupted his undergraduate course and he was forced to take an extra year because of these health problems. Morley only achieved the eighth place in the First Class Honours. To say 'only' here may seem strange since this was an extremely good result in an examination which saw Mathews first and Whitehead fourth. Richmond writes in [4], however:Ill health beyond all doubt had prevented him from doing himself justice, but the disappointment was keen. In middle life he was loath to speak of his student days... Morley graduated from Cambridge with a B.A. in 1884 and taught mathematics at Bath College until 1887. He settled in the United States and was appointed an instructor at the Quaker College in Haverford, Pennsylvania in 1887. The following year he was promoted to professor. At Haverford, Morley worked, not with others at the College, but with the mathematicians Scott and Harkness, both also graduates of Cambridge, England, who were at Bryn Mawr which was close to Haverford. Morley wrote mainly on geometry but also on algebra. His own favourite among his geometry papers was On the Lueroth quartic curve which he published in 1919. He is perhaps best known, however, for a theorem which is now known as Morley's Theorem:If the angles of any triangle be trisected, the triangle, formed by the meets of pairs of trisectors, each pair being adjacent to the same side, is equilateral. Morley loved posing mathematical problems and over a period of 50 years, starting in his undergraduate http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morley.html (1 of 2) [2/16/2002 11:23:52 PM]

Morley

days, he published over 60 problems in the Educational Times. Most are of a geometric nature. Here is an example, see [1]:Show that on a chess-board the number of squares visible is 204, and the number of rectangles (including squares) visible is 1296; and that, on a similar board with n squares in each side, the number of squares is the sum of the first n square numbers, and the number of rectangles (including squares) is the sum of the first n cube numbers. In fact Morley was an exceptionally good chess player, so the problem above reflects one of his hobbies. He played at the highest level and beat Lasker on one occasion while Lasker was World Chess Champion. He is described by Cohen in [2] as:... a striking figure in any group. Deliberate in manner and speech, there was a suggestion of shyness about him. He was generally very well informed and interested in a strikingly wide range of subjects. He was of an artistic temperament. While many of his papers and lectures seemed involved to the uninitiated, they all possessed a characteristic artistic charm. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Frank Morley

Mathematicians born in the same country

Honours awarded to Frank Morley (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1919 - 1920

Other Web sites

Clark Kimberling

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Morley.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morley.html (2 of 2) [2/16/2002 11:23:52 PM]

Morse

Harold Calvin Marston Morse Born: 24 March 1892 in Waterville, Maine, USA Died: 1977 in Princeton, New Jersey, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Marston Morse developed variational theory in the large with applications to equilibrium problems in mathematical physics, a theory which is now called Morse theory and forms a vital role in global analysis. Morse received his M.A. from Colby College in 1914 and his Ph. D. from Harvard in 1917 where his dissertation was directed by Birkhoff. His thesis title was Certain Types of Geodesic Motion of a Surface of Negative Curvature. Morse taught briefly at Harvard before entering military service for the period of World War I. He then held posts at Cornell (1920-1925) and Brown University (1925-1926). From 1926 until 1935 he was at Harvard moving to the Institute for Advanced Study Princeton for the rest of his career until he retired in 1962. Morse developed variational theory in the large with applications to equilibrium problems in mathematical physics. This is now called Morse theory. It is important in the field of global analysis which is the study of ordinary and partial differential equations from a global or topological point of view. It builds on the classical results in the calculus developed by Hilbert and his students. Morse's major works include Functional topology and abstract variational theory (1938), Topological methods in the theory of functions of a complex variable (1947) and Lectures on analysis in the large (1947). What distinguishes Morse from many other famous mathematicians is his single-minded persistence with a single theme throughout his life. However this theme, Morse theory, is perhaps the single greatest contribution of American mathematics. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morse.html (1 of 2) [2/16/2002 11:23:54 PM]

Morse

Morse received a number of awards for his work. For example in 1933 the American Mathematical Society awarded him the Bôcher Prize for his memoir The foundations of a theory of the calculus of variations in the large in m-space published in Transactions of the American Mathematical Society in 1929. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles)

A Quotation

A Poster of Marston Morse

Mathematicians born in the same country

Honours awarded to Marston Morse (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1941 - 1942

AMS Colloquium Lecturer

1931

AMS Gibbs Lecturer

1952

AMS Bôcher Prize

Awarded 1933

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Morse.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Morse.html (2 of 2) [2/16/2002 11:23:54 PM]

Mostowski

Andrzej Mostowski Born: 1 Nov 1913 in Lvov, Poland (now Ukraine) Died: 22 Aug 1975 in Vancouver, Canada

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Andrzej Mostowski entered Warsaw University in 1931. He was influenced by Kuratowski, Lindenbaum and Tarski. His Ph.D. came in 1939, officially directed by Kuratowski but in practice directed by Tarski who was a young lecturer at that time. His studies at Warsaw had been broken by studies in Vienna (he attended a course by Gödel) and Zurich (courses by Polya, Weyl and Bernays). He became an accountant after the Nazi invasion of Poland but continued working in the 'Underground Warsaw University'. After the uprising of 1944 the Nazis tried to put him in a concentration camp. With the help of some Polish nurses he escaped to a hospital, choosing to take bread with him rather than his notebook containing his research. some of the research he reconstructed after the War but much was lost. This work was largely on recursion theory and undecidability. From 1946 until his death he worked at the University of Warsaw. Much of work during that time was on first order logic and model theory. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mostowski.html (1 of 2) [2/16/2002 11:23:56 PM]

Mostowski

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mostowski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mostowski.html (2 of 2) [2/16/2002 11:23:56 PM]

Motzkin

Theodore Samuel Motzkin Born: 26 March 1908 in Berlin, Germany Died: 15 Dec 1970 in Los Angeles, California, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Theodore Samuel Motzkin's father was Leo Motzkin. Born in Russia into a Jewish family, Leo Motzkin went to Berlin when he was thirteen years old to study mathematics. He continued to study mathematics through university being accepted as a research student by Kronecker. After starting work on his doctoral thesis, Leo Motzkin left mathematics to work for the Zionist movement. Theodore Motzkin showed remarkable talents for mathematics as a child growing up in Berlin, and he began his university education when only fifteen years old. He followed the usual pattern of German education of his day, spending time at different universities. Among those he studied at were Göttingen, Paris and Berlin. At Berlin he wrote his diploma thesis, under Schur's supervision, on algebraic structures. For his doctoral work Motzkin went to Basel where he studied with Ostrowski writing his dissertation on linear programming. In 1957 he wrote the following about his thesis:In keeping with the habits in central Europe at that time the author, even though encouraged by the editors of Compositio Mathematica to publish the thesis there, issued it as an independent publication. It became almost inaccessible and, although reviewed in the Fortschritte and the Zentralblatt, remained unknown, for example to a group of recent Russian writers who rediscovered some its results. In the United States an ever increasing interest in topics involving linear inequalities led to the simultaneous translation of the thesis, about 1951, ... for A W Tucker's ONR project at Princeton university, and ... for the RAND Corporation in Santa Monica. Ostrowski was in many ways more of a collaborator of Motzkin's than a supervisor. Motzkin already had several publications before his thesis on linear programming was completed in 1934. It is usual for http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Motzkin.html (1 of 3) [2/16/2002 11:23:58 PM]

Motzkin

mathematicians who have early publications before writing their doctoral thesis, to have published material which is being studied as part of the work of the thesis. Motzkin's first publication, however, was not on linear programming but rather on power series. It was written as a partial solution to a problem which had been posed by Ostrowski and it gave Motzkin particular pleasure when he returned to the problem many years later and was able to give a complete solution. Both linear programming and power series were themes which ran through Motzkin's research throughout his life but he was an extremely broad mathematician and there were many other themes. In 1935 Motzkin was appointed to the Hebrew University in Jerusalem. He remained there throughout World War II, working as a cryptographer for the British Government during the war years. During his stay in Jerusalem, he married Naomi Orenstein and their three sons were all born there. As was characteristic of Motzkin throughout his life, he maintained a remarkable mathematical output, writing several papers in Hebrew and helping to create Hebrew mathematical terminology. We spoke of many different themes running through Motzkin's research and one of these was combinatorial analysis. What might be considered his first paper on this topic was written jointly with A Dvoretzky on the ballot problem. Feller, reviewing the paper, wrote:As the authors point out, most of the formally different proofs in reality use the reflection principle, but without the geometric interpretation this principle loses its simplicity and appears as a curious trick. Dvoretzky and Motzkin give a new proof of great simplicity and elegance. they generalise the ballot problem by requiring that at each instant, P have at least a times the votes scored by Q. The paper studies the discrete problem but the authors published a follow-up paper which considered a continuous version of this combinatorial question. Motzkin emigrated to the United States in 1948 and there he spent two years at Harvard and Boston College. One of the first papers which he published after arriving in the United States was on the Euclidean algorithm in principal ideal domains. He proved that there are principal ideal domains which are not Euclidean domains. For example Z[(1+ -19)/2] is such a principal ideal domain. The problem here is not in showing that this is not Euclidean with respect to the standard norm, which is an undergraduate exercise, but rather that it is not Euclidean in any norm. The editors of [1] write:The proof is very typical of Motzkin in that the Euclidean algorithm is given a new formulation, which at first seems to be leading away from the problem at hand, but is suddenly seen to be the decisive key to its solution. In 1950 he was appointed to the Institute of Numerical Analysis at the University of California, Los Angeles and ten years later he became Professor of Mathematics there. One of the themes which he worked on at UCLA was approximation theory. On this topic many of his publications were joint ones coming from a collaboration with J L Walsh. In many publications on this topic Motzkin examined a wide variety of different ideas, including new measure of closeness of approximation. He examined the zeros of polynomials of best approximation and produced results which were analogues to properties of the Chebyshev polynomials. Other themes which run through Motzkin's work is geometric problems, some involving Ramsey theory, and he wrote many papers on graph theory. Convex polyhedra interested him and are studied in several of his papers which combine his geometric and graph theory interests. A beautiful description of the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Motzkin.html (2 of 3) [2/16/2002 11:23:58 PM]

Motzkin

excitement of Motzkin's work is given in [1]:During many of his years at UCLA, Motzkin conducted seminars that were very exciting to the students and faculty members who participated in them. Some of Motzkin's most beautiful and important work made its first appearance here. For example, he once decided to present a seminar talk on Eberhard's conjecture that if every face of a trivalent convex polyhedron P has edge-number divisible by 3, then the number of edges of P is even. To the astonishment of the audience, he proceeded in the talk to prove the conjecture, using properties of the group SL(2, 3) of order 24, which at first seemed to be completely unrelated to the problem. In [1] a summary of Motzkin's contribution is given:Motzkin was a mathematician of great erudition, versatility, and ingenuity. Exceptionally broad, the range of his work included beautiful and important contributions to the theory of linear inequalities and programming, approximation theory, convexity, combinatorics, algebraic geometry, number theory, algebra, function theory, and numerical analysis. ... The many areas in which he worked were, however, unified by the thread of his own characteristic approach and style. If it is possible to speak of passion in one so mild-mannered, then his was a passion for meticulous precision and order. In his hands this precision became a powerful creative tool. As to his teaching skills:His unique teaching style gained him the admiration and affection of the many talented undergraduate and graduate students who were attracted to his lectures and seminars. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Motzkin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Motzkin.html (3 of 3) [2/16/2002 11:23:58 PM]

Moufang

Ruth Moufang Born: 10 Jan 1905 in Darmstadt, Germany Died: 26 Nov 1977 in Frankfurt, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ruth Moufang was supervised by Dehn and obtained a Ph.D. in 1931 on projective geometry. From 1931 to 1937 she studied projective planes introducing Moufang planes and non-associative systems called Moufang loops. In [2] Chandler and Magnus describe her contributions to geometry, putting them into context as follows:A large part of her work is dedicated to the foundations of geometry. Her most outstanding contribution to this field is a result which adds a third important discovery to two others made previously by Hilbert (1901 and 1930). Reversing a development going from Euclid to Descartes in which geometry is replaced by algebra as a fundamental discipline of mathematics, Hilbert had shown that a subset of his axioms for plane geometry (essentially the incidence axioms) together with the incidence theorem of Desargues permits the introduction of coordinates on a straight line which are elements of a skew field. If Desargues' theorem is replaced by that of Pappus, the coordinates become elements of a field. Moufang (1933) showed that another incidence theorem, called the theorem of the complete quadrilateral (or of the invariance of the fourth harmonic point), allows one to introduce coordinates which are elements of an alternating division algebra. This and a subsequent paper had the effect of stimulating further research of these algebras and of other nonassociative algebraic structures (Moufang loops). Her work is based both on a powerful geometric intuition and on the development of difficult algebraic techniques. It is supplemented by a sequence of papers on continuum mechanics. The Nazis, to be precise Hitler's minister of education, refused Moufang permission to teach (because she

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moufang.html (1 of 2) [2/16/2002 11:23:59 PM]

Moufang

was a woman), so from 1937 she became an industrial mathematician working on elasticity theory. In fact this gives Moufang the unique position of being the first German woman with a doctorate to be employed in industry. She may actually be the first ever such woman anywhere. Moufang published only one paper on group theory which was published in 1937. In this paper, which is motivated by the two papers of Hilbert on geometry mentioned above (published in 1901 and 1930), she examines the group M = F/F'', the free metabelian group on two generators. She proves that the rational group algebra of this group can be embedded in an ordered division ring. As a consequence it is easy to show that M contains a copy of the free semigroup on two generators. Moufang also gives applications of the result to number theory, knot theory and the foundations of geometry. Moufang taught at Frankfurt from 1946 where she became a professor but published nothing further. Again she holds a unique position here as the first German woman professor. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of Ruth Moufang

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Moufang.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Moufang.html (2 of 2) [2/16/2002 11:23:59 PM]

Mouton

Gabriel Mouton Born: 1618 in Lyon, France Died: 28 Sept 1694 in Lyon, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gabriel Mouton obtained his doctorate in theology in Lyon. However he spent much of his spare time studying mathematics and astronomy. He took holy orders and spent his whole career in St Paul's Church in Lyon. His most famous work Observationes diametrorum solis et lunae apparentium published in 1670 studied interpolation and a standard of measurement based on the pendulum. His methods of interpolation were similar to those used by Briggs in the construction of his logarithm tables. Mouton also produced 10 place tables of logarithmic sines and cosines and an astronomical pendulum of remarkable precision. Mouton was the first to propose the decimal system. He also suggested (1670) a standard linear measurement based on the length of the arc of one minute of longitude on the Earth's surface and divided decimally. As an astronomical observer he made accurate observations of the apparent diameter of the sun. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mouton.html (1 of 2) [2/16/2002 11:24:00 PM]

Search Form

Mouton

Glossary index

Quotations index

Mathematicians of the day JOC/EFR December 1996

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mouton.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mouton.html (2 of 2) [2/16/2002 11:24:00 PM]

Muir

Thomas Muir Born: 25 Aug 1844 in Stonebyres, Falls of Clyde, Lanarkshire, Scotland Died: 21 March 1934 in Rondebosch, South Africa

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Muir was educated at the University of Glasgow. He showed great ability in Greek but Thomson (later Lord Kelvin) persuaded him to study mathematics. He became a tutor at St Andrews University, then travelled on the Continent, meeting many of the top mathematicians. His visit to Berlin was especially profitable. In 1871 Muir became an assistant at Glasgow University, then from 1874 to 1892 he was mathematics and science master at Glasgow High School. In 1882 he published Treatise on the theory of determinants while in 1890 he published History of determinants which was to become the first part of a major five volume life's work. Also in 1890 Muir was elected to a Fellow of the Royal Society, then two years later he went to the Cape, South Africa, as Superintendent General of Education. There he was to introduce many educational reforms. Muir was knighted in 1910, four years after he had reworked his 1890 publication to become Volume 1 of History of determinants which covered the origins to Leibniz in 1840. The remaining volumes were Volume 2 1840-1860 (1911), Volume 3 1860-1880 (1920), Volume 4 1880-1900 (1923), Volume 5 1900-1920 (1929). Muir was working on Volume 6 1920-1940 at the time of his death was was somewhat optimistic since he would have been over 100 years of age before it could have been completed. In 1931, at the age of 87, he showed surprising abilities to keep with the modern flavour of his subject when he wrote that he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Muir.html (1 of 2) [2/16/2002 11:24:02 PM]

Muir

welcomed the light matrix proofs in contrast to the heavy footed methods of thirty-five years ago. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to Thomas Muir (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1900

Fellow of the Royal Society of Edinburgh

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Muir.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Muir.html (2 of 2) [2/16/2002 11:24:02 PM]

Mumford

David Bryant Mumford Born: 11 June 1937 in Worth, Sussex, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

David Mumford's father, William Mumford, was English and [2]:... a visionary with an international perspective, who started an experimental school in Tasmania based on the idea of appropriate technology... Mumford's father worked for the United Nations from its foundations in 1945 and this was his job while Mumford was growing up. Mumford's mother was American and the family lived on Long Island Sound in the United States, a semi-enclosed arm of the North Atlantic Ocean with the New York- Connecticut shore on the north and Long Island to the south. After attending Exeter School, Mumford entered Harvard University. It was at Harvard that Mumford first became interested in algebraic varieties. He relates in [2]:... a classmate said "Come with me to hear Professor Zariski's first lecture, even though we won't understand a word" and Oscar Zariski bewitched me. When he spoke the words "algebraic variety", there was a certain resonance in his voice that said distinctly that he was looking into a secret garden. I immediately wanted to be able to do this too. It led me to 25 years of struggling to make this world tangible and visible. After graduating from Harvard, Mumford was appointed to the staff there. He was appointed professor of mathematics in 1967 and, ten years later, he became Higgins Professor. He was chairman of the Mathematics Department at Harvard from 1981 to 1984 and MacArthur Fellow from 1987 to 1992. Mumford's greatest honour was being awarded a Fields Medal at the International Congress in Vancouver in 1974. Tate describes the work that Mumford was awarded the Fields Medal for in [6]. He writes:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mumford.html (1 of 3) [2/16/2002 11:24:04 PM]

Mumford

Mumford's major work has been a tremendously successful multi-pronged attack on problems of the existence and structure of varieties of moduli, that is, varieties whose points parameterise isomorphism classes of some type of geometric object. Besides this he has made several important contributions to the theory of algebraic surfaces. ... Mumford has carried forward, after Zariski, the project of making algebraic and rigorous the work of the Italian school on algebraic surfaces. He has done much to extend Enriques' theory of classification to characteristic p > 0, where many new difficulties appear. Tate goes on to explain in more technical terms Mumford's work in [6] and it is also described in [4]. In the 1980's however, the direction of Mumford's work changed dramatically. He writes in [1] that, after his first wife Erika died:... I turned from algebraic geometry to an old love - is there a mathematical approach to understanding thought and the brain? This is applied mathematics and I have to say that I don't think theorems are very important here. I met remarkable people who showed me the crucial role played by statistics, Grenander, Geman and Diaconis. The article [3] is a survey of this new area that Mumford has worked on. It makes fascinating reading and we quote here some comments from the introduction which give some idea of the scope of the ideas covered:The term "Pattern Theory" was introduced by Ulf Grenander in the 70's as a name for a field of applied mathematics which gave a theoretical setting for a large number of related ideas, techniques and results from fields such as computer vision, speech recognition, image and acoustic signal processing, pattern recognition and its statistical side, neural nets and parts of artificial intelligence. ... The problem that "Pattern Theory" aims to solve ... may be described as follows "the analysis of the patterns generated by the world in any modularity, with all their naturally occurring complexity and ambiguity, with the goal of reconstructing the processes, objects and events that produced them and of predicting these patterns when they reoccur". Mumford has received many honours in addition to the Fields Medal. He received an honorary D.Sc. from the University of Warwick in 1983 and he is a member of the National Academy of Sciences. He was elected President of the International Mathematical Union in 1995, a position he will hold until 1999. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1970 to 1980

Honours awarded to David Mumford (Click a link below for the full list of mathematicians honoured in this way) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mumford.html (2 of 3) [2/16/2002 11:24:04 PM]

Mumford

Fields' Medal

Awarded 1974

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mumford.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mumford.html (3 of 3) [2/16/2002 11:24:04 PM]

Mydorge

Claude Mydorge Born: 1585 in Paris, France Died: July 1647 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Claude Mydorge trained as a lawyer but really had little need to work as he came from a wealthy family. He was able to devote most of his life to research in mathematics without having the problems of earning a salary. Mydorge studied geometry and physics. He published books on optics and conic sections, for example De sectionibus conicis contains a wealth of new examples and ideas which were used by many later geometers. His work simplifies many of Apollonius's proofs. He was interested in mathematical recreations and edited Récréations Mathématique. Mydorge's book Examen du livre des récréations mathématiques was published in 1630 and later books, such as one by Denis Henrion (1659) were often based on it. Mydorge left an unpublished manuscript of over 1000 geometric problems and their solutions. It was not only mathematical problems which interested Mydorge. He also worked on light and refraction in particular. His interest in optics also fitted in with an interest in making astronomical observations. He was a close friend of Descartes and made a large number of optical instruments for him; the two shared a strong interest in explaining vision and the instruments and lenses were to help develop theories. One of Mydorge's most famous results was an extremely accurate measurement of the latitude of Paris. He was also interested in methods of determining longitude and was appointed to a committee to determine the whether Morin's methods for determining longitude from the Moon's motion was practical. Hérigone and Etienne Pascal served with him on this committee. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mydorge.html (1 of 2) [2/16/2002 11:24:05 PM]

Mydorge

Cross-references to History Topics

Longitude and the Académie Royale

Other references in MacTutor

Chronology: 1625 to 1650

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mydorge.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mydorge.html (2 of 2) [2/16/2002 11:24:05 PM]

Mytropolshy

Yurii Alekseevich Mytropolshy Born: 3 Jan 1917 in Shyshaky, Myrhorod, Poltava gubernia, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yurii Mytropolshy attended the Kazakh University in Alma-Ata (renamed Almaty in 1991). Alma-Ata is Soviet version of the Kazakh name Almaty for the capital of Kazakhstan, meaning "Father of Apples". It took this name in 1921 having previously been named Verny. Kazakh Al-Farabi State University was very new when Mytropolshy entered it, the University being founded in 1934. Mytropolshy graduated from the Kazakh University in 1942. He was appointed to the Institute of Constructive Mechanics of the Academy of Sciences of the Ukraine in 1946, moving to the Institute of Mathematics of the Academy of Sciences of the Ukraine in 1951. From 1951 Mytropolshy taught at Kiev University and continued teaching there when made Director of the Institute of Mathematics in 1958. He held the post of Director of the Institute for 30 years, expanding the work of the Institute. Petryshyn writes in [19]:During Yu Mytropolshy's directorship (1958-88), the institute experienced a great expansion in research personnel and mathematical disciplines, and an improvement in the quality of research. Samoilenko worked with Mytropolshy on many joint mathematical projects and when Mytropolshy retired from the directorship of the Institute in 1988 Samoilenko took over the directorship. In [20] Petryshyn summarises Mytropolshy's work as follows:Mytropolshy has made major contributions to the theory of oscillations and nonlinear mechanics as well as the qualitative theory of differential equations. He further developed asymptotic methods and applied them to the solution of practical problems. He extended the Krylov Bogoluibov symbolic method to nonlinear systems and extended asymptotic methods http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mytropolshy.html (1 of 2) [2/16/2002 11:24:07 PM]

Mytropolshy

in the theory of nonlinear mechanics. Using a method of successive substitutes, he constructed a general solution for a system of nonlinear equations and studied its behaviour in the neighbourhood of the quasi-periodic solution. He also successfully applied the averaging method to the study of oscillating systems with slowly varying parameters. In 1955 Mytropolshy and Bogoluibov published a monograph on asymptotic methods in nonlinear oscillations. This work was to lead to further advances by the Kiev school, in particular they applied asymptotic methods to partial and functional differential equations. Mytropolshy was elected to the Academy of Sciences of the Ukraine in 1961 and to the Academy of Sciences of the USSR in 1984. He was also honoured by the award of the A M Lyapunov Gold Medal in 1987. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (29 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Mytropolshy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Mytropolshy.html (2 of 2) [2/16/2002 11:24:07 PM]

Naimark

Mark Aronovich Naimark Born: 5 Dec 1909 in Odessa, Ukraine Died: 30 Dec 1978 in Moscow, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mark Naimark studied mathematics on his own for four years from the age of fifteen completing a university course on analysis. He entered the Odessa Institute of National Education in 1929. Then, in 1933, he went to Odessa State University to undertake graduate studies. Naimark defended his candidate's dissertation in 1936 the moved to the University of Moscow in 1938. In 1941 he received his doctorate from the Steklov Mathematical Institute of the USSR Academy of Sciences. He was appointed to a chair at the Seismological Institute of the USSR Academy of Sciences. During World War II he undertook military work in a number of different places, returning to Moscow at the end of the War. In 1954 he was appointed professor at the Moscow Physical-Technical Institute. Then in 1962 he became professor at the Steklov Mathematical Institute, a post he remained in until his death. Naimark's first work for his candidate's thesis was on the separation of roots of algebraic equations. Once he had established himself in Moscow he worked on functional analysis and group representations. In 1943 he proved the Gelfand-Naimark theorem on self-adjoint algebras of operators in Hilbert space. In the same year he generalised von Neumann's spectral theorem to locally compact abelian groups. He made a detailed analysis of the infinite-dimensional representations of the semisimple Lie groups. His important treatise with Gelfand on irreducible representations of the classical matrix groups was published in 1950. This work formed the basis for later work by Harish-Chandra on representations of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Naimark.html (1 of 2) [2/16/2002 11:24:08 PM]

Naimark

semisimple Lie groups. Naimark also contributed to Banach spaces. He wrote the famous text Normed Rings in 1956. He had written the book Linear differential operators two years earlier in 1954. In all Naimark wrote 123 papers and 5 books. His last book was Theory of group representations published in 1976. By the time this book was written Naimark was suffering from heart disease. Too ill to sit up, he dictated the text to his wife. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1940 to 1950

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Naimark.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Naimark.html (2 of 2) [2/16/2002 11:24:08 PM]

Napier

John Napier Born: 1550 in Merchiston Castle, Edinburgh, Scotland Died: 4 April 1617 in Edinburgh, Scotland

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Napier's father, Archibald Napier, was an important man in late 16th century Scotland. His family had owned the Merchiston estate from the 1430s when one of his ancestors acquired the estate, becoming the first Napare of Merchiston. (We shall comment shortly on the different spellings of Napier's name.) The family also owned estates at Lennox and at Menteith and a residence at Gartness. Archibald Napier married Janet Bothwell, the sister of the Bishop of Orkney, in 1549 when he was only 15 years old. Their son John Napier was born the following year. Archibald Napier was a justice-depute and was knighted in 1565. He was appointed Master of the Mint in 1582. Before continuing we should comment on the spelling of John Napier. The name John is most easily dealt with as John Napier, and almost everyone else around his time, used the old spelling "Jhone". His surname appears in a large variety of different spellings. The forms Napeir, Nepair, Nepeir, Neper, Napare, Naper, Naipper are all seen but John Napier would most commonly have been written Jhone Neper at that time. The only form of Napier that we are sure would not have been used in Napier's lifetime was the present modern spelling "Napier"! Little is known about John Napier's early years. One of the few scraps of information that we have is from a letter from the Bishop of Orkney, John's uncle, to Archibald Napier written when John was eleven years old:I pray you, sir, to send your son John to school; over to France or Flanders; for he cannot learn well at home nor get profit in this most perilous world - that he may be saved in it; that he may seek honour and profit as I do not doubt that he will... This is a translation of the old Scots that the Bishop of Orkney actually wrote. For those interested the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Napier.html (1 of 6) [2/16/2002 11:24:10 PM]

Napier

original version reads:I pray you, schir, to send your son Jhone to the schuyllis; oyer to France or Flandaris; for he can leyr na guid at hame, nor get na proffeitt in this maist perullous worlde ... Napier was educated at St Andrews University, entering the university in 1563 at the age of 13. His mother arranged for him to live in St Salvator's College and special arrangements were made for the Principal of the University, John Rutherford, to take care of him personally. Napier's name appears on the matriculation roll of St Salvator's College for 1563. Shortly after Napier matriculated his mother died. We know that Napier spent some time at St Andrews University and he wrote himself many years later that it was in St Andrews that he first became passionately interested in theology. However Napier's name does not appear in the list of those being awarded degrees in the subsequent years so he must have left St Andrews to study in Europe before completing a degree. Of other facts we can also be certain. Napier did not acquire his knowledge of higher mathematics at St Andrews nor did he acquire his deep knowledge of classical literature there. Both these must have been acquired during his studies in Europe but no record exists to show where he studied, although the University of Paris is highly likely and it is also probable that he spent some time in Italy and the Netherlands. By 1571 Napier had returned to Scotland for he was present at his father's second marriage which took place in that year. It was in 1571 that Napier himself began to make arrangements for his own marriage but it was at nearly two years before that took place. In 1572 most of the estates of the Napier family were made over to John Napier and a castle was planned for the estate at Gartness. When the castle was completed in 1574, Napier and his wife took up residence there. Napier devoted himself to running his estates. This task he took very seriously and, being a great genius as an inventor, he applied his skills to these tasks. He approached agriculture in a scientific way and he experimented with:... improving and manuring of all sorts of field land with common salts, whereby the same may bring forth in more abundance, both of grass and corn of all sorts, and far cheaper than by the common way of dunging used heretofore in Scotland. The above is quoted in [11] without reference to its origin. Napier took part in the religious controversies of the time. He was a fervent Protestant and published, what he considered his most important work, the Plaine Discovery of the Whole Revelation of St. John (1593). Napier had been a fanatical Protestant from his days as an undergraduate at St Andrews. He wrote the Plaine Discovery of the Whole Revelation of St. John according to his preface:... for preventing the apparent danger of Papistry arising within this Island... In fact there were good reasons why Napier thought that a change in the religious situation in Scotland might occur, for there had, for some time, been rumours that Philip of Spain might invade Scotland. The Plaine Discovery of the Whole Revelation of St. John did gain Napier quite a reputation, not only within Scotland, but also on the Continent after the work was translated into Dutch, French and German. Gibson, in [11], remarks however:... I suppose that there are few indeed of the present generation who have read, or even heard of, the book; whatever its merits may have been they do not appeal to the modern mind... http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Napier.html (2 of 6) [2/16/2002 11:24:10 PM]

Napier

Napier's study of mathematics was only a hobby and in his mathematical works he writes that he often found it hard to find the time for the necessary calculations between working on theology. He is best known, however, for his invention of logarithms but his other mathematical contributions include a mnemonic for formulas used in solving spherical triangles, two formulas known as Napier's analogies used in solving spherical triangles and an invention called Napier's bones used for mechanically multiplying dividing and taking square roots and cube roots. Napier also found exponential expressions for trigonometric functions, and introduced the decimal notation for fractions. Much of Napier's work on logarithms seems to have been done while he was living at Gartness. The Statistical Account (Vol. xvi, page 108) contains the following:Adjoining the mill at Gartness are the remains of an old house in which John Napier of Merchiston, Inventor of Logarithms, resided a great part of his time (some years) when he was making his calculations. It is reported that the noise of the cascade, being constant, never gave him uneasiness, but that the clack of the mill, which was only occasional, greatly disturbed his thoughts. He was therefore, when in deep study, sometimes under the necessity of desiring the miller to stop the mill that the train of his ideas might not be interrupted. Napier's discussion of logarithms appears in Mirifici logarithmorum canonis descriptio in 1614. Two years later an English translation of Napier's original Latin text was published, translated by Edward Wright. In the preface of the book Napier explains his thinking behind his great discovery (we quote from the English translation of 1616 of the original Latin of 1614):Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter. But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three. Unlike the logarithms used today, Napier's logarithms are not really to any base although in our present terminology it is not unreasonable (but perhaps a little misleading) to say that they are to base 1/e. Certainly they involve a constant 107 which arose from the construction in a way that we will now explain. Napier did not think of logarithms in an algebraic way, in fact algebra was not well enough developed in Napier's time to make this a realistic approach. Rather he thought by dynamical analogy. Consider two lines AB of fixed length and A'X of infinite length. Points C and C' begin moving simultaneously to the right, starting at A and A' respectively with the same initial velocity; C' moves with uniform velocity and C with a velocity which is equal to the distance CB. Napier defined A'C' (= y) as the logarithm of BC (= x), that is y = Nap.log x. Napier chose the length AB to be 107, based on the fact that the best tables of sines available to him were given to seven decimal places and he thought of the argument x as being of the form 102.sin X.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Napier.html (3 of 6) [2/16/2002 11:24:10 PM]

Napier

The fact that Nap.log 1 does not equal 0 is a major difficulty which make Nap.logs much less convenient for calculations than our logs. A change to logs with log 1 = 0 came about in discussions between Napier and Briggs. Briggs read Napier's 1614 Latin text and, on the 10 March 1615 wrote in a letter to a friend:Napper, lord of Markinston, hath set my head and hands a work with his new and admirable logarithms. I hope to see him this summer, if it please God, for I never saw a book which pleased me better or made me more wonder. In fact Briggs did make the difficult journey from London to Edinburgh to see Napier in the summer of 1615 (would he have dreamed that now it takes 4 hours by train, rather than at least 4 days by horse and coach in those times). A description of their meeting was told by John Marr to William Lilly who writes the following (see [11]):Mr Briggs appoints a certain day when to meet at Edinburgh; but failing thereof, Merchiston was fearful he would not come. It happened one day as John Marr and the Lord Napier were speaking of Mr Briggs, "Oh! John," saith Merchiston, "Mr Briggs will not come now"; at the very instant one knocks at the gate, John Marr hastened down and it proved to be Mr Briggs to his great contentment. He brings Mr Briggs into my Lord's chamber, where almost one quarter of an hour was spent, each beholding other with admiration, before one word was spoken. At last Mr Briggs began, -"My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy, viz. the Logarithms ... Briggs had suggested to Napier in a letter sent before their meeting that logs should be (in our terminology) to base 10 and Briggs had begun to construct tables. Napier replied that he had the same idea but ([11]):... he could not, on account of ill-health and for other weighty reasons undertake the construction of new tables. At their meeting Napier suggested to Briggs the new tables should be constructed with base 10 and with log 1 = 0, and indeed Briggs did construct such tables. In fact Briggs spent a month with Napier on his first visit of 1615, made a second journey from London to Edinburgh to visit Napier again in 1616 and would have made yet a third visit the following year but Napier died in the spring before the planned summer visit. Napier presented a mechanical means of simplifying calculations in his Rabdologiae published in 1617. He described a method of multiplication using "numbering rods" with numbers marked off on them. The reason for publishing the work is given by Napier in the dedication, where he says that so many of his friends, to whom he had shown the numbering rods, were so pleased with them that they were already becoming widely used, even beginning to be used in foreign countries. Napier's numbering rods were made of ivory, so that they looked like bones which explains why they are now known as Napier's bones. To multiply numbers the bones were placed side by side and the appropriate products read off. Glaisher described how to use Napier's bones in an article he wrote for Encyclopaedia Britannica and this description is quoted in [10]. Napier's bones are also described in [5], [15] and [18]. It would be surprising if a man of such great an intellect as Napier did not appear rather strange to his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Napier.html (4 of 6) [2/16/2002 11:24:10 PM]

Napier

contemporaries and, given the superstitious age in which he lived, strange stories began to circulate. Many traditions suggest that Napier was ... in league with the powers of darkness... and these are taken seriously in the biased biography [6] written by Mark Napier, one of John Napier's descendants. Mark Napier suggests that John Napier deliberately played upon the primitive beliefs of his servants by going round with a cock which he had covered in soot. Even the Statistical Account (quoted above) says:[Napier] used frequently to walk out in his nightgown and cap. This, with some things which to the vulgar appear rather odd, fixed on him the character of a warlock. It was formerly believed and currently reported that he was in compact with the devil; and that the time he spent in study was spent in learning the black art and holding conversation with Old Nick. Napier, however, will be remembered for making one of the most important contributions to the advance of knowledge. It was through the use of logarithms that Kepler was able to reduce his observations and make his breakthrough which then in turn underpinned Newton's theory of gravitation. In the preface to the Mirifici logarithmorum canonis descriptio, quoted above, Napier says he hoped that his logarithms will save calculators much time and free them from the slippery errors of calculations. Laplace, 200 year later, agreed, saying that logarithms:...by shortening the labours, doubled the life of the astronomer . Article by: J J O'Connor and E F Robertson List of References (18 books/articles)

Some Quotations (3)

A Poster of John Napier

Mathematicians born in the same country

Some pages from publications

Title page of Mirifici logarithmorum canonis descriptio (1614) Title page of Rhabdologia (1617)

Cross-references to History Topics

An overview of the history of mathematics

Other references in MacTutor

1. A comment about Napier 2. Chronology: 1600 to 1625

Honours awarded to John Napier (Click a link below for the full list of mathematicians honoured in this way) Lunar features Other Web sites

Crater Naper 1. The Galileo Project 2. Nijmegen, Netherlands (An html version of a 1606 English version of Mirifici logarithmorum canonis descriptio) 3. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Napier.html (5 of 6) [2/16/2002 11:24:10 PM]

Napier

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Napier.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Napier.html (6 of 6) [2/16/2002 11:24:10 PM]

Narayana

Narayana Pandit Born: about 1340 in India Died: about 1400 in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Narayana was the son of Nrsimha (sometimes written Narasimha). We know that he wrote his most famous work Ganita Kaumudi on arithmetic in 1356 but little else is known of him. His mathematical writings show that he was strongly influenced by Bhaskara II and he wrote a commentary on the Lilavati of Bhaskara II called Karmapradipika. Some historians dispute that Narayana is the author of this commentary which they attribute to Madhava. In the Ganita Kaumudi Narayana considers the mathematical operation on numbers. Like many other Indian writers of arithmetics before him he considered an algorithm for multiplying numbers and he then looked at the special case of squaring numbers. One of the unusual features of Narayana's work Karmapradipika is that he gave seven methods of squaring numbers which are not found in the work of other Indian mathematicians. He discussed another standard topic for Indian mathematicians namely that of finding triangles whose sides had integral values. In particular he gave a rule of finding integral triangles whose sides differ by one unit of length and which contain a pair of right-angled triangles having integral sides with a common integral height. In terms of geometry Narayana gave a rule for a segment of a circle. Narayana [4]:... derived his rule for a segment of a circle from Mahavira's rule for an 'elongated circle' or an ellipse-like figure. Narayana also gave a rule to calculate approximate values of a square root. He did this by using an indeterminate equation of the second order, Nx2 + 1 = y2, where N is the number whose square root is to be calculated. If x and y are a pair of roots of this equation with x < y then N is approximately equal to y/x. To illustrate this method Narayana takes N = 10. He then finds the solutions x = 6, y = 19 which give the approximation 19/6 = 3.1666666666666666667, which is correct to 2 decimal places. Narayana then gives the solutions x = 228, y = 721 which give the approximation 721/228 = 3.1622807017543859649, correct to four places. Finally Narayana gives the pair of solutions x = 8658, y = 227379 which give the approximation 227379/8658 = 3.1622776622776622777, correct to eight decimal places. Note for comparison that 10 is, correct to 20 places, 3.1622776601683793320. See [3] for more information. The thirteenth chapter of Ganita Kaumudi was called Net of Numbers and was devoted to number sequences. For example, he discussed some problems concerning arithmetic progressions. The fourteenth chapter (which is the last one) of Naryana's Ganita Kaumudi contains a detailed

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Narayana.html (1 of 2) [2/16/2002 11:24:11 PM]

Narayana

discussion of magic squares and similar figures. Narayana gave the rules for the formation of doubly even, even and odd perfect magic squares along with magic triangles, rectangles and circles. He used formulas and rules for the relations between magic squares and arithmetic series. He gave methods for finding "the horizontal difference" and the first term of a magic square whose square's constant and the number of terms are given and he also gave rules for finding "the vertical difference" in the case where this information is given. Article by: J J O'Connor and E F Robertson List of References (12 books/articles) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Narayana.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Narayana.html (2 of 2) [2/16/2002 11:24:11 PM]

Nash

John Forbes Nash Born: 13 June 1928 in Bluefield, West Virginia, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John F Nash's father, also called John Forbes Nash so we shall refer to him as John Nash Senior, was a native of Texas. John Nash Senior was born in 1892 and had an unhappy childhood from which he escaped when he studied electrical engineering at Texas Agricultural and Mechanical. After military service in France during World War I, John Nash Senior lectured on electrical engineering for a year at the University of Texas before joining the Appalacian Power Company in Bluefield, West Virginia. John F Nash's mother, Margaret Virginia Martin, was known as Virginia. She had a university education, studying languages at the Martha Washington College and then at West Virginia University. She was a school teacher for ten years before meeting John Nash Senior, and the two were married on 6 September 1924. Johnny Nash, as he was called by his family, was born in Bluefield Sanatorium and baptised into the Episcopal Church. He was [2]:... a singular little boy, solitary and introverted ... but he was brought up in a loving family surrounded by close relations who showed him much affection. After a couple of years Johnny had a sister when Martha was born. He seems to have shown a lot of interest in books when he was young but little interest in playing with other children. His mother responded by enthusiastically encouraging Johnny's education, both by seeing that he got good schooling and also by teaching him herself. Johnny's teachers at school certainly did not recognise his genius, and it would appear that he gave them little reason to realise that he had extraordinary talents. They were more conscious of his lack of social skills and, because of this, labelled him as backward. Although it is easy to be wise after the event, it now would appear that he was extremely bored at school. By the time he was about twelve years old he was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nash.html (1 of 6) [2/16/2002 11:24:13 PM]

Nash

showing great interest in carrying out scientific experiments in his room at home. It is fairly clear that he learnt more at home than he did at school. Martha seems to have been a remarkably normal child while Johnny seemed different from other children. She wrote later in life (see [2]):Johnny was always different. [My parents] knew he was different. And they knew he was bright. He always wanted to do thinks his way. Mother insisted I do things for him, that I include him in my friendships. ... but I wasn't too keen on showing off my somewhat odd brother. Nash first showed an interest in mathematics when he was about 14 years old. Quite how he came to read E T Bell's Men of mathematics is unclear but certainly this book inspired him. He tried, and succeeded, in proving for himself results due to Fermat which Bell stated in his book. The excitement that Nash found here was in contrast to the mathematics that he studied at school which failed to interest him. He entered Bluefield College in 1941 and there he took mathematics courses as well as science courses, in particular studying chemistry which was a favourite topic. He began to show abilities in mathematics, particularly in problem solving, but still with hardly any friends and behaving in a somewhat eccentric manner, this only added to his fellow pupils view of him as peculiar. He did not considered a career in mathematics at this time, however, which is not surprising since it was an unusual profession. Rather he assumed that he would study electrical engineering and follow his father but he continued to conduct his own chemistry experiments and was involved in making explosives which led to the death of one of his fellow pupils. Nash won a scholarship in the George Westinghouse Competition and was accepted by the Carnegie Institute of Technology (now Carnegie-Mellon University) which he entered in June 1945 with the intention of taking a degree in chemical engineering. Soon, however, his growing interest in mathematics had him take courses on tensor calculus and relativity. There he came in contact with John Synge who had recently been appointed as Head of the Mathematics Department and taught the relativity course. Synge and the other mathematics professors quickly recognised Nash's remarkable mathematical talents and persuaded him to become a mathematics specialist. They realised that he had the talent to become a professional mathematician and strongly encouraged him. Nash quickly aspired to great things in mathematics. He took the William Lowell Putnam Mathematics Competition twice but, although he did well, he did not make the top five. It was a failure in Nash's eyes and one which he took badly. The Putnam Mathematics Competition was not the only thing going badly for Nash. Although his mathematics professors heaped praise on him, his fellow students found him a very strange person. Physically he was strong and this saved him from being bullied, but his fellow students took delight in making fun of Nash who they saw as an awkward immature person displaying childish tantrums. One of his fellow students wrote:We tormented poor John. We were very unkind. We were obnoxious. We sensed he had a mental problem. Nash received a BA and an MA in mathematics in 1948. By this time he had been accepted into the mathematics programme at Harvard, Princeton, Chicago and Michigan. Now he felt that Harvard was the leading university and so he wanted to go there, but on the other hand their offer to him was less generous than that of Princeton. Nash felt that Princeton were keen that he went there while he felt that his lack of success in the Putnam Mathematics Competition meant that Harvard were less enthusiastic. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nash.html (2 of 6) [2/16/2002 11:24:13 PM]

Nash

He took a while to make his decision, while he was encouraged by Synge and his other professors to accept Princeton. When Lefschetz offered him the most prestigious Fellowship that Princeton had, Nash made his decision to study there. In September 1948 Nash entered Princeton where he showed an interest in a broad range of pure mathematics: topology, algebraic geometry, game theory and logic were among his interests but he seems to have avoided attending lectures. Usually those who decide not to learn through lectures turn to books but this appears not to be so for Nash who decided not to learn mathematics "second-hand" but rather to develop topics himself. In many ways this approach was successful for it did contribute to him developing into one of the most original of mathematicians who would attack a problem in a totally novel way. In 1949, while studying for his doctorate, he wrote a paper which 45 years later was to win a Nobel prize for economics. During this period Nash established the mathematical principles of game theory. P Ordeshook wrote:The concept of a Nash equilibrium n-tuple is perhaps the most important idea in noncooperative game theory. ... Whether we are analysing candidates' election strategies, the causes of war, agenda manipulation in legislatures, or the actions of interest groups, predictions about events reduce to a search for and description of equilibria. Put simply, equilibrium strategies are the things that we predict about people. Milnor, who was a fellow student, describes Nash during his years at Princeton in [6]:He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well. However, my most vivid memory of this time is of the many games which were played in the common room. I was introduced to Go and Kriegspiel, and also to an ingenious topological game which we called Nash in honor of the inventor. In fact the game "Nash" was almost identical to Hex which had been invented independently by Piet Hein in Denmark. In 1950 Nash received his doctorate from Princeton with a thesis entitled Non-cooperative Games. In the summer of that year he worked for the RAND Corporation where his work on game theory made him a leading expert on the Cold War conflict which dominated RAND's work. He worked there from time to time over the next few years as the Corporation tried to apply game theory to military and diplomatic strategy. Back at Princeton in the autumn of 1950 he began to work seriously on pure mathematical problems. It might seem that someone who had just introduced ideas which would, one day, be considered worthy of a Nobel Prize would have no problems finding an academic post. However, Nash's work was not seen at the time to be of outstanding importance and he saw that he needed to make his mark in other ways. We should also note that it was not really a move towards pure mathematics for he had always considered himself a pure mathematician. He had already obtained results on manifolds and algebraic varieties before writing his thesis on game theory. His famous theorem, that any compact real manifold is diffeomorphic to a component of a real-algebraic variety, was thought of by Nash as a possible result to fall back on if his work on game theory was not considered suitable for a doctoral thesis. In 1952 Nash published Real algebraic manifolds in the Annals of Mathematics. The most important result in this paper is that two real algebraic manifolds are equivalent if and only if they are analytically homeomorphic. Although publication of this paper on manifolds established him as a leading http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nash.html (3 of 6) [2/16/2002 11:24:13 PM]

Nash

mathematician, not everyone at Princeton was prepared to see him join the Faculty there. This was nothing to do with his mathematical ability which everyone accepted as outstanding, but rather some mathematicians such as Artin felt that they could not have Nash as a colleague due to his aggressive personality. From 1952 Nash taught at the Massachusetts Institute of Technology but his teaching was unusual (and unpopular with students) and his examining methods were highly unorthodox. His research on the theory of real algebraic varieties, Riemannian geometry, parabolic and elliptic equations was, however, extremely deep and significant in the development of all these topics. His paper C1 isometric imbeddings was published in 1954 and Chern, in a review, noted that it:... contains some surprising results on the C1-isometric imbedding into an Euclidean space of a Riemannian manifold with a positive definite C0-metric. Nash continued to develop this work in the paper The imbedding problem for Riemannian manifolds published in 1956. This paper contains his famous deep implicit function theorem. After this Nash worked on ideas that would appear in his paper Continuity of solutions of parabolic and elliptic equations which was published in the American Journal of Mathematics in 1958. Nash, however, was very disappointed when he discovered that E De Giorgi has proved similar results by completely different methods. The outstanding results which Nash had obtained in the course of a few years put him into contention for a 1958 Fields' Medal but with his work on parabolic and elliptic equations was still unpublished when the Committee made their decisions he did not make it. One imagines that the Committee would have expected him to be a leading contender, perhaps even a virtual certainty, for a 1962 Fields' Medal but mental illness destroyed his career long before those decisions were made. During his time at MIT Nash began to have personal problems with his life which were in addition to the social difficulties he had always suffered. He met Eleanor Stier and they had a son, John David Stier, who was born on 19 June 1953. Nash did not want to marry Eleanor although she tried hard to persuade him. In the summer of 1954, while working for RAND, Nash was arrested in a police operation to trap homosexuals. He was dismissed from RAND. One of Nash's students at MIT, Alicia Larde, became friendly with him and by the summer of 1955 they were seeing each other regularly. In 1956 Nash's parents found out about his continuing affair with Eleanor and about his son John David Stier. The shock may have contributed to the death of Nash's father soon after but even if it did not Nash may have blamed himself. In February of 1957 Nash married Alicia; by the autumn of 1958 she was pregnant but, a couple of months later near the end of 1958, Nash's mental state became very disturbed. Norbert Wiener was one of the first to recognize that Nash's extreme eccentricities and personality problems were actually symptoms of a medical disorder. A long sad episode followed which included periods of hospital treatment, temporary recovery, then further treatment. Alicia eventually divorced Nash, although she continued to try to help him, and after a period of extreme mental torture he appeared to become lost to the world, removed from ordinary society, although he spent much of his time in the Mathematics Department at Princeton. The book [2] is highly recommended for its moving account of Nash's mental sufferings.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nash.html (4 of 6) [2/16/2002 11:24:13 PM]

Nash

Slowly over many years Nash recovered. He delivered a paper at the tenth World Congress of Psychiatry in 1996 describing his illness; it is reported in [3]. He was described in 1958 as the:... most promising young mathematician in the world ... but he soon began to feel that:... the staff at my university, the Massachusetts Institute of Technology, and later all of Boston were behaving strangely towards me. ... I started to see crypto-communists everywhere ... I started to think I was a man of great religious importance, and to hear voices all the time. I began to hear something like telephone calls in my head, from people opposed to my ideas. ...The delirium was like a dream from which I seemed never to awake. Despite spending periods in hospital because of his mental condition, his mathematical work continued to have success after success. He said:I would not dare to say that there is a direct relation between mathematics and madness, but there is no doubt that great mathematicians suffer from maniacal characteristics, delirium and symptoms of schizophrenia. In the 1990s Nash made a recovery from the schizophrenia from which he had suffered since 1959. His ability to produce mathematics of the highest quality did not totally leave him. He said:I would not treat myself as recovered if I could not produce good things in my work. Nash was awarded (jointly with Harsanyi and Selten) the 1994 Nobel Prize in Economic Science for his work on game theory. In 1999 he was awarded the Leroy P Steele Prize by the American Mathematical Society:... for a seminal contribution to research. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country Honours awarded to John F Nash (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1994

Other Web sites

1. Nobel prize winners 2. Nobel prizes site (An autobiography of Nash and his Nobel prize presentation speech) 3. AMS 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nash.html (5 of 6) [2/16/2002 11:24:13 PM]

Nash

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2001 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Nash.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nash.html (6 of 6) [2/16/2002 11:24:13 PM]

Navier

Claude Louis Marie Henri Navier Born: 10 Feb 1785 in Dijon, France Died: 21 Aug 1836 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Claude-Louis Navier's father was a lawyer who was a member of the National Assembly in Paris during the time of the French Revolution. However Navier's father died in 1793 when Navier was only eight years old. At this time the family were living in Paris but after Navier's father died, his mother returned to her home town of Chalon-sur-Saône and left Navier in Paris to be cared for by her uncle Emiland Gauthey.• Emiland Gauthey was a civil engineer who worked at the Corps des Ponts et Chaussées in Paris. He was considered the leading civil engineer in France and he certainly gave Navier an interest in engineering. Despite encouraging Navier to enter the Ecole Polytechnique, Gauthey seems not to have been that successful in teaching Navier, who may just have been a late developer, for he only just scraped into to Ecole Polytechnique in 1802. However, from almost bottom place on entry, Navier made such progress in his first year at the Ecole Polytechnique that he was one of the top ten students at the end of the year and chosen for special field work in Boulogne in his second year. During this first year at the Ecole Polytechnique, Navier was taught analysis by Fourier who had a remarkable influence on the young man. Fourier became a life-long friend of Navier as well as his teacher, and he took an active interest in Navier's career from that time on. In 1804 Navier entered the Ecole des Ponts et Chaussées and graduated as one of the top students in the school two years later. It was not long after Navier's graduation that his granduncle Emiland Gauthey died and Navier, who had left Paris to undertake field work, returned to Paris, at the request of the Corps des Ponts et Chaussées, to take on the task of editing Gauthey's works. Anderson writes in [3]:Over the next 13 years, Navier became recognised as a scholar of engineering science. He http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Navier.html (1 of 3) [2/16/2002 11:24:15 PM]

Navier

edited the works of his granduncle, which represented the traditional empirical approach to numerous applications in civil engineering. In that process, on the basis of his own research in theoretical mechanics, Navier added a somewhat analytical flavour to the works of Gauthey. That, in combination with textbooks that Navier wrote for practicing engineers, introduced the basic principles of engineering science to a field that previously had been almost completely empirical. Navier took charge of the applied mechanics courses at the Ecole des Ponts et Chaussées in 1819, being named as professor there in 1830. He did not just carry on the traditional teaching in the school, but rather he changed the syllabus to put much more emphasis on physics and on mathematical analysis. In addition, he replaced Cauchy as professor at the Ecole Polytechnique from 1831. His ideas for teaching were not shared by all, however, and soon after his appointment to the professorship at the Ecole Polytechnique Navier became involved in a dispute with Poisson over the teaching of Fourier's theory of heat. A specialist in road and bridge building, he was the first to develop a theory of suspension bridges which before then had been built to empirical principles. His major project to build a suspension bridge over the Seine was, however, to end in failure. The real reason that the project ran into difficulties was that the Municipal Council never supported it. Despite this it went ahead but, when the bridge was almost complete, a sewer ruptured at one end causing a movement of one of the bridge supports. The problem was not considered a major one by the Corps des Ponts et Chaussées who reported that repairs were straightforward, but the Municipal Council were looking for an excuse to stop the project and they had the bridge dismantled. Navier is remembered today, not as the famous builder of bridges for which he was known in his own day, but rather for the Navier-Stokes equations of fluid dynamics. He worked on applied mathematics topics such as engineering, elasticity and fluid mechanics and, in addition, he made contributions to Fourier series and their application to physical problems. He gave the well known Navier-Stokes equations for an incompressible fluid in 1821 while in 1822 he gave equations for viscous fluids. We should note, however, that Navier derived the Navier-Stokes equations despite not fully understanding the physics of the situation which he was modelling. He did not understand about shear stress in a fluid, but rather he based his work on modifying Euler's equations to take into account forces between the molecules in the fluid. Although his reasoning is unacceptable today, as Anderson writes in [3]:The irony is that although Navier had no conception of shear stress and did not set out to obtain equations that would describe motion involving friction, he nevertheless arrived at the proper form for such equations. Navier received many honours, perhaps the most important of which was election to the Académie des Sciences in Paris in 1824. He became Chevalier of the Legion of Honour in 1831. Finally we should say a little of Navier's political position. Of course he lived through a period when there was great political movements throughout Europe and in France in particular. The two men who had the most influence on Navier's political thinking were Auguste Comte, the French philosopher known as the founder of sociology and of positivism, and Henri de Saint-Simon who started the Saint-Simonian movement which proposed a socialist ideology based on society taking advantage of science and technology. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Navier.html (2 of 3) [2/16/2002 11:24:15 PM]

Navier

Comte had been educated at the Ecole Polytechnique, entering in 1814, where he had studied mathematics. Navier appointed him as one of his assistants at the Ecole Polytechnique and this connection was to see Navier become an ardent supporter of the ideas of Comte and Saint-Simon. Navier believed in an industrialised world in which science and technology would solve most of the problems. He also took a stand against war and against the bloodletting of the French Revolution and the military aggression of Napoleon. From 1830 Navier was employed as a consultant by the government to advise on how science and technology could be used to better the country. He advised on policies of road transport, the construction of both roads and railways. His many reports show both his remarkable abilities as an engineer coupled with his strong political views on building an industrialised society for the advantage of all. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1820 to 1830

Honours awarded to Claude-Louis Navier (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Navier (17th Arrondissement)

Commemorated on the Eiffel Tower

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Navier.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Navier.html (3 of 3) [2/16/2002 11:24:15 PM]

Neile

William Neile Born: 7 Dec 1637 in Bishopsthorpe (near York), England Died: 24 Aug 1670 in White Waltham, Berkshire, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Neile was born at Bishopsthrope in the house of his grandfather who was Archbishop of York. He entered Wadham College, Oxford in 1652 where he was taught mathematics by John Wilkins and Seth Ward. In 1657 he became a pupil of law at the Middle Temple in London. He went on to become a member of the privy council of King Charles II. In 1657 he became the first to find the arc length of an algebraic curve when he rectified the cubical parabola. He communicated his results to Brouncker and Wren at the Gresham College Society, the Society based at Gresham College, London, which a few years later became the Royal Society. Neile's work on this appeared in Wallis's De Cycloide in 1659. Neile was elected a Fellow of the Royal Society in 1663, one of the first members of this Society. In 1666 he became a member of the Council of the Royal Society. He sent a work on the theory of motion to the Society in 1669. As well as his mathematical work Neile made astronomical observations using instruments on the roof of his father's house, the 'Hill House' at White Waltham in Berkshire. He died in this house at the age of 32. Thomas Hearne, the English historian and antiquarian who was himself born in White Waltham 8 years after Neile's death, describes him as follows:He was a virtuous, sober pious man, and had such a powerful genius to mathematical leaning that had he not been cut off in the prime of his years, in all probability he would have equalled, if not excelled, the celebrated men of that profession. deep melancholy hastened his end, through his love for a maid of honour, to marry whom he could not obtain his father's consent. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neile.html (1 of 2) [2/16/2002 11:24:16 PM]

Neile

Cross-references to Famous Curves

Neile's semi-cubical parabola

Other references in MacTutor

Chronology: 1650 to 1675

Honours awarded to William Neile (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1663

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Neile.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neile.html (2 of 2) [2/16/2002 11:24:16 PM]

Nekrasov

Aleksandr Ivanovich Nekrasov Born: 9 Dec 1883 in Moscow, Russia Died: 21 May 1957 in Moscow, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aleksandr Nekrasov was educated in Moscow, attending the Fifth Moscow Gymnasium. His work at the school was outstanding and he was awarded the gold medal in 1901. In that year he entered the Faculty of Physics and Mathematics at Moscow University. He graduated from the University of Moscow in 1906, receiving a first class diploma. He was awarded a gold medal for his essay on the Theory of the Satellites of Jupiter. At this stage he decided to aim at an academic appointment so he began working for his Master's Degree. However, to finance his studies he took a post as a secondary school teacher, working in several different schools in Moscow. In fact Nekrasov worked for a Master's degree in both astronomy and mechanics and he qualified for these in 1909 and 1911. Appointed an assistant professor in the Department of Astronomy and Geodesy in 1912, he became an assistant professor in the Department of Theoretical Mechanics in the following year. He taught and undertook research in Moscow for the rest of his life but he did this at a number of different institutions. In addition to Moscow University he worked at the Higher Technical School, the Central Aerohydrodynamics Institute, the Sergo Orjonikidze Aviation Institute, and the Institute of Mechanics at the Academy of Sciences. The two scientists who most influenced Nekrasov most were Zhukovsky and Chaplygin. Zhukovsky had founded the Russian schools of hydromechanics and aeromechanics. He was known as the Father of Russian Aviation. Chaplygin, a student of Zhukovsky, wrote first on hydrodynamics under Zhukovsky's influence, in particular he worked on the mechanics of liquids and gases studying jet stream flow in the 1890's. Zhukovsky and Chaplygin set up the Central Aerohydrodynamic Institute in 1918 where

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nekrasov.html (1 of 2) [2/16/2002 11:24:18 PM]

Nekrasov

Nekrasov worked. After Zhukovsky died in 1921 Chaplygin and Nekrasov continued to build on the foundations which he had put in place. As Grigorian writes in [1]:A fully worthy disciple of and successor to Zhukovsky, Nekrasov enriched Soviet science with his scientific works ... Nekrasov published important work on the theory of waves, the theory of whirlpools, the theory of jet streams and gas dynamics. He also investigated mathematical questions which were related to these applications, in particular writing important works on non-linear integral equations. In fact his deep understanding of mathematical analysis as developed by mathematicians such as Goursat enabled him to succeed in solving a whole range of concrete problems. Grigorian writes in [1] that:Nekrasov was a brilliant representative of the trend in the development of precise mathematical methods in hydromechanics and aeromechanics... He was awarded the N E Zhukovsky Prize in 1922 for his work On smooth-form waves on the surface of a heavy liquid. He was the author of an excellent two volume text on vector mechanics, the first volume being published in 1945 with the second in the following year. In 1947 he published a monograph on aerodynamics which set out the basic theory in a systematic way. In the same year another important work on the applications of integral equations to aerodynamics was published. Several other works by Nekrasov were awarded prizes. He won the State Prize of the U.S.S.R. in 1951 for mainstream work on solving problems in hydromechanics. Elected to the Academy of Sciences of the U.S.S.R. in 1932 he was also given the title of Honoured Worker in Science and Technology in 1947. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Nekrasov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nekrasov.html (2 of 2) [2/16/2002 11:24:18 PM]

Netto

Eugen Otto Erwin Netto Born: 30 June 1848 in Halle, Germany Died: 13 May 1919 in Giessen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eugen Netto's father was Heinrich Netto, an official at the Franckesche Stiftung at Halle. The Stiftung was a Protestant religious institute which included a school for the poor, an orphanage, a medical centre, and publishing house. Eugen's mother was Sophie Neumann. Up to the age of ten Eugen attended a school in Halle, but from that time he went to a Gymnasium in Berlin. Netto was fortunate to have an outstanding teacher of mathematics at the Berlin Gymnasium in Karl Heinrich Schellbach, who had been Eisenstein's mathematics teacher. It was Schellbach who showed Netto the excitment of mathematics and from that time on mathematics was clearly the only topic that he considered. After graduating from the Gymnasium in 1866, Netto entered the University of Berlin to study mathematics. He again had some inspiring teachers in Kronecker, Weierstrass and Kummer. Netto graduated from Berlin in 1870 having worked specifically under Weierstrass and Kummer. It was in fact Weierstrass who examined his final dissertation. There was no immediate university appointment for Netto, however, and he taught in a Gymnasium in Berlin for nine years before being appointed as extraordinary professor at the University of Strasbourg in 1879. It was the French-German war of 1870-71, which ended with Alsace being annexed by the German empire, which had led to a German university being set up in Strasbourg. In 1872 the so-called Kaiser-Wilhelms-Universitat was opened in Strasbourg. The Mathematisches Seminar there was directed by Christoffel and Reye, and Netto took part in this seminar. His involvement is described in [6] where interesting background information about the working conditions and the number of students is given.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Netto.html (1 of 3) [2/16/2002 11:24:20 PM]

Netto

After three years at the University of Strasbourg Weierstrass recommended that Netto be appointed an extraordinary professor at the University of Berlin and he took up the appointment in 1882. There he taught courses on advanced algebra, the calculus of variations, mechanics, Fourier series, and synthetic geometry. Netto held this post in Berlin until 1888 when he was appointed ordinary professor at the University of Giessen. He held this post for twenty-five years until his retirement in 1913. In 1878 he attempted the second general proof of the invariance of 'dimension' but, like the first by Thomae, it was not completely satisfactory. Despite this, Netto's "proof" was widely accepted as providing a solution to the dimension problem until Jurgens' criticism in 1899 of Netto's proof. Jurgens similarly criticised a proof of the invariance of 'dimension' which had been given by Cantor. These events are fully described in [4]. Cantor showed in 1878 that the unit interval I can be mapped bijectively onto the unit square I2. In the following year Netto showed that such a mapping cannot be a continuous function. These results by Cantor and Netto are starting points for the investigations of space-filling curves which are an active research area today. Netto made major steps towards abstract group theory when he combined permutation group results and groups in number theory. He did not however include matrix groups! He published this work in his book Substitutionentheorie und ihre Anwendung auf die Algebra in Berlin in 1882 described by Biermann in [1] as:... a milestone in the development of abstract group theory. He further contributed to the development of group theory in other papers. In particular, in 1877 Netto had given new proofs of the Sylow's theorems.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

The development of group theory

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Netto.html (2 of 3) [2/16/2002 11:24:20 PM]

Netto

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Netto.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Netto.html (3 of 3) [2/16/2002 11:24:20 PM]

Neuberg

Joseph Jean Baptiste Neuberg Born: 30 Oct 1840 in Luxembourg City, Luxembourg Died: 22 March 1926 in Liège, Belgium Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Neuberg attended the Athénée de Luxembourg where his work was outstanding, then progressed to the Ecole normale des Sciences of the Facultée de Sciences at the University of Ghent in 1859. He graduated in 1862. The first part of Neuberg's career was spent in various colleges. He was professor at the Ecole Normale de Nivelle from 1862 to 1865, then at the Athénée Royal d'Arlon from 1865 to 1867, and then at the Athénée Royal and the Ecole Normale at Bruges from 1868 to 1878. Neuberg was professor at the Athénée royal of Liège from 1878 to 1884 and then extraordinary professor at the university in Liège from 1884 to 1887, being ordinary professor from 1887 until he retired in 1910. At Liège he taught analysis, higher algebra, descriptive geometry, projective geometry, analytic geometry, and the foundations of mathematics. Although he was a citizen of Luxembourg, Neuberg took Belgium nationality in 1866 and was elected to the Belgium Royal Academy. In fact he achieved the honour of being elected as President of the Academy in 1911 after he retired from his chair at Liège. Quetelet and Garnier edited the Belgium publication Correspondance mathématique et physique. In 1874 Neuberg, together with Catalan and Mansion, founded a journal Nouvelle correspondance mathématique named to honour the earlier Correspondance mathématique et physique. This journal founded by Mansion, Catalan and Neuberg was published between 1874 and 1880. After this Catalan encouraged Mansion and Neuberg to collaborate in publishing a new journal and, indeed, they did precisely this, publishing Mathesis from 1881 onwards. Neuberg worked on the geometry of the triangle, discovering many interesting new details but no large new theory. Pelseneer writes in [1]:The considerable body of his work is scattered among a large number of articles for journals; in it the influence of A Möbius is clear. Neuberg's life and work are discussed in [2]. In addition to his membership of the Belgium Royal Academy discussed above, he was a member of many other learned societies, including the Institute of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neuberg.html (1 of 2) [2/16/2002 11:24:21 PM]

Neuberg

Science of Luxembourg, the Royal Society of Science of Liège, and the Mathematical Society of Amsterdam. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Neuberg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neuberg.html (2 of 2) [2/16/2002 11:24:21 PM]

Neugebauer

Otto Neugebauer Born: 26 May 1899 in Innsbruck, Austria Died: 19 Feb 1990

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Otto Neugebauer became interested in mathematics while at school but, in 1917, he joined the Austrian army to avoid having to take his final school examinations. In 1918 the war ended and he became a prisoner of the Italians. He was held in a prison camp in Italy along with another Austrian who has achieved world-wide fame, namely Ludwig Wittgenstein. After he was released from the prison camp, Neugebauer moved around until he settled in Göttingen in 1922. There he began a serious study of mathematics having become friends with Courant, Harald Bohr, and Aleksandrov. His friendship with Harald Bohr developed into a mathematical collaboration and they wrote a joint paper on almost periodic functions. It was to be Neugebauer's first and last paper on mathematics as such for his work at this point took a definite turn. Neugebauer was an expert in languages and he had studied Egyptian. It was natural, therefore, for Harald Bohr to ask his friend to review a publication on the Rhind papyrus. Once he had begun to study the work, Neugebauer realised that the subject which he wanted to work in was the history of mathematics. He approached Courant and Hilbert to see if he could work for his doctorate on the history of Egyptian unit fractions. They agreed to supervise such a project and Neugebauer received his doctorate for a dissertation on this topic in 1926. In 1927 Neugebauer was appointed to the staff at Göttingen and he began to lecture on the history of ancient mathematics. One student who attended this first lecture course was Bartel van der Waerden and, as a result, he also developed an interest in ancient mathematics and was to publish works of major importance throughout his life. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neugebauer.html (1 of 4) [2/16/2002 11:24:23 PM]

Neugebauer

However, in 1927 Neugebauer decided that he wanted to research into Babylonian mathematics and, to enable him to do so, he learnt Akkadian which is the language in which the Babylonians wrote their tablets. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets, the earliest dating from around 1700 BC, had survived and Neugebauer knew that they were held by various museums but at that time little work had been undertaken to study them and to evaluate the Babylonian contribution. Gray writes in [4]:Neugebauer was to publish his 3-volume collection on mathematical tablets in the mid-1930s. They established the great richness of Babylonian mathematics, far exceeding anything one could have guessed from Greek or Egyptian sources. Another project which Neugebauer became involved in was the building of a new Mathematical Institute at Göttingen. This was completed in 1929, with support from the Rockefeller Foundation, and Courant and Neugebauer jointly directed the Institute until 1932. However, Neugebauer had before this started the first of two projects which would be among the most important contributions anyone has made to mathematics. He persuaded Springer-Verlag to publish a journal reviewing all mathematical publications, which would complement their reviewing journals in other topics. In 1931 the first issue of Zentralblatt für Matematik appeared, edited by Neugebauer. Zentralblatt für Matematik rapidly became an indispensable tool for all mathematicians. However, the political situation in Germany as the Nazis came to power was to bring about changes which completely changed the course of Neugebauer's career. As Boas writes in [2]:He opposed the National Socialists in Germany from the beginning and was forced out of his academic position as a consequence. Davis [3] recalls Neugebauer saying:If you never heard the sound of Nazi boots below you in the street, you cannot understand the history of the period. I'm sure that Neugebauer is right, yet his very quote may aid us a little in our understanding of the situation. Fortunately Neugebauer had a good friend in Harald Bohr, and he invited Neugebauer to move to the University of Copenhagen in January 1934. Neugebauer took the editorial office of Zentralblatt für Matematik to Copenhagen with him and from 1934 until 1938 Zentralblatt continued to flourish from its headquarters there. The struggle to produce the reviewing journal became more difficult throughout this period, however, for the Nazis tried more and more to influence the editorial policy of the journal. Sadly some fine mathematicians were seduced by the Nazi ideas and mathematicians such as Blaschke attacked the journal. Matters came to a head in 1938 when Springer-Verlag insisted that Zentralblatt für Matematik be produced in accordance with Nazi principles. Levi-Civita, who was on the editorial board, was dismissed and Neugebauer, together with almost the whole of the editorial board, resigned. Neugebauer destroyed all the records of the journal except for the cumulative index. Neugebauer was a highly respected historian of mathematics, and the world of mathematics could not afford to lose the reviewing journal that it had come to depend on in only a few years. Veblen arranged for Brown University to offer Neugebauer a chair and the American Mathematical Society saw the chance to support Neugebauer in founding a new reviewing journal. Neugebauer sailed to the United

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neugebauer.html (2 of 4) [2/16/2002 11:24:23 PM]

Neugebauer

States and [2]:... the index to the Zentralblatt came with Neugebauer, although the U.S. customs almost confiscated it as potentially subversive, and it survives to this day. In a remarkably short time Neugebauer had Mathematical Reviews up and running. The journal started reviewing articles which appeared from July 1939 and the first issue appeared in January 1940. Neugebauer continued as editor of Mathematical Reviews until 1945 when a full-time executive editor was appointed. Neugebauer policy regarding reviews was an interesting one. He [1]:... always insisted that the length of the review was not intended to be directly proportional to the importance of the paper; indeed, a bad paper needed to have a review sufficiently detailed so that nobody needed to look at the paper itself, whereas a really important paper needed only to be called to the world's attention. In 1947 Neugebauer was appointed Professor of the History of Mathematics at Brown University. His contributions to the history of ancient mathematics and astronomy continued to astound. Gray writes in [4] that:... his greatest pleasure was in entirely reshaping and extending our knowledge of the history of science. Indeed, the message that Babylonians knew more (and, he impishly insisted, the Egyptians knew less) than most people believe still needs amplification today. The high levels of scholarship that now prevails in the subject gives every prospect that received opinion will change, and that high level is largely due to the standards he set himself, his organisational skills, and the support he was able to attract. Neugebauer received many awards, prizes, and honorary degrees. He was elected to membership of the leading academies around the world. Boas writes in [1] that his greatest pleasure was in the award he received in 1979 from the Mathematical Association of America when they gave his their Award for Distinguished service to Mathematics. The article [2] gives Neugebauer's main contributions which led to the award and, although his distinguished work on the history of science is praised, the main reason for the honour was the contribution:... is that he founded, and for many years edited, first the Zentralblatt für Matematik ..., and later, Mathematical Reviews, and so gave mathematics the essential tool of a working abstracting service. Article by: J J O'Connor and E F Robertson List of References (9 books/articles)

A Quotation

Mathematicians born in the same country Cross-references to History Topics

1. Greek Astronomy 2. A history of Zero

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neugebauer.html (3 of 4) [2/16/2002 11:24:23 PM]

Neugebauer

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Neugebauer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neugebauer.html (4 of 4) [2/16/2002 11:24:23 PM]

Neumann_Bernhard

Bernhard Hermann Neumann Born: 15 Oct 1909 in Berlin, Germany

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bernhard Neumann's father was Richard Neumann, an engineer who worked for the electricity company AEG. The family lived in a wealthy district of Berlin. Bernhard attended school in Berlin spending three years in primary school followed by nine years at the Herderschule. As one might imagine mathematics was his best subject but at first he did not enjoy his other subjects very much at all. He found the teaching at the Herderschule rather uninspiring particularly the lessons on French and Latin. However, later in his school career things changed and Latin in particular became one of his favourite subjects. He began reading Latin for pleasure and found some Latin texts on scientific topics of particular interest. Neumann entered the University of Freiburg to study mathematics in 1928 and spent two semesters there before moving to the Friedrich-Wihelms University in Berlin. There he was influenced by an impressive collection of teachers including Schmidt, Robert Remak and Schur, together with his assistant Alfred Brauer, and near contemporaries of Neumann such as Hurt Hirsch, Richard Rado and Helmut Wielandt. In fact it was Remak, more than any of the others, who influenced Neumann to turn towards group theory for at first he intended to become a topologist. Hopf had given him a love of topology and this seemed the topic on which he would undertake research. However his reading on the topic involved studying a paper by Jakob Nielsen, and he realised how a certain result on the number of generators of a group could be strengthened. Schur advised him to apply the same methods to prove results on wreath products of groups and his doctoral dissertation followed on naturally from this first excursion into group theory. In November 1931 Neumann submitted his doctoral dissertation which was examined by Schur and Schmidt. He was awarded his doctorate by the University of Berlin in July 1932 and after that remained there attending lectures and acting as an unpaid assistant in the experimental physics laboratory. At the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Bernhard.html (1 of 4) [2/16/2002 11:24:25 PM]

Neumann_Bernhard

University of Berlin at this time he met Hanna von Caemmerer who was an undergraduate. A particularly difficult time was approaching, however, which would have a major effect on his life and that of Hanna von Caemmerer who later became Hanna Neumann after they married. When Hitler came to power in 1933, only a couple of months after Bernhard and Hanna first met, life in Germany became very hard for those of Jewish origin. Neumann realised immediately the dangers of remaining in Germany and quickly left the country, going first to Amsterdam before being advised that Cambridge was the best place for a mathematician to go. At the University of Cambridge he registered for a Ph.D. despite already holding a doctorate. In taking this course he followed the route adopted by most of those arriving in Cambridge fleeing from the Nazis, but in doing so he went against the advice of Hardy who said all that was necessary was to produce top quality mathematics. Registering for a doctorate, Neumann was assigned Philip Hall as a supervisor. Hall gave him a research problem on rings of polynomials but Neumann did not make much progress on it. Returning to questions in group theory which he had studied while in Berlin, he made rapid progress and was awarded his second doctorate in 1935. Even a mathematician as outstanding as Neumann was not guaranteed a lecturing post at that time and he spent two years unemployed. He remained at Cambridge for a year teaching a preparatory course to give students the right background to take Olga Taussky-Todd's algebraic number theory course. He was appointed to an assistant lectureship in Cardiff in 1937, the post being a temporary one for three years. He was joined there by Hanna Neumann who left Germany in 1938 and the two were married in July of that year. The year 1939 saw the start of World War II and now Neumann's position as a German in England became a difficult one despite having fled there to escape from the Nazis. He was briefly interned as an enemy alien but, in 1940, he was released. The University of Cardiff had not requested that he return there (if they had he would have been released earlier) so he joined the Pioneer Corps. Later he joined the Royal Artillery, and lastly the Intelligence Corps for the duration of the war. After the war ended Neumann volunteered for service in Germany with the Intelligence Corps and he was able to make contact with his wife's family at that time. Turning down an offer to return to Cardiff on the grounds that they had not helped him when he was interned, Neumann searched for an academic appointment again, and this time was appointed as a lecturer at Hull in 1946. The Neumann's were fortunate in that Hanna Neumann, who by this time had obtained her doctorate, was soon able to join him on the staff as an assistant lecturer in Hull. In 1948 Neumann was appointed to the University of Manchester, after being approached by Max Newman, although he continued to live in Hull where Hanna still worked. In 1958 Hanna was appointed to a post in Manchester and the Neumanns then moved to a house in Manchester in which they lived for three years before Bernhard accepted an offer from the Australian National University of a professorship and the Head of the Mathematics Department at the Institute of Advanced Studies. He retired in 1974 but continued to live in Canberra. Neumann is one of the leading figures in group theory who has influenced the direction of the subject in many different ways. While still in Berlin he published his first group theory paper on the automorphism group of a free group. However his doctoral thesis at Cambridge introduced a new major area into group theory research. In his thesis he initiated the study of varieties of groups, that is classes of groups defined which are by a collection of laws which must hold when any group elements are substituted into them. One of the questions raised in Neumann's thesis was the finite basis problem:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Bernhard.html (2 of 4) [2/16/2002 11:24:25 PM]

Neumann_Bernhard

Can each variety be defined by a finite set of laws? Neumann himself made many contributions to this question over many years but the answer to the problem was not given until 1969 when Ol'sanskii proved that the problem had a negative answer. An indication of some of the topics which interested Neumann can be seen from looking at the material covered in Lectures on topics in the theory of infinite groups (1960). The notes provide an introduction to universal algebras, groups, presentations, word problems, free groups, varieties of groups, cartesian products and wreath products. He goes into greater detail when discussing varieties of groups, embedding theorems for groups and amalgamated products of groups. His methods here are based on wreath products and permutational products. Then he studies embeddings of nilpotent and soluble groups and finally looks at Hopfian groups. Frank Levin, reviewing the work, writes:The author's leisurely but informative style make these notes a pleasure to read and profitable even for the novice with no background in infinite groups. Among the many important concepts which Neumann introduced we should note in particular that of an HNN extension, which appears in the paper Embedding theorems for groups (1949) written jointly with Hanna Neumann and Graham Higman. Their results proved that every countable group can be embedded in a 2-generator group. One of Neumann's many research students, Gilbert Baumslag, began a paper dedicated to Neumann on his 70th birthday with the paragraph:In 1955 when I first arrived in Manchester to work with B H Neumann he suggested that I read his paper 'Ascending derived series' which had only just been submitted for publication. This was a beautifully crafted paper, filled with ideas and very stimulating. The present note, written in gratitude, affection and esteem, in Bernhard Neumann's honour, comprises some simple variations on the themes of that paper. The history of mathematics first interested Neumann when he was at Manchester. At that time he was given access to the papers which had come from Augusta Ada Lovelace. These papers were loaned to him by the Lovelace family and he was particularly interested in the correspondence between Lovelace and her mathematics tutor De Morgan. In 1973 Neumann published a paper Byron's daughter in the Mathematical Gazette which gives an account of the mathematical activities of Ada Lovelace, her correspondence with De Morgan, and details of her friendship with Babbage. Following on from this work he later wrote a fascinating account of De Morgan's life which was published in the Bulletin of the London Mathematical Society in 1984. But Bernhard Neumann's contribution to mathematics goes far beyond his leadership in research. As the authors of [3] write:... he always had a very deep appreciation of the need to serve the mathematical community in other ways. He was a member of the Council of the London Mathematical Society between 1954 and 1961, and its Vice-President between 1957 and 1959. His scholarly influence stretched much further than Britain... His contributions to mathematics in Australia have been many and varied. Not only did he form a department of very able mathematicians at the ANU specialising in group theory and functional analysis, he also took a deep interest in the Australian Mathematical Society. Neumann served the Australian Mathematical Society as Vice-President on a number of occasions and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Bernhard.html (3 of 4) [2/16/2002 11:24:25 PM]

Neumann_Bernhard

was President during 1966-68. Neumann worked to set up the Bulletin of the Australian Mathematical Society and was editor for ten years after it was founded in 1969. He played a crucial role in setting up the Australian Association of Mathematics Teachers and the New Zealand Mathematical Society, see [2]. Many honours have been given to Neumann for his outstanding contribution and continue to be awarded. He received the Wiskundig Genootschap te Amsterdam Prize in 1949, and the Adams Prize from the University of Cambridge. He was elected a Fellow of the Royal Society in 1959 and a Fellow of the Australian Academy of Science in 1963. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Bernhard Neumann (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1959

Other Web sites

Australian Academy of Science (An interview with Bernhard Neumann) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Neumann_Bernhard.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Bernhard.html (4 of 4) [2/16/2002 11:24:25 PM]

Neumann_Carl

Carl Gottfried Neumann Born: 7 May 1832 in Königsberg, Germany (now Kaliningrad, Russia) Died: 27 March 1925 in Leipzig, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Carl Neumann was the son of Franz Neumann who has a biography in this archive. His mother was Bessel's sister-in-law. Carl was born and received his school education at Königsberg where his father was the Professor of Physics. Neumann entered the University of Königsberg where he became close friends with two of his teachers, Otto Hesse and F J Richelot who taught mathematical analysis. After graduating with a qualification to teach mathematics in secondary schools, Neumann continued to study at Königsberg for his doctorate which was awarded in 1855. After receiving his doctorate, Neumann studied for his habilitation and he submitted his thesis to the University of Halle. He received his habilitation giving him the right to lecture in 1858 when he became a Privatdozent at Halle. He was promoted to extraordinary professor in 1863. Neumann did not remain at Halle for long after his promotion for he was offered a professorship at the University of Basel. Arriving in Basel in 1863 he only spent two years at the university there before being offered a professorship at the University of Tübingen. However, during these two years in Basel he married Mathilde Elise Kloss in 1875. A slightly longer time, namely three years, spent in Tübingen, from 1865 to 1868, and then Neumann was on the move again, this time to a chair at the University of Leipzig. Appointed to Leipzig in the autumn of 1868 he gave his inaugural lecture, called an Antrittsvorlesung, in 1869 with the title On the principles of the Galileian-Newtonian theory of mechanics. The German text of this lecture is given in [2]. Neumann held the chair at Leipzig until he retired in 1911 but sadly his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Carl.html (1 of 2) [2/16/2002 11:24:27 PM]

Neumann_Carl

wife died in 1875. Wussing writes in [1]:Neumann, who led a quite life, was a successful university teacher and a productive researcher. More than two generations of future Gymnasium teachers received their basic mathematical education from him. He worked on a wide range of topics in applied mathematics such as mathematical physics, potential theory and electrodynamics. He also made important pure mathematical contributions. He studied the order of connectivity of Riemann surfaces. During the 1860s Neumann wrote papers on the Dirichlet principle and the 'logarithmic potential', a term he coined. In 1890 Emile Picard used Neumann's results to develop his method of successive approximation which he used to give existence proofs for the solutions of partial differential equations. This is discussed in detail in [4]. In addition to his research and teaching, Neumann made another important contribution to mathematics as an editor of Mathematische Annalen. He was honoured with membership of several academies and societies, including the Berlin Academy and the societies in Göttingen, Munich and Leipzig. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Neumann_Carl.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Carl.html (2 of 2) [2/16/2002 11:24:27 PM]

Neumann_Franz

Franz Ernst Neumann Born: 11 Sept 1798 in Joachimsthal, Germany (now Jachymov, Czech Republic) Died: 23 May 1895 in Königsberg, Germany (now Kaliningrad, Russia)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Franz Neumann's father, Ernst Neumann, was a farmer who later gave up farming and became an estate agent. Franz's mother was a Countess, who had been divorced, and her parents did not allow her to marry Ernst Neumann, the Countess's factotum, since he was a commoner. It was impossible for Franz's mother even to acknowledge that Franz was her son. Therefore Franz was brought up by Ernst Neumann's parents and he did not meet his mother until he was ten years old. While we are discussing the Neumann family we should note that Franz Neumann was the father of Carl Neumann who also has a biography in this archive. Franz Neumann studied in Berlin Gymnasium where he showed some talent for mathematics. However, he did not complete his studies there because of the problems caused by wars between the Prussians and the French. In 1814, at the age of sixteen, Neumann left the Gymnasium and volunteered for the Prussian army. It was a period of victories for Prussia and its allies as the French army under Napoleon was driven back. By May 1814 the allies had taken Paris, the Treaty of Paris was signed and Napoleon was sent into exile. However, on 1 March 1815, Napoleon escaped and landed at Cannes with a detachment of his guard. The allied armies began to assemble on the borders of France, and Neumann was with the Prussian forces. Napoleon assembled an army and marched into Belgium where he defeated the Prussians at Ligny on 16 June 1815. Neumann took part in the battle, was seriously wounded, and was taken to a hospital in Düsseldorf. He therefore missed fighting in the battle of Waterloo at which Napoleon was finally http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Franz.html (1 of 3) [2/16/2002 11:24:28 PM]

Neumann_Franz

defeated two days after the Battle of Ligny. Being wounded was not the only misfortune to befall Neumann, for his father lost everything he possessed in a fire leaving little to support the rest of Neumann's education. Despite the financial problems, Neumann returned to the Berlin Gymnasium to complete his course and entered the University of Berlin in 1817. He did not enrol for a degree in science, however, for he followed the wishes of his father and began to study theology. Neumann moved from the University of Berlin to continue his studies at Jena in April 1818. At Jena, his interests turned to more scientific courses and he took up the study of mineralogy and crystallography. He became friendly with one of his teachers at Jena, Christian Weiss, and Weiss arranged for some financial support to allow Neumann to make field trips to Silesia to study geology. He made his first trip in the summer of 1820 and he planned to make further trips in 1822 and 1823. However, Neumann's father died and, concerned about his mother's health and well-being, he took the year 1822-23 out from his studies to manage his mother's farm. Taking a year out to manage the farm did not stop Neumann writing his first paper on crystallography which was published in 1823. Then he returned to the University of Berlin where he worked as curator of the mineral cabinet while completing the research for his doctorate. Neumann obtained his doctorate in November 1825, and in the following May, together with Jacobi, he was appointed as a Privatdozent at the University of Königsberg. In 1828 Neumann was appointed as a lecturer at Königsberg, being appointed to the chair of minerology and physics in the following year. When Neumann arrived in Königsberg the science courses were being taught by Karl Hagen, and Neumann took over teaching some of these courses. In 1830, he married Luise Florentine Hagen, Karl Hagen's daughter but Luise died in 1838 having given birth to five children. Five years later Neumann married again, this time to Wilhelmina Hagen, the first cousin of his first wife. At Königsberg in 1833, Neumann and Jacobi together started up a mathematics-physics seminar which was used to introduce their students to methods of research. One such student was Kirchhoff who attended the Neumann-Jacobi seminar from 1843 to 1846. In 1847 Neumann came into money through an inheritance from his second wife's parents. This enabled him to build a physics laboratory for himself next to his home. He had tried for years to persuade the university authorities to build a Physics Institute but, only after Neumann retired was the Physics Institute built. The physics laboratory which he built for himself was also used by his students. Neumann's early work was in crystallography. He used least squares methods of error analysis of instruments giving new precision to measurements. However, his work at Königsberg was influenced by Bessel and Jacobi, and he turned towards a study of mathematical physics. In 1831 he formulated a law on molecular heat, namely that the molecular heat of a compound is equal to the sum of the atomic heats of its constituents. In a second paper he studied why when hot water and cold water are mixed the result does not have a temperature which is the average of the two temperatures. His explanation is broadly correct, namely the specific heat of water increases with temperature, but he failed to realise that this was only true over a certain range of temperatures. In 1832 Neumann investigated the wave theory of light, obtaining results similar to those of Cauchy and Fresnel. He published his theory of electrical induction in two papers, the first in 1845 and the second in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Franz.html (2 of 3) [2/16/2002 11:24:28 PM]

Neumann_Franz

1847. These assumed action at a distance and deduced the mathematical laws for induction of electric currents. He also discovered the Neumann lines, now named after him, thin straight scratches that appear when some iron meteorites are cut open and the exposed surface polished. Neumann only published a fraction of his work, and major portions of his discoveries were contained in his lectures at Königsberg but never published. Carl Neumann, Franz Neumann's son, claimed that the unpublished researches of his father, presented in his Königsberg lectures before 1850, the year of the first memoir of Clausius on thermodynamics, prove his priority as one of the founders of the mechanical theory of heat. Carl Neumann prepared his father's lectures for publication in 1895 but they never appeared in print. The article [3] contains extracts from the third lecture given by Franz Neumann in session 1854-1855. Article by: J J O'Connor and E F Robertson List of References (7 books/articles)

A Quotation

Mathematicians born in the same country Honours awarded to Franz Neumann (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1862

Royal Society Copley Medal

Awarded 1886

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Neumann_Franz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Franz.html (3 of 3) [2/16/2002 11:24:28 PM]

Neumann_Hanna

Hanna Neumann Born: 12 Feb 1914 in Berlin, Germany Died: 14 Nov 1971 in Ottawa, Canada

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hanna Neumann's maiden name was Caemmerer. She attended Berlin University and there she was taught analytic and projective geometry by Bieberbach, differential and integral calculus by Schmidt and number theory by Schur. Bieberbach was an inspiring, if unorganised, lecturer and Hanna almost became a geometer. However after the Nazis came to power in 1933 Hanna became unhappy. She left Berlin University in 1934 to marry Bernhard Neumann. As Bernhard was a Jew, they were forced to leave Germany in 1938. They settled in England. Hanna taught at Hull, then UMIST in Manchester until 1963. The year 1961-62 she spent with Bernhard at the Courant Institute in New York. While they were in New York the invitation arrived for both of them to set up mathematics at the Australian National University. In 1963 Hanna and Bernhard went to Australia where she was to spend the rest of her career. In 1971 she undertook a lecture tour of Canada. After lecturing in a number of universities Hanna reached Carleton University, Ottawa. There she became ill and died two days later. Like her husband, Hanna worked in group theory. Her thesis examined free products with amalgamation. Later she worked on varieties of groups and her book Varieties of Groups (1967) is a classic. A letter from two of her students, published after her death, shows her character: We will remember her not only as a mathematician, she was a friend who always had a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Hanna.html (1 of 2) [2/16/2002 11:24:30 PM]

Neumann_Hanna

sympathetic ear for any student, and was never too busy. We will always miss her tremendous dedication and sincerity, and the friendliness of her presence. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) A Poster of Hanna Neumann

Mathematicians born in the same country

Other Web sites

Academy of Sciences, Australia (Obituary)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Neumann_Hanna.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neumann_Hanna.html (2 of 2) [2/16/2002 11:24:30 PM]

Nevanlinna

Rolf Herman Nevanlinna Born: 22 Oct 1895 in Joensuu, Finland Died: 28 May 1980 in Helsinki, Finland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Rolf Nevanlinna's interests at school were firstly classics, secondly mathematics. However he read Lindelöf's Introduction to Higher Analysis before going to university and became an enthusiastic analyst for the whole of his life. (Lindelöf was a cousin of Nevanlinna's father.) Rolf entered Helsinki University in 1913. There he was inspired by Lindelöf's teaching. He avoided military service since he was too light and continued his studies throughout the war years. His thesis was presented in 1919, and he applied the care and attention to detail in this which was to typify all his work. University posts were not available in Finland in 1919 so Nevanlinna became a school teacher. In 1920 he received an invitation from Landau to go to Göttingen but he did not accept immediately. He became a lecturer at Helsinki University in 1922 but he did not give up school teaching until he was appointed professor at Helsinki in 1926. He remained there for the rest of his life except for extensive travels he made to many countries. His first visit was to belatedly accept Landau's invitation to go to Göttingen which he did in 1924. In addition to Landau he met Hilbert, Courant and Emmy Noether in Göttingen. Later visits included one to Paris where he met Hadamard, Montel and he also visited Bloch in a mental hospital. Nevanlinna was offered Weyl's chair when he left Zurich but refused. He continued to work at Helsinki, becoming rector there in 1941. His most important work was on harmonic measure (which he invented). He also developed the theory of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nevanlinna.html (1 of 2) [2/16/2002 11:24:32 PM]

Nevanlinna

value distribution named after him. The main results of Nevanlinna theory appeared in a 100 page paper in 1925. This paper was described by Weyl as one of the few great mathematical events of our century. A full version of the theory appeared in a monograph in 1929. Another great piece of work was his invention of harmonic measure in 1936. The article [8] lists nearly 200 papers written by Nevanlinna. His working habits may explain this very high output of work. These are described in [8] as follows:Rolf never needed much sleep. In the late 1930's he used to rise at 4, join his family at 10 till lunch time, then work again till 7, and spend the evening with his family. Since 1982, an award, the Rolf Nevanlinna Prize, is presented at the International Congress of Mathematicians. The one prize per congress is for young mathematicians dealing with the mathematical aspects of information science. Article by: J J O'Connor and E F Robertson List of References (19 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Nevanlinna.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nevanlinna.html (2 of 2) [2/16/2002 11:24:32 PM]

Newcomb

Simon Newcomb Born: 12 March 1835 in Wallace, Nova Scotia, Canada Died: 11 July 1909 in Washington, D.C., USA

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Simon Newcomb had no formal education but, in about 1854 after he joined his father who had moved to Maryland USA, he began to study mathematics in the libraries at Washington. He obtained a job (1857) in the American Nautical Almanac Office (in Cambridge, Mass. at that time). He studied at Harvard graduating in 1858. In 1861 Newcomb was appointed to the Naval Observatory at Washington. He spent the next10 years determining the positions of celestial objects using various telescopes including a 26-inch refractor telescope which had just been built. In 1877 Newcomb became director of the American Nautical Almanac Office (by this time in Washington). He then started his most important work which, in his own words, gave ... a systematic determination of the constants of astronomy from the best existing data, a reinvestigation of the theories of the celestial motions, and the preparation of tables, formulae, and precepts for the construction of ephemerides, and for other applications of the same results. The reason he undertook this work was because of the ... confusion which pervaded the whole system of exact astronomy, arising from the diversity of the fundamental data made use of by the astronomers of foreign countries and various institutions in their work. Newcomb was professor of mathematics and astronomy at Johns Hopkins (1884-1893). He was an editor of the American Journal of Mathematics for many years. He was also a founding member and first president (1899-1905) of the American Astronomical Society. He served as president of the American Mathematical Society from 1897 to 1898.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newcomb.html (1 of 3) [2/16/2002 11:24:34 PM]

Newcomb

Although most of Newcomb's work was in mathematical astronomy, some of his papers were purely theoretical. He wrote a paper showing how the coordinates of a planet might be represented by trigonometric series. He also wrote on non-euclidean geometry and Cayley commented on one of this theorems saying:... from the boldness of the conception and beauty of the result a very remarkable one, and constitutes an important addition to theoretical dynamics. In [5] Newcomb's interests outside mathematics and astronomy are described:He spoke French and German fluently and knew sufficient of the languages of Italy and Sweden to be able to travel in these countries with comfort. Accustomed from childhood to long walks he continued this form of exercise throughout his life, walking daily several miles between the close of office hours and dinner. On Sundays the walks were much longer. Nothing delighted him more than his walking trips to Switzerland while he was abroad. Even when he was seventy years old he climbed to the chalet high up the side of the Matterhorn, a feat almost unprecedented for a man of his age. He was a lover of travel. .. he was full of fun and loved to romp with his children, when they were young. He read history and other literature extensively and could recite page after page of poetry. He delighted in art... he was an expert chess player... Newcomb received many honours, in fact it requires almost two pages in [5] to list them. Among these honours, he was elected a Fellow of the Royal Society in 1877 he received its Copley Medal in 1890. He was elected an honorary member of most of the major scientific societies of the world and received many awards and prizes for his work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles)

A Quotation

Mathematicians born in the same country Cross-references to History Topics

Orbits and gravitation

Honours awarded to Simon Newcomb (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1877

Royal Society Copley Medal

Awarded 1890

American Maths Society President

1897 - 1898

ASP Bruce Medallist

1898

Lunar features

Crater Newcomb

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newcomb.html (2 of 3) [2/16/2002 11:24:34 PM]

Newcomb

Other Web sites

1. University of Chicago 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Newcomb.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newcomb.html (3 of 3) [2/16/2002 11:24:34 PM]

Newman

Maxwell Herman Alexander Newman Born: 7 Feb 1897 in Chelsea, London, England Died: 22 Feb 1984 in Comberton (near Cambridge), Cambridgeshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Max Newman's father was German and he had the family name of Neumann until he changed it in 1916. He was educated at St John's College Cambridge, entering the university in 1915. From 1916 until 1919 he undertook work related to the war, doing various jobs such as army paymaster and schoolmaster. After Max returned to Cambridge he graduated in 1921, then became a Fellow of St John's College in 1923. During a visit to Vienna in session 1922-23 he was strongly influenced by Reidemeister. From 1927 he was a lecturer at Cambridge in addition to his Fellowship. Newman visited Princeton in 1928-9, then returned in 1937-8. In 1939 Newman was elected a Fellow of the Royal Society. He was later (1958) to receive the Sylvester medal from the Royal Society:... in recognition of his distinguished contributions to combinatory topology, Boolean algebras and mathematical logic. In 1942 he joined the Government Code and Cipher School and worked there with Turing. At the end of the War he was appointed to a chair at Manchester and, 3 years later, he appointed Turing to the post of Reader in Mathematics at Manchester. Along with Hodge and Henry Whitehead, Newman set up the British Mathematical Colloquium. His mathematical work was in the field of combinatorial topology where he greatly influenced his friend http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newman.html (1 of 3) [2/16/2002 11:24:35 PM]

Newman

Henry Whitehead. Newman also wrote an important paper on theoretical computer science. Newman only wrote one book Elements of the topology of plane sets of points. Writing in [2] Peter Hilton says ... this is the only text in general topology which can be wholeheartedly recommended without qualification. It is beautifully written in the limpid style one would expect of one who combined clarity of thought, breadth of view, depth of understanding and mastery of language. Newman saw, and presented, topology as part of the whole of mathematics, not as an isolated discipline: and many must wish he had written more. In 1962 Newman was presented with the De Morgan Medal from the London Mathematical Society. The President of the Society, Mary Cartwright, gave a tribute to Newman's work which is reported in [1]:His early work on Combinatory Topology has excercised a decisive influence on the development of that subject. At a time when the study of manifolds was based on a number of different combinatory concepts, he established a simple combinatory system of simplicial complexes with an equivalence relation based on elementary moves. ... He has proved two important results about fixed points. The first was an early inroad on Hilbert's Fifth Problem, in which he proved that abelian continuous groups do not have arbitrarily small subgroups, the second was a simplified proof of a difficult fixed point theorem of Cartwright and Littlewood arising in the study of differential equations. ... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Max Newman (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1939

Royal Society Sylvester Medal

Awarded 1958

London Maths Society President

1949 - 1951

LMS De Morgan Medal

Awarded 1962

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newman.html (2 of 3) [2/16/2002 11:24:35 PM]

Newman

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Newman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newman.html (3 of 3) [2/16/2002 11:24:35 PM]

Newton

Sir Isaac Newton Born: 4 Jan 1643 in Woolsthorpe, Lincolnshire, England Died: 31 March 1727 in London, England

Click the picture above to see twenty larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Isaac Newton's life can be divided into three quite distinct periods. The first is his boyhood days from 1643 up to his appointment to a chair in 1669. The second period from 1669 to 1687 was the highly productive period in which he was Lucasian professor at Cambridge. The third period (nearly as long as the other two combined) saw Newton as a highly paid government official in London with little further interest in mathematical research. Isaac Newton was born in the manor house of Woolsthorpe, near Grantham in Lincolnshire. Although by the calendar in use at the time of his birth he was born on Christmas Day 1642, we give the date of 4 January 1643 in this biography which is the "corrected" Gregorian calendar date bringing it into line with our present calendar. (The Gregorian calendar was not adopted in England until 1752.) Isaac Newton came from a family of farmers but never knew his father, also named Isaac Newton, who died in October 1642, three months before his son was born. Although Isaac's father owned property and animals which made him quite a wealthy man, he was completely uneducated and could not sign his own name. You can see a picture of Woolsthorpe Manor as it is now. Isaac's mother Hannah Ayscough remarried Barnabas Smith the minister of the church at North Witham, a nearby village, when Isaac was two years old. The young child was then left in the care of his grandmother Margery Ayscough at Woolsthorpe. Basically treated as an orphan, Isaac did not have a happy childhood. His grandfather James Ayscough was never mentioned by Isaac in later life and the fact that James left nothing to Isaac in his will, made when the boy was ten years old, suggests that there was no love lost between the two. There is no doubt that Isaac felt very bitter towards his mother and his step-father Barnabas Smith. When examining his sins at age nineteen, Isaac listed:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html (1 of 9) [2/16/2002 11:24:38 PM]

Newton

Threatening my father and mother Smith to burn them and the house over them. Upon the death of his stepfather in 1653, Newton lived in an extended family consisting of his mother, his grandmother, one half-brother, and two half-sisters. From shortly after this time Isaac began attending the Free Grammar School in Grantham. Although this was only five miles from his home, Isaac lodged with the Clark family at Grantham. However he seems to have shown little promise in academic work. His school reports described him as 'idle' and 'inattentive'. His mother, by now a lady of reasonable wealth and property, thought that her eldest son was the right person to manage her affairs and her estate. Isaac was taken away from school but soon showed that he had no talent, or interest, in managing an estate. An uncle, William Ayscough, decided that Isaac should prepare for entering university and, having persuaded his mother that this was the right thing to do, Isaac was allowed to return to the Free Grammar School in Grantham in 1660 to complete his school education. This time he lodged with Stokes, who was the headmaster of the school, and it would appear that, despite suggestions that he had previously shown no academic promise, Isaac must have convinced some of those around him that he had academic promise. Some evidence points to Stokes also persuading Isaac's mother to let him enter university, so it is likely that Isaac had shown more promise in his first spell at the school than the school reports suggest. Another piece of evidence comes from Isaac's list of sins referred to above. He lists one of his sins as:... setting my heart on money, learning, and pleasure more than Thee ... which tells us that Isaac must have had a passion for learning. We know nothing about what Isaac learnt in preparation for university, but Stokes was an able man and almost certainly gave Isaac private coaching and a good grounding. There is no evidence that he learnt any mathematics, but we cannot rule out Stokes introducing him to Euclid's Elements which he was well capable of teaching (although there is evidence mentioned below that Newton did not read Euclid before 1663). Anecdotes abound about a mechanical ability which Isaac displayed at the school and stories are told of his skill in making models of machines, in particular of clocks and windmills. However, when biographers seek information about famous people there is always a tendency for people to report what they think is expected of them, and these anecdotes may simply be made up later by those who felt that the most famous scientist in the world ought to have had these skills at school. Newton entered his uncle's old College, Trinity College Cambridge, on 5 June 1661. He was older than most of his fellow students but, despite the fact that his mother was financially well off, he entered as a sizar. A sizar at Cambridge was a student who received an allowance toward college expenses in exchange for acting as a servant to other students. There is certainly some ambiguity in his position as a sizar, for he seems to have associated with "better class" students rather than other sizars. Westfall (see [23] or [24]) has suggested that Newton may have had Humphrey Babington, a distant relative who was a Fellow of Trinity, as his patron. This reasonable explanation would fit well with what is known and mean that his mother did not subject him unnecessarily to hardship as some of his biographers claim. Newton's aim at Cambridge was a law degree. Instruction at Cambridge was dominated by the philosophy of Aristotle but some freedom of study was allowed in the third year of the course. Newton studied the philosophy of Descartes, Gassendi, Hobbes, and in particular Boyle. The mechanics of the Copernican astronomy of Galileo attracted him and he also studied Kepler's Optics. He recorded his thoughts in a book which he entitled Quaestiones Quaedam Philosophicae (Certain Philosophical Questions). It is a fascinating account of how Newton's ideas were already forming around 1664. He http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html (2 of 9) [2/16/2002 11:24:38 PM]

Newton

headed the text with a Latin statement meaning "Plato is my friend, Aristotle is my friend, but my best friend is truth" showing himself a free thinker from an early stage. How Newton was introduced to the most advanced mathematical texts of his day is slightly less clear. According to de Moivre, Newton's interest in mathematics began in the autumn of 1663 when he bought an astrology book at a fair in Cambridge and found that he could not understand the mathematics in it. Attempting to read a trigonometry book, he found that he lacked knowledge of geometry and so decided to read Barrow's edition of Euclid's Elements. The first few results were so easy that he almost gave up but he:... changed his mind when he read that parallelograms upon the same base and between the same parallels are equal. Returning to the beginning, Newton read the whole book with a new respect. He then turned to Oughtred's Clavis Mathematica and Descartes' La Géométrie. The new algebra and analytical geometry of Viète was read by Newton from Frans van Schooten's edition of Viète's collected works published in 1646. Other major works of mathematics which he studied around this time was the newly published major work by van Schooten Geometria a Renato Des Cartes which appeared in two volumes in 1659-1661. The book contained important appendices by three of van Schooten disciples, Jan de Witt, Johan Hudde, and Hendrick van Heuraet. Newton also studied Wallis's Algebra and it appears that his first original mathematical work came from his study of this text. He read Wallis's method for finding a square of equal area to a parabola and a hyperbola which used indivisibles. Newton made notes on Wallis's treatment of series but also devised his own proofs of the theorems writing:Thus Wallis doth it, but it may be done thus ... It would be easy to think that Newton's talent began to emerge on the arrival of Barrow to the Lucasian chair at Cambridge in 1663 when he became a Fellow at Trinity College. Certainly the date matches the beginnings of Newton's deep mathematical studies. However, it would appear that the 1663 date is merely a coincidence and that it was only some years later that Barrow recognised the mathematical genius among his students. Despite some evidence that his progress had not been particularly good, Newton was elected a scholar on 28 April 1664 and received his bachelor's degree in April 1665. It would appear that his scientific genius had still not emerged, but it did so suddenly when the plague closed the University in the summer of 1665 and he had to return to Lincolnshire. There, in a period of less than two years, while Newton was still under 25 years old, he began revolutionary advances in mathematics, optics, physics, and astronomy. While Newton remained at home he laid the foundations for differential and integral calculus, several years before its independent discovery by Leibniz. The 'method of fluxions', as he termed it, was based on his crucial insight that the integration of a function is merely the inverse procedure to differentiating it. Taking differentiation as the basic operation, Newton produced simple analytical methods that unified many separate techniques previously developed to solve apparently unrelated problems such as finding areas, tangents, the lengths of curves and the maxima and minima of functions. Newton's De Methodis Serierum et Fluxionum was written in 1671 but Newton failed to get it published and it did not appear in print until John Colson produced an English translation in 1736. When the University of Cambridge reopened after the plague in 1667, Newton put himself forward as a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html (3 of 9) [2/16/2002 11:24:38 PM]

Newton

candidate for a fellowship. In October he was elected to a minor fellowship at Trinity College but, after being awarded his Master's Degree, he was elected to a major fellowship in July 1668 which allowed him to dine at the Fellows' Table. In July 1669 Barrow tried to ensure that Newton's mathematical achievements became known to the world. He sent Newton's text De Analysi to Collins in London writing:[Newton] brought me the other day some papers, wherein he set down methods of calculating the dimensions of magnitudes like that of Mr Mercator concerning the hyperbola, but very general; as also of resolving equations; which I suppose will please you; and I shall send you them by the next. Collins corresponded with all the leading mathematicians of the day so Barrow's action should have led to quick recognition. Collins showed Brouncker, the President of the Royal Society, Newton's results (with the author's permission) but after this Newton requested that his manuscript be returned. Collins could not give a detailed account but de Sluze and Gregory learnt something of Newton's work through Collins. Barrow resigned the Lucasian chair in 1669 to devote himself to divinity, recommending that Newton (still only 27 years old) be appointed in his place. Shortly after this Newton visited London and twice met with Collins but, as he wrote to Gregory:... having no more acquaintance with him I did not think it becoming to urge him to communicate anything. Newton's first work as Lucasian Professor was on optics and this was the topic of his first lecture course begun in January 1670. He had reached the conclusion during the two plague years that white light is not a simple entity. Every scientist since Aristotle had believed that white light was a basic single entity, but the chromatic aberration in a telescope lens convinced Newton otherwise. When he passed a thin beam of sunlight through a glass prism Newton noted the spectrum of colours that was formed. He argued that white light is really a mixture of many different types of rays which are refracted at slightly different angles, and that each different type of ray produces a different spectral colour. Newton was led by this reasoning to the erroneous conclusion that telescopes using refracting lenses would always suffer chromatic aberration. He therefore proposed and constructed a reflecting telescope. In 1672 Newton was elected a fellow of the Royal Society after donating a reflecting telescope. Also in 1672 Newton published his first scientific paper on light and colour in the Philosophical Transactions of the Royal Society. The paper was generally well received but Hooke and Huygens objected to Newton's attempt to prove, by experiment alone, that light consists of the motion of small particles rather than waves. The reception that his publication received did nothing to improve Newton's attitude to making his results known to the world. He was always pulled in two directions, there was something in his nature which wanted fame and recognition yet another side of him feared criticism and the easiest way to avoid being criticised was to publish nothing. Certainly one could say that his reaction to criticism was irrational, and certainly his aim to humiliate Hooke in public because of his opinions was abnormal. However, perhaps because of Newton's already high reputation, his corpuscular theory reigned until the wave theory was revived in the 19th century. Newton's relations with Hooke deteriorated further when, in 1675, Hooke claimed that Newton had stolen some of his optical results. Although the two men made their peace with an exchange of polite letters, Newton turned in on himself and away from the Royal Society which he associated with Hooke http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html (4 of 9) [2/16/2002 11:24:38 PM]

Newton

as one of its leaders. He delayed the publication of a full account of his optical researches until after the death of Hooke in 1703. Newton's Opticks appeared in 1704. It dealt with the theory of light and colour and with (i) investigations of the colours of thin sheets (ii) 'Newton's rings' and (iii) diffraction of light. To explain some of his observations he had to use a wave theory of light in conjunction with his corpuscular theory. Another argument, this time with the English Jesuits in Liège over his theory of colour, led to a violent exchange of letters, then in 1678 Newton appears to have suffered a nervous breakdown. His mother died in the following year and he withdrew further into his shell, mixing as little as possible with people for a number of years. Newton's greatest achievement was his work in physics and celestial mechanics, which culminated in the theory of universal gravitation. By 1666 Newton had early versions of his three laws of motion. He had also discovered the law giving the centrifugal force on a body moving uniformly in a circular path. However he did not have a correct understanding of the mechanics of circular motion. Newton's novel idea of 1666 was to imagine that the Earth's gravity influenced the Moon, counterbalancing its centrifugal force. From his law of centrifugal force and Kepler's third law of planetary motion, Newton deduced the inverse-square law. In 1679 Newton corresponded with Hooke who had written to Newton claiming:... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall ... M Nauenberg writes an account of the next events:After his 1679 correspondence with Hooke, Newton, by his own account, found a proof that Kepler's areal law was a consequence of centripetal forces, and he also showed that if the orbital curve is an ellipse under the action of central forces then the radial dependence of the force is inverse square with the distance from the centre. This discovery showed the physical significance of Kepler's second law. In 1684 Halley, tired of Hooke's boasting [M Nauenberg]:... asked Newton what orbit a body followed under an inverse square force, and Newton replied immediately that it would be an ellipse. However in De Motu.. he only gave a proof of the converse theorem that if the orbit is an ellipse the force is inverse square. The proof that inverse square forces imply conic section orbits is sketched in Cor. 1 to Prop. 13 in Book 1 of the second and third editions of the Principia, but not in the first edition. Halley persuaded Newton to write a full treatment of his new physics and its application to astronomy. Over a year later (1687) Newton published the Philosophiae naturalis principia mathematica or Principia as it is always known. The Principia is recognised as the greatest scientific book ever written. Newton analysed the motion of bodies in resisting and non-resisting media under the action of centripetal forces. The results were

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html (5 of 9) [2/16/2002 11:24:38 PM]

Newton

applied to orbiting bodies, projectiles, pendulums, and free-fall near the Earth. He further demonstrated that the planets were attracted toward the Sun by a force varying as the inverse square of the distance and generalised that all heavenly bodies mutually attract one another. Further generalisation led Newton to the law of universal gravitation:... all matter attracts all other matter with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. Newton explained a wide range of previously unrelated phenomena: the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis, and motion of the Moon as perturbed by the gravity of the Sun. This work made Newton an international leader in scientific research. The Continental scientists certainly did not accept the idea of action at a distance and continued to believe in Descartes' vortex theory where forces work through contact. However this did not stop the universal admiration for Newton's technical expertise. James II became king of Great Britain on 6 February 1685. He had become a convert to the Roman Catholic church in 1669 but when he came to the throne he had strong support from Anglicans as well as Catholics. However rebellions arose, which James put down but he began to distrust Protestants and began to appoint Roman Catholic officers to the army. He then went further, appointing only Catholics as judges and officers of state. Whenever a position at Oxford or Cambridge became vacant, the king appointed a Roman Catholic to fill it. Newton was a staunch Protestant and strongly opposed to what he saw as an attack on the University of Cambridge. When the King tried to insist that a Benedictine monk be given a degree without taking any examinations or swearing the required oaths, Newton wrote to the Vice-Chancellor:Be courageous and steady to the Laws and you cannot fail. The Vice-Chancellor took Newton's advice and was dismissed from his post. However Newton continued to argue the case strongly preparing documents to be used by the University in its defence. However William of Orange had been invited by many leaders to bring an army to England to defeat James. William landed in November 1688 and James, finding that Protestants had left his army, fled to France. The University of Cambridge elected Newton, now famous for his strong defence of the university, as one of their two members to the Convention Parliament on 15 January 1689. This Parliament declared that James had abdicated and in February 1689 offered the crown to William and Mary. Newton was at the height of his standing - seen as a leader of the university and one of the most eminent mathematicians in the world. However, his election to Parliament may have been the event which let him see that there was a life in London which might appeal to him more than the academic world in Cambridge. After suffering a second nervous breakdown in 1693, Newton retired from research. The reasons for this breakdown have been discussed by his biographers and many theories have been proposed: chemical poisoning as a result of his alchemy experiments; frustration with his researches; the ending of a personal friendship with Fatio de Duillier, a Swiss-born mathematician resident in London; and problems resulting from his religious beliefs. Newton himself blamed lack of sleep but this was almost certainly a symptom of the illness rather than the cause of it. There seems little reason to suppose that the illness was anything other than depression, a mental illness he must have suffered from throughout most of his life, perhaps made worse by some of the events we have just listed. Newton decided to leave Cambridge to take up a government position in London becoming Warden of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html (6 of 9) [2/16/2002 11:24:38 PM]

Newton

the Royal Mint in 1696 and Master in 1699. However, he did not resign his positions at Cambridge until 1701. As Master of the Mint, adding the income from his estates, we see that Newton became a very rich man. For many people a position such as Master of the Mint would have been treated as simply a reward for their scientific achievements. Newton did not treat it as such and he made a strong contribution to the work of the Mint. He led it through the difficult period of recoinage and he was particularly active in measures to prevent counterfeiting of the coinage. In 1703 he was elected president of the Royal Society and was re-elected each year until his death. He was knighted in 1705 by Queen Anne, the first scientist to be so honoured for his work. However the last portion of his life was not an easy one, dominated in many ways with the controversy with Leibniz over which had invented the calculus. Given the rage that Newton had shown throughout his life when criticised, it is not surprising that he flew into an irrational temper directed against Leibniz. We have given details of this controversy in Leibniz's biography and refer the reader to that article for details. Perhaps all that is worth relating here is how Newton used his position as President of the Royal Society. In this capacity he appointed an "impartial" committee to decide whether he or Leibniz was the inventor of the calculus. He wrote the official report of the committee (although of course it did not appear under his name) which was published by the Royal Society, and he then wrote a review (again anonymously) which appeared in the Philosophical Transactions of the Royal Society. Newton's assistant Whiston had seen his rage at first hand. He wrote:Newton was of the most fearful, cautious and suspicious temper that I ever knew. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (271 books/articles)

Some Quotations (13)

A Poster of Isaac Newton

Mathematicians born in the same country

Some pages from publications

The title page of Philosophiae naturalis principia mathematica (The Principia 1687) The title page of Analysis per quantitatum series, fluxiones (1711)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html (7 of 9) [2/16/2002 11:24:38 PM]

Newton

Cross-references to History Topics

1. Orbits and gravitation 2. Elliptic functions. 3. Longitude and the Académie Royale 4. A brief history of cosmology 5. English attack on the Longitude Problem 6. General relativity 7. An overview of the history of mathematics 8. The rise of the calculus 9. Special relativity 10. A chronology of pi 11. An overview of Indian mathematics 12. The Bakhshali manuscript

Cross-references to Famous Curves

1. Cartesian ovals 2. Cissoid 3. Conchoid 4. Cycloid 5. Epicycloid 6. Epitrochoid 7. Hypocycloid 8. Hypotrochoid 9. Kappa curve 10. Serpentine 11. Newton's diverging parabolas 12. Trident of Newton

Other references in MacTutor

1. Julian and Gregorian calendars 2. Newton-Raphson method 3. Chronology: 1650 to 1675 4. Chronology: 1675 to 1700 5. Chronology: 1700 to 1720

Honours awarded to Isaac Newton (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1672

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html (8 of 9) [2/16/2002 11:24:38 PM]

Newton

Lucasian Professor of Mathematics

1669

Lunar features

Crater Newton

Paris street names

Rue Newton (16th Arrondissement)

Other Web sites

1. Cambridge University Library 2. Isaac Newton Institute, Cambridge 3. Rouse Ball 4. Bob Bruen 5. The Galileo Project 6. Andrew McNab (Many links) 7. Isaac Newton Home page 8. G Don Allen 9. Glasgow University 10. High Altitude Observatory 11. Kevin Brown (More about Newton's birthday) 12. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Newton.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html (9 of 9) [2/16/2002 11:24:38 PM]

Neyman

Jerzy Neyman Born: 16 April 1894 in Bendery, Moldavia Died: 5 Aug 1981 in Oakland, California, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jerzy Neyman was originally named Splawa-Neyman, but he dropped the first part of his name at the age of 30. He studied at Kharkov University and wrote on Lebesgue integration. Sergi Bernstein influenced him, encouraging him to read Pearson's Grammar of Science. In Warsaw he lectured in mathematics and statistics and received a doctorate in 1924. Receiving a fellowship to work with Pearson in London, he was disappointed to discover that Pearson was ignorant of modern mathematics. E S Pearson, writing in [3], describes Neyman at this time:What I remember ... is a week-end which we spent together in the spring of 1926 at our family holiday cottage (the Old School House) at Coldharbour on Leith Hill in Surrey. It was then that I listened with fascination to an account of his early life in Russia and of the experiences which he had later undergone in the shadow of those disruptive forces, set in train throughout Central Europe by war and the Russian Revolution. In Paris in 1927 Neyman attended lectures by Lebesgue and Hadamard but his interest in statistics was stimulated again by Pearson's son, E S Pearson, who sought a general principle from which Gosset's tests could be derived. Neyman went on to produce fundamental results on hypothesis testing. Neyman returned to Poland in May 1927 and immediately tried to set up a biometric laboratory in Warsaw. In June 1927 Neyman wrote:Certainly [the laboratory] is not yet sure, especially as our loan in America is not yet

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neyman.html (1 of 3) [2/16/2002 11:24:40 PM]

Neyman

signed. However by 1928 he had managed to set up a Biometric Laboratory at the Nencki Institute for Experimental Biology in Warsaw. In 1931 he wrote:... we have in Poland a terrific crisis in everything. Accordingly the money from the Government given usually to the Nencki Institute will be diminished considerably and I shall have difficulties in feeding my pups. Neyman's 'pups' were the research workers in his laboratory! By 1932 things seemed worse for Neyman who wrote:I simply cannot work, the crisis and the struggle for existence takes all my time and energy. Between 1928 and 1933 Neyman and E S Pearson wrote a number of important papers such as On the problem of the most efficient tests of statistical hypotheses (1933) and The testing of statistical hypotheses in relation to probabilities a priori (1933). Neyman came in England in 1934 to fill a temporary post in E S Pearson's department. The following year the post was made permanent and Neyman held it until 1938. In 1938 Neyman emigrated to the United States where he worked in Berkeley for the rest of his life. Neyman's work on hypothesis testing, confidence intervals and survey sampling revolutionised statistics. Article by: J J O'Connor and E F Robertson List of References (15 books/articles) Mathematicians born in the same country Honours awarded to Jerzy Neyman (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1979

Other Web sites

1. University of Minnesota 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neyman.html (2 of 3) [2/16/2002 11:24:40 PM]

Neyman

JOC/EFR February 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Neyman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Neyman.html (3 of 3) [2/16/2002 11:24:40 PM]

Nicolson

Phyllis Nicolson Born: 21 Sept 1917 in Macclesfield, England Died: 6 Oct 1968

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Phyllis Nicolson's maiden name was Lockett. She was educated at Stockport High School and received the degrees of B.Sc. (1938) and M.Sc. (1939) and Ph.D. in Physics (1946) from Manchester University and was a research student (1945-46) and research fellow (1946-49) at Girton College, Cambridge. In 1942 she married Malcolm Nicolson. She had a strong wish to have her first child before reaching thirty, and she achieved this ambition with a day to spare. After her husband's untimely death in a train crash in 1952, she was appointed to fill his lectureship in Physics at Leeds University. In 1956 she married Malcolm McCaig, who was also a physicist. During the period 1940-45 she was a member of a research group in Manchester University directed by Douglas Hartree, working on wartime problems for the Ministry of Supply, one being concerned with magnetron theory and performance. Phyllis Nicolson is best known for her joint work with John Crank on the heat equation, where a continuous solution u(x,t) is required which satisfies the second order partial differential equation ut - uxx = 0 for t > 0, subject to an initial condition of the form u(x,0) = f(x) for all real x. They considered numerical methods which find an approximate solution on a grid of values of x and t, replacing ut(x,t) and uxx(x,t) by finite difference approximations. One of the simplest such replacements was proposed by L F Richardson in 1910. Richardson's method yielded a numerical solution which was very easy to compute, but alas was numerically unstable and thus useless. The instability was not recognised until lengthy

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nicolson.html (1 of 2) [2/16/2002 11:24:42 PM]

Nicolson

numerical computations were carried out by Crank, Nicolson and others. Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level. Article by: G M Phillips, St Andrews Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Nicolson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nicolson.html (2 of 2) [2/16/2002 11:24:42 PM]

Nicomachus

Nicomachus of Gerasa Born: about 60 in Gerasa, Roman Syria (now Jarash, Jordan) Died: about 120

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nicomachus of Gerasa is mentioned in a small number of sources and we can date him fairly accurately from the information given. Nicomachus himself refers to Thrasyllus who died in 36 AD so this gives lower limits on his dates. On the other hand Apuleius, the Platonic philosopher, rhetorician and author whose dates are 124 AD to about 175 AD, translated Nicomachus's Introduction to Arithmetic into Latin so this gives an upper limit on his dates. One of the most interesting references is by Lucian, the rhetorician, pamphleteer and satirist who was born about 120 AD, who makes one of his characters say:You calculate like Nicomachus. Clearly Nicomachus had achieved fame for his arithmetical work! In the paper [7] Dillon argues that Nicomachus died in 196 AD. His argument is based on the fact that Marinus claimed that Proclus believed that he was the reincarnation of Nicomachus. Since Proclus was born in 412 AD and there was a belief among Pythagoreans that reincarnations occurred with an interval of 216 years, the date fits. Although 196 AD is not ruled out by his translator dying in 175 AD (although it comes close) the most serious objection to Dillon's theory seems to be the lack of evidence that Proclus himself believed in the 216 year interval. Let us move from conjectures to more certain ground, and record that Nicomachus was a Pythagorean. This is obvious from his writings on numbers and music, but we are also told this by Porphyry who says that he was one of the leading members of the Pythagoreans School. Nicomachus wrote Arithmetike eisagoge (Introduction to Arithmetic) which was the first work to treat arithmetic as a separate topic from geometry. Unlike Euclid, Nicomachus gave no abstract proofs of his theorems, merely stating theorems and illustrating them with numerical examples.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nicomachus.html (1 of 4) [2/16/2002 11:24:44 PM]

Nicomachus

However Introduction to Arithmetic does contain quite elementary errors which show that Nicomachus chose not to give proofs of his results because he did not in general have such proofs. Many of the results were known by Nicomachus to be true since they appeared with proofs in Euclid, although in a geometrical formulation. Sometimes Nicomachus stated a result which is simply false and then illustrated it with an example which happens to have the properties described in the result. We must deduce from this that some of the results are merely guesses based on the evidence of the numerical examples (and in some cases perhaps even based on one example!). An example of this we look more closely at the results which Nicomachus quotes on perfect numbers. He states that the nth perfect number has n digits, and that all perfect numbers end in 6 and 8 alternately. These statements must be merely false deductions from the fact that there were four perfect numbers known to Nicomachus, namely 6, 28, 496 and 8128. The work contains the first multiplication table in a Greek text. It is also remarkable in that it contains Arabic numerals, not Greek ones. However, in many respects the book is old fashioned in its style since it appears more in tune with the number theoretic ideas of Pythagoras with his mystical approach, rather than a true mathematical approach. To illustrate Nicomachus's rather strange approach to numbers, giving the moral properties, we look at his description of abundant numbers and deficient numbers. An abundant number has the sum of its proper divisors greater than the number, while a deficient number has the sum of its proper divisors less than the number. Nicomachus writes of these numbers in Introduction to Arithmetic (see [6], or [3] for a different translation):In the case of the too much, is produced excess, superfluity, exaggerations and abuse; in the case of too little, is produced wanting, defaults, privations and insufficiencies. And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort - of which the most exemplary form is that type of number which is called perfect. He then continues his description of abundant numbers as resembling an animal:... with ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands.... while a deficient number is like an animal:... with a single eye, ... one armed or one of his hands has less than five fingers, or if he does not have a tongue... For over 1000 years Introduction to Arithmetic was the standard arithmetic text. In view of the comments we have made regarding the work, this may seem a surprising fact. Mathematicians disliked the work, in particular Pappus is said to have despised it. However, several people including Boethius translated Introduction to Arithmetic into Latin and it was used as a school book. How then could a poor book become so popular. Heath tries to explain the apparent contradiction in [4], suggesting that:... it was at first read by philosophers rather than mathematicians, and afterwards became generally popular at a time when there were no mathematicians left, but only philosophers who incidentally took an interest in mathematics. Arab translations of Nicomachus's Introduction to Arithmetic were important and in [5] Brentjes studies the influence of these Arabic translations. She concludes that most Arabic texts on number theory written by mathematicians were influenced by both Euclid and Nicomachus, but were mainly influenced by

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nicomachus.html (2 of 4) [2/16/2002 11:24:44 PM]

Nicomachus

Euclid. However, texts by non-mathematicians were most strongly influenced by Nicomachus. This research in [5] tends to support the views of Heath on this subject. Nicomachus also wrote two volumes Theologoumena arithmetikes (The Theology of Numbers) which was completely concerned with mystic properties of numbers. However Heath writes [4]:The curious farrago which has come down to us under that title and which was edited by Ast [published in Leipzig in 1817] is, however, certainly not by Nicomachus; for among the authors from whom it gives extracts is Anatolius, Bishop of Laodicaea (270 AD); but it contains quotations from Nicomachus which appear to come from the genuine work. Another work by Nicomachus which has survived is Manual of Harmonics which is a work on music. Again Nicomachus shows the influence of Pythagoras but also Aristotle's theories of music. The work looks at musical notes and the octave. The principles of tuning a stretched string are studied as is an extension of the octave to the two-octave range. The influences of Pythagoras's theory of music are seen from Nicomachus' (see [1]):... assignment of number and numerical ratios to notes and intervals, his recognition of the indivisibility of the octave and the whole tone... But, unlike Euclid, who attempts to prove musical propositions through mathematical theorems, Nicomachus seeks to show their validity by measurement of the lengths of strings. Both Porphyry and Iamblichus wrote biographies of Pythagoras which quote from Nicomachus. From this evidence some historians have conjectured that Nicomachus also wrote a biography of Pythagoras and, although there is no direct evidence, it is indeed quite possible. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles)

Some Quotations (2)

Mathematicians born in the same country Some pages from publications

Extract from a Greek multiplication table printed in 1538.

Cross-references to History Topics

Perfect numbers

Other references in MacTutor

Chronology: 1AD to 500

Other Web sites

1. Jay Kappraff in the NNJ 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nicomachus.html (3 of 4) [2/16/2002 11:24:44 PM]

Nicomachus

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Nicomachus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nicomachus.html (4 of 4) [2/16/2002 11:24:44 PM]

Nicomedes

Nicomedes Born: about 280 BC in Greece Died: about 210 BC

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

We know nothing of Nicomedes' life. To make a guess at dating his life we have some limits which are given by references to his work. Nicomedes himself criticised the method that Eratosthenes used to duplicate the cube and we have made a reasonably accurate guess at Eratosthenes's life span (276 BC 194 BC). A less certain piece of information comes from Apollonius choosing to name a curve 'sister of the conchoid' which is assumed to be a name he has chosen to compliment Nicomedes' discovery of the conchoid. Since Apollonius lived from about 262 BC to 190 BC these two pieces of information give a fairly accurate estimate of Nicomedes' dates. However, as we remarked the second of these pieces of information cannot be relied on but nevertheless, from what we know of the mathematics of Nicomedes, the deduced dates are fairly convincing.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nicomedes.html (1 of 3) [2/16/2002 11:24:46 PM]

Nicomedes

Nicomedes is famous for his treatise On conchoid lines which contain his discovery of the curve known as the conchoid of Nicomedes. How did Nicomedes define the curve. Consider the diagram. We are given a line XY and a point A not on the line. ABC is drawn perpendicular to XY cutting it at B and having the length BC some fixed value, say b. Then rotate ABC about A so that, in an arbitrary position ADE then DE = b. So if a point P on the conchoid is joined to A then PA cuts the line XY in a point at distance b from P. Nicomedes recognised three distinct forms in this family but the sources do not go into detail on this point. It is believed that they must be the three branches of the curve. The conchoid can be used in solving both the problems of trisection of an angle and of the duplication of a cube. Both these problems were solved by Nicomedes using the conchoid, in fact as Toomer writes in [1]:As far as is known, all applications of the conchoid made in antiquity were developed by Nicomedes himself. It was not until the late sixteenth century, when the works of Pappus and Eutocius describing the curve became generally known, that interest in it revived... As indicated in this quote Pappus also wrote about Nicomedes, in particular he wrote about his solution to the problem of trisecting an angle (see for example [2]):Nicomedes trisected any rectilinear angle by means of the conchoidal curves, the construction, order and properties of which he handed down, being himself the discoverer of their peculiar character. Nicomedes also used the quadratrix, discovered by Hippias, to solve the problem of squaring the circle. Pappus tells us (see for example [2]):For the squaring of the circle there was used by Dinostratus, Nicomedes and certain other later persons a certain curve which took its name from this property, for it is called by them square-forming in other words the quadratrix. Eutocius tells us that Nicomedes [2]:... prided himself inordinately on his discovery of this curve, contrasting it with Eratosthenes's mechanism for finding any number of mean proportionals, to which he objected formally and at length on the ground that it was impracticable and entirely outside the spirit of geometry. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nicomedes.html (2 of 3) [2/16/2002 11:24:46 PM]

Nicomedes

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Squaring the circle 2. Trisecting an angle

Cross-references to Famous Curves

1. Conchoid 2. Quadratrix of Hippias

Other references in MacTutor

Chronology: 500BC to 1AD

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Nicomedes.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nicomedes.html (3 of 3) [2/16/2002 11:24:46 PM]

Nielsen

Niels Nielsen Born: 2 Dec 1865 in Orslev, Denmark Died: 16 Sept 1931 in Copenhagen, Denmark Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Niels Nielsen's father was a farmer and the family were quite poor. Nielsen entered the University of Copenhagen in 1885 and graduated from there in 1891. In fact he had begun to teach in secondary schools in 1887 while still working for his degree at the University of Copenhagen and he continued teaching while working for his doctorate which was awarded in 1895. He gave preparatory courses for the Polytechnic Institute beginning in 1900 and from 1903 until 1906 he was on the University Inspectorate for Secondary Schools. Nielsen became a university teacher in 1905 and he succeeded Petersen as professor at Copenhagen in 1909. He wrote on special functions, particularly the gamma function, building on theory introduced by Jensen. His works were textbooks containing little original mathematics but they were well written texts with much skill in organising the material. In 1917 he suffered a breakdown from which he never fully recovered. He turned to number theory and studied Bernoulli numbers and Fermat's equation writing good textbooks on these topics. He also wrote two books on the history of Danish mathematics and two books on the history of French mathematics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other Web sites

1. Theseus 2. Nielsen-Ramanujan Constants

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nielsen.html (1 of 2) [2/16/2002 11:24:47 PM]

Nielsen

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Nielsen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nielsen.html (2 of 2) [2/16/2002 11:24:47 PM]

Nielsen_Jakob

Jakob Nielsen Born: 15 Oct 1890 in Mjels, Als, Schleswig (now Denmark) Died: 3 Aug 1959 in Elsinore, Denmark

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jakob Nielsen's mother died when he was three years old. From the age of 10 he lived with his aunt at Rendsborg where he attended a Realgymnasium. There he studied religion, Latin and modern languages, history and geography, and mathematics and science. His relations with his aunt deteriorated, however, and Nielsen left her home when he was fourteen and he continued at school but earned his living by tutoring. In 1907 he was expelled from school for breaking school rules by founding a pupil's club. Nielsen did not let this damage his education but continued to study on his own. He entered the University of Kiel in 1908, studying a wide range of subjects. Among the teachers at Kiel who had a profound influence on him was Landsberg who suggested the topics which were to form Nielsen's doctoral thesis, although Landsberg died before the thesis was completed. Nielsen spent the summer term of 1910 at the University of Berlin but returned to Kiel where Dehn was appointed in 1911. Dehn greatly influenced Nielsen and introduced him to the newest ideas in topology and group theory. In particular it was through Dehn that he became interested in free groups and his paper he published on this topic in 1921 contained ideas which came from his association with Dehn in Kiel. After completing his doctorate Nielsen served in the German marines, continuing in military service in different parts of the world throughout World War I. In particular he served in Belgium and then Constantinople where he advised the Turkish government on how to defend the entrance to the Black Sea. At the end of the war Nielsen returned from Turkey to Germany through Russia and Poland and he later published an account of his travels from a diary which he had kept. Nielsen married Carola von Pieverling early in 1919. She was a German medical doctor and the couple http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nielsen_Jakob.html (1 of 4) [2/16/2002 11:24:49 PM]

Nielsen_Jakob

had a happy marriage bringing up one son and two daughters. Nielsen spent time in 1919 in Göttingen where he met Hecke. When Hecke was appointed to Hamburg in 1919, Nielsen went as his assistant, but the following year Nielsen was appointed to a chair at Breslau. This allowed Nielsen to become a colleague of Dehn who was a professor there. Two inaugural lectures which Nielsen gave in Breslau in 1921 are published for the first time in [1]. The Treaty of Versailles at the end of World War I declared that part of Schleswig should revert to Denmark in keeping with the principle of self-determination. The boundary was determined by a plebiscite in 1920. In 1921 Nielsen, who was born in Schleswig, elected to became a Danish citizen and he resigned his chair in Breslau and returned to Denmark to teach in Copenhagen. He taught mathematics at the Royal Veterinary and Agricultural University for four years. In 1925 he succeeded Juel as professor of theoretical mechanics at the Technical University of Copenhagen. In 1919 Nielsen purchased a house on the island of Als on which he was born. A few years later Harald Bohr also purchased a house on the island. Fenchel writes in [2]:Year after year, in the summer vacation, young and old, Danish and foreign, gathered about [Nielsen and Harald Bohr]. Apart from normal holiday activities, the study of mathematics was pursued. Not a few advances and discoveries were presented in Bohr's little half-timbered house, in the study remarkable for its blackboard - unforgettable experiences which are remembered with gratitude by all who had the privilege of attending. Nielsen wrote a new text on theoretical mechanics which was published in two volumes in 1933-34. Hansen writes in [6]:The book was to a large extent original pedagogical work on an advanced level for its time, and in his exposition, Nielsen made extensive use of mathematical tools such as vectors and matrices, which were then relatively new concepts in textbooks. The text is not very easy, and Jacob Nielsen's lectures were rather demanding on the part of the students. He was, however, well known for his ability to express himself with great clarity and intensity. There were some fears for Nielsen's safety during World War II. Principally there were worries that since he had chosen to leave his post in a German university and take Danish citizenship in 1921 the Nazis might give him a difficult time. These fears were not realised and Nielsen was not targeted by the Nazis. Nielsen taught a course on aerodynamics in 1941 and the course formed the basis of a third volume of his theoretical mechanics text published in 1952. In 1951 Nielsen succeeded Harald Bohr as professor of mathematics at the University of Copenhagen. However he resigned in 1955 and spent the last years of his life in the Royal Danish Academy of Sciences residence. He had been elected to membership of the Academy in 1926. It might seem a strange decision for Nielsen to resign this prestigious chair, but he was sixty-five years of age in 1955 and he felt that his international commitments were too much to allow him to do full justice to all his tasks. These international commitments related mostly to UNESCO, the United Nations Educational, Scientific and Cultural Organisation which had been set up in 1948. He had been elected to the executive board of UNESCO in 1952. It was a role he carried out until 1958 and he made an outstanding contribution as someone of the highest personal integrity. Nielsen is well known for work in several areas. His work on group theory is important as he was one of the founders of combinatorial group theory. In particular he proved that a subgroup of a finitely http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nielsen_Jakob.html (2 of 4) [2/16/2002 11:24:49 PM]

Nielsen_Jakob

generated free group is free in 1921. (In 1926 Schreier proved that the finitely generated assumption is not necessary.) His work on topological transformations is also important. Nielsen studied the mapping class group of a torus in his thesis of 1913. He went on to examine surfaces of genus 1 as well as surfaces of higher genus. He also produced work on fixed point theory related to that of Dehn and the theory of discontinuous groups of isometries of the hyperbolic plane. Bernhard Neumann, reviewing Nielsen's collected papers which were published in two volumes in 1986, writes:Jakob Nielsen initiated much of the topology of surfaces and of combinatorial group theory, and for this reason alone he occupies an important place in the history of 20th century mathematics. However, his work is not merely of historical interest, but is still (or again) much used and quoted today. The publication of his collected mathematical papers is, therefore, timely and very welcome; the more so as the two volumes under review make the papers more readily accessible to today's researchers. Thus, for example, not many mathematicians will in the past have had access to Nielsen's doctoral dissertation, presented to the Universitat Kiel in 1913. Other papers appeared in various Scandinavian periodicals or conference proceedings, many of them in Danish: the majority of them, and also many of Nielsen's papers in German, have for this collection been translated into English. Among them is his fundamental paper, in the Matematisk Tidsskrift in 1921, on free groups, in which the Nielsen-Schreier theorem (or rather Nielsen's part of it) is proved for the first time: now there is no further excuse for misquoting this paper, as has happened repeatedly in the past. There was however, some work by Nielsen which was not published in his collected works. This was work which he had undertaken with his student and friend Werner Fenchel, the author of the articles [2] and [3], and was based on a course of lectures Nielsen gave on the theory of discontinuous groups of isometries of the hyperbolic plane in 1938-39. In fact Fenchel contributed the final article [1] on this topic. This piece of mathematics, now known as Fenchel-Nielsen theory, had a rather strange history. Fenchel and Nielsen decided to write a monograph on the theory but neither were happy with the first draft of the manuscript which they produced. They were working on a revision when Nielsen died in 1959 after an illness lasting about eight months. Matters were certainly not helped by the fact that the original manuscript was stolen from Fenchel's car but he continued to feel that the results were unsatisfactory. Eventually Fenchel wrote a book Elementary geometry in hyperbolic space which was intended to put give an approach with would make presentation of Fenchel-Nielsen theory much clearer. Fenchel died in 1988 with the whole project close to completion. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nielsen_Jakob.html (3 of 4) [2/16/2002 11:24:49 PM]

Nielsen_Jakob

Other Web sites

Theseus

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Nielsen_Jakob.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nielsen_Jakob.html (4 of 4) [2/16/2002 11:24:49 PM]

Nightingale

Florence Nightingale Born: 12 May 1820 in Florence, Italy Died: 13 August 1910 in East Wellow, England

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Florence Nightingale is best remembered for her work as a nurse during the Crimean War and her contribution towards the reform of the sanitary conditions in military field hospitals. However, what is less well known about this amazing woman is her love of mathematics, especially statistics, and how this love played an important part in her life's work. Named after the city of her birth, Nightingale was born at the Villa Colombia in Florence, Italy, on the 12th May 1820. Her parents, William Edward Nightingale and his wife Frances, were touring Europe for the first two years of their marriage. Nightingale's elder sister had been born in Naples the year before. The Nightingales' gave their first born the Greek name for the city, which was Panthenope. William Nightingale had been born with the surname Shore. He had changed it to Nightingale after inheriting from a rich relative, Peter Nightingale of Lea, near Matlock, Derbyshire. The girls grew up in the country spending much of their time at Lea Hurst in Derbyshire. When Nightingale was about five years old her father bought a house called Embley near Romsey in Hampshire. This now meant that the family spent the summer months in Derbyshire, while the rest of the year was spent at Embley. Between these moves there were trips to London, the Isle of Wight and to relatives. The early education of Panthenope and Florence was placed in the hands of governesses, later their Cambridge educated father took over responsibility. Nightingale loved her lessons and had a natural ability for studying. Under her father's influence Nightingale became aquainted with the classics, Euclid, Aristotle, the Bible and political matters. In 1840, Nightingale begged her parents to let her study mathematics instead of:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nightingale.html (1 of 5) [2/16/2002 11:24:52 PM]

Nightingale

worsted work and practising quadrilles, but her mother did not approve of this idea. Although Mr. Nightingale loved mathematics and had bequeathed this love to his daughter, he urged her to study subjects more appropriate for a woman. After many long emotional battles, Nightingale's parents finally gave their permission and allowed her to be tutored in mathematics. These included Sylvester, who developed the theory of invariants with Cayley. Nightingale was said to be Sylvester's most distinguished pupil. Lessons included learning arithmetic, geometry and algebra and prior to Nightingale entered nursing, she spent time tutoring children in these subjects. Nightingale's interest in mathematics extended beyond the subject matter. One of the people who also influenced Nightingale was Belgian scientist Quetelet. He had applied statistical methods to data from several fields, including moral statistics or social sciences. Religion played an important part in Nightingale's life. Her unbiased view on religion, unusual at the time, was owed to the liberal outlook Nightingale found in her home. Although her parents were from a Unitarian background, Mrs Nightingale found a more conventional denomination more preferable and the girls were brought up as members of the Church of England. On 7 February 1837 Nightingale believed she heard her calling from God, whilst walking in the garden at Embley, although at this time though she did not know what this calling was. Nightingale developed an interest in the social issues of the time, but in 1845 her family was firmly against the suggestion of Nightingale gaining any hospital experience. Until then the only nursing that she had done was looking after sick friends and relatives. During the mid nineteenth century nursing was not considered a suitable profession for a well-educated woman. Nurses of the time were lacking in training and they also had the reputation of being coarse, ignorant women, given to promiscuity and drunkenness. While Nightingale was on a tour of Europe and Egypt, starting in 1849, with family friends Charles and Selina Bracebridge, allowing her the chance to study the different hospital systems. In early 1850 Nightingale began her training as a nurse at the Institute of St Vincent de Paul in Alexandra, Egypt, which was a hospital run by the Roman Catholic Church. Nightingale visited Pastor Theodor Flidener's hospital at Kaiserwerth, near Dussledorf, in July 1850. Nightingale returned to Kaiserwerth, in 1851, to undertake 3 months of nursing training at the Institute for Protestant Deaconesses and from Germany she moved to a hospital in St. Germain, near Paris, run by the Sisters of Mercy. On returning to London in 1853 Nightingale took up the unpaid position as the Superintendent at the Establishment for Gentlewomen during Illness at No 1 Harley Street. March of 1854 brought the start of the Crimean War, with Britain, France and Turkey declaring war on Russia. Although the Russians were defeated at the battle of the Alma River, on 20 September 1854, The Times newspaper criticised the British medical facilities. In response to this Nightingale was asked in a letter from her friend Sidney Herbert, the British Secretary for War, to become a nursing administrator to oversee the introduction of nurses to military hospitals. Her official title was Superintendent of the Female Nursing Establishment of the English General Hospitals in Turkey. Nightingale arrived in Scutari, an Asian suburb of Constantinople, (now Istanbul), with 38 nurses on 4 November 1854. Although being female meant Nightingale had to fight against the military authorities at every step, she went about reforming the hospital system. With conditions like soldiers lying on bare floors surrounded

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nightingale.html (2 of 5) [2/16/2002 11:24:52 PM]

Nightingale

by vermin and the unhygienic operations taking place it is not surprising that when Nightingale first arrived in Scutari, diseases such as cholera and typhus were rife in the hospitals. This meant that injured soldiers were 7 times more likely to die from disease in hospital, than on the battlefield. Whilst in Turkey, Nightingale collected data and organised a record keeping system, this information was then used as a tool to improve city and military hospitals. Nightingale's knowledge of mathematics became evident when she used her collected data to calculate the mortality rate in the hospital. These calculations showed that an improvement of the sanitary methods employed would result in a decrease in the number of deaths. By February 1855 the mortality rate had dropped from 60% to 42.7%. Through the establishment of a fresh water supply as well as using her own funds to buy fruit, vegetables and standard hospital equipment, the mortality rate in the spring had dropped further to 2.2%. Nightingale used this statistical data to create her Polar Area Diagram, or "coxcombs" as she called them. These were used to give a graphical representation of the mortality figures during the Crimean War (1854 - 56). The area of each coloured wedge, measured from the centre as a common point, is in proportion to the statistic it represents. The blue outer wedges represent the deaths from:preventable or mitigable zymotic diseases or in other words contagious diseases such as cholera and typhus. The central red wedges show the deaths from wounds. The black wedges in between represent deaths from all other causes. Deaths in the British field hospitals reached a peak during January 1855, when 2,761 soldiers died of contagious diseases, 83 from wounds and 324 from other causes making a total of 3,168. The army's average manpower for that month was 32,393. Using this information, Nightingale computed a mortality rate of 1,174 per 1,000 with 1,023 per 1,000 being from zymotic diseases. If this rate had continued, and troops had not been replaced frequently, then disease alone would have killed the entire British army in the Crimea. These unsanitary conditions, however, were not only limited to military hospitals in the field. On her return to London in August 1856, four months after the signing of the peace treaty, Nightingale

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nightingale.html (3 of 5) [2/16/2002 11:24:52 PM]

Nightingale

discovered that soldiers during peacetime, aged between 20 and 35 had twice the mortality rate of civilians. Using her statistics, she illustrated the need for sanitary reform in all military hospitals. While pressing her case, Nightingale gained the attention of Queen Victoria and Prince Albert as well as that of the Prime Minister, Lord Palmerston. Her wishes for a formal investigation were granted in May 1857 and led to the establishment of the Royal Commission on the Health of the Army. Nightingale hid herself from public attention, and became concerned for the army stationed in India. In 1858, for her contributions to army and hospital statistics Nightingale became the first woman to be elected to be a Fellow of the Royal Statistical Society. In 1860, the Nightingale Training School and Home for Nurses based at St. Thomas' Hospital in London, opened with 10 students. It was financed by the Nightingale Fund, a fund of public contributions set up during Nightingale's time in the Crimea and had raised a total of £50,000. It was based around two principals. Firstly that the nurses should have practical training in hospitals specially organised for that purpose. The other was that the nurses should live in a home fit to form a moral life and discipline. Due to the foundation of this school Nightingale had achieved the transformation of nursing from its disreputable past into a responsible and respectable career for women. Nightingale responded to the British war office's request for advice on army medical care in Canada and was also a consultant to the United States government on army health during the American Civil War. For most of the remainder of her life Nightingale was bedridden due to an illness contracted in the Crimea, which prevented her from continuing her own work as a nurse. This illness did not stop her, however, campaigning to improve health standards, she published 200 books, reports and pamphlets. One of these publications was a book titled Notes on Nursing (1860). This was the first textbook specifically for use in the teaching of nurses and was translated into many languages. Nightingale's other published works included Notes on Hospitals (1859) and Notes on Nursing for the Labouring Classes (1861). Florence Nightingale deeply believed that her work had been her calling from God. In 1874 she became an honorary member of the American Statistical Association and in 1883 Queen Victoria awarded Nightingale the Royal Red Cross for her work. She also became the first woman to receive the Order of Merit from Edward VII in 1907. Nightingale died on 13 August 1910 aged 90. She is buried at St Margaret's Church, East Wellow, near Embley Park. Nightingale never married, although this was not from lack of opportunity. She believed, however, that God had decided she was one whom he:... had clearly marked out ... to be a single woman. The Crimean Monument, erected in 1915 in Waterloo Place, London, was done so in honour of the contribution Florence Nightingale had made to this war and the health of the army. Article by: J J O'Connor and E F Robertson based on a project by Suzanne Davidson. List of References (5 books/articles)

A Quotation

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nightingale.html (4 of 5) [2/16/2002 11:24:52 PM]

Nightingale

Other Web sites

1. Agnes Scott College 2. Spartacus Schoolnet 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Nightingale.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nightingale.html (5 of 5) [2/16/2002 11:24:52 PM]

Nilakantha

Nilakantha Somayaji Born: 14 June 1444 in Trkkantiyur (near Tirur), Kerala, India Died: 1544 in India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nilakantha was born into a Namputiri Brahmin family which came from South Malabar in Kerala. The Nambudiri is the main caste of Kerala. It is an orthodox caste whose members consider themselves descendants of the ancient Vedic religion. He was born in a house called Kelallur which it is claimed coincides with the present Etamana in the village of Trkkantiyur near Tirur in south India. His father was Jatavedas and the family belonged to the Gargya gotra, which was a Indian caste that prohibits marriage to anyone outside the caste. The family followed the Ashvalayana sutra which was a manual of sacrificial ceremonies in the Rigveda, a collection of Vedic hymns. He worshipped the personified deity Soma who was the "master of plants" and the healer of disease. This explains the name Somayaji which means he was from a family qualified to conduct the Soma ritual. Now Nilakantha studied astronomy and Vedanta, one of the six orthodox systems of Indian Hindu philosophy, under the teacher Ravi. He was also taught by Damodra who was the son of Paramesvara. Now Paramesvara was a famous Indian astronomer and Damodra followed his father's teachings. This led Nilakantha also to become a follower of Paramesvara. A number of texts on mathematical astronomy written by Nilakantha have survived. In all he wrote about ten treatises on astronomy. The Tantrasamgraha is his major astronomy treatise written in 1501. It consists of 432 Sanskrit verses divided into 8 chapters, and it covers various aspects of Indian astronomy. It is based on the epicyclic and eccentric models of planetary motion. The first two chapters deal with the motions and longitudes of the planets. The third chapter Treatise on shadow deals with various problems related with the sun's position on the celestial sphere, including the relationships of its expressions in the three systems of coordinates, namely ecliptic, equatorial and horizontal coordinates. The fourth and fifth chapters are Treatise on the lunar eclipse and On the solar eclipse and these two chapters treat various aspects of the eclipses of the sun and the moon. The sixth chapter is On vyatipata and deals with the complete deviation of the longitudes of the sun and the moon. The seventh chapter On visibility computation discusses the rising and setting of the moon and planets. The final chapter On elevation of the lunar cusps examines the size of the part of the moon which is illuminated by the sun and gives a graphical representation of it.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nilakantha.html (1 of 3) [2/16/2002 11:24:53 PM]

Nilakantha

The Tantrasamgraha is very important in terms of the mathematics Nilakantha uses. In particular he uses results discovered by Madhava and it is an important source of the remarkable mathematical results which he discovered. However, Nilakantha does not just use Madhava's results, he extends them and improves them. An anonymous commentary entitled Tantrasangraha-vakhya appeared and, somewhat later in about 1550, Jyesthadeva published a commentary entitled Yuktibhasa that contained proofs of the earlier results by Madhava and Nilakantha. This is quite unusual for an Indian text in giving mathematical proofs. The series

/4 = 1 - 1/3 + 1/5 - 1/7 + ... is a special case of the series representation for arctan, namely

tan-1 x = x - x3/3 + x5/5 - x7/7 + ... It is well known that one simply puts x = 1 to obtain the series for /4. The author of [4] reports on the appearance of these series in the work of Leibniz and James Gregory from the 1670s. The contributions of the twom European mathematicians to this series are well known but in [4] the results on this series in the work of Madhava nearly three hundred years earlier as presented by Nilakantha in the Tantrasamgraha is also discussed. Nilakantha derived the series expansion tan-1 x = x - x3/3 + x5/5 - x7/7 + ... by obtaining an approximate expression for an arc of the circumference of a circle and then considering the limit. An interesting feature of his work was his introduction of several additional series for /4 that converged more rapidly than /4 = 1 - 1/3 + 1/5 - 1/7+ ... . The author of [4] provides a reconstruction of how he may have arrived at these results based on the assumption that he possessed a certain continued fraction representation for the tail series 1/(n+2) - 1/(n+4) + 1/(n+6) - 1/(n+8) + .... . The Tantrasamgraha is not the only work of Nilakantha of which we have the text. He also wrote Golasara which is written in fifty-six Sanskrit verses and shows how mathematical computations are used to calculate astronomical data. The Siddhanta Darpana is written in thirty-two Sanskrit verses and describes a planetary model. The Candracchayaganita is written in thirty-one Sanskrit verses and explains the computational methods used to calculate the moon's zenith distance. The head of the Nambudiri caste in Nilakantha's time was Netranarayana and he became Nilakantha's patron for another of his major works, namely the Aryabhatiyabhasya which is a commentary on the Aryabhatiya of Aryabhata I. In this work Nilakantha refers to two eclipses which he observed, the first on 6 March 1467 and the second on 28 July 1501 at Anantaksetra. Nilakantha also refers in the Aryabhatiyabhasya to other works which he wrote such as the Grahanirnaya on eclipses which have not survived. Article by: J J O'Connor and E F Robertson List of References (7 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nilakantha.html (2 of 3) [2/16/2002 11:24:53 PM]

Nilakantha

Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Nilakantha.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nilakantha.html (3 of 3) [2/16/2002 11:24:53 PM]

Niven

William Davidson Niven Born: 1843 in Peterhead, Scotland Died: 29 May 1917 in Sidcup, Kent, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Niven was one of three distinguished mathematical brothers. He graduated from the University of Aberdeen in Scotland which was his local university as Peterhead is not far from Aberdeen. From there, as was the tradition of the Scottish Universities at that time, Niven went to study at the University of Cambridge. At Cambridge Niven studied mathematics at Trinity College, where he graduated as third Wrangler in 1866. The following year he was elected to a fellowship at Trinity College. Niven left Cambridge to take up an appointment as professor at Woolwich. There he worked on gunnery and ballistics. However the attraction of Cambridge was great and he returned there becaming a firm friend of Maxwell. After Maxwell's death, Niven helped to edit the second edition of Maxwell's Electricity and Magnetism. Inspired by Maxwell and his mathematics, Niven turned increasingly towards the study of spherical and ellipsoidal harmonics. In 1882 Niven was appointed to a chair at the Royal Naval College in Greenwich. In the same year he was honoured by being elected a Fellow of the Royal Society. Niven was an active member and staunch supporter of the London Mathematical Society and he served as its President from 1908 until 1910. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Niven.html (1 of 2) [2/16/2002 11:24:55 PM]

Niven

Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country Honours awarded to William Niven (Click a link below for the full list of mathematicians honoured in this way) London Maths Society President

1908 - 1910

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Niven.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Niven.html (2 of 2) [2/16/2002 11:24:55 PM]

Noether_Emmy

Emmy Amalie Noether Born: 23 March 1882 in Erlangen, Bavaria, Germany Died: 14 April 1935 in Bryn Mawr, Pennsylvania, USA

Click the picture above to see eight larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Emmy Noether's father Max Noether was a distinguished mathematician and a professor at Erlangen. Her mother was Ida Kaufmann, from a wealthy Cologne family. Both Emmy's parents were of Jewish origin and Emmy was the eldest of their four children, the three younger children being boys. Emmy Noether attended the Höhere Töchter Schule in Erlangen from 1889 until 1897. She studied German, English, French, arithmetic and was given piano lessons. She loved dancing and looked forward to parties with children of her father's university colleagues. At this stage her aim was to become a language teacher and after further study of English and French she took the examinations of the State of Bavaria and, in 1900, became a certificated teacher of English and French in Bavarian girls schools. However Noether never became a language teacher. Instead she decided to take the difficult route for a woman of that time and study mathematics at university. Women were allowed to study at German universities unofficially and each professor had to give permission for his course. Noether obtained permission to sit in on courses at the University of Erlangen during 1900 to 1902. Then, having taken and passed the matriculation examination in Nürnberg in 1903, she went to the University of Göttingen. During 1903-04 she attended lectures by Blumenthal, Hilbert, Klein and Minkowski. In 1904 Noether was permitted to matriculate at Erlangen and in 1907 was granted a doctorate after working under Paul Gordan. Hilbert's basis theorem of 1888 had given an existence result for finiteness of invariants in n variables. Gordan, however, took a constructive approach and looked at constructive methods to arrive at the same results. Noether's doctoral thesis followed this constructive approach of Gordan and listed systems of 331 covariant forms.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Noether_Emmy.html (1 of 4) [2/16/2002 11:24:57 PM]

Noether_Emmy

Having completed her doctorate the normal progression to an academic post would have been the habilitation. However this route was not open to women so Noether remained at Erlangen, helping her father who, particularly because of his own disabilities, was grateful for his daughter's help. Noether also worked on her own research, in particular she was influenced by Fischer who had succeeded Gordan in 1911. This influence took Noether towards Hilbert's abstract approach to the subject and away from the constructive approach of Gordan. Noether's reputation grew quickly as her publications appeared. In 1908 she was elected to the Circolo Matematico di Palermo, then in 1909 she was invited to become a member of the Deutsche Mathematiker Vereinigung and in the same year she was invited to address the annual meeting of the Society in Salzburg. In 1913 she lectured in Vienna. In 1915 Hilbert and Klein invited Noether to return to Göttingen. They persuaded her to remain at Göttingen while they fought a battle to have her officially on the Faculty. In a long battle with the university authorities to allow Noether to obtain her habilitation there were many setbacks and it was not until 1919 that permission was granted. During this time Hilbert had allowed Noether to lecture by advertising her courses under his own name. For example a course given in the winter semester of 1916-17 appears in the catalogue as:Mathematical Physics Seminar: Professor Hilbert, with the assistance of Dr E Noether, Mondays from 4-6, no tuition. Emmy Noether's first piece of work when she arrived in Göttingen in 1915 is a result in theoretical physics sometimes referred to as Noether's Theorem, which proves a relationship between symmetries in physics and conservation principles. This basic result in the general theory of relativity was praised by Einstein in a letter to Hilbert when he referred to Noether's penetrating mathematical thinking. It was her work in the theory of invariants which led to formulations for several concepts of Einstein's general theory of relativity. At Göttingen, after 1919, Noether moved away from invariant theory to work on ideal theory, producing an abstract theory which helped develop ring theory into a major mathematical topic. Idealtheorie in Ringbereichen (1921) was of fundamental importance in the development of modern algebra. In this paper she gave the decomposition of ideals into intersections of primary ideals in any commutative ring with ascending chain condition. Lasker (the world chess champion) had already proved this result for polynomial rings. In 1924 B L van der Waerden came to Göttingen and spent a year studying with Noether. After returning to Amsterdam van der Waerden wrote his book Moderne Algebra in two volumes. The major part of the second volume consists of Noether's work. From 1927 on Noether collaborated with Helmut Hasse and Richard Brauer in work on noncommutative algebras. In addition to teaching and research, Noether helped edit Mathematische Annalen. Much of her work appears in papers written by colleagues and students, rather than under her own name.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Noether_Emmy.html (2 of 4) [2/16/2002 11:24:57 PM]

Noether_Emmy

Further recognition of her outstanding mathematical contributions came with invitations to address the International Mathematical Congress at Bologna in 1928 and again at Zurich in 1932. In 1932 she also received, jointly with Artin, the Alfred Ackermann-Teubner Memorial Prize for the Advancement of Mathematical Knowledge. In 1933 her mathematical achievements counted for nothing when the Nazis caused her dismissal from the University of Göttingen because she was Jewish. She accepted a visiting professorship at Bryn Mawr College in the USA and also lectured at the Institute for Advanced Study, Princeton in the USA. Weyl in his Memorial Address [28] said:Her significance for algebra cannot be read entirely from her own papers, she had great stimulating power and many of her suggestions took shape only in the works of her pupils and co-workers. In [26] van der Waerden writes:For Emmy Noether, relationships among numbers, functions, and operations became transparent, amenable to generalisation, and productive only after they have been dissociated from any particular objects and have been reduced to general conceptual relationships. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (28 books/articles) A Poster of Emmy Noether

Mathematicians born in the same country

Cross-references to History Topics

General relativity

Other references in MacTutor

Chronology: 1920 to 1930

Honours awarded to Emmy Noether (Click a link below for the full list of mathematicians honoured in this way) Lunar features Other Web sites

Crater Noether 1. Agnes Scott College 2. Simon Fraser University 3. WISE project 4. Pass Magazine 5. UCLA (her work in physics) 6. Clark Kimberling 7. San Diego 8. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Noether_Emmy.html (3 of 4) [2/16/2002 11:24:57 PM]

Noether_Emmy

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Noether_Emmy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Noether_Emmy.html (4 of 4) [2/16/2002 11:24:57 PM]

Noether_Max

Max Noether Born: 24 Sept 1844 in Mannheim, Baden, Germany Died: 13 Dec 1921 in Erlangen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Max Noether suffered an attack of polio when he was 14 years old and it left him with a handicap for the rest of his life. He attended the University of Heidelberg from 1865 and obtained a doctorate from there in 1868. After this he lectured at Heidelberg and moved from Heidelberg to a chair at Erlangen where he remained for the rest of his life. Max Noether was one of the leaders of nineteenth century algebraic geometry. He was influenced by Abel, Riemann, Cayley and Cremona. Following Cremona, Max Noether studied the invariant properties of an algebraic variety under the action of birational transformations. In 1873 he proved an important result on the intersection of two algebraic curves. Nine years later, in 1882, his daughter Emmy Noether was born. Emmy became interested in many similar topics to her father and generalised some of his theorems. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) A Poster of Max Noether

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Noether_Max.html (1 of 2) [2/16/2002 11:24:59 PM]

Noether_Max

Other Web sites

Evansville

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Noether_Max.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Noether_Max.html (2 of 2) [2/16/2002 11:24:59 PM]

Novikov

Petr Sergeevich Novikov Born: 15 Aug 1901 in Moscow, Russia Died: 9 Jan 1975 in Moscow, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Petr Sergeevich Novikov was the son of Sergi Novikov, a Moscow merchant, and Alexandra Novikov. He attended school in Moscow then, in September 1919, he entered the Faculty of Physics and Mathematics at Moscow University. However, even before Novikov entered university, the Russian nation had been plunged into civil war. The Red Army had been formed in February 1918 with Trotsky as its leader. The Red Army opposed the White Army formed of anticommunists led by former imperial officers. In the spring of 1920, with the Civil war still raging, Novikov joined the Red Army. He served with this army until July 1922 when he returned to Moscow University to complete his studies. He graduated in 1925 then, remaining at Moscow University, he undertook research under Luzin's supervision. Novikov graduated in 1929 and then taught at the Moscow Chemical Technology Institute until he joined the Department of Real Function Theory at the Steklov Institute in 1934. He was awarded his doctorate in 1935 and, in 1939, he was promoted to full professor. Novikov married Ludmila Vsvolodovna Keldysh in 1935. They had five children; one of their sons Sergi Novikov was awarded a Fields Medal in 1970. Novikov headed the Department of Analysis at Moscow State Teachers Training Institute from 1944. In 1957 Novikov set up a new department at the Steklov Institute, namely the Department of Mathematical Logic, and he was appointed as the first head of that department. He held the two posts, one at the Moscow State Teachers Training Institute and the other at the Steklov Institute, until he retired in 1972 and 1973 respectively. After early work on set theory, influenced by Luzin and his school, he began to publish results in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Novikov.html (1 of 3) [2/16/2002 11:25:01 PM]

Novikov

mathematical physics from 1938. Perhaps his most fundamental result in this area was that [1]:... any two solids having the same constant density must coincide if they both are star-shaped relative to a common point and have the same external gravitational potential. He began to study mathematical logic and the theory of algorithms just before 1940. He studied consistency of arithmetic, proving that formal arithmetic with recursive definitions is consistent. He also examined the consistency of certain propositions in Gödel's system of axiomatic set theory. Novikov showed, in 1952, that the word problem for groups is insoluble. The word problem asks the fundamental question of whether there is an algorithm to determine whether a word in a group given by a presentation consisting of a finite number of generators and relations is trivial. The problem was first posed by Dehn in 1912 and Novikov was able to show that no such algorithm exists in general. Research into questions of this type is still of major importance in combinatorial group theory. Novikov was awarded the Lenin Prize in 1957 for this outstanding piece of work. In fact Boone published another proof of this result in 1957, the same year that Novikov received his prize. The word problem was not the only problem of major importance in combinatorial group theory which Novikov solved. Jointly with Adian he showed that the problem of the finiteness of periodic groups proposed by Burnside in 1902 had a negative solution. Although in 1959 Novikov announced that for every n > 71 there exists a finitely generated infinite group with every element of order dividing n, his proof was not quite correct. Let us state the problem more precisely. The Burnside problem asks whether, for fixed d and n, the group B(d, n) having d generators and in which every element x satisfies xn = 1, is finite. Novikov's argument of 1959 was correct in general terms but the details were not, and in putting the arguments right it was found that one required larger values of n. In 1968 Novikov and Adian jointly published a proof B(d, n) is infinite for every d > 1 and every n > 4380. They continued to work on improving the result and, in 1979, published a book The Burnside problem and identities in groups which they improved the result to n > 664. There is still a large gap, however, between those values of n for which B(d, n) is known to be finite and those for which it is known to be infinite. It is really easy to show the B(d, 2) is finite. Burnside himself showed that B(d, 3) is finite, Sanov showed B(d, 4) is finite and Marshall Hall showed B(d, 6) is finite. However, it is still an open question as to whether B(2, 5) is finite. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Other references in MacTutor

1. Chronology: 1950 to 1960 2. Chronology: 1960 to 1970

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Novikov.html (2 of 3) [2/16/2002 11:25:01 PM]

Novikov

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Novikov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Novikov.html (3 of 3) [2/16/2002 11:25:01 PM]

Novikov_Sergi

Sergi Petrovich Novikov Born: 20 March 1938 in Gorky (now Nizhny Novgorod), Russia

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sergi Novikov's father was Petr Sergeevich Novikov and his mother, Ludmila Vsevolodovna Keldysh, was also an outstanding mathematician. Sergi Novikov entered the Faculty of Mathematics and Mechanics of Moscow University in 1955. He obtained his first degree in 1960 and then became a research student at the Steklov Institute of Mathematics in Moscow. His research there was supervised by M M Postnikov and Novikov was awarded his doctorate in 1964. In 1963 Novikov was appointed to the staff of the Steklov Institute of Mathematics and, the following year, he was also appointed to the Department of Differential Geometry at Moscow University. In 1971 Novikov became head of the Mathematics Division at the L D Landau (Lev Landau) Institute for Theoretical Physics of the Academy of Sciences of the USSR. Novikov also became head of the Department of Higher Geometry and Topology of Moscow University in 1983 and, the following year he became head of the Department of Geometry and Topology of the Mathematical Institute of the Academy of Sciences of the USSR. From 1985 to 1996 he was President of the Moscow Mathematical Society. Since 1996 he has been working at the University of Maryland in the United States. Novikov's work up to 1971 was on differential topology; in particular he studied calculating stable homotopy groups and classifying smooth simply-connected manifolds of dimension greater than 4. He also studied the topological invariance of rational Pontryagin classes.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Novikov_Sergi.html (1 of 3) [2/16/2002 11:25:02 PM]

Novikov_Sergi

In 1965 Novikov stated the conjecture, now known as the Novikov conjecture, concerning the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group. It is one of the most fundamental problems in topology. Novikov's original motivation was the theory, in the simply connected case, of Browder-Novikov and Wall, which led to the classification of manifolds in high dimensions. Novikov talked about his conjecture at the 1970 International Congress of Mathematicians in Nice where he received a Fields Medal. After 1971 Novikov became interested in mathematical physics and dynamical systems. He studied a wide variety of applications of mathematics such as dynamical systems in the theory of homogeneous cosmological models, the theory of solitons, the spectral theory of linear operators, quantum field theory and string theory. Novikov has received many honours for his outstanding work. Perhaps the most important of these awards has been the Fields Medal, referred to above, which he received in 1970. In 1981 he became a full member of the Academy of Sciences of the USSR, receiving the Lobachevsky Prize of the Academy in the same year. Many societies have honoured Novikov with membership such as the London Mathematical Society in 1987 and the Accademia dei Lincei in 1991. He was elected to the National Academy of Sciences of United States in 1994. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1960 to 1970

Honours awarded to Sergi Novikov (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1970

Other Web sites

1. Another picture of Novikov 2. Dmitrii Millionshchikov 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Novikov_Sergi.html (2 of 3) [2/16/2002 11:25:02 PM]

Novikov_Sergi

Glossary index

Quotations index

Mathematicians of the day JOC/EFR December 1997

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Novikov_Sergi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Novikov_Sergi.html (3 of 3) [2/16/2002 11:25:02 PM]

Nunez

Pedro Nunez Salaciense Born: 1502 in Alcácer do Sal, Portugal Died: 11 Aug 1578 in Coimbra, Portugal

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pedro Nunez or Nunes studied at Salamanca from 1521 until 1522. He then attended the University of Lisbon where he obtained a degree in medicine in 1525. He continued his medical studies but held various teaching posts within the University of Lisbon. He was appointed to the chair of moral philosophy in 1529 then to the chair of logic in 1530 then to the chair if metaphysics in 1532. Not bad going for a medical student! Nunez moved to the University of Coimbra to the chair of mathematics in 1537, a post he held until 1562. This was a new post in the University of Coimbra and it was set up to provide instruction in the technical requirements for navigation, clearly a topic of great importance in Portugal at this period when control of sea trade was the chief source of Portuguese wealth. In addition to this post he was appointed Royal Cosmographer in 1529 and Chief Royal Cosmographer in 1547. He held this post until his death. Nunez worked in geometry and spherical trigonometry publishing Treatise on the Sphere. He also did important work in algebra and published Algebra in Spanish. Outside mathematics he worked in geography, physics, cosmology and he wrote poetry. He also made many important contributions to navigation writing Navigandi Libri Duo in 1546. Nunez devised a system to allow fractional parts of a degree to be measured. He describes it as:drawing on the face of a quadrant for measuring angles 45 concentric arcs, one of which was divided into 90 equal parts or degrees, and the remainder into 89, 88, 87, 86, etc., successively, the last being divided into 46 equal parts. When the index did not exactly cut http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nunez.html (1 of 2) [2/16/2002 11:25:04 PM]

Nunez

one of the divisions of the arc of degrees, it passed through or near to one of the divisions of one or other of the other arcs; and by noting the place of that division the fractional parts of a degree were calculated. While at Coimbra Nunez taught Clavius. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Longitude and the Académie Royale 2. Squaring the circle

Other Web sites

1. The Catholic Encyclopedia 2. The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Nunez.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Nunez.html (2 of 2) [2/16/2002 11:25:04 PM]

Ockham

William of Ockham Born: 1288 in Ockham (near Ripley, Surrey), England Died: 9 April 1348 in Munich, Bavaria (now Germany)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William of Ockham's early Franciscan education concentrated on logic. He studied theology at Oxford and between 1317 and 1319 he lectured on the Sentences, the standard theology text used in universities up to 16 C. His opinions aroused strong oposition and he left Oxford without his Master's Degree. He continued studying mathematical logic and made important contributions to it. He considered a three valued logic where propositions can take one of three truth values. This became important for mathematics in the 20th Century but it is remarkable that it was first studied by Ockham 600 years earlier. Ockham went to France and was denounced by the Pope. He was excommunicated and in 1328 he fled seeking the protection of Louis IV in Bavaria (Louis had also been excommunicated!). He continued to attack papal power always employing logical reasoning in his arguments. Article by: J J O'Connor and E F Robertson List of References (22 books/articles)

A Quotation

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ockham.html (1 of 2) [2/16/2002 11:25:06 PM]

Ockham

Other Web sites

1. Dave Beckett 2. The Catholic Encyclopedia 3. Internet Encyclopedia of Philosophy 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ockham.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ockham.html (2 of 2) [2/16/2002 11:25:06 PM]

Oenopides

Oenopides of Chios Born: about 490 BC in Chios (now Khios), Greece Died: about 420 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Very little is known about the life of Oenopides of Chios except that his place of birth was the island of Chios. We believe that Oenopides was in Athens when a young man but there is only circumstantial evidence for this. In Plato's Erastae Oenopides is described as (see for example [1]):... having acquired a reputation for mathematics... and Plato also describes a scene where Socrates comes across two young men in the school of Dionysius who was Plato's teacher. The young men were discussing a question in mathematical astronomy which had been tackled by Oenopides and Anaxagoras. This question was certainly that of the angle that the ecliptic makes with the celestial equator. Bulmer-Thomas writes in [1]:... it was probably [Oenopides] who settled on the value of 24 , which was accepted in Greece until refined by Eratosthenes. Indeed, if Oenopides did not fix on this or some other figure it is difficult to know in what his achievement consisted, for the Babylonians no less than the Pythagoreans and Egyptians must have realised from early days that the apparent path of the sun was inclined to the celestial equator. However, in contrast to these claims, Heath writes [2]:It does not appear that Oenopides made any measurement of the obliquity of the ecliptic. Another major contribution to mathematical astronomy made by Oenopides was his discovery of the period of the Great Year. Originally the "Great Year" was the period after which the motions of the sun and moon came to repeat themselves. Later it came to mean the period after which the motions of the sun, moon and planets all repeated themselves so in the period of one Great Year all should have returned to their positions at the beginning of the Great Year. Oenopides gave a value of the Great Year as 59 years. Heath writes [2]:His Great Year clearly had reference to the sun and moon only; he merely sought to find the least integral number of complete years which would contain an exact number of lunar months. Based on this Paul Tannery [5] showed that Oenopides' result leads to a lunar month of 29.53013 days

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Oenopides.html (1 of 3) [2/16/2002 11:25:07 PM]

Oenopides

which is remarkably close to the modern value of 29.53059 days. However many historians doubt whether Oenopides could have collected sufficient good quality data to enable him to obtain a value as accurate as this. To collect the data for even one period requires 59 years and this makes it almost impossible for someone to gather the data in their own lifetime. Toomer believed that in fact despite Oenopides' Great Year of 59 years, he did not have this accurate value for the length of the month, and later calculations were made using better data than would have been available to Oenopides to give this very accurate value for the length of the month, more accurate than Oenopides could ever have known. Paul Tannery in [5] makes another claim however when he states that Oenopides considered some of the planets as well as the sun and moon as part of his 59 year Great Year. The data works well for some of the planets, for example Saturn is only 2 from its starting position at the end of the 59 year cycle. Tannery is forced to conclude that not all the planets could have been taken into account by Oenopides, however, as some of the planets would be in the wrong sign of the Zodiac after the period ended. Proclus attributes two theorems which appear in Euclid's Elements to Oenopides. These are to draw a perpendicular to a line from a given point not on the line, and to construct on a line from a given point a line at a given angle to the first line. These are elementary results but Heath believes that their significance might be that Oenopides set out for the first time the explicit 'ruler and compass' type of allowable construction. He writes [2]:... [Oenopides] may have been the first to lay down the restriction of the means permissible in constructions t the ruler and compasses which became a canon of Greek geometry for all plane constructions... Oenopides also developed a theory to account for the Nile floods. He suggested that heat stored in the ground during the winter dries up the underground water so that the river shrinks. In the summer the heat disappears, as testing the temperature of deep wells suggests, and water flows up into the river so causing floods. This theory, which of course is false, did not prove popular as other rivers in Libya were subject to similar conditions but did not behave in the same way. We have some other indications of the philosophy of Oenopides. He is said to have believed in fire and air as basic elements and thought of the world as a living being with God as its soul. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Greek Astronomy 2. Squaring the circle

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Oenopides.html (2 of 3) [2/16/2002 11:25:07 PM]

Oenopides

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Oenopides.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Oenopides.html (3 of 3) [2/16/2002 11:25:07 PM]

Ohm

Georg Simon Ohm Born: 16 March 1789 in Erlangen, Bavaria (now Germany) Died: 6 July 1854 in Munich, Bavaria, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Simon Ohm came from a Protestant family. His father, Johann Wolfgang Ohm, was a locksmith while his mother, Maria Elizabeth Beck, was the daughter of a tailor. Although his parents had not been formally educated, Ohm's father was a rather remarkable man who had educated himself to a high level and was able to give his sons an excellent education through his own teachings. Had Ohm's brothers and sisters all survived he would have been one of a large family but, as was common in those times, several of the children died in their childhood. Of the seven children born to Johann and Maria Ohm only three survived, Georg Simon, his brother Martin who went on to become a well-known mathematician, and his sister Elizabeth Barbara. When they were children, Georg Simon and Martin were taught by their father who brought them to a high standard in mathematics, physics, chemistry and philosophy. This was in stark contrast to their school education. Georg Simon entered Erlangen Gymnasium at the age of eleven but there he received little in the way of scientific training. In fact this formal part of his schooling was uninspired stressing learning by rote and interpreting texts. This contrasted strongly with the inspired instruction that both Georg Simon and Martin received from their father who brought them to a level in mathematics which led the professor at the University of Erlangen, Karl Christian von Langsdorf, to compare them to the Bernoulli family. It is worth stressing again the remarkable achievement of Johann Wolfgang Ohm, an entirely self-taught man, to have been able to give his sons such a fine mathematical and scientific education. In 1805 Ohm entered the University of Erlangen but he became rather carried away with student life. Rather than concentrate on his studies he spent much time dancing, ice skating and playing billiards.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ohm.html (1 of 5) [2/16/2002 11:25:10 PM]

Ohm

Ohm's father, angry that his son was wasting the educational opportunity that he himself had never been fortunate enough to experience, demanded that Ohm leave the university after three semesters. Ohm went (or more accurately, was sent) to Switzerland where, in September 1806, he took up a post as a mathematics teacher in a school in Gottstadt bei Nydau. Karl Christian von Langsdorf left the University of Erlangen in early 1809 to take up a post in the University of Heidelberg and Ohm would have liked to have gone with him to Heidelberg to restart his mathematical studies. Langsdorf, however, advised Ohm to continue with his studies of mathematics on his own, advising Ohm to read the works of Euler, Laplace and Lacroix. Rather reluctantly Ohm took his advice but he left his teaching post in Gottstadt bei Nydau in March 1809 to become a private tutor in Neuchâtel. For two years he carried out his duties as a tutor while he followed Langsdorf's advice and continued his private study of mathematics. Then in April 1811 he returned to the University of Erlangen. His private studies had stood him in good stead for he received a doctorate from Erlangen on 25 October 1811 and immediately joined the staff as a mathematics lecturer. After three semesters Ohm gave up his university post. He could not see how he could attain a better status at Erlangen as prospects there were poor while he essentially lived in poverty in the lecturing post. The Bavarian government offered him a post as a teacher of mathematics and physics at a poor quality school in Bamberg and he took up the post there in January 1813. This was not the successful career envisaged by Ohm and he decided that he would have to show that he was worth much more than a teacher in a poor school. He worked on writing an elementary book on the teaching of geometry while remaining desperately unhappy in his job. After Ohm had endured the school for three years it was closed down in February 1816. The Bavarian government then sent him to an overcrowded school in Bamberg to help out with the mathematics teaching. On 11 September 1817 Ohm received an offer of the post of teacher of mathematics and physics at the Jesuit Gymnasium of Cologne. This was a better school than any that Ohm had taught in previously and it had a well equipped physics laboratory. As he had done for so much of his life, Ohm continued his private studies reading the texts of the leading French mathematicians Lagrange, Legendre, Laplace, Biot and Poisson. He moved on to reading the works of Fourier and Fresnel and he began his own experimental work in the school physics laboratory after he had learnt of Oersted's discovery of electromagnetism in 1820. At first his experiments were conducted for his own educational benefit as were the private studies he made of the works of the leading mathematicians. The Jesuit Gymnasium of Cologne failed to continue to keep up the high standards that it had when Ohm began to work there so, by 1825, he decided that he would try again to attain the job he really wanted, namely a post in a university. Realising that the way into such a post would have to be through research publications, he changed his attitude towards the experimental work he was undertaking and began to systematically work towards the publication of his results [1]:Overburdened with students, finding little appreciation for his conscientious efforts, and realising that he would never marry, he turned to science both to prove himself to the world and to have something solid on which to base his petition for a position in a more stimulating environment. In fact he had already convinced himself of the truth of what we call today "Ohm's law" namely the relationship that the current through most materials is directly proportional to the potential difference http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ohm.html (2 of 5) [2/16/2002 11:25:10 PM]

Ohm

applied across the material. The result was not contained in Ohm's firsts paper published in 1825, however, for this paper examines the decrease in the electromagnetic force produced by a wire as the length of the wire increased. The paper deduced mathematical relationships based purely on the experimental evidence that Ohm had tabulated. In two important papers in 1826, Ohm gave a mathematical description of conduction in circuits modelled on Fourier's study of heat conduction. These papers continue Ohm's deduction of results from experimental evidence and, particularly in the second, he was able to propose laws which went a long way to explaining results of others working on galvanic electricity. The second paper certainly is the first step in a comprehensive theory which Ohm was able to give in his famous book published in the following year. What is now known as Ohm's law appears in this famous book Die galvanische Kette, mathematisch bearbeitet (1827) in which he gave his complete theory of electricity. The book begins with the mathematical background necessary for an understanding of the rest of the work. We should remark here that such a mathematical background was necessary for even the leading German physicists to understand the work, for the emphasis at this time was on a non-mathematical approach to physics. We should also remark that, despite Ohm's attempts in this introduction, he was not really successful in convincing the older German physicists that the mathematical approach was the right one. To some extent, as Caneva explains in [1], this was Ohm's own fault:... in neither the introduction nor the body of the work, which contained the more rigorous development of the theory, did Ohm bring decisively home either the underlying unity of the whole or the connections between fundamental assumptions and major deductions. For example, although his theory was conceived as a strict deductive system based on three fundamental laws, he nowhere indicated precisely which of their several mathematical and verbal expressions he wished to be taken as the canonical form. It is interesting that Ohm's presents his theory as one of contiguous action, a theory which opposed the concept of action at a distance. Ohm believed that the communication of electricity occurred between "contiguous particles" which is the term Ohm himself uses. The paper [8] is concerned with this idea, and in particular with illustrating the differences in scientific approach between Ohm and that of Fourier and Navier. A detailed study of the conceptual framework used by Ohm in formulating Ohm's law is given in [6]. As we described above, Ohm was at the Jesuit Gymnasium of Cologne when he began his important publications in 1825. He was given a year off work in which to concentrate on his research beginning in August 1826 and although he only received the less than generous offer of half pay, he was able to spend the year in Berlin working on his publications. Ohm had believed that his publications would lead to his receiving an offer of a university post before having to return to Cologne but by the time he was due to begin teaching again in September 1827 he was still without such an offer. Although Ohm's work strongly influenced theory, it was received with little enthusiasm. Ohm's feeling were hurt, he decided to remain in Berlin and, in March 1828, he formally resigned his position at Cologne. He took some temporary work teaching mathematics in schools in Berlin. He accepted a position at Nüremberg in 1833 and although this gave him the title of professor, it was still not the university post for which he had strived all his life. His work was eventually recognised by the Royal Society with its award of the Copley Medal in 1841. He became a foreign member of the Royal http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ohm.html (3 of 5) [2/16/2002 11:25:10 PM]

Ohm

Society in 1842. Other academies such as those in Berlin and Turin elected him a corresponding member, and in 1845 he became a full member of the Bavarian Academy. This belated recognition was welcome but there remains the question of why someone who today is a household name for his important contribution struggled for so long to gain acknowledgement. This may have no simple explanation but rather be the result of a number of different contributary factors. One factor may have been the inwardness of Ohm's character while another was certainly his mathematical approach to topics which at that time were studied in his country a non-mathematical way. There was undoubtedly also personal disputes with the men in power which did Ohm no good at all. He certainly did not find favour with Johannes Schultz who was an influential figure in the ministry of education in Berlin, and with Georg Friedrich Pohl, a professor of physics in that city. Electricity was not the only topic on which Ohm undertook research, and not the only topic in which he ended up in controversy. In 1843 he stated the fundamental principle of physiological acoustics, concerned with the way in which one hears combination tones. However the assumptions which he made in his mathematical derivation were not totally justified and this resulted in a bitter dispute with the physicist August Seebeck. He succeeded in discrediting Ohm's hypothesis and Ohm had to acknowledge his error. See [10] for details of the dispute between Ohm and Seebeck. In 1849 Ohm took up a post in Munich as curator of the Bavarian Academy's physical cabinet and began to lecture at the University of Munich. Only in 1852, two years before his death, did Ohm achieve his lifelong ambition of being appointed to the chair of physics at the University of Munich. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country Honours awarded to Georg Simon Ohm (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1842

Royal Society Copley Medal

Awarded 1841

Lunar features

Crater Ohm

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ohm.html (4 of 5) [2/16/2002 11:25:10 PM]

Ohm

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ohm.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ohm.html (5 of 5) [2/16/2002 11:25:10 PM]

Oka

Kiyoshi Oka Born: 19 April 1901 in Osaka, Wakayama Prefecture, Japan Died: 1 March 1978 in Nara, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kiyoshi Oka entered the Imperial University of Kyoto in 1922 to study physics. However in 1923 he changed topic to study mathematics, graduating with a degree in mathematics in 1925. In the same year he was appointed as a lecturer in the Faculty of Science at the Imperial University of Kyoto. In 1929 he was promoted to assistant professor of mathematics. In fact 1929 was a very significant year for Oka for in that year he took sabbatical leave and went to the University of Paris. He became interested in unsolved problems in the theory of functions of several complex variables while working in Paris. The reason that his work took this direction was that in Paris he became acquainted with Julia. Oka remained on the staff at the Imperial University of Kyoto while he was on leave in Paris but on his return to Japan in 1932 he accepted a position as assistant professor in the Faculty of Science of Hiroshima University. In 1938 Oka went to Kimitoge in Wakayama where he studied, presented his doctoral thesis to the University of Kyoto in 1940. After obtaining his doctorate, Oka was a research assistant at Hokkaido University during 1941/41, then, with the support of the Huju-kai Foundation under the chairmanship of Takagi, he spent the next seven years again at Kimitoge in Wakayama. Oka was appointed professor at the Nara University for Women in 1949, a post he held until 1964. From 1969 until his death in 1978, he was professor of mathematics at the Industrial University of Kyoto. His most famous work was published in a 25 year period from 1936, when he solved a number of important problems. He worked on the theory of functions of several complex variables. Henri Cartan describes in [1] the way that Oka came into the subject:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Oka.html (1 of 2) [2/16/2002 11:25:11 PM]

Oka

The publication in 1934 of a monograph by Behnke-Thullen marked a crucial stage in the development of the theory of analytic functins of several complex variables. By giving a list of the open problems in the area, this work played an important role in deciding the direction of Oka's research. He set himself the almost super-human task of solving these difficult problems. One could say that he was successful, overcoming one after the other the obstacles he encountered on the way. However, the technical aspects of his proofs and the way he presents his results make his work difficult to read. It is only with considerable effort that one can appreciate the considerable strength of his results. This is why, even today, it is worth collecting his work as a tribute to its creator, Kiyoshi Oka. Oka was also interested in Japanese poetry and Zen philosophy. Oka received a number of important honours. He was awarded the Medal of the Japan Academy of Science in 1951, the Asahi-bunka-sho Prize of Culture in 1954 and the Bunka-kunsho Medal of Culture in 1960. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Oka.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Oka.html (2 of 2) [2/16/2002 11:25:11 PM]

Olivier

Théodore Olivier Born: 21 Jan 1793 in France Died: 5 Aug 1853 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Théodore Olivier was a student at the Ecole Polytechnique and spent the years from 1821 to 1825 in Sweden. He became one of the founders of Ecole Centrale des Arts et Manufactures in 1829. It was set up to handle to commercial and industrial side of engineering and Olivier became professor there in 1829. Olivier was a follower of Monge and worked and lectured on descriptive geometry and mechanics. From the 1840's Olivier wrote textbooks. Article by: J J O'Connor and E F Robertson

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Olivier.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Olivier.html [2/16/2002 11:25:12 PM]

Oresme

Nicole d' Oresme Born: 1323 in Allemagne (near Caen), France Died: 11 July 1382 in Lisieux, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

After studying theology in Paris, Nicole Oresme was bursar in the University of Paris, then canon and later dean of Rouen. In 1370 he was appointed chaplain to King Charles V and advised him on financial matters. Oresme invented coordinate geometry before Descartes, finding the logical equivalence between tabulating values and graphing them. He proposed the use of a graph for plotting a variable magnitude whose value depends on another. It is possible that Descartes was influenced by Oresme's work since it was reprinted several times over the 100 years following its first publication. Another work by Oresme contains the first use of a fractional exponent, although, of course, not in modern notation. Oresme also worked on infinite series. Oresme also opposed the theory of a stationary Earth as proposed by Aristotle and taught motion of the Earth, 200 years before Copernicus. However he rather spoilt this fine piece of thinking by rejecting his own ideas in the end. He wrote Questiones Super Libros Aristotelis de Anima dealing with the nature of light, reflection of light, and the speed of light. Oresme's work on light is discussed in detail in [16]. Article by: J J O'Connor and E F Robertson List of References (23 books/articles) A Poster of Nicole Oresme

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Oresme.html (1 of 2) [2/16/2002 11:25:14 PM]

Oresme

Some pages from publications

A page from Tractatus de latitudinibus formarum (1505)

Other references in MacTutor

Chronology: 1300 to 1500

Honours awarded to Nicole Oresme (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Oresme

Other Web sites

1. Lisieux library (in French) 2. The Catholic Encyclopedia 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Oresme.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Oresme.html (2 of 2) [2/16/2002 11:25:14 PM]

Orlicz

Wladyslaw Orlicz Born: 24 May 1903 in Okocim, Galicia, Austria-Hungary (now Poland) Died: 9 Aug 1990 in Poznan, Poland

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wladyslaw Roman Orlicz was born in Okocim, a village near Cracow. His parents, Franciszek and Maria née Rossknecht, had five sons. His father died when he was only four years old. In 1919 Orlicz's family moved to Lvov (Lwów in Polish), where he completed his secondary education and then studied mathematics at the Jan Kazimierz University in Lvov having Stefan Banach, Hugo Steinhaus and Antoni Lomnicki as teachers. From 1922 to 1929 he worked as a teaching assistent at the Department of Mathematics of Jan Kazimierz University in Lvov. In 1928 he wrote his doctoral thesis Some problems in the theory of orthogonal series under the supervision of Eustachy Zylinski. In the same year he married Zofia Krzysik. In the late twenties and early thirties Orlicz worked as a teacher in private secondary schools and in a military school. Orlicz spent the academic year 1929/30 at Göttingen University on a scholarship in theoretical physics, not in mathematics. During his stay in Göttingen he started his collaboration with Zygmunt Wilhelm Birnbaum (also from Lvov). They published two papers in Studia Mathematica in 1930 and 1931. Their results became a starting point for Orlicz to consider and investigate in 1932 and 1936 function spaces more general than Lp spaces which later on became known as Orlicz spaces. It should be emphasized that from the functional analysis point of view (that is, as function spaces) Orlicz spaces appeared for the first time in 1932 in Orlicz's paper: Uber eine gewisse Klasse von Räumen vom Typus B in Bull. Int. Acad. Polon. Sci. A 1932, 8/9, 207-220 with an additional condition on the function ( the so called 2-condition for large u), and in full generality (that is, without the 2-condition) in 1936. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Orlicz.html (1 of 5) [2/16/2002 11:25:17 PM]

Orlicz

In the years 1931-1937 Orlicz worked at the Lvov Technical University. In 1934 he was granted the habilitation (veniam legendi) for a thesis entitled Investigations of orthogonal systems. Working in Lvov Orlicz participated in the famous meetings at the Scottish Café (Kawiarnia Szkocka) where Stefan Banach, Hugo Steinhaus, Stanislaw Ulam, Stanislaw Mazur, Marek Kac, Juliusz Schauder, Stefan Kaczmarz and many others talked about mathematical problems and looked for their solutions. The group gained international recognition and was later described as the Lvov School of Mathematics. A collection of 193 mathematical problems from meetings at the Scottish Café appeared later on as the Scottish Book. Orlicz is the author or co-author of 14 problems there. (R D Mauldin edited The Scottish Book, Mathematics from the Scottish Café (Birkhäuser 1981) which contains problems and also commentaries on them by specialists). You can see a picture of the Scottish Café. In 1968 when presenting the mathematical output of Steinhaus (in an article published in Wiadom. Mat. in 1969), Orlicz wrote: In Lvov under the leadership of our dear masters Banach and Steinhaus we were practising intricacies of mathematics. In 1937 Orlicz became a professor at Poznan University (now Adam Mickiewicz University) and spent the Second World War in Lvov. He was professor at the State University of Iwan Franko from January 1940 to June 1941 and from August 1944 to February 1945 he also taught at the school of commerce and handicrafts and lectured at forestry courses. In March 1945 Orlicz went back to Poland and from May 1945 he returned to University of Poznan. In July 1948 Orlicz was promoted to an ordinary professorship. Until his retirement in 1974 he worked both at the University of Poznan and the Mathematical Institute of the Polish Academy of Sciences, Poznan Branch. Orlicz continued his seminar Selected Problems of Functional Analysis until 1989. The seminar ran every Wednesday from half past twelve to two o'clock in Mathematical Institute. He was interested in works of other mathematicians and in branches far removed from functional analysis. Orlicz collaborated with several mathematicians. Collaboration with Mazur was especially fruitful. They wrote a dozen joint papers and their results are now considered classical theorems. When in 1960 Steinhaus was writing about Banach he emphasised this fact (Nauka Polska 8 (4) (1960), 157 or Wiadom. Mat. 4 (1961), 257 or [2], 242): Mazur and Orlicz are direct pupils of Banach; they represent the theory of operations today in Poland and their names on the cover of "Studia Mathematica" indicate direct continuation of Banach's scientific program. Altogether Orlicz published 171 mathematical papers, about half of them in cooperation with several authors. He was the supervisor of 39 doctoral dissertations and over 500 master's theses. Orlicz participated in congresses of mathematics in Oslo (1936), Edinburgh (1958), Stockholm (1962) and Warsaw (1983), and in many scientific conferences. He was invited to universities in Canada, China, Germany and Israel. His book Linear Functional Analysis, (Peking 1963, 138 pp - in Chinese), based on a series of lectures

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Orlicz.html (2 of 5) [2/16/2002 11:25:17 PM]

Orlicz

delivered in German on selected topics of functional analysis at the Institute of Mathematics of Academia Sinica in Beijing in 1958, was translated into English and published in 1992 by World Scientific, Singapore. Orlicz is also a co-author of two school textbooks. Orlicz was the editor of Commentationes Mathematicae (1955 - 1990), and of Studia Mathematica (1962 - 1990), and President of the Polish Mathematical Society (1977 - 1979). In 1956 Orlicz was elected a corresponding-member of the Polish Academy of Sciences and in 1961 its full member. Three universities (York University in Canada, Poznan Technical University and Adam Mickiewicz University in Poznan) conferred upon him the title of doctor honoris causa, in 1974, 1978 and 1983, respectively. Orlicz was awarded many high state decorations, prizes as well as medals of scientific institutions and societies, including the Stefan Banach Prize of the Polish Mathematical Society (1948), the Golden Cross of Merit (1954), the Commander's Cross of Polonia Restituta Order (1958), Honorary Membership of the Polish Mathematical Society (1973), the Alfred Jurzykowski Foundation Award (1973), Copernicus Medal of the Polish Academy of Sciences (1973), Order of Distinguished Teacher (1977), Waclaw Sierpinski Medal of the Warsaw University (1979), Medal of the Commission for National Education (1983) and the Individual State Prizes (second degree in 1952, first degree in 1966). Orlicz's contribution is important in the following areas in mathematics: function spaces (mainly Orlicz spaces), orthogonal series, unconditional convergence in Banach spaces, summability, vector-valued functions, metric locally convex spaces, Saks spaces, real functions, measure theory and integration, polynomial operators and modular spaces. Orlicz spaces L |x(t)|)d (t)
0:

) such that

(

(x) > 0 with the Orlicz norm:

x 0 = sup { |x(t)y(t)|d (t) : or the Luxemburg-Nakano norm: x

L0( , ,

*

(|y(t)|)d (t) 1 }

(|x(t)|/ )d (t) 1 }

Orlicz spaces L are a natural generalization of Lp spaces. They have very rich topological and geometrical structure; they may possess peculiar properties that do not occur in ordinary Lp spaces. Orlicz's ideas have inspired the research of many mathematicians. In recent decades those spaces have been used in analysis, constructive theory of functions, differential equations, integral equations, probability, mathematical statistics, etc. (cf. monographs on Orlicz spaces: M A Krasnoselskii and Ya B Rutickii, Convex Functions and Orlicz Spaces (Groningen 1961), J Lindenstrauss and L Tzafriri, Classical Banach Spaces I, II (Springer 1977, 1979), C Wu and T Wang, Orlicz Spaces and their Applications, (Harbin 1983 - Chinese), A C Zaanen, Riesz Spaces II, (North-Holland 1983), C Wu, T Wang, S Chen and Y Wang, Theory of Geometry of Orlicz Spaces (Harbin 1986 - Chinese), L Maligranda, Orlicz Spaces and Interpolation, (Campinas 1989), M M Rao and Z D Ren, Theory of Orlicz Spaces (Marcel Dekker 1991) and S Chen, Geometry of Orlicz Spaces (Dissertationes Math. 356, 1996). The term Orlicz spaces appeared in the sixties in the Mathematics Subject Classification index of the American Mathematical Society in Section 4635, which is now 46E30, Spaces of measurable functions

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Orlicz.html (3 of 5) [2/16/2002 11:25:17 PM]

Orlicz

(Lp-spaces, Orlicz spaces, etc. ). To emphasize the importance of Orlicz spaces in a jocular way, Professor Orlicz used to say that when he was occasionally asked: Why are Orlicz spaces "better" than Lp spaces? he liked to answer: Tell me first why Lp spaces are "better" than L2? Here is another anecdote in connection with Orlicz spaces: Professor Orlicz had a small apartment and he once applied to the city administration for a bigger one. The answer of an employee was: Your apartment is really small but we cannot accept your claim since we know that you have your own spaces ! Orlicz's name is associated not only with the Orlicz spaces but also with the Orlicz-Pettis theorem, Orlicz property, Orlicz theorem on unconditional convergence in Lp, Mazur-Orlicz bounded consistency theorem, Mazur-Orlicz theorem on inequalities, Mazur-Orlicz theorem on uniform boundedness in F-spaces, Orlicz category theorem, Orlicz interpolation theorem, Orlicz norm, Orlicz function, convexity in the sense of Orlicz, F-norm of Mazur-Orlicz, Drewnowski-Orlicz theorem on representation of orthogonal additive functionals and modulars, Orlicz theorem on Weyl multipliers, Matuszewska-Orlicz indices, Hardy-Orlicz spaces, Marcinkiewicz-Orlicz spaces, Musielak-Orlicz spaces, Orlicz-Sobolev spaces and Orlicz-Bochner spaces. For example, the Orlicz-Pettis theorem says that in Banach spaces the classes of weakly subseries convergent and norm unconditionally convergent series coincide. The Orlicz theorem on unconditional convergence in Lpis: If 1 p
and < being due to Harriot at almost the same time. Oughtred is best known for his invention of an early form of the slide rule. Edmund Gunter (1620) plotted a logarithmic scale along a single straight two foot long ruler. He added and subtracted lengths by using a pair of dividers, operations that were equivalent to multiplying and dividing. In 1630 Oughtred invented a circular slide rule. In 1632 he used two Gunter rulers so that he could do away with the dividers. He published Circles of Proportion and the Horizontal Instrument in 1632 describing slide rules and sundials. There was a dispute however regarding priority over the invention of the circular slide rule. Delamain certainly published a description of a circular slide rule before Oughtred. His Grammelogia, or the Mathematicall ring was published in 1630. It may well be that both invented this instrument independently. Unfortunately a very heated argument ensued and to some extent this formed a cloud over the later years of Oughtred's life. The present form of the slide rule was designed in 1850 by a French army officer, Amedee Mannheim. Oughtred's other works were Trigonometrie (1657), one of the first works on trigonometry to use concise symbolism, and a number of more minor works on watchmaking, solving spherical triangles by the planisphere and methods to determine the position of the sun. Article by: J J O'Connor and E F Robertson List of References (8 books/articles) A Poster of William Oughtred

Mathematicians born in the same country

Some pages from publications

Page from Clavis mathematicae (1652) showing + and and the notation for decimal fractions

Cross-references to History Topics

1. The trigonometric functions 2. Pi through the ages

Other references in MacTutor Other Web sites

Chronology: 1625 to 1650 1. The Galileo Project 2. G Pastori (Slide rule history) 3. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Oughtred.html (2 of 3) [2/16/2002 11:25:27 PM]

Oughtred

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Oughtred.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Oughtred.html (3 of 3) [2/16/2002 11:25:27 PM]

Ozanam

Jacques Ozanam Born: 1640 in Bouligneux, Bresse, France Died: 3 April 1717 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jacques Ozanam studied theology but appears to have been more interested in mathematics and chemistry than in theology. Basically he was self taught in mathematics. It would appear that he was pressed by his father to be a priest since, as soon as his father died, he gave up his study of theology and became a teacher of mathematics. Ozanam taught mathematics in Lyon, then later in Paris. He appears to have made just enough money to live on in Lyon but prospered rather more in Paris. Ozanam was interested in recreational mathematics and analysis. The precursor of books to follow for the next 200 years, he published in four volumes in 1694 his Récréations mathématique et physiques which went through many editions. Ozanam based his book on earlier books by Bachet, Mydorge, Leurechon, and Schwenter. It was later revised and enlarged by Montucla, then translated into English by Hutton (1803, 1814). It contains an early example of a problem about orthogonal Latin squares. Arrange the sixteen court cards so that each row and each column contains one of each suit and one of each value. Ozanam was also interested in cartography and military engineering. He wrote many works on mathematics and its applications other than his famous Récréations mathématique et physiques including Méthod de lever les plans et les cartes de terre et de mer (1691), Traité de la fortification régulière et irrégulière (1691), Méthod facile pour arpenter et mesurer toutes sortes de superficies (1699), La perspective théorique et practique (1711) and La géographie et cosmographie qui traite de la sphere (1711). Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

Mathematical games and recreations

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ozanam.html (1 of 2) [2/16/2002 11:25:28 PM]

Ozanam

Honours awarded to Jacques Ozanam (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Place Ozanam (6th Arrondissement)

Other Web sites

1. The Catholic Encyclopedia 2. The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ozanam.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ozanam.html (2 of 2) [2/16/2002 11:25:28 PM]

Pacioli

Luca Pacioli Born: 1445 in Sansepolcro, Italy Died: 1517 in Sansepolcro, Italy

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Luca Pacioli's father was Bartolomeo Pacioli, but Pacioli does not appear to have been brought up in his parents house. He lived as a child with the Befolci family in Sansepolcro which was the town of his birth. This town is very much in the centre of Italy about 60 km north of the city of Perugia. As far as Pacioli was concerned, perhaps the most important feature of this small commercial town was the fact that Piero della Francesca had a studio and workshop in there and della Francesca spent quite some time there despite frequent commissions in other towns Although we know little of Pacioli's early life, the conjecture that he may have received at least a part of his education in the studio of della Francesca in Sansepolcro must at least have a strong chance of being correct. One reason that this seems likely to be true is the extensive knowledge that Pacioli had of the work of Piero della Francesca and Pacioli's writings were very strongly influenced by those of Piero. Pacioli moved away from Sansepolcro while he was still a young lad. He moved to Venice to enter the service of the wealthy merchant Antonio Rompiansi whose house was in the highly desirable Giudecca district of that city. One has to assume that Pacioli was already well educated in basic mathematics from studies in Sansepolcro and he certainly must have been well educated generally to have been chosen as a tutor to Rompiansi's three sons. However, Pacioli took the opportunity to continue his mathematical studies at a higher level while in Venice, studying mathematics under Domenico Bragadino. During this time Pacioli gained experience both in teaching, from his role as tutor, and also in business from his role helping with Rompiansi's affairs. It was during his time in Venice that Pacioli wrote his first work, a book on arithmetic which he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pacioli.html (1 of 5) [2/16/2002 11:25:30 PM]

Pacioli

dedicated to his employer. This was completed in 1470 probably in the year that Rompiansi died. Pacioli certainly seemed to know all the right people for he left Venice and travelled to Rome where he spent several months living in the house of Leone Battista Alberti who was secretary in the Papal Chancery. As well as being an excellent scholar and mathematician, Alberti was able to provide Pacioli with good religious connections. At this time Pacioli then studied theology and, at some time during the next few years, he became a friar in the Franciscan Order. In 1477 Pacioli began a life of travelling, spending time at various universities teaching mathematics, particularly arithmetic. He taught at the University of Perugia from 1477 to 1480 and while there he wrote a second work on arithmetic designed for the classes that he was teaching. He taught at Zara (now called Zadar or Jadera in Croatia but at that time in the Venetian Empire) and there wrote a third book on arithmetic. None of the three arithmetic texts were published, and only the one written for the students in Perugia has survived. After Zara, Pacioli taught again at the University of Perugia, then at the University of Naples, then at the University of Rome. Certainly Pacioli become acquainted with the duke of Urbino at some time during this period. Pope Sixtus IV had made Federico da Montefeltro the duke of Urbino in 1474 and Pacioli seems to have spent some time as a tutor to Federico's son Guidobaldo who was to become the last ruling Montefeltro when his father died in 1482. The court at Urbino was a notable centre of culture and Pacioli must have had close contact with it over a number of years. In 1489, after two years in Rome, Pacioli returned to his home town of Sansepolcro. Not all went smoothly for Pacioli in his home town, however. He had been granted some privileges by the Pope and there was a degree of jealousy among the men from the religious orders in Sansepolcro. In fact Pacioli was banned from teaching there in 1491 but the jealousy seemed to be mixed with a respect for his learning and scholarship for in 1493 he was invited to preach the Lent sermons. During this time in Sansepolcro, Pacioli worked on one of his most famous books the Summa de arithmetica, geometria, proportioni et proportionalita which he dedicated to Guidobaldo, the duke of Urbino. Pacioli travelled to Venice in 1494 to publish the Summa. The work gives a summary of the mathematics known at that time although it shows little in the way of original ideas. The work studies arithmetic, algebra, geometry and trigonometry and, despite the lack of originality, was to provide a basis for the major progress in mathematics which took place in Europe shortly after this time. As stated in [1] the Summa was:... not addressed to a particular section of the community. An encyclopaedic work (600 pages of close print, in folio) written in Italian, it contains a general treatise on theoretical and practical arithmetic; the elements of algebra; a table of moneys, weights and measures used in the various Italian states; a treatise on double-entry bookkeeping; and a summary of Euclid's geometry. He admitted to having borrowed freely from Euclid, Boethius, Sacrobosco, Fibonacci, ... The geometrical part of Pacioli's Summa is discussed in detail in [6]. The authors write:The geometrical part of L Pacioli's Summa [Venice, 1494] in Italian is one of the earliest printed mathematical books. Pacioli broadly used Euclid's Elements, retelling some parts of it. He referred also to Leonardo of Pisa (Fibonacci). Other interesting aspects of the Summa was the fact that it studied games of chance. Pacioli studied the problem of points, see [9], although the solution he gave is incorrect.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pacioli.html (2 of 5) [2/16/2002 11:25:30 PM]

Pacioli

Ludovico Sforza was the second son of Francesco Sforza, who had made himself duke of Milan. When Francesco died in 1466, Ludovico's elder brother Galeazzo Sforza became duke of Milan. However, Galeazzo was murdered in 1476 and his seven year old son became duke of Milan. Ludovico, after some political intrigue, became regent to the young man in 1480. With very generous patronage of artists and scholars, Ludovico Sforza set about making his court in Milan the finest in the whole of Europe. In 1482 Leonardo da Vinci entered Ludovico's service as a court painter and engineer. In 1494 Ludovico became the duke of Milan and, around 1496, Pacioli was invited by Ludovico to go to Milan to teach mathematics at Ludovico Sforza's court. This invitation may have been made at the prompting of Leonardo da Vinci who had an enthusiastic interest in mathematics. At Milan Pacioli and Leonardo quickly became close friends. Mathematics and art were topics which they discussed at length, both gaining greatly from the other. At this time Pacioli began work on the second of his two famous works, Divina proportione and the figures for the text were drawn by Leonardo. Few mathematicians can have had a more talented illustrator for their book! The book which Pacioli worked on during 1497 would eventually form the first of three books which he published in 1509 under the title Divina proportione (see for example [3]). This was the first of the three books which finally made up this treatise, and it studied the 'Divine Proportion' or 'golden ratio' which is the ratio a : b = b : (a + b). It contains the theorems of Euclid which relate to this ratio, and it also studies regular and semiregular polygons (see in particular [4] for a discussion of Pacioli's work on regular polygons). Clearly the interest of Leonardo in this aesthetically satisfying ratio both from a mathematical and artistic point of view was an important influence on the work. The golden ratio was also of importance in architectural design and this topic was to form the second part of the treatise which Pacioli wrote later. The third book in the treatise was a translation into Italian of one of della Francesca's works. Louis XII became king of France in 1498 and, being a descendant of the first duke of Milan, he claimed the duchy. Venice supported Louis against Milan and in 1499 the French armies entered Milan In the following year Ludovico Sforza was captured when he attempted to retake the city. Pacioli and Leonardo fled together in December 1499, three months after the French captured Milan. They stopped first at Mantua, where they were the guests of Marchioness Isabella d'Este, and then in March 1500 they continued to Venice. From Venice they returned to Florence, where Pacioli and Leonardo shared a house. The University of Pisa had suffered a revolt in 1494 and had moved to Florence. Pacioli was appointed to teach geometry at the University of Pisa in Florence in 1500. He remained in Florence, teaching geometry at the university, until 1506. Leonardo, although spending ten months away working for Cesare Borgia, also remained in Florence until 1506. Pacioli, like Leonardo, had a spell away from Florence when he taught at the University of Bologna during 1501-02. During this time Pacioli worked with Scipione del Ferro and there has been much conjecture as to whether the two discussed the algebraic solution of cubic equations. Certainly Pacioli discussed this topic in the Summa and some time after Pacioli's visit to Bologna, del Ferro solved one of the two cases of this classic problem. During his time in Florence Pacioli was involved with Church affairs as well as with mathematics. He was elected the superior of his Order in Romagna and then, in 1506, he entered the monastery of Santa Croce in Florence. After leaving Florence, Pacioli went to Venice where he was given the sole rights to publish his works there for the following fifteen years. In 1509 he published the three volume work Divina proportione and also a Latin translation of Euclid's Elements. The first printed edition of Euclid's

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pacioli.html (3 of 5) [2/16/2002 11:25:30 PM]

Pacioli

Elements was the thirteenth century translation by Campanus which had been published in printed form in Venice in 1482. Pacioli's edition was based on that of Campanus but it contained much in the way of annotation by Pacioli himself. In 1510 Pacioli returned to Perugia to lecture there again. He also lectured again in Rome in 1514 but by this time Pacioli was 70 years of age and nearing the end of his active life of scholarship and teaching. He returned to Sansepolcro where he died in 1517 leaving unpublished a major work De viribus amanuensis on recreational problems, geometrical problems and proverbs. This work makes frequent reference to Leonardo da Vinci who worked with him on the project, and many of the problems in this treatise are also in Leonardo's notebooks. Again it is a work for which Pacioli claimed no originality, describing it as a compendium. Despite the lack of originality in Pacioli's work, his contributions to mathematics are important, particularly because of the influence which his book were to have over a long period. In [10] the importance of Pacioli's work is discussed, in particular his computation of approximate values of a square root (using a special case of Newton's method), his incorrect analysis of certain games of chance (similar to those studied by Pascal which gave rise to the theory of probability), his problems involving number theory (similar problems appeared in Bachet's compilation), and his collection of many magic squares. In 1550 there appeared a biography of Piero della Francesca written by Giorgio Vasari. This biography accused Pacioli of plagiarism and claimed that he stole della Francesca's work on perspective, on arithmetic and on geometry. This is an unfair accusation, for although there is truth that Pacioli relied heavily on the work of others, and certainly on that of della Francesca in particular, he never attempted to claim the work as his own but acknowledged the sources which he used. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles)

A Quotation

A Poster of Luca Pacioli

Mathematicians born in the same country

Some pages from publications

A page from Summa de arithmetica, geometria proportioni et propornionalita (1494)

Cross-references to History Topics

1. Quadratic, cubic and quartic equations 2. Perfect numbers 3. An overview of the history of mathematics

Other references in MacTutor Other Web sites

Chronology: 1300 to 1500 1. The Catholic Encyclopedia 2. George W Hart (Pacioli's polyhedra)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pacioli.html (4 of 5) [2/16/2002 11:25:30 PM]

Pacioli

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pacioli.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pacioli.html (5 of 5) [2/16/2002 11:25:30 PM]

Pade

Henri Eugène Padé Born: 17 Dec 1863 in France Died: 9 July 1953

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henri Padé was educated at the Lycée St Louis and Ecole Normale Supérieure in Paris and then taught in secondary schools. In 1889 and 1890 he studied at Leipzig and Göttingen under Klein and Schwarz. He returned to France and obtained a doctorate under Hermite's supervision. He then taught at Lille, Poitiers and Bordeaux. In 1906 he received the Grand Prize of the French Academy and, two years later, he became Rector of Besançon. In 1917 he became Rector of the Academy of Dijon and from 1923 until he retired in 1934 he was Rector at Aix-Marseille. He wrote 41 papers, 29 of which were on continued fractions and Padé approximates. He also wrote an elementary algebra book and translated Klein's Erlangen programme into French. His thesis and related papers on Padé approximates became well known after Borel included them in his 1901 book on divergent series. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Henri Padé

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pade.html (1 of 2) [2/16/2002 11:25:32 PM]

Pade

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pade.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pade.html (2 of 2) [2/16/2002 11:25:32 PM]

Padoa

Alessandro Padoa Born: 14 Oct 1868 in Venice, Italy Died: 25 Nov 1937 in Genoa, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alessandro Padoa attended secondary school in Venice, then studied engineering at the University of Padua. He was awarded a mathematics degree from the University of Turin in 1895. After graduating, Padoa became a secondary school teacher, teaching in Pinerolo, Rome and Cagliari. However, Padoa gave many lectures at universities and lectures at congresses. Beginning in 1898 he gave a series of lectures at the Universities of Brussels, Pavia, Berne, Padua, Cagliari and Genoa. He lectured at congresses in Paris, Cambridge, Livorno, Parma, Padua and Bologna. From 1909 he taught at the Technical Institute in Genoa. Padua belonged to Peano's school of mathematical logic, popularising this type of work. He gave the important lecture Essay of an algebraic theory of whole numbers, preceded by a logical introduction to any deductive theory at the International Congress of Philosophy in Paris in 1900. He had discovered an important method in the theory of definition which became even more important when model theory was developed and Tarski proved Padoa's method in 1924. Padoa was the first to present a method to prove that a primative term of a theory cannot be defined within the system using the remaining primative terms. This result was first made public in his lecture at the Paris Congress referred to above. Padoa believed, correctly, that his result was of major importance and wrote in this paper:We can now settle completely (and, we believe, for the first time) a question of the greatest logical importance. Immediately following the Congress of Philosophy in Paris, the Second International Congress of Mathematicians took place. Padoa spoke on A new system of definitions for Euclidean geometry but began with a summary of his lecture at the Philosophy Congress. Charlotte Scott found this one of the most interesting talks at the Congress but wrote:Mr Padoa did not get beyond this definition, possibly because he had entered so minutely into the details of the proof of the independance of the seven postulates that he had exhausted his allowance of time. Halsted also attended the Congress and wrote that Padoa was:... among the most interesting personalities present. In 1934 Padoa was awarded the mathematics prize of the Accademia dei Lincei.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Padoa.html (1 of 2) [2/16/2002 11:25:33 PM]

Padoa

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Padoa.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Padoa.html (2 of 2) [2/16/2002 11:25:33 PM]

Painleve

Paul Painlevé Born: 5 Dec 1863 in Paris, France Died: 29 Oct 1933 in Paris, France

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Painlevé received a doctorate in mathematics from Paris in 1887, then taught at Lille and the Ecole Polytechnique in Paris. He worked on differential equations and mechanics. He solved, using Painlevé functions, differential equations which Poincaré and Emile Picard had failed to solve. Painlevé took a special interest in aviation, applying his theoretical skills to study the theory of flight. He was Wilbur Wright's first passenger making a record 1 hour 10 minute flight, then the following year 1909 he created the first university course in aeronautical mechanics. Although less skilled in politics than mathematics he began a political career in 1906 which led to two periods as French Prime Minister. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Paul Painlevé

Mathematicians born in the same country

Honours awarded to Paul Painlevé (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Place and Square Paul Painlevé (5th Arrondissement)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Painleve.html (1 of 2) [2/16/2002 11:25:35 PM]

Painleve

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Painleve.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Painleve.html (2 of 2) [2/16/2002 11:25:35 PM]

Paley

Raymond Edward Alan Christopher Paley Born: 7 Jan 1907 in England Died: 7 April 1933 in Banff, Alberta, Canada Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Raymond Paley was educated at Eton. From there he entered Trinity College, Cambridge where he showed himself the most brilliant student among a remarkable collection of fellow undergraduates. He was taught at Cambridge by Hardy and Littlewood and it was under Littlewood's supervision that he undertook research. While Paley was undertaking research at Cambridge, Zygmund spent the academic year 1930-31 there. Paley had already proved impressive results on Fourier series and had collaborated with Littlewood, his supervisor. Zygmund discovered Paley's extraordinary talent and the two worked jointly on existence proofs, brilliantly applying ideas from Borel's Calcul des probabilités dénombrables. Zygmund's book Trigonometric Series published in 1935 owes a debt to the joint work that he carried out with Paley. Norbert Wiener was proving important results in areas of interest to Paley so he applied for an International Research Fellowship to allow him to travel to the United States to collaborate with him. Norbert Wiener wrote in [1]:Soon after his arrival in America, however, certain studies of lacunary series which Paley had already begun suggested a new attack on the theory of interpolation and allied trigonometrical problems. These results led successively to the study of quasi-analytic functions, of entire functions of order one-half, and of many related questions. For a young man of 26, Paley had collaborated with a remarkable group of mathematicians. In addition to Littlewood, Zygmund and Norbert Wiener, he had also collaborated with Pólya. As Norbert Wiener wrote in [1]:Possessed of an extraordinary capacity for making friends and for scientific collaboration, Paley believed that the inspiration of continual interchange of ideas stimulates each collaborator to accomplish more than he would alone. Already with a reputation remarkable for one so young, Paley stood on the brink of becoming one of the very first rank of research mathematicians. However, in 1933 while working in the United States, he went to Canada for a skiing holiday. While skiing near Banff he was killed by avalanche. Had he lived to continue his mathematical work, one feels sure that his name would today be as well known as the mathematicians with whom he collaborated. Norbert Wiener gave the Colloquium Lectures of the American Mathematical Society in 1934 and spoke

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Paley.html (1 of 2) [2/16/2002 11:25:36 PM]

Paley

on Paley's work. Paley was to have been a Colloquium Lecturer himself. Norbert Wiener wrote in [1]:... he was already recognised as the ablest of the group of young English mathematicians who have been inspired by the genius of G H Hardy and J E Littlewood. In a group notable for its brilliant technique, no one had developed this technique to a higher degree than Paley. Nevertheless he should not be though of primarily as a technician, for with this ability he combined creative power of the first order. As he himself was wont to say, technique without 'rugger tactics' will not get one far, and these rugger tactics he practised to a degree that was characteristic of his forthright and vigorous nature. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to Raymond Paley (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1934

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Paley.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Paley.html (2 of 2) [2/16/2002 11:25:36 PM]

Paman

Roger Paman Born: not known Died: 1748 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

We know nothing of Roger Paman's date of birth, place of birth, or his parents. Almost all we know of him comes from information which he himself gives in the preface to, what appears to be, his only published book. Other details of the events in which he participated are known, however, so we can indirectly work out quite a few details using these other sources. Although there is no record of Paman being a student at the University of Cambridge, we know that Mr Frank of St John's College, Cambridge gave him a copy of George Berkeley's The Analyst (1734). Paman wrote a paper on this treatise which he called The Harmony of the Ancient and Modern Geometry Asserted. This paper was communicated to several members of The Royal Society, and it kept circulating until 1739. On 18 September 1740 Paman set off with George Anson on his journey around the world. Eight ships set out from St Helen's but only one ship, the Centurion, succeeded in making the journey around the world. It returned to England, reaching Spithead on 15 June 1744 but, since Paman was back in England long before this, we know that he was not on board the Centurion. Five of the ships, the Gloucester, the Wager, the Tryal, the Anna and the Industry, were destroyed during the journey, three of them while rounding Cape Horn, so Paman was not on board any of these ships. We can therefore deduce that Paman must have been on one of the two remaining ships, either the Severn or the Pearl. The Severn and Pearl, with Paman on board, left England together with the other ships. They set off across the Atlantic but they were poorly manned and they were ill-equipped. However, they anchored near the coast of Patagonia (Southern Argentina) on 18 February 1741. On 7 March they passed the Straits of Le Maire, still with all the other ships. However, on 10 April they lost sight of the other ships, and on 25 April they even lost sight of each other. The two ships, with Paman certainly on one of them, made contact again on 21 May. The weather had been terrible and most of the men were ill, and both ships had to wait before going on. Although we know nothing specific about Paman, he must have suffered along with the other members of the two crews. On 4 July 1741, the ships arrived in Rio de Janeiro, and Captain Legge, the captain of the Severn, wrote:And I arrived by the great mercy of Almighty God safe in this port the 6th of June, not having above thirty men in the ship, myself, lieutenants, officers and servants (besides three men I had at sea from the Pearl) that were able to assist to the working of the ship; and all of us so weak and so much reduced that we could hardly walk along the deck. The two ships remained in Rio for a long time. The crew attempted to get their ships repaired and the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Paman.html (1 of 5) [2/16/2002 11:25:38 PM]

Paman

extended stop allowed the men to regain their health. However the ships were not happy places and there was much quarrelling over what to do next. On 5 February 1742, they arrived in Barbados, by this time on their way home to England. It seems that much of the blame for making Anson's voyage relatively unsuccessful, was given to these two ships. For instance, John Campbell wrote:The scheme which Commodore Anson was sent to execute, was certainly well laid; and if the two ships that repassed the Streights of Le Maire, and thereby exposed themselves to greater dangers, than they could have met with by continuing their voyage, had either proceeded with the Commodore, or had followed him to the island of Juan Fernandez, he would have had men enough to have undertaken something of consequence either in Chile or Peru ... Given these feelings it is clear that Paman and the other the men from the Severn and the Pearl would not have been treated as heroes when they returned to England. It is reasonable to ask exactly what Paman's role had been on the voyage. Perhaps the best indication of this is given in an advertisement for a book which Paman intended to write. This book would have given:... the height of the Mercury in the Thermometer every Day at Noon, during the Months of February, March, April and May, between the Latitudes of 40 and 60 South. An accurate Account of the Variations of the Needle, at different Distances, on the same Parallels from the Coasts of Brazil, Patagonia, and Terra del Fuego. With such Curious Particulars relating to different Parts of South America, as the Author had an Opportunity of remarking himself, or procuring from Persons of Credit and Distinction, during Seven Months that he lived in Brazil. It is possible that Paman wrote this book, but no trace of it has been found nor is there any evidence that it was published. The most likely explanation is that there was insufficient interest generated by the advertisement. Before setting out on the voyage, Paman had given his paper, The Harmony of the Ancient and Modern Geometry Asserted, to his friend Dr Hartley, and when he returned, in February 1742, Paman sent it to the Royal Society. This paper must have been the main reason for his election as a Fellow of the Royal Society. He was recommended by Abraham de Moivre, R Barker and G Scott on 10 February 1743, with the following text (preserving the old English spelling):Mr Roger Paman of London: A Gentleman Extremely well versed in all the Parts of the higher Mathematicks desiring to be a member of this Society we recommend him as personally known to us and likely to become a usefull Member thereof. He was elected a Fellow on 12 May 1743. Paman's paper was published the paper as a book in 1745. The book was titled, as the paper, The Harmony of the Ancient and Modern Geometry asserted. The preface was dated 1 August 1745 while the postscript to the preface is dated 24 August 1745. The book probably appeared in October of that year, as The Gentleman's Magazine includes this book in its list of "Books and Pamphlets published this Month":The harmony of the ancient and modern geometry asserted; in answer to the Analyst, etc. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Paman.html (2 of 5) [2/16/2002 11:25:38 PM]

Paman

pr. 7s. 6d. sew'd. As this states Paman had written his work to answer the criticisms of the calculus presented by Berkeley in The Analyst published in 1734. His reasons are clearly stated in the preface:Having undertaken to cultivate the Discoveries of the Moderns upon the Principles of the Ancients, without any Considerations of Velocity, Time or Motion of Indivisibility or Infinity, in such a Manner that, whilst I omitted those Considerations, I might not neglect the Design ... of introducing them first into Geometry, and that whilst I aimed at the Rigour of the Ancients, I might avoid the Tedium and Perplexity of their Demonstrations ad absurdum. The way that Paman attempted to avoid "considerations of velocity, time or motion of indivisibility or infinity" was to introduce certain new concepts. First he defines "radical quantity":I call one Expression the radical Quantity of another; when the latter is compos'd of any Power or Powers of the former, their Parts or Multiples. A "radical Quantity" is close to what today would be called a "variable", even though Paman implies that an expression can be composed of this variable only by taking powers of it, and by multiplicating by scalars, which means that Paman is thinking of polynomials or power series. He next defines the "first State of x" which would in today's terminology be a neighbourhood of 0 in the positive real numbers. He also defines concepts he calls "Maximinus" and "Minimajus":If one Expression be less (greater) than another, in the first State of their radical Quantity, and yet no Quantity of the same Kind can be added to (subtracted from) the former, without making the Sum (Remainder) greater (less) than the latter in the first State of their radical Quantity; then I call the former the first Maximinus (Minimajus) of the latter. Paman himself explains this concept in the preface:... all that is understood by a Maximinus, is such a Quantity as being less than another, cannot be augmented by any Quantity of the same Kind, that is by any Part of itself, without becoming greater; thus A is the Maximinus of B, when it is the greatest of all those Quantities of the same Kind that are less than B. And all that is understood by a Minimajus is such a Quantity, as being greater than another, cannot be diminished by any Quantity of the same Kind without becoming less. Paman's explanation looks like our present definition of infimum and supremum, but whereas our infimum and supremum given a set (of real numbers) gives a real number, the Maximinus and Minimajus of a quantity is a quantity of the same kind. With these concepts Paman defines fluxions. Paman's terminology will of course seem strange at first glance. But as all of the terms he introduces cover concepts developed by Paman, we cannot blame him for using new names. But we must consider whether these concepts are actually useful. In a limited sense, they certainly are. Paman managed to give foundations to the calculus using these concepts, and he needed all of them. But we would not be able to define the derivative using Paman's terms since we consider more complicated functions than the polynomials or power series which Paman considered. However, the states of x are closely related to the very important concept of neighbourhoods (the latter of course being used far more generally than just on R), and the Maximinus and Minimajus are cousins of lim inf and lim sup (although more powerful).

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Paman.html (3 of 5) [2/16/2002 11:25:38 PM]

Paman

It must therefore be said that far from introducing concepts for their own sake, Paman introduced interesting new concepts that were useful to him and that would have been useful to mathematics if other mathematicians had noticed them. There was another approach to answering Berkeley due to Maclaurin who published A Treatise of Fluxions in 1742. This was after Paman's first version of his paper, but before the publication of Paman's book. Certainly we know that Paman read Maclaurin's treatise for he wrote in his book that the two books at times agreed strongly with each other:... at my return I found Mr MacLaurin had published his Treatise of Fluxions, in which I have the pleasure to see my agreement with that ingenious gentleman ... There seems no way of discovering whether Paman made major changes to his manuscript after reading Maclaurin's treatise. In fact there were three authors, Robins, Maclaurin and Paman, who attempted to answer Berkeley in different ways. Robins gave an explanation of Newton's theories, with clearer definitions and rigorous proofs, while still depending on intuitive concepts. Maclaurin managed to give foundations using neither infinitesimals nor motion or velocities. Given his position in the learned world, it was only natural that his answer was the one to be remembered by most, especially when his work also included much interesting mathematics. However, it is interesting to see Paman's remarkably modern work - with concepts resembling our neighbourhood concept and lim inf and lim sup. Sadly, his work was apparently not studied at the time, and did not influence the later developments. It only in 1989, when Breidert and Sageng made detailed studies of Paman's work, that its contents became known. Thanks to Bjorn Smestad for his permission to use material from his article on Paman. See the Web site below. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other Web sites

Hammerfest, Norway

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Paman.html (4 of 5) [2/16/2002 11:25:38 PM]

Paman

Mathematicians of the day JOC/EFR January 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Paman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Paman.html (5 of 5) [2/16/2002 11:25:38 PM]

Panini

Panini Born: about 520 BC in Shalatula (near Attock), now Pakistan Died: about 460 BC in India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Panini was born in Shalatula, a town near to Attock on the Indus river in present day Pakistan. The dates given for Panini are pure guesses. Experts give dates in the 4th, 5th, 6th and 7th century BC and there is also no agreement among historians about the extent of the work which he undertook. What is in little doubt is that, given the period in which he worked, he is one of the most innovative people in the whole development of knowledge. We will say a little more below about how historians have gone about trying to pinpoint the date when Panini lived. Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology. Sanskrit was the classical literary language of the Indian Hindus and Panini is considered the founder of the language and literature. It is interesting to note that the word "Sanskrit" means "complete" or "perfect" and it was thought of as the divine language, or language of the gods. A treatise called Astadhyayi (or Astaka ) is Panini's major work. It consists of eight chapters, each subdivided into quarter chapters. In this work Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe Sanskrit grammar. Starting with about 1700 basic elements like nouns, verbs, vowels, consonants he put them into classes. The construction of sentences, compound nouns etc. is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways Panini's constructions are similar to the way that a mathematical function is defined today. Joseph writes in [2]:[Sanskrit's] potential for scientific use was greatly enhanced as a result of the thorough systemisation of its grammar by Panini. ... On the basis of just under 4000 sutras [rules expressed as aphorisms], he built virtually the whole structure of the Sanskrit language, whose general 'shape' hardly changed for the next two thousand years. ... An indirect consequence of Panini's efforts to increase the linguistic facility of Sanskrit soon became apparent in the character of scientific and mathematical literature. This may be brought out by comparing the grammar of Sanskrit with the geometry of Euclid - a particularly apposite comparison since, whereas mathematics grew out of philosophy in ancient Greece, it was ... partly an outcome of linguistic developments in India. Joseph goes on to make a convincing argument for the algebraic nature of Indian mathematics arising as a consequence of the structure of the Sanskrit language. In particular he suggests that algebraic http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Panini.html (1 of 3) [2/16/2002 11:25:39 PM]

Panini

reasoning, the Indian way of representing numbers by words, and ultimately the development of modern number systems in India, are linked through the structure of language. Panini should be thought of as the forerunner of the modern formal language theory used to specify computer languages. The Backus Normal Form was discovered independently by John Backus in 1959, but Panini's notation is equivalent in its power to that of Backus and has many similar properties. It is remarkable to think that concepts which are fundamental to today's theoretical computer science should have their origin with an Indian genius around 2500 years ago. At the beginning of this article we mentioned that certain concepts had been attributed to Panini by certain historians which others dispute. One such theory was put forward by B Indraji in 1876. He claimed that the Brahmi numerals developed out of using letters or syllables as numerals. Then he put the finishing touches to the theory by suggesting that Panini in the eighth century BC (earlier than most historians place Panini) was the first to come up with the idea of using letters of the alphabet to represent numbers. There are a number of pieces of evidence to support Indraji's theory that the Brahmi numerals developed from letters or syllables. However it is not totally convincing since, to quote one example, the symbols for 1, 2 and 3 clearly do not come from letters but from one, two and three lines respectively. Even if one accepts the link between the numerals and the letters, making Panini the originator of this idea would seem to have no more behind it than knowing that Panini was one of the most innovative geniuses that world has known so it is not unreasonable to believe that he might have made this step too. There are other works which are closely associated with the Astadhyayi which some historians attribute to Panini, others attribute to authors before Panini, others attribute to authors after Panini. This is an area where there are many theories but few, if any, hard facts. We also promised to return to a discussion of Panini's dates. There has been no lack of work on this topic so the fact that there are theories which span several hundreds of years is not the result of lack of effort, rather an indication of the difficulty of the topic. The usual way to date such texts would be to examine which authors are referred to and which authors refer to the work. One can use this technique and see who Panini mentions. There are ten scholars mentioned by Panini and we must assume from the context that these ten have all contributed to the study of Sanskrit grammar. This in itself, of course, indicates that Panini was not a solitary genius but, like Newton, had "stood on the shoulders of giants". Now Panini must have lived later than these ten but this is absolutely no help in providing dates since we have absolutely no knowledge of when any of these ten lived. What other internal evidence is there to use? Well of course Panini uses many phrases to illustrate his grammar any these have been examined meticulously to see if anything is contained there to indicate a date. To give an example of what we mean: if we were to pick up a text which contained as an example "I take the train to work every day" we would know that it had to have been written after railways became common. Let us illustrate with two actual examples from the Astadhyayi which have been the subject of much study. The first is an attempt to see whether there is evidence of Greek influence. Would it be possible to find evidence which would mean that the text had to have been written after the conquests of Alexander the Great? There is a little evidence of Greek influence, but there was Greek influence on this north east part of the Indian subcontinent before the time of Alexander. Nothing http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Panini.html (2 of 3) [2/16/2002 11:25:39 PM]

Panini

conclusive has been identified. Another angle is to examine a reference Panini makes to nuns. now some argue that these must be Buddhist nuns and therefore the work must have been written after Buddha. A nice argument but there is a counter argument which says that there were Jaina nuns before the time of Buddha and Panini's reference could equally well be to them. Again the evidence is inconclusive. There are references by others to Panini. However it would appear that the Panini to whom most refer is a poet and although some argue that these are the same person, most historians agree that the linguist and the poet are two different people. Again this is inconclusive evidence. Let us end with an evaluation of Panini's contribution by Cardona in [1]:Panini's grammar has been evaluated from various points of view. After all these different evaluations, I think that the grammar merits asserting ... that it is one of the greatest monuments of human intelligence. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. An overview of Indian mathematics 2. Indian numerals

Other references in MacTutor

Chronology: 500BC to 1AD

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR November 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Panini.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Panini.html (3 of 3) [2/16/2002 11:25:39 PM]

Papin

Denis Papin Born: 22 Aug 1647 in Blois, France Died: 1712 in London, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Denis Papin attended a Jesuit school in Blois then, in 1661, he began his studies at the University of Angers. He graduated with a medical degree in 1669. Papin assisted Huygens with air pump experiments from 1671 to 1674, during which time he lived in Huygens's apartments in the Royal Library in Paris. Papin went to London in 1675 to work with Boyle. He remained in this post until 1679 when he became Hooke's assistant at the Royal Society. Papin was elected a Fellow of the Royal Society in 1680. In 1681 Papin left for Italy where he was director of experiments at the Accademia publicca di scienze in Venice until 1684. There was an attempt to turn the Accademia in Venice into a Society modelled on the Royal Society in London and the Académie Royale in Paris but lack of financial support ended the attempt. There were religious reasons why Papin could not return to France. He was a Calvinist, born into a Huguenot family, and after the Edict of Nantes which had granted religious liberty to the Huguenots was revoked by Louis XIV in 1685, he became an exile. Papin returned to London in 1684 working again with the Royal Society until 1687. After this Papin left England and went to Hesse-Kassel where he was appointed professor of mathematics at the University of Marburg. He held this post until 1696 when he worked for the Landgrave of Hesse-Kassel until 1707. This time in Hesse-Kassel was not a successful one for Papin who found himself in disagreement with his colleagues.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Papin.html (1 of 2) [2/16/2002 11:25:41 PM]

Papin

Papin is best known for his work as an inventor, particularly his work on the steam engine. In 1679 he invented the pressure cooker and, in 1690 he published his first work on the steam engine in De novis quibusdam machinis. The purpose of the steam engine was to raise water to a canal between Kassel and Karlshaven. He also used a steam engine to pump water to a tank on the roof of the palace to supply water for the fountains in the grounds. In 1705, when Leibniz sent Papin a sketch of a steam engine, Papin began working on that topic again and wrote The New Art of Pumping Water by using Steam (1707). He designed a safety valve to prevent the pressure of steam building up to dangerous levels. Other inventions which Papin worked on were the construction of a submarine, an air gun and a grenade launcher. He tried to build up a glass industry in Hesse-Kassel and also experimented with preserving food both with chemicals and using a vacuum. In 1707 Papin built the first paddle boat and that same year he returned to London where he lived in obscurity and poverty until his death. The date given for his death is only a guess since no records seem to exist of his last years in London. His last known letter is dated 23 January 1712. Article by: J J O'Connor and E F Robertson List of References (11 books/articles) Mathematicians born in the same country Honours awarded to Denis Papin (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1682

Paris street names

Rue Papin (3rd Arrondissement)

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Papin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Papin.html (2 of 2) [2/16/2002 11:25:41 PM]

Pappus

Pappus of Alexandria Born: about 290 in Alexandria, Egypt Died: about 350 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pappus of Alexandria is the last of the great Greek geometers and one of his theorems is cited as the basis of modern projective geometry. Our knowledge of Pappus's life is almost nil. There appear in the literature one or two references to dates for Pappus's life which must be wrong. There is a reference in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Pappus was a contemporary of Theon of Alexandria (see for example [1]):Pappus, of Alexandria, philosopher, lived about the time of the Emperor Theodosius the Elder [379 AD - 395 AD], when Theon the Philosopher, who wrote the Canon of Ptolemy, also flourished. This would seem convincing but there is a chronological table by Theon of Alexandria which, when being copied, has had inserted next to the name of Diocletian (who ruled 284 AD - 305 AD) "at that time wrote Pappus". Similar insertions give the dates for Ptolemy, Hipparchus and other mathematical astronomers. Clearly both of these cannot be correct, and the known inaccuracy of the Suda led historians to favour dates for Pappus which would have him writing in the period 284 AD - 305 AD, as suggested by the insertion into Theon's chronological table. Heath in [4] is completely convinced saying that [4]:Pappus lived at the end of the third century AD. However, we now know that both the above sources are wrong, for Rome (see [6]) showed that it can be deduced from Pappus's commentary on the Almagest that he observed the eclipse of the sun in Alexandria which took place on 18 October 320. This fixes clearly the date of 320 for Pappus's commentary on Ptolemy's Almagest. Other than this accurate date we know little else about Pappus. He was born and appears to have lived in Alexandria all his life. We know that he dedicated works to Hermodorus, Pandrosion and Megethion but other than knowing that Hermodorus was Pappus's son, we have no further knowledge of these men. Again Pappus refers to a friend who was also a philosopher, named Hierius, but other than knowing that he encouraged Pappus to study certain mathematical problems, we know nothing else about him either. Finally a reference to Pappus in Proclus's writings says that he headed a school in Alexandria. Pappus's major work in geometry is Synagoge or the Mathematical Collection which is a collection of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pappus.html (1 of 6) [2/16/2002 11:25:43 PM]

Pappus

mathematical writings in eight books thought to have been written in around 340 (although some historians believe that Pappus had completed the work by 325 AD). Heath in [4] describes the Mathematical Collection as follows:Obviously written with the object of reviving the classical Greek geometry, it covers practically the whole field. It is, however, a handbook or guide to Greek geometry rather than an encyclopaedia; it was intended, that is, to be read with the original works (where still extant) rather than to enable them to be dispensed with. It seems likely that this work was not originally written as a single treatise but rather was written as a series of books dealing with different topics. Each book has its own introduction and often a valuable historical account of the topic, particularly in the case where such an account is not readily available from other sources. Book I covered arithmetic (and is lost) while Book II is partly lost but the remaining part deals with Apollonius's method for dealing with large numbers. The method expresses numbers as powers of a myriad, that is as powers of 10000. Book III is divided by Pappus into four parts. The first part looks at the problem of finding two mean proportionals between two given straight lines. The second part gives a construction of the arithmetic, geometric and harmonic means. The third part describes a collection of geometrical paradoxes which Pappus says are taken from a work by Erycinus. Other than what is included in this part, we know nothing of Erycinus or his work. The final part shows how each of the five regular polyhedra can be inscribed in a sphere. The authors of [9] discuss the muddle Pappus made in Book III of the problem of displaying the arithmetic, geometric and harmonic means of two segments in one circle. Book IV contains properties of curves including the spiral of Archimedes and the quadratrix of Hippias and includes his trisection methods. Pappus introduces the various types of curves that he will consider:There are, we say, three types of problem in geometry, the so-called 'plane', 'solid', and 'linear' problems. Those that can be solved with straight line and circle are properly called 'plane' problems, for the lines by which such problems are solved have their origin in a plane. Those problems that are solved by the use of one or more sections of the cone are called 'solid' problems. For it is necessary in the construction to use surfaces of solid figures, that is to say, cones. There remain the third type, the so-called 'linear' problem. For the construction in these cases curves other than those already mentioned are required, curves having a more varied and forced origin and arising from more irregular surfaces and from complex motions. Of this character are the curves discovered in the so-called 'surface loci' and numerous others even more involved ... . These curves have many wonderful properties. More recent writers have indeed considered some of them worthy of more extended treatment, and one of the curves is called 'the paradoxical curve' by Menelaus. Other curves of the same type are spirals, quadratrices, cochloids, and cissoids. Pappus introduces some of the ideas of Book V by describing how bees construct honeycombs. He concludes his discussion of honeycombs and introduces the aims of his work as follows (see for example [3] or [4]):Bees, then, know just this fact which is useful to them, that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material in constructing each. But we, claiming a greater share in wisdom than the bees, will http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pappus.html (2 of 6) [2/16/2002 11:25:43 PM]

Pappus

investigate a somewhat wider problem, namely that, of all equilateral and equiangular plane figures having an equal perimeter, that which has the greater number of angles is always the greater, and the greatest of then all is the circle having its perimeter equal to them. Also in Book V Pappus discusses the thirteen semiregular solids discovered by Archimedes. He compares the areas of figures with equal perimeters and volumes of solids with equal surface areas, proving a result due to Zenodorus that the sphere has greater volume than any regular solid with equal surface area. He also proves the related result that, for two regular solids with equal surface area, the one with the greater number of faces has the greater volume. Books VI and VII consider books of other authors (Theodosius, Autolycus, Aristarchus, Euclid, Apollonius, Aristaeus and Eratosthenes). Book VI deals with the books on astronomy which were collected into the Little Astronomy so-called in contrast to Ptolemy's Almagest or Greater Astronomy. As well as reviewing these works, Pappus points out errors which have somehow entered the texts. In Book VII Pappus writes about the Treasury of Analysis (see for example [3]):The so-called "Treasury of Analysis", my dear Hermodorus, is, in short, a special body of doctrine furnished for the use of those who, after going through the usual elements, wish to obtain power to solve problems set to then involving curves, and for this purpose only is it useful. It is the work of three men, Euclid the writer of the "Elements", Apollonius of Perga and Aristaeus the elder, and proceeds by the method of analysis and synthesis. Pappus then goes on to explain the different approaches of analysis and synthesis [3]:... in analysis we suppose that which is sought to be already done, and inquire what it is from which this comes about, and again what is the antecedent cause of the latter, and so on until, by retracing our steps, we light upon something already known or ranking as a first principle... But in synthesis, proceeding in the opposite way, we suppose to be already done that which was last reached in analysis, and arranging in their natural order as consequents what were formerly antecedents and linking them one with another, we finally arrive at the construction of what was sought... The article [13] is a wide ranging discussion of analysis and synthesis, taking this work by Pappus as a starting point. It is in Book VII that the 'Pappus problem' appears. This problem had a major impact on the development of geometry. It was discussed by Descartes and Newton and what is now known as Guldin's theorem is was proved by Pappus in Book VII of the Mathematical Collection. See [7] for a discussion of whether Guldin knew of Pappus's result when he published his work in 1640. In Book VIII Pappus deals with mechanics. We quote Pappus's own description of the subject (see for example [3]):The science of mechanics, my dear Hermodorus, has many important uses in practical life, and is held by philosophers to be worthy of the highest esteem, and is zealously studied by mathematicians, because it takes almost first place in dealing with the nature of the material elements of the universe. for it deals generally with the stability and movement of bodies about their centres of gravity, and their motions in space, inquiring not only into the causes

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pappus.html (3 of 6) [2/16/2002 11:25:43 PM]

Pappus

of those that move in virtue of their nature, but forcibly transferring others from their own places in a motion contrary to their nature; and it contrives to do this by using theorems appropriate to the subject matter. The whole work does not show a great deal of originality but it does show that Pappus has a deep understanding of a whole range of mathematical topics and that he had mastered all the major available mathematical techniques. He writes well, shows great clarity of thought and the Mathematical Collection is a work of very great historical importance in the study of Greek geometry. Of Pappus's commentary on Ptolemy's Almagest only the part on Books 5 and 6 has survived. We cannot be certain that Pappus wrote a commentary which extended to the whole 13 books, but it seems highly probable that he did. Certainly there is evidence that his commentary covered Books 1, 3 and 4 since traces exist or are quoted by other commentators on the Almagest. This commentary seems to be of much poorer quality to Pappus's geometrical work. Neugebauer [5] writes:.. the dullness and pomposity of these school treatises is only too evident. When Ptolemy in the chapter on the apparent diameter of the sun, moon and shadow simply remarks that the tangential cones in question contact the spheres within a negligible error in great circles, then Pappus refers to Euclid's "Optics" to show that the circle of contact has a smaller diameter than the sphere, only to add a lengthy argument to demonstrate that the error committed in Ptolemy's construction is nevertheless negligible. Or, when Ptolemy says that some phenomenon cannot take place, neither for the same clima nor for different geographical latitudes, Pappus feels obliged to explain "same clima" by "either in clima 3, or in 4, or in any other clima", and to illustrate "different" by referring to "Rome or Alexandria". Neugebauer also points out that, in addition to these pointless comments, there are also comments by Pappus which are simply incorrect. In case it might be thought that the quality of the Mathematical Collection and the commentary on Ptolemy's Almagest as of such different quality that Pappus may not have written both, then this is ruled out by his references which he makes in the Mathematical Collection (see for example [1]):... as Archimedes showed, and as is proved by us in the commentary on the first book of the ["Almagest"] by a theorem of our own. Of course Pappus did not write "Almagest" but the Greek title of the work. Other commentaries which Pappus wrote include one on Euclid's Elements. Proclus, in his own commentary on the Elements refers three times to Pappus's commentary and Eutocius also refers to Pappus's commentary. Part of Pappus's commentary may exist in an Arabic translation, namely that on Book X of the Elements. However, the commentary is very different in style to that of the Mathematical Collection and if indeed Pappus is the author it is a commentary which fails to show the depth of understanding that he shows in other parts of his work. Marinus claims that Pappus also wrote a commentary on Euclid's Data of which nothing has survived. That Pappus wrote on Geography is stated in the Suda and a work which claims to be written by Moses of Khoren in the fifth century seems to be largely based on Pappus's Geography. Moses writes (see for example [1]):We shall begin therefore after the Geography of Pappus of Alexandria, who followed the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pappus.html (4 of 6) [2/16/2002 11:25:43 PM]

Pappus

circle or special map of Claudius Ptolemy. Another reference to Pappus in this work states:Having spoken of geography in general, we shall now begin to explain each of the countries according to Pappus of Alexandria. Other works which could have been written by Pappus include one on music and one on hydrostatics. Certainly an instrument to measure liquids is attributed to him.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles)

A Quotation

Mathematicians born in the same country Some pages from publications

Title page of Collection (printed in 1588) The first page of Collection, Book 3 A page from Collection

Cross-references to History Topics

1. Squaring the circle 2. Trisecting an angle 3. Arabic mathematics : forgotten brilliance?

Cross-references to Famous Curves

1. spiral of Archimedes 2. quadratrix of Hippias

Other references in MacTutor

1. 2. 3. 4.

Pappus's theorem The focus and directrix of an ellipse Pappus's trisection method Chronology: 1AD to 500

Other Web sites

1. Vatican exhibition 2. Karen H Parshall 3. G Don Allen 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pappus.html (5 of 6) [2/16/2002 11:25:43 PM]

Pappus

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pappus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pappus.html (6 of 6) [2/16/2002 11:25:43 PM]

Paramesvara

Paramesvara Born: about 1370 in Alattur, Kerala, India Died: about 1460 in India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paramesvara was an Indian astronomer and mathematician who wrote many commentaries on earlier works as well as making many observations. Although his father has not been identified, we know that Paramesvara was born into a Namputiri Brahmana family who were astrologers and astronomers. The family home was Vatasseri (sometimes called Vatasreni) in the village of Alattur. This village was in Kerala and Paramesvara himself gives its coordinates with respect to Ujjain. This puts it at latitude 10 51' north. It is on the north bank of the river Nila at its mouth. From Paramesvara's writing we know that Rudra was his teacher, and Nilakantha, who knew Paramesvara personally, tells us that Paramesvara's teachers included Madhava and Narayana. We can be fairly confident that the dates we have given for Paramesvara are roughly correct since he made eclipe observations over a period of 55 years. We will say a little more about these observations below. He played an important part in the remarkable developments in mathematics which took place in Kerala in the late 14th and early part of the 15th century. The commentaries by Paramesvara on a number of works have been published. For example the Karmadipika is a commentary on the Mahabhaskariyam, an astronomical and mathematical work by Bhaskara I, and its text is given in [3]. In [2] the text of Paramesvara's commentary on the Laghubhaskariyam of Bhaskara I is given. Munjala wrote the astronomical work Laghumanasam in the year 932 and Paramesvara wrote a commentary (see [4]). It is a work containing typical topics for Indian mathematical astronomy works of this period: the mean motions of the heavenly bodies; the true motions of the heavenly bodies; miscellaneous mathematical rules; the systems of coordinates, direction, place and time; eclipses of the sun and the moon; and the operation for apparent longitude. Aryabhatya gave a rule for determining the height of a pole from the lengths of its shadows in the Aryabhatiya. Paramesvara gave several illustrative examples of the method in his commentary on the Aryabhatiya. Like many mathematicians from Kerala, Madhava clearly had a very strong influence on Paramesvara. One can see throughout his work that it is teachings by Madhava which direct much of Paramesvara's mathematical ideas. One of Paramesvara's most remarkable mathematical discoveries, no doubt influenced by Madhava, was a version of the mean value theorem. He states the theorem in his

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Paramesvara.html (1 of 3) [2/16/2002 11:25:44 PM]

Paramesvara

commentary Lilavati Bhasya on Bhaskara II's Lilavati. There are other examples of versions of the mean value theorem in Paramesvara's work which we now consider. The Siddhantadipika by Paramesvara is a commentary on the commentary of Govindasvami on Bhaskara I's Mahabhaskariya. Paramesvara gives some of his eclipse observations in this work including one made at Navaksetra in 1422 and two made at Gokarna in 1425 and 1430. This work also contains a mean value type formula for inverse interpolation of the sine. It presents a one-point iterative technique for calculating the sine of a given angle. In the Siddhantadipika Paramesvara also gives a more efficient approximation that works using a two-point iterative algorithm which turns out to be essentially the same as the modern secant method. See [8] and [9] for further details. The expression for the radius of the circle in which a cyclic quadrilateral is inscribed, given in terms of the sides of the quadrilateral, is usually attributed to Lhuilier in 1782. However Paramesvara described the rule 350 years earlier. If the sides of the cyclic quadrilateral are a, b, c and d then the radius r of the circumscribed circle was given by Paramesvara as: r2 = x/y where x = (ab + cd) (ac + bd) (ad + bc) and y = (a + b + c - d) (b + c + d - a) (c + d + a - b) (d + a + b - c). The original text by Paramesvara describing the rule is given in [7]. Paramesvara made a series of eclipse observations between 1393 and 1432 which we have referred to above. The last observation which we know he made was in 1445 but Nilakantha quotes a verse by Paramesvara in which he claims to have made observations spanning 55 years. The known observatons by Paramesvara do not quite square with this statement, there being a discrepancy of three years. Although we do not know when Paramesvara died we do know, again from Nilakantha, that the two knew each other personally. Since we have a definite date for Nilakantha's birth of 1444 it is hard to believe that Paramesvara died before 1460. Using his observations, Paramesvara made revisions of the planetary parameters and, like many other Indian astronomers, he constantly attempted to compare the theoretically computed positions of the planets with those which he actually observed. He was keen to improve the theoretical model to bring it into as close an agreement with observations as possible. Article by: J J O'Connor and E F Robertson List of References (10 books/articles) Mathematicians born in the same country Cross-references to History Topics

Previous

An overview of Indian mathematics

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Paramesvara.html (2 of 3) [2/16/2002 11:25:44 PM]

Paramesvara

Previous

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Paramesvara.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Paramesvara.html (3 of 3) [2/16/2002 11:25:44 PM]

Pars

Leopold Alexander Pars Born: 2 Jan 1896 in Whittlesford (8km south of Cambridge), Cambridgeshire, England Died: 28 Jan 1985 in Acton, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leopold Pars was educated at Latymer Upper School in Hammersmith (his family, who called him Leo, moved to London in 1899). In 1914 he won a scholarship to study mathematics and physics at Jesus College, Cambridge. Despite ill health which affected him throughout his undergraduate days, he produced outstanding results. Alan, as he was known as an undergraduate, then took a London University M.Sc. and, in 1921, he was the First Smith's Prizeman. His prize work was entitled Vector and Tensor Fields and was in two parts. Part 1 was Geometrical Vector Theory and the Restricted Principle of Relativity and Part 2 was On the General Theory of Relativity. On the strength of this work he was awarded a Fellowship at Jesus College, Cambridge in 1921 which he held for 64 years. Pars's first publications, influenced by Larmor and Eddington, were on relativity and were part of his prize essay. Thereafter he published little, although he did publish a valuable textbook Introduction to dynamics in 1953, until he published a monumental 650 page work Treatise on analytical dynamics in 1965. This book, published 4 years after he retired, was his life work in a single book. A colleague, writing after Pars's death, described the book in these terms:A Treatise on analytic dynamics is a massive, scholarly, systematic 650-page survey of the classical canon of mathematical dynamics. It is an exposition of enormous elegance, rigour and clarity, honed by decades of teaching undergraduate Tripos candidates and broadened by a perspective which encompassed all the major developments in dynamics from Galileo http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pars.html (1 of 2) [2/16/2002 11:25:46 PM]

Pars

to the mid-twentieth century. Within its context it is a pedagogical masterpiece. Pars's aim in writing the book is described in [1] as:... to give a compact, consistent and reasonably complete account of the subject as it then stood. He based his treatment on the theorem of Lagrange that he called the fundamental equation, which he proceeded to translate into six different forms, each exploited in appropriate contexts. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pars.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pars.html (2 of 2) [2/16/2002 11:25:46 PM]

Parseval

Marc-Antoine Parseval des Chênes Born: 27 April 1755 in Rosières-aux-Saline, France Died: 16 Aug 1836 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Very little is known of Antoine Parseval's life. We know that he was born into a family of high standing in France and he describes himself as a squire, certainly suggesting that his family were wealthy land-owners. One of the few pieces of information which exists is that he married Ursule Guerillot in 1795. The marriage certainly did not last long and the pair were soon divorced. The starting point of the historical events which were to play a major role in Parseval' life was the storming of the Bastille on 14 July 1789. From this point the monarchy of Louis 16th was in major difficulties as the majority of Frenchmen put aside their differences and united behind an attempt to destroy the privileged establishment of the church and the state. Parseval, perhaps not surprisingly since he was of noble birth, was a royalist so for him the increasing problems for the monarchy meant that his life was more and more in danger. In 1792 Louis 16th gave up attempts at a compromise with opponents of the monarchy and tried to flee from France. He did not make it but was arrested and brought back to Paris. It was a time of great danger for royalist supporters and indeed it proved so for Parseval who was imprisoned in 1792. Following the execution of the King on 21 January 1793 there followed a reign of terror with many political trials. By the end of 1793 there were 4595 political prisoners held in Paris. However France began to have better times as their armies, under the command of Napoleon Bonaparte, won victory after victory. This may have been good news for France in general but royalists like Parseval, despite being freed from prison, remained in fear of their lives. Napoleon became 1st Consul in 1800 and then Emperor in 1804. Parseval should have kept his head down if he wanted at avoid trouble but it was a time when it was almost impossible not to get drawn into the political events. Parseval published poetry against Napoleon's regime and, not surprisingly, had to flee from France when Napoleon ordered his arrest. He was successful in avoiding arrest and was able to return to Paris. Parseval had only five publications, all presented to the Académie des Sciences. The first was Mémoire sur la résolution des équations aux différences partielle linéaires du second ordre dated 5 May 1798, the second was Mémoire sur les séries et sur l'intégration complète d'une équation aux differences partielle linéaires du second ordre, à coefficiens constans dated 5 April 1799, the third was Ingégration générale et complète des équations de la propogation du son, l'air étant considéré avec les trois dimensions dated http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Parseval.html (1 of 3) [2/16/2002 11:25:47 PM]

Parseval

5 July 1801, the fourth was Ingégration générale et complète de deux équations importantes dans la méchanique des fluides dated 16 August 1803, and finally Méthode générale pour sommer, par le moyen des intégrales définies, la suite donnée par le théorème de M Lagrange, au moyen de laquelle il trouve une valeur qui satisfait à une équation algébrique ou transcendente dated 7 May 1804. It was the second of these, dated 5 April 1799, which contains the result known today as Parseval's theorem. Today this theorem is seen in the context of Fourier series, and often also in more abstract settings which are quite far removed from Parseval's original ideas. The original theorem was concerned with summing infinite series. Parseval thought the result was obvious and only remarked that it followed by using de Moivre's result for (cos x + i sin x)n. It also only worked, he noted, when certain imaginary parts of two complex numbers cancelled out. This he reasonably suggested was unfortunate and he hoped to remove this problem later. Indeed he did remove the problem and added a note to this effect in his 1801 publication. The improved version, as given in 1801, states that if two series M=

anxn and m =

bnxn

are given then, substituting x = cos u + i sin u, and separating the answers into real and imaginary parts M = p+iq, m = r+is, then 2a1b1 + ab + a3b3 + ab + ... = (2/ ) pr du where the integral is taken from 0 to . Of course we have modernised the notation, for example subscript notation was not used in Parseval's time, and we have also corrected his theorem for he omitted the first 2 on the left hand side. The error may well have been a typographical error in printing the article. Parseval's result was not published until his five papers were all published by the Académie des Sciences in 1806. Before that it was known by members of the Academy and appeared in works by Lacroix and Poisson before Parseval's papers were printed. Parseval was never honoured with election to the Académie des Sciences. He was proposed on five separate occasions, namely in 1796, 1799, 1802, 1813 and 1828. He was never particularly close although he did come third in 1799, the year that Lacroix was elected. It would not be unfair to say that Parseval has fared well in having a well known result, which is quite far removed from his contribution, named after him. However he remains a somewhat shadowy figure and it is hoped that research will one day provide a better understanding of his life and achievements. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other Web sites

Parseval family

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Parseval.html (2 of 3) [2/16/2002 11:25:47 PM]

Parseval

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Parseval.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Parseval.html (3 of 3) [2/16/2002 11:25:47 PM]

Pascal

Blaise Pascal Born: 19 June 1623 in Clermont (now Clermont-Ferrand), Auvergne, France Died: 19 Aug 1662 in Paris, France

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Blaise Pascal was the third of Etienne Pascal's children and his only son. Blaise's mother died when he was only three years old. In 1632 the Pascal family, Etienne and his four children, left Clermont and settled in Paris. Blaise Pascal's father had unorthodox educational views and decided to teach his son himself. Etienne Pascal decided that Blaise was not to study mathematics before the age of 15 and all mathematics texts were removed from their house. Blaise however, his curiosity raised by this, started to work on geometry himself at the age of 12. He discovered that the sum of the angles of a triangle are two right angles and, when his father found out, he relented and allowed Blaise a copy of Euclid. At the age of 14 Blaise Pascal started to accompany his father to Mersenne's meetings. Mersenne belonged to the religious order of the Minims, and his cell in Paris was a frequent meeting place for Gassendi, Roberval, Carcavi, Auzout, Mydorge, Mylon, Desargues and others. Soon, certainly by the time he was 15, Blaise came to admire the work of Desargues. At the age of sixteen, Pascal presented a single piece of paper to one of Mersenne's meetings in June 1639. It contained a number of projective geometry theorems, including Pascal's mystic hexagon. In December 1639 the Pascal family left Paris to live in Rouen where Etienne had been appointed as a tax collector for Upper Normandy. Shortly after settling in Rouen, Blaise had his first work, Essay on Conic Sections published in February 1640. Pascal invented the first digital calculator to help his father with his work collecting taxes. He worked on

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html (1 of 5) [2/16/2002 11:25:50 PM]

Pascal

it for three years between 1642 and 1645. The device, called the Pascaline, resembled a mechanical calculator of the 1940's. This, almost certainly, makes Pascal the second person to invent a mechanical calculator for Schickard had manufactured one in 1624. There were problems faced by Pascal in the design of the calculator which were due to the design of the French currency at that time. There were 20 sols in a livre and 12 deniers in a sol. The system remained in France until 1799 but in Britain a system with similar multiples lasted until 1971. Pascal had to solve much harder technical problems to work with this division of the livre into 240 than he would have had if the division had been 100. However production of the machines started in 1642 but, as Adamson writes in [3], By 1652 fifty prototypes had been produced, but few machines were sold, and manufacture of Pascal's arithmetical calculator ceased in that year. Events of 1646 were very significant for the young Pascal. In that year his father injured his leg and had to recuperate in his house. He was looked after by two young brothers from a religious movement just outside Rouen. They had a profound effect on the young Pascal and he became deeply religious. From about this time Pascal began a series of experiments on atmospheric pressure. By 1647 he had proved to his satisfaction that a vacuum existed. Descartes visited Pascal on 23 September. His visit only lasted two days and the two argued about the vacuum which Descartes did not believe in. Descartes wrote, rather cruelly, in a letter to Huygens after this visit that Pascal ...has too much vacuum in his head. In August of 1648 Pascal observed that the pressure of the atmosphere decreases with height and deduced that a vacuum existed above the atmosphere. Descartes wrote to Carcavi in June 1647 about Pascal's experiments saying:It was I who two years ago advised him to do it, for although I have not performed it myself, I did not doubt of its success ... In October 1647 Pascal wrote New Experiments Concerning Vacuums which led to disputes with a number of scientists who, like Descartes, did not believe in a vacuum. Etienne Pascal died in September 1651 and following this Blaise wrote to one of his sisters giving a deeply Christian meaning to death in general and his father's death in particular. His ideas here were to form the basis for his later philosophical work Pensées. From May 1653 Pascal worked on mathematics and physics writing Treatise on the Equilibrium of Liquids (1653) in which he explains Pascal's law of pressure. Adamson writes in [3]:This treatise is a complete outline of a system of hydrostatics, the first in the history of science, it embodies his most distinctive and important contribution to physical theory. He worked on conic sections and produced important theorems in projective geometry. In The Generation of Conic Sections (mostly completed by March 1648 but worked on again in 1653 and 1654) Pascal considered conics generated by central projection of a circle. This was meant to be the first part of a treatise on conics which Pascal never completed. The work is now lost but Leibniz and Tschirnhaus made notes from it and it is through these notes that a fairly complete picture of the work is now possible.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html (2 of 5) [2/16/2002 11:25:50 PM]

Pascal

Although Pascal was not the first to study the Pascal triangle, his work on the topic in Treatise on the Arithmetical Triangle was the most important on this topic and, through the work of Wallis, Pascal's work on the binomial coefficients was to lead Newton to his discovery of the general binomial theorem for fractional and negative powers. In correspondence with Fermat he laid the foundation for the theory of probability. This correspondence consisted of five letters and occurred in the summer of 1654. They considered the dice problem, already studied by Cardan, and the problem of points also considered by Cardan and, around the same time, Pacioli and Tartaglia. The dice problem asks how many times one must throw a pair of dice before one expects a double six while the problem of points asks how to divide the stakes if a game of dice is incomplete. They solved the problem of points for a two player game but did not develop powerful enough mathematical methods to solve it for three or more players. Through the period of this correspondence Pascal was unwell. In one of the letters to Fermat written in July 1654 he writes ... though I am still bedridden, I must tell you that yesterday evening I was given your letter. However, despite his health problems, he worked intensely on scientific and mathematical questions until October 1654. Sometime around then he nearly lost his life in an accident. The horses pulling his carriage bolted and the carriage was left hanging over a bridge above the river Seine. Although he was rescued without any physical injury, it does appear that he was much affected psychologically. Not long after he underwent another religious experience, on 23 November 1654, and he pledged his life to Christianity. After this time Pascal made visits to the Jansenist monastery Port-Royal des Champs about 30 km south west of Paris. He began to publish anonymous works on religious topics, eighteen Provincial Letters being published during 1656 and early 1657. These were written in defence of his friend Antoine Arnauld, an opponent of the Jesuits and a defender of Jansenism, who was on trial before the faculty of theology in Paris for his controversial religious works. Pascal's most famous work in philosophy is Pensées, a collection of personal thoughts on human suffering and faith in God which he began in late 1656 and continued to work on during 1657 and 1658. This work contains 'Pascal's wager' which claims to prove that belief in God is rational with the following argument. If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing. With 'Pascal's wager' he uses probabilistic and mathematical arguments but his main conclusion is that ...we are compelled to gamble... His last work was on the cycloid, the curve traced by a point on the circumference of a rolling circle. In 1658 Pascal started to think about mathematical problems again as he lay awake at night unable to sleep for pain. He applied Cavalieri's calculus of indivisibles to the problem of the area of any segment of the cycloid and the centre of gravity of any segment. He also solved the problems of the volume and surface area of the solid of revolution formed by rotating the cycloid about the x-axis. Pascal published a challenge offering two prizes for solutions to these problems to Wren, Laloubère, Leibniz, Huygens, Wallis, Fermat and several other mathematicians. Wallis and Laloubère entered the competition but Laloubère's solution was wrong and Wallis was also not successful. Sluze, Ricci,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html (3 of 5) [2/16/2002 11:25:50 PM]

Pascal

Huygens, Wren and Fermat all communicated their discoveries to Pascal without entering the competition. Wren had been working on Pascal's challenge and he in turn challenged Pascal, Fermat and Roberval to find the arc length, the length of the arch, of the cycloid. Pascal published his own solutions to his challenge problems in the Letters to Carcavi. After that time on he took little interest in science and spent his last years giving to the poor and going from church to church in Paris attending one religious service after another. Pascal died at the age of 39 in intense pain after a malignant growth in his stomach spread to the brain. He is described in [3] as:... a man of slight build with a loud voice and somewhat overbearing manner. ... he lived most of his adult life in great pain. He had always been in delicate health, suffering even in his youth from migraine ... His character is described as:... precocious, stubbornly persevering, a perfectionist, pugnacious to the point of bullying ruthlessness yet seeking to be meek and humble ... In [1] the following assessment is given:At once a physicist, a mathematician, an eloquent publicist in the Provinciales ... Pascal was embarrassed by the very abundance of his talents. It has been suggested that it was his too concrete turn of mind that prevented his discovering the infinitesimal calculus, and in some of the Provinciales the mysterious relations of human beings with God are treated as if they were a geometrical problem. But these considerations are far outweighed by the profit that he drew from the multiplicity of his gifts, his religious writings are rigorous because of his scientific training... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (53 books/articles)

Some Quotations (44)

A Poster of Blaise Pascal

Mathematicians born in the same country

Cross-references to History Topics

1. An overview of the history of mathematics 2. Jaina mathematics

Cross-references to Famous Curves

1. parabola 2. Pearls of Sluze

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html (4 of 5) [2/16/2002 11:25:50 PM]

Pascal

Other references in MacTutor

1. 2. 3. 4.

Pascal's Mystic Hexagram Pascal's calculator: the Pascaline Another picture of it Chronology: 1625 to 1650

5. Chronology: 1650 to 1675 Honours awarded to Blaise Pascal (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Pascal

Paris street names

Rue Pascal (3rd and 5th Arrondissements)

Other Web sites

1. The Galileo Project 2. Rouse Ball 3. Fourmilab (Pascal's Calculator) 4. Yahoo (More links) 5. Notre Dame 6. The Catholic Encyclopedia 7. Simon Fraser University 8. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pascal.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html (5 of 5) [2/16/2002 11:25:50 PM]

Pascal_Etienne

Etienne Pascal Born: 2 May 1588 in Clermont (now Clermont-Ferrand), Auvergne, France Died: 24 Sept 1651 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Etienne Pascal was the father of Blaise Pascal. Etienne was trained in law in Paris, receiving his degree in 1610. He returned to his home town of Clermont were he bought the post of counsellor for Bas-Auvergne, the region around Clermont. He came from a wealthy family so had ample money without earning a living. In 1631 he went to Paris so that his son could have the best education and he devoted himself the Blaise's education there. In 1634 he was appointed to a committee set up by Cardinal Richelieu to judge whether Morin's scheme for determining longitude from the Moon's motion was practical. From about this time he became involved with Mersenne's meetings. Through these he collaborated with Roberval, Desargues and Mydorge. Etienne also held a number of government appointments and a court official. Etienne is famed as the discoverer of the curve the Limaçon of Pascal. The curve, so named by Roberval, can be used to trisect an angle. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Cross-references to History Topics

Longitude and the Académie Royale

Cross-references to Famous Curves

Limaçon of Pascal

Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pascal_Etienne.html (1 of 2) [2/16/2002 11:25:51 PM]

Pascal_Etienne

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pascal_Etienne.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pascal_Etienne.html (2 of 2) [2/16/2002 11:25:51 PM]

Pasch

Moritz Pasch Born: 8 Nov 1843 in Breslau, Germany (now Wroclaw, Poland) Died: 20 Sept 1930 in Bad Homburg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Moritz Pasch studied in Berlin and then taught in Giessen. He worked on the foundations of geometry. He found a number of assumptions in Euclid that nobody had noticed before. Pasch's Axiom is that if a line enters a triangle ABC through the side AB and does not pass through C then it must leave the triangle either between B and C or between C and A. Pasch argued in 1882 that geometers rely too heavily on physical intuition. In his view an argument in mathematics should not depend on the physical interpretation of the terms involved but upon purely formal axioms. Pash claimed that the principle of duality contradicted physical intuition about points and lines, nobody believed that these terms were interchangeable. Hilbert was to be influenced by these ideas of Pasch. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) A Poster of Moritz Pasch

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pasch.html (1 of 2) [2/16/2002 11:25:53 PM]

Pasch

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pasch.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pasch.html (2 of 2) [2/16/2002 11:25:53 PM]

Patodi

Vijay Kumar Patodi Born: 12 March 1945 in Guna, Madhya Pradesh, India Died: 21 Dec 1976 in Bombay, India

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vijay Patodi attended the Government Higher Secondary School in Guna before entering Vikram University in Ujjain. After obtaining his B.Sc. degree from Vikram University, Patodi moved to Banares Hindu University were he studied for his Master's Degree in Mathematics. This was awarded in 1966 and Patodi then spent a year at the Centre for Advanced Study at the University of Bombay. In 1967 Patodi joined the School of Mathematics of the Tata Institute of Fundamental Research in Bombay and he was to remain on the staff there until his death at the distressingly early age of 31. Mathematical fame for Patodi came early in his career with papers of great importance coming for the work of his Ph.D. His doctoral thesis, Heat equation and the index of elliptic operators, was supervised by M S Narasimhan and S Ramanan and the degree was awarded by the University of Bombay in 1971. Patodi's first paper Curvature and the eigenforms of the Laplace operator was part of his thesis and the contents of this paper are described in [2]:An analytic approach, via the heat equation yields easily a formula for the index of an elliptic operator on a compact manifold: but, the formula involves an integrand containing too many derivatives of the symbol, while from the Atiyah-Singer index theorem one would expect only two derivatives to figure. ... Patodi's first contribution was to prove that such a fantastic cancellation of higher derivatives did indeed take place. The second paper which came from his thesis was An analytic proof of the Riemann- Roch- Hirzebruch theorem for Kaehler manifolds which extended the methods of his first paper to a much more complicated situation. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Patodi.html (1 of 2) [2/16/2002 11:25:54 PM]

Patodi

The years 1971 to 1973 were ones which Patodi spent on leave at the Institute for Advanced Study at Princeton. There he worked with M F Atiyah and made several visits to work with others in his field at various centres in the United States and England. During this time he also collaborated with R Bott and I M Singer. On his return to Bombay and the Tata Institute in 1973 Patodi was promoted to associate professor. He was promoted to full professor in 1976 but by this time his health was very poor. He had in fact had to overcome health problems for most of his career, making his achievements the more remarkable. Patodi's publications, in addition to the two mentioned above, include a number of joint ones with Atiyah and Singer. These papers introduce a spectral invariant of a compact Riemannian manifold. In another paper he studies the relationship between Riemannian structures and triangulations. Other work gives a combinatorial formula for Pontryagin classes. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Patodi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Patodi.html (2 of 2) [2/16/2002 11:25:54 PM]

Pauli

Wolfgang Ernst Pauli Born: 25 April 1900 in Vienna, Austria Died: 15 Dec 1958 in Zurich, Switzerland

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wolfgang Pauli was the son of Wolfgang Joseph and Berta Camilla Schütz. Wolfgang Joseph had trained as a medical doctor in Prague. After qualifying, he practiced as a doctor in Vienna and quickly became popular. In 1898 he changed his name to Wolfgang Joseph Pauli and, in the following year, converted from Judaism to become a Roman Catholic. He married Berta Schütz in May 1899 but by this time he had given up his medical practice for research in chemistry and physics, becoming a university professor. Wolfgang Joseph had been inspired to study science by Ernst Mach, and when his first child was born he named him Wolfgang Ernst Pauli, giving him the middle name of Ernst in honour of Mach. Not only did Pauli's middle name come from Mach, but Mach was also his godfather giving him a silver cup when he was christened on 31 May 1900. Wolfgang attended school in Vienna where he began a deep study of mathematics and physics at the Döblingen Gymnasium. He was certainly not a typical pupil for he read Einstein's papers on relativity while he was still at the Gymnasium. School work was boring to the brilliant Pauli and he hid Einstein's papers under his school desk and studied them during the lessons. Not paying attention in class did not hold Pauli back, for he graduated from the Gymnasium in July 1918 with distinction. After leaving the Gymnasium he entered the Ludwig-Maximilian university of Munich. Within two months of leaving school he had submitted his first paper on the theory of relativity. While still an undergraduate at Munich he wrote two further articles on the theory of relativity. At Munich, Pauli was taught by Sommerfeld who quickly recognised his genius. Sommerfeld asked Pauli to write a review

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html (1 of 6) [2/16/2002 11:25:56 PM]

Pauli

article on relativity for the Encyclopädie der mathematischen Wissenschaften when he had only been two years at university, a mark of the high regard in which he held Pauli. The respect was mutual, for Pauli showed more respect for Sommerfeld both as a person and a scientist than he did for any other. Pauli, writing about his days as a student at Munich, wrote (see the extracts from Pauli's Nobel Prize lecture in 1945 given in [16]):I was not spared the shock which every physicist accustomed to the classical way of thinking experienced when he came to know Bohr's basic postulate of quantum theory for the first time. He wrote his first paper on quantum physics in June 1920, a work on the magnetic properties of matter. The year 1920 was when Heisenberg arrived in Munich, also to become a student of Sommerfeld. In [20] Pais quotes from Heisenberg's description of Pauli's way of life at this time:Wolfgang was a typical night bird. He preferred the town, liked to spend evenings in some café, and would thereafter work on his physics with great intensity and great success. To Sommerfeld's dismay he would therefore rarely attend morning lectures and would not turn up until about noon. Pauli received his doctorate, which had been supervised by Sommerfeld, in July 1921 for a thesis on the quantum theory of ionised molecular hydrogen. In his report on the thesis Sommerfeld wrote that it showed:... like his many already published smaller investigations and his larger encyclopedia article, the full command of the tools of mathematical physics. Sommerfeld was certainly right to heap much praise on the thesis but it had been a disappointment to Pauli since the theoretical results he had proved did not agree with experimental evidence. Looking at it now one can see that it showed that quantum theory, as then formulated, was not in itself going to provide the necessary structure on which to build a logical theory of atomic structure which agreed with experimental evidence. Two months after the award of his doctorate Pauli's survey of the theory of relativity appeared, by this time having grown into a work of 237 pages. His genius was immediately recognised by Einstein who, after reading Pauli's monograph on relativity, wrote a review [20]:Whoever studies this mature and grandly conceived work might not believe that its author is a twenty-one year old man. One wonders what to admire most, the psychological understanding for the development of ideas, the sureness of mathematical deduction, the profound physical insight, the capacity for lucid, systematical presentation, the knowledge of the literature, the complete treatment of the subject matter, or the sureness of critical appraisal. Pauli was then appointed to Göttingen as Born's assistant from October 1921. It was in Göttingen that he first met Niels Bohr in person and he said (see for example [16]):... a new phase of my scientific life began when I met Niels Bohr personally for the first time. This was in 1922, when he gave a series of guest lectures at Göttingen when he reported on his theoretical investigations on the periodic system of elements. During these meetings, Bohr asked me whether I could come to Copenhagen for a year. Pauli eagerly accepted the invitation and spent the year 1922-23 at Bohr's Institute [16]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html (2 of 6) [2/16/2002 11:25:56 PM]

Pauli

Following Bohr's invitation, I went to Copenhagen in the autumn of 1922, where I made a serious effort to explain the so-called 'anomalous Zeeman effect', ... a type of splitting of the spectral lines in a magnetic field which is different from the normal triplet. In 1923, Pauli was appointed a privatdozent at Hamburg [16]:Very soon after my return to the university of Hamburg, in 1923, I gave there my inaugural lecture as privatdozent on the periodic system of elements. The contents of the lecture appeared very unsatisfactory to me, since the problem of the closing of the electronic shells had been clarified no further. In 1924 Pauli proposed a quantum spin number for electrons. He is best known for the Pauli exclusion principle , proposed in 1925, which states that no two electrons in an atom can have the same four quantum numbers. Less than a year after this Heisenberg submitted his article on quantum mechanics which was to change the whole approach to the topic. Pauli, who before that had begun to feel that further advances could not be made with the theory as it then existed, quickly made progress using Heisenberg's new ideas and before the end of 1925 he had derived the hydrogen spectrum from the new theory. The year 1927 saw personal tragedy for Pauli when his mother, to whom he had been very close, committed suicide. In the following year his father remarried making an even more unhappy situation for Pauli who referred to his father's new wife as "the evil step-mother". On 6 May 1929 Pauli left the Roman Catholic Church, but his reasons for this are not entirely clear. Further unhappiness was to follow when he married Käthe Margarethe Deppner in Berlin on 23 December 1929. The marriage was never a success even in the first few months and they were divorced in Vienna on 29 November 1930. Despite the personal problem, Pauli's career progressed well. In 1928 he was appointed Professor of Theoretical Physics at the Federal Institute of Technology in Zurich and soon made some remarkable progress. He predicted mathematically, in 1931, that conservation laws required the existence of a new particle which he proposed to call the "neutron". He first mentioned his theoretical evidence for this particle in a letter written on 4 December 1930 and his public announcement came at a conference in Pasadena on 16 June 1931. The New York Times of 17 June reported:A new inhabitant of the heart of the atom was introduced to the world of physics today when Dr W Pauli of the Institute of Technology in Zurich, Switzerland, postulated the existence of particles or entities which he christened "neutrons". The existence and properties of the particle were still not clear to Pauli, however, and it was not until 1933 that he published his prediction in print. At that time he made the claim, for the first time, that the particle had zero mass. The particle which we now know as the neutron had been discovered by Chadwick in 1932. Pauli's particle was named the neutrino by Fermi in 1934 and at that time he correctly stated that it was not a constituent of the nucleus of an atom. It was later found experimentally. This period of scientific discovery by Pauli coincided with a period of increasing personal difficulties for him. Perhaps as a consequence of his disastrous marriage, he began drinking and as a result consulted the psychologist Carl Gustave Jung. He was not treated by Jung, rather one of his assistants helped Pauli. However, Pauli detailed over 1000 dreams which he sent to Jung over many years and Jung published work based on some of the dreams. Pauli clearly believed in psychology as much as he did physics. He wrote later in his life in a letter to Pais (see for example [20]):It is my personal opinion that in the science of the future reality will neither be "psychic" http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html (3 of 6) [2/16/2002 11:25:56 PM]

Pauli

nor "physical" but somehow both and somehow neither. Things went better for Pauli after he married Franciska Bertram on 4 April 1934. In contrast with his first disastrous marriage his second marriage proved a great support to him. After his death Franciska Pauli said this of her late husband:He was very easily hurt and therefore would let down a curtain. He tried to live without admitting reality. And his unworldliness stemmed precisely from his belief that this was possible. In 1931 Pauli was visiting Professor at the University of Michigan, then in 1935-1936 he was visiting Professor at the Institute for Advanced Study, Princeton. He returned to Zurich but after the Second World War broke out in 1939 he found himself in an awkward situation since Germany having annexed Austria in 1938 had made him a German citizen. In 1940 he was greatly relieved to receive an offer from Princeton and he was appointed to the chair of theoretical physics there, spending 1941 as visiting Professor at the University of Michigan, and 1942 as visiting Professor at Purdue University. Pauli worried that fascism might bring about the end of scientific life in Europe. For this reason he actively encouraged scientific developments in the United States and also in the Soviet Union. He was keen to participate in conferences in the Soviet Union, attending the All-Union physics conference in Odessa in 1939 and the All-Union physics conference in Moscow in 1937. Pauli also tried to encourage those scientists who could remain in Italy and Germany to do so, for he believed this might ensure that scientific culture survived after the War. Pauli did not remain in the United States but he returned to Zurich after World War II. It was not an easy decision for him but basically he always felt European and never quite felt that he fitted in the United States. Pauli was awarded the Nobel Prize in 1945 for his:... decisive contribution through his discovery in 1925 of a new law of Nature, the exclusion principle or Pauli principle. He had been nominated for the prize by Einstein. He did not go to Stockholm for the prize ceremony in 1945 but there was special ceremony at Princeton for him on 10 December. In Stockholm Professor I Waller delivered a presentation speech in Pauli's absence. He explained the importance of the exclusion principle:Pauli based his investigation on a profound analysis of the experimental and theoretical knowledge in atomic physics at the time. He found that four quantum numbers are in general needed in order to define the energy state of an electron. He then pronounced his principle, which can be expressed by saying that there cannot be more than one electron in each energy state when this state is completely defined. Three quantum numbers only can be related to the revolution of the electron round the nucleus. The necessity of a fourth quantum number proved the existence of interesting properties of the electron. Other physicists found that these properties may be interpreted by stating that the electron has a "spin", i.e. that it behaves to some extent as if it were rapidly rotating round an axis through its centre of gravity. Pauli showed himself that the electronic configuration is made fully intelligible by the exclusion principle, which is therefore essential for the elucidation of the characteristic physical and chemical properties of different elements. Among those important phenomena for the explanation of which the Pauli principle is indispensable, we mention the electric conductivity of metals and the magnetic properties of matter. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html (4 of 6) [2/16/2002 11:25:56 PM]

Pauli

In 1925 and 1926 essential progress of another kind was made in the quantum theory, which is the foundation of atomic physics. New and revolutionary methods were developed for the description of the motion of particles. The spin proposal, which gave meaning to Pauli's fourth quantum number, which first suggested by Uhlenbeck in 1925. Pauli delived his Nobel Lecture in Stockholm on 13 December in the following year. In [19] Laurikainen writes about other directions which Pauli's work took him in the years following World War II:During the last 10-15 years of his life, Pauli spent much time studying the history and philosophy of science. His starting point was the philosophy of quantum mechanics, but this led him to psychology, the history of ideas and many other fields, not least the relation of religion to natural science. Pauli received many honours for his work in addition to the Nobel Prize. He was elected a Fellow of the Royal Society of London in 1953 and he was also elected a member of the Swiss Physical Society, the American Physical Society and the American Association for the Advancement of Science. He was awarded the Lorentz Medal in Amsterdam in October 1931.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles)

Some Quotations (3)

Mathematicians born in the same country Cross-references to History Topics

The quantum age begins

Honours awarded to Wolfgang Pauli (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1945

Fellow of the Royal Society

Elected 1953

Lunar features

Crater Pauli

Other Web sites

1. Nobel prizes site (A biography of Pauli and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html (5 of 6) [2/16/2002 11:25:56 PM]

Pauli

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Famous curves

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Pauli.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html (6 of 6) [2/16/2002 11:25:57 PM]

Peacock

George Peacock Born: 9 April 1791 in Denton, England Died: 8 Nov 1858 in Ely, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Peacock was educated at home by his father until he was 17 years old, then he attended a school in Richmond, Yorkshire (one of the nearest towns to Denton) to prepare for entering Cambridge. In 1809 he became a student at Trinity College, Cambridge. As an undergraduate at Cambridge he made friends with John Herschel and Charles Babbage. In 1812 he graduated, placed second to John Herschel in the examinations. He also won the second Smith's prize. In 1814 Peacock was awarded a fellowship and, in the following year, he became a tutor and lecturer in Trinity College. While undergraduates Peacock, Herschel and Babbage planned to bring reforms to Cambridge and, in 1815, they formed the Analytical Society whose aims were to bring the advanced continental methods of calculus to Cambridge. In 1816 the Analytical Society produced a translation of a book of Lacroix in the differential and integral calculus. The following year Peacock became an examiner and used his position to further his reforms. He wrote to one of his friends saying I assure you that I shall never cease to exert myself to the utmost in the cause of reform, and that I will never decline any office which may increase my power to effect it. ... It is by silent perseverance only, that we can hope to reduce the many-headed monster of prejudice and make the University answer her character as the loving mother of good learning and science. Peacock published Collection of Examples of the Application of the Differential and Integral Calculus in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peacock.html (1 of 3) [2/16/2002 11:25:58 PM]

Peacock

1820, a publication which sold well and helped further the aims of the Analytical Society. In 1830 he published Treatise on Algebra which attempted to give algebra a logical treatment comparable to Euclid's Elements. He has two types of algebra, arithmetical algebra and symbolic algebra. In the book he describes symbolic algebra as the science which treats the combinations of arbitrary signs and symbols by means defined through arbitrary laws. He also said We may assume any laws for the combination and incorporation of such symbols, so long as our assumptions are independent, and therefore not inconsistent with each other. Peacock extended the rules of arithmetic using what he called the principle of the permanence of equivalent forms to give his symbolic algebra, so he was not as bold in practice as the abstract ideas for symbolic algebra which he gives in theory. He investigated the basic properties of numbers, such as the distributive property, that underlie the subject of algebra. In 1831 the British Association for the Advancement of Science was set up. One of its first aims was to obtain reports on the state and progress of various sciences from leaders in their fields. Hamilton was asked to prepare a report on mathematics but he declined. Peacock was then asked and he accepted although he restricted his report to Algebra, Trigonometry and the Arithmetic of Sines. He read his report at the 1833 meeting of the Association in Cambridge and the report was subsequently printed. In 1836 he was appointed Lowndean professor of astronomy and geometry at Cambridge and three years later was appointed dean of Ely cathedral, spending the last 20 years of his life there. Peacock was a reformer for his whole life. He worked hard to reform the statutes of Cambridge University and, when the Government set up a Commission to propose reforms, he was appointed to it. Although he attended meetings of the Commission, he died before the report was finished. Article by: J J O'Connor and E F Robertson List of References (7 books/articles) A Poster of George Peacock Other references in MacTutor

Mathematicians born in the same country 1. Chronology: 1810 to 1820 2. Chronology: 1830 to 1840

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peacock.html (2 of 3) [2/16/2002 11:25:58 PM]

Peacock

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Peacock.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peacock.html (3 of 3) [2/16/2002 11:25:58 PM]

Peano

Giuseppe Peano Born: 27 Aug 1858 in Cuneo, Piemonte, Italy Died: 20 April 1932 in Turin, Italy

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giuseppe Peano's parents worked on a farm and Giuseppe was born in the farmhouse 'Tetto Galant' about 5 km from Cuneo. He attended the village school in Spinetta then he moved up to the school in Cuneo, making the 5km journey there and back on foot every day. His parents bought a house in Cuneo but his father continued to work the fields at Tetto Galant with the help of a brother and sister of Giuseppe, while his mother stayed in Cuneo with Giuseppe and his older brother. Giuseppe's mother had a brother who was a priest and lawyer in Turin and, when he realised that Giuseppe was a very talented child, he took him to Turin in 1870 for his secondary schooling and to prepare him for university studies. Giuseppe took exams at Ginnasio Cavour in 1873 and then was a pupil at Liceo Cavour from where he graduated in 1876 and, in that year, he entered the University of Turin. Among Peano's teachers in his first year at the University of Turin was D'Ovidio who taught him analytic geometry and algebra. In his second year he was taught calculus by Angelo Genocchi and descriptive geometry by Giuseppe Bruno. Peano continued to study pure mathematics in his third year and found that he was the only student to do so. The others had continued their studies at the Engineering School which Peano himself had originally intended to do. In his third year Francesco Faà di Bruno taught him analysis and D'Ovidio taught geometry. Among his teachers in his final year were again D'Ovidio with a further geometry course and Francesco Siacci with a mechanics course. On 29 September 1880 Peano graduated as doctor of mathematics. Peano joined the staff at the University of Turin in 1880, being appointed as assistant to D'Ovidio. He http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peano.html (1 of 6) [2/16/2002 11:26:00 PM]

Peano

published his first mathematical paper in 1880 and a further three papers the following year. Peano was appointed assistant to Genocchi for 1881-82 and it was in 1882 that Peano made a discovery that would be typical of his style for many years, he discovered an error in a standard definition. Genocchi was by this time quite old and in relatively poor health and Peano took over some of his teaching. Peano was about to teach the students about the area of a curved surface when he realised that the definition in Serret's book, which was the standard text for the course, was incorrect. Peano immediately told Genocchi of his discovery to be told that Genocchi already knew. Genocchi had been informed the previous year by Schwarz who seems to have been the first to find Serret's error. In 1884 there was published a text based on Genocchi's lectures at Turin. This book Course in Infinitesimal Calculus although based on Genocchi's lectures was edited by Peano and indeed it has much in it written by Peano himself. The book itself states on the title page that it is:... published with additions by Dr Giuseppe Peano. Genocchi seemed somewhat unhappy that the work came out under his name for he wrote:... the volume contains important additions, some modifications, and various annotations, which are placed first. So that nothing will be attributed to me which is not mine, I must declare that I have had no part in the compilation of the aforementioned book and that everything is due to that outstanding young man Dr Giuseppe Peano ... Peano received his qualification to be a university professor in December 1884 and he continued to teach further courses, some for Genocchi whose health had not recovered sufficiently to allow him to return to the University. In 1886 Peano proved that if f(x,y) is continuous then the first order differential equation dy/dx = f(x, y) has a solution. The existence of solutions with stronger hypothesis on f had been given earlier by Cauchy and then Lipschitz. Four years later Peano showed that the solutions were not unique, giving as an example the differential equation dy/dx=3y2/3, with y(0) = 0. In addition to his teaching at the University of Turin, Peano began lecturing at the Military Academy in Turin in 1886. The following year he discovered, and published, a method for solving systems of linear differential equations using successive approximations. However Emile Picard had independently discovered this method and had credited Schwarz with discovering the method first. In 1888 Peano published the book Geometrical Calculus which begins with a chapter on mathematical logic. This was his first work on the topic that would play a major role in his research over the next few years and it was based on the work of Schröder, Boole and Charles Peirce. A more significant feature of the book is that in it Peano sets out with great clarity the ideas of Grassmann which certainly were set out in a rather obscure way by Grassmann himself. This book contains the first definition of a vector space given with a remarkably modern notation and style and, although it was not appreciated by many at the time, this is surely a quite remarkable achievement by Peano. In 1889 Peano published his famous axioms, called Peano axioms, which defined the natural numbers in terms of sets. These were published in a pamphlet Arithmetices principia, nova methodo exposita which, according to [5] were:... at once a landmark in the history of mathematical logic and of the foundations of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peano.html (2 of 6) [2/16/2002 11:26:00 PM]

Peano

mathematics. The pamphlet was written in Latin and nobody has been able to give a good reason for this, other than [5]:... it appears to be an act of sheer romanticism, perhaps the unique romantic act in his scientific career. Genocchi died in 1889 and Peano expected to be appointed to fill his chair. He wrote to Casorati, who he believed to be part of the appointing committee, for information only to discover that there was a delay due to the difficulty of finding enough members to act on the committee. Casorati had been approached but his health was not up to the task. Before the appointment could be made Peano published another stunning result. He invented 'space-filling' curves in 1890, these are continuous surjective mappings from [0,1] onto the unit square. Hilbert, in 1891, described similar space-filling curves. It had been thought that such curves could not exist. Cantor had shown that there is a bijection between the interval [0,1] and the unit square but, shortly after, Netto had proved that such a bijection cannot be continuous. Peano's continuous space-filling curves cannot be 1-1 of course, otherwise Netto's theorem would be contradicted. Hausdorff wrote of Peano's result in Grundzüge der Mengenlehre in 1914:This is one of the most remarkable facts of set theory. In December 1890 Peano's wait to be appointed to Genocchi's chair was over when, after the usual competition, Peano was offered the post. In 1891 Peano founded Rivista di matematica, a journal devoted mainly to logic and the foundations of mathematics. The first paper in the first part is a ten page article by Peano summarising his work on mathematical logic up to that time. Peano had a great skill in seeing that theorems were incorrect by spotting exceptions. Others were not so happy to have these errors pointed out and one such was his colleague Corrado Segre. When Corrado Segre submitted an article to Rivista di matematica Peano pointed out that some of the theorems in the article had exceptions. Segre was not prepared to just correct the theorems by adding conditions that ruled out the exceptions but defended his work saying that the moment of discovery was more important than a rigorous formulation. Of course this was so against Peano's rigorous approach to mathematics that he argued strongly:I believe it new in the history of mathematics that authors knowingly use in their research propositions for which exceptions are known, or for which they have no proof... It was not only Corrado Segre who suffered from Peano's outstanding ability to spot lack of rigour. Of course it was the precision of his thinking, using the exactness of his mathematical logic, that gave Peano this clarity of thought. Peano pointed out an error in a proof by Hermann Laurent in 1892 and, in the same year, reviewed a book by Veronese ending the review with the comment:We could continue at length enumerating the absurdities that the author has piled up. But these errors, the lack of precision and rigour throughout the book take all value away from it. From around 1892, Peano embarked on a new and extremely ambitious project, namely the Formulario Mathematico. He explained in the March 1892 part of Rivista di matematica his thinking:Of the greatest usefulness would be the publication of collections of all the theorems now

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peano.html (3 of 6) [2/16/2002 11:26:00 PM]

Peano

known that refer to given branches of the mathematical sciences ... Such a collection, which would be long and difficult in ordinary language, is made noticeably easier by using the notation of mathematical logic ... In many ways this grand idea marks the end of Peano's extraordinary creative work. It was a project that was greeted with enthusiasm by a few and with little interest by most. Peano began trying to convert all those around him to believe in the importance of this project and this had the effect of annoying them. However Peano and his close associates, including his assistants, Vailati, Burali-Forti, Pieri and Fano soon became deeply involved with the work. When describing a new edition of the Formulario Mathematico in 1896 Peano writes:Each professor will be able to adopt this Formulario as a textbook, for it ought to contain all theorems and all methods. His teaching will be reduced to showing how to read the formulas, and to indicating to the students the theorems that he wishes to explain in his course. When the calculus volume of the Formulario was published Peano, as he had indicated, began to use it for his teaching. This was the disaster that one would expect. Peano, who was a good teacher when he began his lecturing career, became unacceptable to both his students and his colleagues by the style of his teaching. One of his students, who was actually a great admirer of Peano, wrote:But we students knew that this instruction was above our heads. We understood that such a subtle analysis of concepts, such a minute criticism of the definitions used by other authors, was not adapted for beginners, and especially was not useful for engineering students. We disliked having to give time and effort to the "symbols" that in later years we might never use. The Military Academy ended his contract to teach there in 1901 and although many of his colleagues at the university would have also liked to stop his teaching there, nothing was possible under the way that the university was set up. The professor was a law unto himself in his own subject and Peano was not prepared to listen to his colleagues when they tried to encourage him to return to his old style of teaching. The Formulario Mathematico project was completed in 1908 and one has to admire what Peano achieved but although the work contained a mine of information it was little used. However, perhaps Peano's greatest triumph came in 1900. In that year there were two congresses held in Paris. The first was the International Congress of Philosophy which opened in Paris on 1 August. It was a triumph for Peano and Russell, who attended the Congress, wrote in his autobiography:The Congress was the turning point of my intellectual life, because there I met Peano. I already knew him by name and had seen some of his work, but had not taken the trouble to master his notation. In discussions at the Congress I observed that he was always more precise than anyone else, and that he invariably got the better of any argument on which he embarked. As the days went by, I decided that this must be owing to his mathematical logic. ... It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years ... The day after the Philosophy Congress ended the Second International Congress of Mathematicians began. Peano remained in Paris for this Congress and listened to Hilbert's talk setting out ten of the 23 problems which appeared in his paper aimed at giving the agenda for the next century. Peano was particularly interested in the second problem which asked if the axioms of arithmetic could be proved

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peano.html (4 of 6) [2/16/2002 11:26:00 PM]

Peano

consistent. Even before the Formulario Mathematico project was completed Peano was putting in place the next major project of his life. In 1903 Peano expressed interest in finding a universal, or international, language and proposed an artificial language "Latino sine flexione" based on Latin but stripped of all grammar. He compiled the vocabulary by taking words from English, French, German and Latin. In fact the final edition of the Formulario Mathematico was written in Latino sine flexione which is another reason the work was so little used. Peano's career was therefore rather strangely divided into two periods. The period up to 1900 is one where he showed great originality and a remarkable feel for topics which would be important in the development of mathematics. His achievements were outstanding and he had a modern style quite out of place in his own time. However this feel for what was important seemed to leave him and after 1900 he worked with great enthusiasm on two projects of great difficulty which were enormous undertakings but proved quite unimportant in the development of mathematics. Of his personality Kennedy writes in [5]:... I am fascinated by his gentle personality, his ability to attract lifelong disciples, his tolerance of human weakness, his perennial optimism. ... Peano may not only be classified as a 19th century mathematician and logician, but because of his originality and influence, must be judged one of the great scientists of that century. Although Peano is a founder of mathematical logic, the German mathematical philosopher Gottlob Frege is today considered the father of mathematical logic. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles)

Some Quotations (2)

A Poster of Giuseppe Peano

Mathematicians born in the same country

Some pages from publications

Extract from Rivista di matematica (1895) showing the original Peano axioms.

Cross-references to History Topics

1. The beginnings of set theory 2. Abstract linear spaces

Other references in MacTutor

1. Space filling curves 2. Chronology: 1880 to 1890 3. Chronology: 1890 to 1900

Other Web sites

1. Interactive Real Analysis 2. An article in Italian by Beppo Levi (In Italian) 3. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peano.html (5 of 6) [2/16/2002 11:26:00 PM]

Peano

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Peano.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peano.html (6 of 6) [2/16/2002 11:26:00 PM]

Pearson

Karl Pearson Born: 27 March 1857 in London, England Died: 27 April 1936 in London, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karl Pearson graduated from Cambridge University in 1879, then spent most of his career at University College, London. He was the first Galton professor of eugenics, holding the chair from 1911 to 1933. His book The Grammar of Science (1892), was remarkable in that it anticipated some of the ideas of relativity theory. It was wide ranging and attempted to extend the influence of science into all aspects. Pearson then became interested in developing mathematical methods for studying the processes of heredity and evolution. He applied statistics to biological problems of heredity and evolution. From 1893-1912 he wrote 18 papers entitled Mathematical Contribution to the Theory of Evolution which contain his most valuable work. These papers contain contributions to regression analysis, the correlation coefficient and includes the chi-square test of statistical significance (1900). His chi-square test was produced in an attempt to remove the normal distribution from its central position. Pearson coined the term 'standard deviation' in 1893. His work was influenced by the work of Edgeworth and in turn influenced the work of Yule. Pearson had a long dispute with Fisher. Pearson used large sample which he measured and tried to deduce correlations. Fisher, on the other hand, followed Gosset in trying to use small samples and, rather than deduce correlations, to find causes. The dispute was bad enough to have Fisher turn down the post of Chief Statistician at the Galton Laboratory in 1919 since it would have meant working under Pearson.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pearson.html (1 of 2) [2/16/2002 11:26:02 PM]

Pearson

He was a co-founder, with Weldon and Galton, of the statistical journal Biometrika. Article by: J J O'Connor and E F Robertson List of References (12 books/articles)

Some Quotations (3)

A Poster of Karl Pearson

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1890 to 1900

Honours awarded to Karl Pearson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1896

Fellow of the Royal Society of Edinburgh Other Web sites

1. University of Minnesota 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR December 1996 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Pearson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pearson.html (2 of 2) [2/16/2002 11:26:02 PM]

Pearson_Egon

Egon Sharpe Pearson Born: 11 Aug 1895 in Hampstead (near London), England Died: 12 June 1980 in Midhurst, Sussex, England

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Egon Pearson's father was Karl Pearson, whose biography is given in this archive, and his mother was Maria Sharpe. Egon was the middle child of three in the family; Sigrid Loetitia was three years older and Helga Sharpe three years younger than Egon. Even as a child at the age of five he was aware of his father's work and his efforts to bring the journal Biometrika into existence. Later in life Egon recalled creating his own journal as a five years old:... which was all scrawls with chalk. He attended school, first at Dragon School Oxford from 1907 to 1909, then going to Winchester College from which he graduated in 1914. World War I began in 1914, and had Pearson's health been good he would have found himself in military service. However, his health had never been good as a child and he had a heart murmur which now prevented him from enlisting. He therefore went to Trinity College Cambridge to begin his university studies. These studies were interrupted by influenza which hit him hard and he was unable to study from August 1914 until the end of that year. At the end of one year of study, Pearson left Cambridge in 1915 determined to make a contribution to the war effort, and he worked for the Admiralty and the Ministry of Shipping. Pearson never took up his undergraduate studies at Cambridge again after the war but was awarded his B.A. in 1920 after taking the Military Special Examination in 1919 which had been set up to cope with those who had their studies disrupted by the war. He then began research at Cambridge, but not on statistics as one might have expected, rather on solar physics. However, even the astronomy lectures he attended, which were given by Eddington, involved him in statistics since Eddington was lecturing on the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pearson_Egon.html (1 of 4) [2/16/2002 11:26:05 PM]

Pearson_Egon

theory of errors. Pearson also attended astronomy lectures by F J M Stratton and undertook work with F L Engledow and G U Yule. In 1921 Pearson joined his father's Department of Applied Statistics at University College London as a lecturer. However, despite being a lecturer, his father seems to have kept him away from lecturing. Instead Pearson attended all of his father's lectures and began to produce a stream of high quality research publications on statistics. In 1924 Pearson became an assistant editor of Biometrika but perhaps one of the most important events for his future research happened in the following year. Neyman had a two year Rockefeller Research Fellowship for the years 1925-27. He spent the first of these two years in University College London, and the second of them in Paris. When Neyman met Pearson in his father's Department in 1925 he did not realise that he was going through a sort of crisis. Karl Pearson's work had been under attack from R A Fisher for a number of years and Egon later explained (see for example [2]):... I had to go through the painful stage of realising that K.P. [his father] could be wrong ... and I was torn between conflicting emotions: a. finding it difficult to understand R.A.F. [Fisher], b. hating him for his attacks on my paternal 'god', c. realising that in some things at least he was right. In [2] the friendship that developed between Neyman and Pearson during 1926 is described. It paints a picture of Pearson, and his difficulties, at this time:Pearson was an introverted young man who felt inferior for a number of reasons. He had grown up in the great shadow of K.P. [Karl Pearson], "lovingly protected" in his childhood and kept out of the war in his youth. At Cambridge he had felt cut off from classmates of his own generation, all veterans of the conflict. He suspected that K.P. was disappointed in him, for he had not gone on to his second mathematics tripos but had taken his degree on the basis of work he had done during the war. After joining the staff of K.P.'s laboratory, he had continued to live at home and to have almost all his social contacts with relatives. It is perhaps worth noting that 1926 was the year when Karl Pearson allowed his son to begin lecturing at University College. Even then it took place simply because Karl Pearson's health prevented him teaching, rather than for positive reasons. Pearson and Neyman agreed to undertake a joint research project in June 1926, just before Neyman left for Paris. Their joint research was carried on by letters, but there were meetings such as in the spring of 1927 when Pearson visited Neyman in Paris. During the visit they mapped out their first joint paper and planned their future research. Neyman describes their collaboration in [8]:The initiative for cooperative studies was Egon's. Also, at least during the early stages, he was the leader. Our cooperative work was conducted through voluminous correspondence and at sporadic get-togethers, some in England, others in France and some others in Poland. This cooperation continued over the decade 1928-38. Neyman also describes in [8] the aims of their work:My joint work with Egon was concerned with problems of testing hypotheses. One aspect of this work was philosophical ... The decade-long joint work of Egon and myself was aimed at

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pearson_Egon.html (2 of 4) [2/16/2002 11:26:05 PM]

Pearson_Egon

building a mathematical theory of tests the use of which could minimise the frequency of erroneous conclusions regarding the hypotheses considered ... As H A David writes in [5]:The Neyman-Pearson theory of testing statistical hypotheses has become an integral part of every statistician's education and vocabulary. There was a second major correspondence which Pearson carried out over much the same period as the Neyman-Pearson collaboration. This was one which Pearson carried out with Gosset and Gosset's ideas played a big role in the discussions between Pearson and Neyman. Pearson visited the United States in 1931 and, in addition to lecturing in Iowa, he held discussions with Shewhart in the Bell Telephone Laboratories in New York. The following year Shewhart visited Pearson in London and their discussions on quality control in industry led to the creation of the Industrial and Agricultural Research Section of the Royal Statistical Society. In 1933 Karl Pearson retired from the Galton Chair of Statistics which he had held in University College London. Against his wishes the University authorities decided to split the Department into two separate departments; the Galton Chair and Head of the Department of Eugenics went to Fisher, while Egon Pearson was appointed Reader and became Head of the Department of Applied Statistics. If Karl Pearson did not like that arrangement, then certainly neither did Fisher. Neyman came to work in Pearson's Department in 1934 and in that same year Pearson married Eileen. They would have two daughters. Family commitments, further administrative duties and assuming the role of Managing Editor of Biometrika on his father's death in 1936 all reduced the time that Pearson could devote to research. He had been awarded the Weldon prize and medal in 1935, mainly for his work with Neyman, but despite having Neyman as a colleague, Pearson's efforts began to be directed towards revising his father's two volume work Tables for Statisticians and Biometricians. He worked on this project with H O Hartley, but it was some considerable time before the revision was completed; volume 1 appeared in 1954 and volume 2 in 1972. H A David [5] writes:With their attractive layout, easy means for interpolation, and extensive, helpful introductory material, these tables have been widely recognised as models of their kind. The advent of World War II in 1939 led to a shift in Pearson's work. Neyman had already left London in 1938 for a post in Berkeley and with the outbreak of war Pearson began to undertake war work for the Ordinance Board undertaking statistical analysis of the fragmentation of shells hitting aircraft and similar work. In fact, Pearson enjoyed this period:... despite bombings, V1 flying bombs, and V2 rockets. He was later awarded a C.B.E. for his war service. Pearson had found the years working in the same institution as Fisher very difficult. Karl Pearson had attacked Fisher aggressively and been attacked aggressively by Fisher. Egon Pearson had a very different personality from his father and held Fisher's work in high regard. However, even after Karl Pearson died, Fisher kept up his attack on him in print and Egon Pearson must have found it difficult to share a building with Fisher. After Fisher moved away from London in 1939, especially after he began to work at Cambridge from 1943, Pearson must have found the atmosphere in London a much happier one.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pearson_Egon.html (3 of 4) [2/16/2002 11:26:05 PM]

Pearson_Egon

He did suffer a great tragedy in 1949 however when his wife died of pneumonia. He continued to contribute greatly to statistics, however, retiring from University College in 1961 and retiring as Managing Editor of Biometrika in 1966. It was in this year that he was elected a Fellow of the Royal Society, rather belatedly as some of his biographers have noted. Article by: J J O'Connor and E F Robertson List of References (8 books/articles) Mathematicians born in the same country Other Web sites

University of Minnesota

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Pearson_Egon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pearson_Egon.html (4 of 4) [2/16/2002 11:26:05 PM]

Peirce_Benjamin

Benjamin Peirce Born: 4 April 1809 in Salem, Massachusetts, USA Died: 6 Oct 1880 in Cambridge, Massachusetts, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Benjamin Peirce was a graduate of Harvard and became a tutor there in 1831, two years after graduating. He worked on a wide range of mathematical topics from celestial mechanics and geodesy on the applied side to linear associative algebra and number theory on the pure side. In an early number theory paper he proved that there is no odd perfect number with fewer than four distinct prime factors. He also revised and wrote a commentary on Bowditch's translation of the first four volumes of Laplace's Traité de mécanique céleste. Appointed professor at Harvard in 1833 he was to establish the Harvard Observatory. Peirce helped determine the orbit of Neptune (discovered in 1846) and calculated the perturbations produced by Neptune on the orbit of Uranus and on the other planets. In 1870 Peirce published, at his own expense, Linear Associative Algebra a classification of all complex associative algebras of dimension less than seven. He used the, now familiar, tools of idempotent and nilpotent elements. Earlier, in 1852, Peirce had introduced methods into the theory of errors applied to observations which would allow faulty observations to be discarded. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peirce_Benjamin.html (1 of 2) [2/16/2002 11:26:06 PM]

Peirce_Benjamin

List of References (12 books/articles)

Some Quotations (2)

Mathematicians born in the same country Other references in MacTutor

1. Chronology: 1860 to 1870 2. Chronology: 1870 to 1880

Honours awarded to Benjamin Peirce (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Peirce

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Peirce_Benjamin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peirce_Benjamin.html (2 of 2) [2/16/2002 11:26:06 PM]

Peirce_Charles

Charles Sanders Peirce Born: 10 Sept 1839 in Cambridge, Massachusetts, USA Died: 19 April 1914 in Milford, Pennsylvania, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles S Peirce was the son of Benjamin Peirce and studied at Harvard and worked for many years on the Coast and Geodetic Survey. He worked on geodesy but became interested in conformal map projections where he invented a quincuncial map projection using elliptic functions. He was also interested in the Four Colour Problem and problems of knots and linkages studied by Kempe. He then extended his father's work on associative algebras and worked on mathematical logic and set theory. Except for courses on logic he gave at Johns Hopkins University, between 1879 and 1884, he never held an academic post. T S Fiske, writing about the New York Mathematical Society (before it became the American Mathematical Society) in [23], describes Charles Peirce:Conspicuous among those who in the early nineties attended the monthly meetings ... was the famous logician, Charles S Peirce. His dramatic manner, his reckless disregard of accuracy in what he termed 'unimportant details', his clever newspaper articles describing the meetings of our young Society interested and amused us all. ... He was always hard up, living partly on what he could borrow from friends, and partly on what he got from odd jobs such as writing book reviews ... He was equally brilliant, whether under the influence of liquor or otherwise, and his company was prized by the various organisations to which he belonged; and he was never dropped from any of them even though he was unable to pay his dues. He infuriated Charlotte Angas Scott by contributing to the New York Evening Post an unsigned obituary of Arthur Cayley in which he stated upon no grounds, except that

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peirce_Charles.html (1 of 2) [2/16/2002 11:26:08 PM]

Peirce_Charles

Cayley's father had for a time resided in Russia, that Cayley had inherited his genius from a Russian whom his father had married in St Petersburg. Shortly afterwards Miss Scott contributed to the Bulletin a more factual, sober article upon Cayley's life and work...

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (26 books/articles)

Some Quotations (7)

Mathematicians born in the same country Cross-references to History Topics

the four colour theorem

Other Web sites

1. Charles Peirce Studies 2. Stanford Encyclopedia of Philosophy 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Peirce_Charles.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peirce_Charles.html (2 of 2) [2/16/2002 11:26:08 PM]

Pell

John Pell Born: 1 March 1611 in Southwick, Sussex, England Died: 12 Dec 1685 in London, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

After attending Steyning School in Sussex, John Pell entered Trinity College, Cambridge in 1624. He received his B.A. in 1628 and his M.A. in 1630. After leaving Cambridge, Pell became a schoolmaster. He worked first in Horsham, then in Chichester. He spent five years from 1638 teaching mathematics in London. He then went abroad becoming Professor of Mathematics at a College in Amsterdam from 1643 until he took up a similar post the University of Breda in 1646. Pell returned to England in 1652 and was appointed by Oliver Cromwell to a post teaching mathematics in London. He spent the years 1654 to 1658 holding a government post in Zurich. On his return to England became a vicar and remained in this position in the church for the last 20 years of his life. Pell worked on algebra and number theory. He gave a table of factors of all integers up to 100000 in 1668. Pell's equation y2 = ax2 + 1, where a is a non-square integer, was first studied by Brahmagupta and Bhaskara. Its complete theory was worked out by Lagrange, not Pell. It is often said that Euler mistakenly attributed Brouncker's work on this equation to Pell. However the equation appears in a book by Rahn which was certainly written with Pell's help: some say entirely written by Pell. Perhaps Euler knew what he was doing in naming the equation. Pell published a number of works, for example Idea of Mathematics (1638) and Controversiae de vera circuli mensura (1647), this second work being written because of a dispute Pell was involved in over the value of . He also translated Lansberge's tables, which were published in 1632, and also worked on astronomy. Pell was elected to the Royal Society in 1663. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pell.html (1 of 2) [2/16/2002 11:26:09 PM]

Pell

List of References (7 books/articles) Mathematicians born in the same country Cross-references to History Topics

English attack on the Longitude Problem

Other references in MacTutor

1. Pell's equation 2. Chronology: 1650 to 1675

Honours awarded to John Pell (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1663

Other Web sites

1. The Galileo Project 2. Math Forum (Pell's equation)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR December 1996 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Pell.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pell.html (2 of 2) [2/16/2002 11:26:09 PM]

Penney

William George Penney Born: 24 June 1909 in Gibraltar Died: 3 March 1991 in East Hendred (near London), England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bill Penney attended school in Colchester, then spent the years 1924-26 at Sheerness Technical College. After leaving he took a job as a laboratory assistant before being awarded a scholarship to studied at the University of London in 1927. He won the Governor's Prize for Mathematics and graduated with First Class Honours in 1929. After two years at the University of Wisconsin he returned to England and obtained a doctorate from the University of Cambridge in 1935 on the application of quantum mechanics to the physics of crystals. He was then appointed Reader in Mathematics at Imperial College London, a post he held from 1936 to 1945. In 1944-45 he worked on the USA atomic bomb project at the Los Alamos Scientific Laboratory, New Mexico. In particular he worked on the use of the atomic bomb, its effects and in particular the height at which it should be detonated. Returning to England he worked on the British atomic bomb project, saw the project through to the test of the first bomb in 1952. At this point Penney was offered a Chair at the University of Oxford. Always more inclined toward the academic life he was keen to accept this post but he was persuaded that the "national interests" required him to continue as director of atomic-weapons research and development at Aldermaston. From 1954 Penney served on the Board of the Atomic Energy Authority, becoming Chairman in 1964. He retired his post of Chairman in 1967 and accepted an invitation to become Rector of Imperial College. He resigned this post after 6 years and although he led the College successfully the reason for his early http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Penney.html (1 of 3) [2/16/2002 11:26:12 PM]

Penney

retirement may be explained by his own words about this period:These were difficult times .... The students were sometimes uncontrollable, the heads of departments were bewildered and angry, the rest of the staff, including the technicians, wanted a say in how everything was run. He received many honours. In 1946 he was made O.B.E., in 1952 K.B.E. and made Baron Penney of East Hendred in 1967. He received an O.M. in 1969. Other honours include being elected a Fellow of the Royal Society in 1946 and in 1966 he was awarded its Rumford Medal:... in recognition of his distinguished anf paramount personal contribution to the establishment of economic nuclear energy in Great Britain. He was elected a Fellow of the Royal Society of Edinburgh in 1970. He also received honorary degrees from five universities. In [2] he is described as follows:In appearance, he was a rotund, rather owl-like figure, usually casually dressed. He was friendly and amiable, if sometimes rather withdrawn in company, often with an amused smile for he had a keen sense of humour. ... He spoke slowly ... He said little, but what he said was much to the point, and carried authority, and he had a gift for laconic comment. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Bill Penney (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1946

Fellow of the Royal Society of Edinburgh Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Penney.html (2 of 3) [2/16/2002 11:26:12 PM]

Penney

JOC/EFR November 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Penney.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Penney.html (3 of 3) [2/16/2002 11:26:12 PM]

Peres

Joseph Jean Camille Pérès Born: 31 Oct 1890 in Clermont-Ferrand, France Died: 12 Feb 1962 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Pérès was the son of a famous philosopher. He studied at the Ecole Normale Supérieure in Paris, entering in 1908 and graduating three years later. He was awarded a scholarship to support him while he undertook research for his doctorate, and Borel introduced him to Volterra who made many journeys to France to promote scientific collaboration. Pérès went to Italy to work for his doctorate under Volterra but he returned to France and was teaching at the Lycée at Montpellier when he submitted his thesis Sur les fonctions permutable do Volterra in 1915. He taught at Toulouse and then at Strasbourg before being appointed Professor of Rational and Applied Mechanics at Marseilles in 1921. At Marseilles, Pérès founded an institute of fluid mechanics in 1930. However, he only remained at Marseilles for two years after this before he was offered a chair at the Sorbonne. There he balanced his career between teaching and research being active in both while at the same time taking on some major administrative roles such as Dean of the Faculty of Science in Paris from 1954 to 1961. On the research front he was awarded prizes from the Académie des Sciences in 1932, again in 1938 and for yet a third time in 1940. Two years after he received this third prize he was elected to membership of the Académie. Pérès' work on analysis and mechanics was always influenced by Volterra, extending results of Volterra's on integral equations. His work in this area is now of relatively little importance since perhaps even for its day it was somewhat old fashioned. A joint collaboration between Pérès and Volterra led to the first volume of Theorie generale des fonctionnelles published in 1936. Although the project was intended to lead to further volumes only this one was ever published. This work is discussed in [3] where the author points out that the book belongs to an older tradition, being based on ideas introduced by Volterra himself from 1887 onwards. By the time the work was published the ideas it contained were no longer in the mainstream of development of functional analysis since topological and algebraic concepts introduced by Banach, von Neumann, Stone and others were determining the direction of the subject. However, the analysis which Pérès and Volterra studied proved important in developing ideas of mathematical physics rather than analysis and Pérès made good use of them in his applications. He studied the dynamics of viscous fluids and the theory of vortices with applications to aeronautics in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peres.html (1 of 2) [2/16/2002 11:26:13 PM]

Peres

mind. Costabel writes in [1]:To his scientific colleagues [Pérès] remained a circumspect theorist, animator, and promoter. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Peres.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peres.html (2 of 2) [2/16/2002 11:26:13 PM]

Perron

Oskar Perron Born: 7 May 1880 in Frankenthal, Pfalz, Germany Died: 22 Feb 1975 in Munich, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oskar Perron studied classics at school and, despite his father wishing him to continue the family business, he studied mathematics in his spare time. In 1898 Perron entered the University of Munich and, in keeping with the custom of the time to spend semesters at different universities, he also studied at the universities of Berlin, Tübingen and Göttingen. Perron was most influenced by his teachers at Munich. Pringsheim's lectures at Munich made a lasting impression on Perron and their influence extended to Perron work on continued fractions Die Lehre von den Kettenbrüchen which was published in 1913. Geometry became the topic of Perron's doctoral thesis directed by Lindemann and Perron went on to complete his habilitation at Munich and was appointed a lecturer there in 1906. In 1910 Perron accepted the offer of a post as extraordinary professor at Tübingen and then, in 1914, he became an ordinary professor at Heidelberg. However World War I disrupted his career and, in 1915, he undertook war work which was to earn him the Iron Cross. At the end of the war he returned to Heidelberg where he taught until 1922 when he was appointed to a chair at Munich. In [1] 218 publications by Perron are listed in a bibliography which he complied himself. These publications cover a wide range of mathematical topics. His work in analysis is certainly remembered through the Perron integral. However he also worked on differential equations, matrices and other topics in algebra, continued fractions, geometry and number theory. Perron published a number of important texts. In addition to the work on continued fractions mentioned

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Perron.html (1 of 3) [2/16/2002 11:26:15 PM]

Perron

above, which in fact ran to three editions the last being a two volume version in 1954/57, he published an important text on irrational numbers in 1921. This text was designed to require only school level mathematics as a prerequisite and the topic was skilfully developed in a beautiful self-contained way. Again this was a text which ran to several editions and Perron revised the text in 1960 when he was aged 80. A two volume work on algebra first appeared in 1927 with the third revised edition being published in 1951. Perhaps most remarkable of all was his text on non-euclidean geometry which he published at the age of 82. Frank, in [1], writes:This work won the approval of the entire mathematical world due to its great worth and masterful presentation. It is of interest not only to students of mathematics and physics but also especially to teachers of mathematics. Despite the large amount of mathematics which Perron produced over a long career, he also had other interests. These are described in [1] and included:... his love of the mountains of his surroundings. No vacation would have been complete without the mountains. As well as higher mountains, he climbed the 2200 meter Totenkirchl in the Wilder Kaiser more than twenty times, the last time when he was 74. Although Perron formally retired in 1951, he continued to teach certain courses at Munich until 1960. However even when he ended his teaching at the age of 80 he was still able to continue with a vigorous research program, publishing 18 papers between 1964 and 1973. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Other Web sites

F Litton (Oskar Perron in the Third Reich -- in German) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Perron.html (2 of 3) [2/16/2002 11:26:15 PM]

Perron

JOC/EFR June 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Perron.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Perron.html (3 of 3) [2/16/2002 11:26:15 PM]

Perseus

Perseus Born: 2nd century BC Died: 2nd century BC Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

There are only two references to Perseus and these both occur in the writings of Proclus. They give no indication of where he was born or where he lived. His dates can at least be put within certain bounds by the information given, but our knowledge is still almost nil. The first reference says that Perseus is associated with the discovery of the "spiric" curves in the same way as that of Apollonius is with conics. The second reference is taken from Geminus and says that Perseus wrote an epigram on his discovery (see for example [1]):Three curves upon five sections finding, Perseus made offering to the gods... All that can be deduced with certainty is that Perseus must have lived before Geminus. Less certain, but still very reasonable, is the belief that conic sections must have been developed first so he would then have lived after Euclid wrote in say 300 BC. The references do not really give enough details to be able to tell what Perseus discovered. We do know what a spiric section is. Proclus defines a spiric surface as being the surface generated by a circle revolving about a straight line called the axis of revolution and always remaining in the same plane as this axis. There are three distinct types of spiric surfaces depending whether the axis of revolution cuts the circle, it a tangent to the circle, or is outside the circle. A spiric section is then the curve produced when a plane parallel to the axis of revolution cuts the spiric surface. However, it is now difficult to see what the "three curves upon five sections finding ..." means. Paul Tannery in [3] gives a clever argument based on assuming that Proclus made an error and should have written "three curves in addition to five sections finding ...".

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Perseus.html (1 of 3) [2/16/2002 11:26:16 PM]

Perseus

The five types of curve are shown in the diagram. The first is the case of an oval, the second is a transition from the first case to the third. The third case is that in which the curve is closed and narrowest in the middle. The fourth case is that of the hippopede, which was studied by Eudoxus. Finally the fifth case consists of two symmetrically positioned closed curves. Proclus gives the three curves of types 1, 3 and 4 only.

In [1] Bulmer-Thomas prefers the simpler suggestion that Perseus found five sections but only three of these gave new curves, the other two (types 2 and 5) gave curves which were closely related to the others and not considered new. Another possibility, which is not favoured by historians, is that the three spiric curves were one from each of the three different spiric surfaces. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Cross-references to Famous Curves

Spiric Sections

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Perseus.html (2 of 3) [2/16/2002 11:26:16 PM]

Perseus

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Perseus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Perseus.html (3 of 3) [2/16/2002 11:26:16 PM]

Peter

Rózsa Péter Born: 1905 in Hungary Died: 1977 in Hungary

Click the picture above to see three larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Rózsa Péter's original name was Rósa Politzer but in the 1930 she, like many other Hungarians, changed her German style name to a Hungarian one. Rózsa Péter studied at Loránd Eötvös University in Budapest where her interest in mathematics was brought about by Fejér's lectures. Another one to have an important influence on Rózsa Péter was László Kalmár who was a fellow student at Loránd Eötvös University. After graduating in 1927 Péter earned a living tutoring mathematics, unable to obtain a permanent job. Her first post, at the Budapest Teachers Training College, was obtained in 1945. Péter's years at the teacher's college produced the charming book Playing with Infinity (first in German, 1955). When the College closed in 1955 she became a professor at Loránd Eötvös University and remained in this post until she retired in 1975. Her first research topic was number theory but she became discouraged on finding that her results had already been proved by Dickson. For a while Péter wrote poetry but around 1930 she was encouraged to return to mathematics by Kalmár. He suggested Péter examine Gödel's work and in a series of papers she became a founder of recursive function theory. Walter Felscher, in a personal communication to me [EFR], described the context of Péter's work on recursive function theory:Recursive functions were invented during the 1920s in the Hilbert school, but nothing much was proved about them. Developing ideas of Herbrand, Gödel defined the more general 'general' recursive functions (to which Ackermann's function belongs) in his Princeton lectures 1933-34; soon after, the old functions received the name 'primitive recursive', and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peter.html (1 of 3) [2/16/2002 11:26:18 PM]

Peter

the general ones lost their adjective. In a series of articles, beginning in 1934, Péter developed various deep theorems about primitive recursive functions, most of them with an explicit algorithmic content. I admire this work, and it may well be said that she forged, with her bare hands, the theory of primitive recursive functions into existence. [On the other side, it was Kleene who, having attended Gödel's lectures, developed the theory of general (including partial) recursive functions; this is a much more conceptual than computational area.] In 1951 Péter collected what was known by then, including her own work, in the book Rekursive Funtionen. An English translation appeared only in 1967. It was the first book devoted exclusively to this topic, but (1) there had been extensive chapters on this matter earlier in Hilbert-Bernays (1934-1939) where some of Péter's work was quoted, and (2) the English speaking world did not read her book but read, instead, Kleene's book of 1952. In 1952 Kleene described Rózsa Péter in a paper in Bull. Amer. Math. Soc. as the leading contributor to the special theory of recursive functions. From the mid 1950's Péter applied recursive function theory to computers. In 1976 her last book was on this topic Recursive Functions in Computer Theory. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other Web sites

1. Agnes Scott College 2. San Diego

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peter.html (2 of 3) [2/16/2002 11:26:18 PM]

Peter

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Peter.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peter.html (3 of 3) [2/16/2002 11:26:18 PM]

Petersen

Julius Peter Christian Petersen Born: 16 June 1839 in Soro, Denmark Died: 5 Aug 1910 in Copenhagen, Denmark

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Julius Petersen wrote a series of school and undergraduate texts which achieved international acclaim despite being too difficult for all but the ablest pupils. His thesis treated equations soluble by square roots with applications to ruler and compass constructions. His research was on a wide variety of topic from algebra and number theory to analysis and mechanics. He also wrote on mathematical physics, mathematical economics and cryptography. His most important work however was in geometry and work on regular graphs. A paper which he wrote in 1891 marks the birth of the theory of regular graphs. He is best remembered for the Petersen graph. In [4] it is stated that the work of Petersen and Zeuthen is regarded as being responsible for the emergence of Danish mathematics on the international scene towards the end of the 19th century. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Julius Petersen

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Petersen.html (1 of 2) [2/16/2002 11:26:20 PM]

Petersen

Other Web sites

1. Theseus 2. Odense, Denmark (in Danish)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR December 1996 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Petersen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Petersen.html (2 of 2) [2/16/2002 11:26:20 PM]

Peterson

Karl Mikhailovich Peterson Born: 25 May 1828 in Riga, Russia (now Latvia) Died: 19 April 1881 in Moscow, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karl Peterson worked in Moscow as a teacher of mathematics. His main work is in differential geometry and he obtained an honorary doctorate for work on partial differential equations. Largely because he was not at a university his results were not well known but they did influence Egorov in Moscow and Peterson gained an international reputation only when Darboux and Bianchi used his results. A class of surfaces is named after him. In [5] his work is described as follows:Peterson's most important paper was On the ratios and relationships between curved surfaces (1866), devoted to deformation of surfaces, which laid the foundation for a series of papers on the problem of bending on a principal basis, i.e., preserving the conjugacy of a certain net on the surface, the first example of which for deformation of surfaces of revolution on a surface of revolution was found by Minding ... (1838). Peterson's paper "On curves on surfaces" (1867) and the book "Über Curven und Flächen" (1868) were devoted to differential geometry. Some of the results of these papers of Peterson were later duplicated by G Darboux and other foreign geometers, but after E Cosserat's translations of Peterson's main works from 1866-1867 were published in Toulouse in 1905, his work achieved general recognition. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peterson.html (1 of 2) [2/16/2002 11:26:22 PM]

Peterson

List of References (9 books/articles) A Poster of Karl Peterson

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Peterson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peterson.html (2 of 2) [2/16/2002 11:26:22 PM]

Petit

Aléxis Thérèse Petit Born: 2 Oct 1791 in Vesoul, France Died: 21 June 1820 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aléxis Petit entered the Ecole Polytechnique as a student in 1807 and was in the same class as Poncelet. Petit was awarded a doctorate in 1811 for a thesis on capillary action. In 1818 he won the Academy Prize for work on the law of cooling and, in the same year, he published on the general principles of machine theory. The following year he published on the theory of heat. Working with Pierre Louis Dulong he formulated, in 1819, an empirical law concerning the specific heat of elements. The Dulong-Petit law states that the specific heat of all elements is the same on a per atom basis. The law has exceptions and was not fully understood until quantum theory was used. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Aléxis Petit (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Petit

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Petit.html (1 of 2) [2/16/2002 11:26:23 PM]

Search Form

Petit

Glossary index

Quotations index

Mathematicians of the day JOC/EFR December 1996

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Petit.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Petit.html (2 of 2) [2/16/2002 11:26:23 PM]

Petrovsky

Ivan Georgievich Petrovsky Born: 18 Jan 1901 in Sevsk, Orlov guberniya, Russia Died: 15 Jan 1973 in Moscow, USSR

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ivan Petrovsky was a student of Egorov at the University of Moscow. From this time he began to build up his collection of books, described by P S Aleksandrov and O A Oleinik in [2]:From his student days onwards, he was always buying books. His personal library contained over thirty thousand books ... It is remarkable that Petrowsky did not have a catalogue but knew his way round the library very well and was able to find his favourite books quickly. Petrovsky taught at Moscow University and, from 1943, at the Steklov Institute. He became rector of Moscow University and Vice-Director of the Steklov Institute. P S Aleksandrov and O A Oleinik write in [2]:It can be said without exaggeration that among the Rectors of Russian Universities two have a special and prominent place : [Lobachevsky and Petrovsky]. Both of them lived the life of the universities of which they were head, took part in all aspects of this life, and tried to steer it along the best possible path. Petrovsky's main mathematical work was on the theory of partial differential equations, the topology of algebraic curves and surfaces, and probability. Petrovsky also worked on the boundary value problem for the heat equation and this was applied to both probability theory and work of Kolmogorov. P S Aleksandrov and O A Oleinik write in [2]:Petrowsky is one of the few mathematicians whose work shapes the face of modern mathematics. However, he regarded his rectorship as the most important thing in his life, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Petrovsky.html (1 of 2) [2/16/2002 11:26:25 PM]

Petrovsky

even more important than his mathematical research. His career is summed up in [2] as follows:Petrovsky's knowledge was encyclopaedic. He had a thorough understanding of modern science and all its interconnections, was perspicacious and far-sighted, was able to discern long-term trends, and always emphasised them. He also showed a wide general culture, knowledge of diverse branches of science, a deep understanding of state problems in many outstanding public and state activities. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Petrovsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Petrovsky.html (2 of 2) [2/16/2002 11:26:25 PM]

Petryshyn

Volodymyr Petryshyn Born: 22 Jan 1929 in Liashky Murovani, Lvov, Galicia (now Ukraine)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Volodymyr Petryshyn was 10 years old when World War II broke out and his education was severely disrupted, becoming a displaced person at the end of the war. In 1950 he emigrated from Germany to the United States and completed his education there. In 1961 he was awarded his doctorate from Columbia University. From 1964 Petryshyn taught at the University of Chicago, then in 1967 he was appointed to Rutgers University. He was elected to the Shevchenko Scientific Society in 1980 and to the Academy of Sciences of the Ukraine in 1992. He is also an honorary member of the Kiev Mathematical Society, being elected in 1989. Petryshyn's main work in has been in iterative and projective methods, fixed point theorems, nonlinear Friedrichs extension, approximation-proper mapping theorem, and topological degree and index theories for multivalued condensing maps. His mathematical achievements are described by Andrushkov in [1]:Petryshyn's main achievements are in functional analysis. His major results include the development of the theory of iterative and projective methods for the constructive solution of linear and nonlinear abstract and differential equations. The theory of A-proper maps was developed by Petryshyn and this work is described in [1]:Petryshyn is a founder and principal developer of the theory of approximation-proper (A-proper) maps, a new class of maps which attracted considerable attention in the mathematical community. He has shown that the theory of A-proper type maps not only extends and unifies the classical theory of compact maps with some recent theories of condensing and monotone-accretive maps, but also provides a new approach to the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Petryshyn.html (1 of 2) [2/16/2002 11:26:27 PM]

Petryshyn

constructive solution of nonlinear abstract and differential equations. ... The theory has been applied to ordinary and partial differential equations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Petryshyn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Petryshyn.html (2 of 2) [2/16/2002 11:26:27 PM]

Petzval

Józeph Miksa Petzval Born: 6 Jan 1807 in Spisska Bela, Hungary (now in Slovakia) Died: 17 Sept 1891 in Vienna, Austria

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

There are different versions of Józeph Petzval's name, and, in addition to the one given here, he is often known as Jozef Maximilian Petzval. Jozeph was the son of a schoolmaster and he attended schools in Levoca and Kosice. In 1826 he entered the University of Pest to study philosophy and mathematics. Later the town of Pest was to join with the town of Buda on the opposite bank of the Danube to form Budapest. Petzval became an assistant at the University of Pest in 1835. Then, two years later, he accepted a chair of mathematics at the University of Vienna. Petzval worked for much of his life on the Laplace transform. He was influenced by the work of Liouville and wrote both a long paper and a two volume treatise on the Laplace transform and its application to ordinary linear differential equations. His study is thorough but not entirely satisfactory since he was unable to use contour integration to invert the transform. But for a student of Petzval we might today call the Laplace transform the Petzval transform. Petzval fell out with this student who then accused Petzval of plagiarising Laplace's work. Although this was untrue, Boole and Poincaré, influenced no doubt by the quarrel, called the transformation the Laplace transform. Petzval is best remembered for his work on optical lenses and lens aberration done in the early 1840's (Petzval curvature is named after him) which allowed the construction of modern cameras. Petzval produced an achromatic portrait lens that was vastly superior to the simple meniscus lens then in use.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Petzval.html (1 of 2) [2/16/2002 11:26:28 PM]

Petzval

In [1] his work in optics is described as follows:[At the University of Vienna] he studied in detail L M Daguerre's invention, the so-called daguerreotype, and took on shortening its exposure time from minutes to seconds. In 1840, his extraordinary mathematical talent allowed him to assess and build an anastigmatic with six times greater luminosity. This Petzval highly luminous early form of photo lens was used by the enterprising Viennese optician Voigtländer, who launched its mass production and won a silver medal at the World's Exhibition Fair in Paris. Petzval also perfected the telescope and designed the opera glasses. Petzval won many distinctions for his work. In addition to the medal referred to above, he was elected a member of the Academy of Science of Vienna, the Union of Czech Mathematics and he received the platinum medal of Ch Chevalier from France. A street in Vienna bears his name as does a crater on the far side of the Moon. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to Józeph M Petzval (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Petzval

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Petzval.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Petzval.html (2 of 2) [2/16/2002 11:26:28 PM]

Peurbach

Georg Peurbach Born: 30 May 1423 in Peuerbach, Austria Died: 8 April 1461 in Vienna, Austria Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Peurbach studied at Vienna, graduating in 1446. In 1453 he was awarded a Master's Degree then travelled through Europe and lectured in Germany, France and Italy on astronomy. He was appointed court astronomer by King Ladislaus of Hungary in 1454. Peurbach was served as professor of astronomy at the University of Vienna. Peurbach wrote on astronomy and gave tables of eclipse calculations in Tabulae Ecclipsium. He observed Halley's comet in June 1456 and wrote a report on his observations. He made further observations of comets and, together with Regiomontanus, recorded the lunar eclipse of 3 September 1457 from a site near Vienna. Peurbach published further tables, checked by his own eclipse observations, and devised astronomical instruments. In Theoricae Novae Planetarum Peurbach gave Ptolemy's epicycle theory of the planets. Peurbach believed that the planets were in solid crystalline spheres although he believed that their motions were controlled by the Sun. He also constructed a large globe which depicted the stars. He wrote on the computation of sines and chords. His book Algorismus, an elementary textbook on practical calculations, was popular and reprinted several times. Regiomontanus was Peurbach's student and they collaborated on a number of works including Epitome in Ptolemaei Almagestum. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other references in MacTutor

Previous

Chronology: 1300 to 1500

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peurbach.html (1 of 2) [2/16/2002 11:26:30 PM]

Peurbach

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Peurbach.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Peurbach.html (2 of 2) [2/16/2002 11:26:30 PM]

Pfaff

Johann Friedrich Pfaff Born: 22 Dec 1765 in Stuttgart, Württemberg (now Germany) Died: 21 April 1825 in Halle, Saxony (now Germany)

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Friedrich Pfaff's father, Burkhard Pfaff, was chief financial councillor of Württemberg while his mother was the daughter of a member of the exchequer of Württemberg. It was a family with a tradition of working as civil servants for the government of Württemberg. Johann Friedrich was the second of his parents seven sons and, although perhaps the one to attain the greatest fame, he was certainly not the only one to excel in science. The youngest of the family, Johann Wilhelm Pfaff who was born in 1774, also became a mathematician and held chairs in Würtzburg and Erlangen. The second youngest, Christoph Heinrich Pfaff was born in 1773 and, with interests in chemistry, medicine and pharmacy, he worked with Volta on electricity in animals. There was a school in Stuttgart, the Hohe Karlsschule, which was run to train sons of government officials of Württemberg and Johann Friedrich attended this school from the age of nine. It was a rather uninspiring school, strong on discipline but less good academically. Pfaff did not learn much in the way of mathematics there despite attending the school until he was nearly twenty. When he left in the autumn of 1785 he had completed his studies in law, a fitting subject for a civil servant. Despite a lack of training in mathematics at his school, Pfaff had studied mathematics on his own and began to study the works of Euler. He was encouraged to move toward scientific topics by the Duke of Württemberg, and he spent two year studying at the University of Göttingen where he was taught mathematics by Kaestner and he also studied physics. From Göttingen, Pfaff moved to Berlin in the summer of 1787. There he studied astronomy under J E Bode, and Pfaff wrote his first paper which was on a problem in astronomy.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pfaff.html (1 of 3) [2/16/2002 11:26:32 PM]

Pfaff

In the spring of 1788 Pfaff set off on a journey to Vienna but he visited many universities on the way, in particular Halle, Jena, Helmstedt, Gotha, Dresden, and Prague. Klügel was professor of mathematics at Helmstedt and he accepted a chair at Halle leaving the position at Helmstedt vacant. Pfaff's physics professor at Göttingen recommended him for the chair, and Pfaff submitted a dissertation on the occasion of his election as professor of mathematics at the University of Helmstedt. It was a tradition that new professors at the university there submitted an inaugural dissertation. Pfaff's inaugural dissertation was titled Programma inaugurale in quo peculiarem differentialia investigandi rationem ex theoria functionum deducit. It investigates the use of some functional equations in order to calculate the differentials of logarithmic and trigonometrical functions as well as the binomial expansion and Taylor formula. This is studied in detail in [4]. From his appointment in 1788 until 1810 Pfaff held the chair at Helmstedt. His appointment was approved by the Duke of Württemberg but it was not the best of positions, since it was poorly paid. He did good work building the strength of mathematics there and he put much effort into teaching and was successful in increasing the number of students of mathematics. One student who studied at Helmstedt was Gauss. After studying at Göttingen, Gauss came to Helmstedt in 1798. He attended Pfaff's lectures and even lived in his house. Wussing writes in [1]:Pfaff recommended Gauss's doctoral dissertation and, when necessary, greatly assisted him; Gauss always retained a friendly memory of Pfaff both as a teacher and as a man. By the time Gauss studied with Pfaff at Helmstedt, the university was under threat of closure. Pfaff fought hard to prevent this and for a few years he was successful. Pfaff married in 1803 to Caroline Brand but sadly their first child died as an infant. By 1810 Pfaff's attempts to preserve the University of Helmstedt finally failed with the closure of the university. The staff were given a number of different choices as to which university they might move to, and Pfaff chose to move to Halle. He was appointed to the chair of mathematics at Halle in 1810 and in 1812, on the death of Klügel, he took on the directorship of the University Observatory too. Pfaff did important work in analysis working on partial differential equations, special functions and the theory of series. He developed Taylor's Theorem using the form with remainder as given by Lagrange. In 1810 he contributed to the solution of a problem due to Gauss concerning the ellipse of greatest area which could be drawn inside a given quadrilateral. His most important work on Pfaffian forms was published in 1815 when Pfaff was nearly fifty years old but its importance was not recognised until 1827 when Jacobi published a paper on Pfaff's method. This failure to recognise the importance of the work is strange, particularly given the very positive review which Gauss wrote of the work shortly after it was published. In the 1815 paper, which Pfaff submitted to the Berlin Academy on 11 May, he presented a transformation of a first-order partial differential equation into a differential system. This theory of equations in total differentials is undoubtedly Pfaff's most significant contribution. Wussing writes in [1] that this work by Pfaff:... constituted the starting point of a basic theory of integration of partial differential equations which, through the work of Jacobi, Lie, and others, has developed into a modern Cartan calculus of extreme differential forms. Among his other important works are Disquisitiones analytica maxime ad calculum integralem et http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pfaff.html (2 of 3) [2/16/2002 11:26:32 PM]

Pfaff

doctrinam serierum pertinentes (1797), an introductory work written in the style of Euler, and Observationes ad Euleri institutiones calculi integralis. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1810 to 1820

Other Web sites

1. Rouse Ball 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pfaff.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pfaff.html (3 of 3) [2/16/2002 11:26:32 PM]

Pfeiffer

Georgii Yurii Pfeiffer Born: 23 Dec 1872 in Sokyryntsi, Pryluka, Poltava gubernia, Ukraine Died: 10 Oct 1946 in Kiev, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georgii Pfeiffer graduated from the University of Kiev in 1896. He was appointed to the Kiev Polytechnic Institute in 1899 and he taught there until 1909. While holding this appointment he was also appointed to the University of Kiev in 1900 and he taught there until his death in 1946. Elected to the Academy of Sciences of the Ukraine in 1920, Pfeiffer chaired the Commission on Pure Mathematics from that time. Pfeiffer was also attached to the Institute of Mathematics of the Academy of Sciences of the Ukraine in Kiev during two periods, namely 1934 to 1941 and again from 1944 until his death in 1946. In the three years between these, namely 1941-44, Pfeiffer was in Ufa, the capital of the Bashkortostan republic in western Russia. In Ufa, Pfeiffer was Director of the Institute of Mathematics and Physics. Pfeiffer did important work on partial differential equations following on from the methods developed by Lie and Lagrange. He showed how to find integrals of a general system of partial differential equations by using sequential complete systems instead of passing to Jacobian systems. Pfeiffer also constructed all the infinitesimal operators of a system of equations. Summarising Pfeiffer's work in [1], KovAl-chuk says that Pfeiffer's methods greatly expanded the class of integrable systems, but have been neglected over the past half-century as functional-analytic methods have been in fashion. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pfeiffer.html (1 of 2) [2/16/2002 11:26:34 PM]

Pfeiffer

Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of Georgii Pfeiffer

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pfeiffer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pfeiffer.html (2 of 2) [2/16/2002 11:26:34 PM]

Philon

Philon of Byzantium Born: about 280 BC in Byzantium (Turkey) Died: about 220 BC Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Only a few references to Philon of Byzantium exist in the literature. He is mentioned by Vitruvius who was a Roman architect and engineer. Vitruvius (1st century BC) was the author of the famous treatise De architectura (On Architecture) and in this work he gives a list of twelve inventors of machines which include Archytas (second in the list), Archimedes (third in the list), Ctesibius (fourth in the list), and Philon of Byzantium (sixth in the list). Heron of Alexandria mentions a work by Philon On automatic theatres which in fact forms part of his Mechanics treatise. Eutocius also mentions Philon and cites a work by him on the duplication of the cube and this material is again contained in his Mechanics treatise. Perhaps the most information about Philon's life, and this is very little indeed, comes from the only work of his which has survived (at least major parts have survived) Mechanics. In this treatise he writes about the catapult which was recently invented by Ctesibius, who we mentioned above as coming before Philon in the list of inventors given by Vitruvius. From this information we can date Philon fairly accurately and we know that he wrote his treatise Mechanics around 250 BC. Before describing the contents of Philon's masterpiece Mechanics let us give some small details of Philon's life which can be deduced from comments which he makes in this text. Certainly Philon describes journeys he had made to Rhodes and to Alexandria to study catapults. He appears to have discussed military applications of catapults with the rulers of Alexandria. The tone here would suggest that Philon was a wealthy man of independent means able to travel in the pursuit of his studies. On the other hand it is possible that he was considered the right sort of person whose advice should be sought on military matters and he may have been earned his living advising military rulers. What exactly was in Philon's Mechanics treatise? We know that it had nine books: 1. Introduction 2. On the lever 3. On the building of seaports 4. On catapults 5. On pneumatics 6. On automatic theatres 7. On the building of fortresses 8. On besieging and defending towns 9. On stratagems http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Philon.html (1 of 3) [2/16/2002 11:26:35 PM]

Philon

The text of Books 4, 5, 7 and 8 has survived, while the rest has been lost. However Philon has the habit of cross referencing fully in his work so we can learn quite a bit about what was contained in the lost sections by studying the surviving ones. The style of the treatise is rather unusual since the books are composed of many short chapters. For example Book 8 consists of two sections with 75 chapters in the first section and 111 chapters in the second section. This treatise is not just a work on what we would consider today to be applied mathematics. For example Book 8, in addition to describing ways of defending town walls from both land and sea attack, also stresses how important it is to have a good doctor available. Philon argues that those badly injured in attacks so that they cannot work again should be awarded pensions, and that the wives of those killed should be provided for. To capture a town through a siege one must, according to Philon, make proper use of machines such as catapults and other war engines. In addition one must try to starve the inhabitants of the town, bribe suitable people to assist you, use his poison recipes to kill the inhabitants, and also use cryptography to pass secret messages. It would be interesting to have details of his proposed cryptography but unfortunately Philon's work on this topic has been lost. One important mathematical contribution by Philon was to the problem of duplicating the cube. At first sight this seems far removed from the topics we have noted that are in his treatise. However, this is not so for Philon examines the following problem. Given a catapult, how do you make a second catapult which can fire a missile twice as heavy as the first. To do this it is necessary to construct a machine whose linear dimensions are increased exactly the amount necessary for its volume (the cube of the linear dimension) to double. His method of duplicating the cube is similar to that due to Heron. The solution is effectively produced by the intersection of a circle and a rectangular hyperbola. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

Doubling the cube

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Philon.html (2 of 3) [2/16/2002 11:26:35 PM]

Philon

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Philon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Philon.html (3 of 3) [2/16/2002 11:26:35 PM]

Picard_Emile

Charles Emile Picard Born: 24 July 1856 in Paris, France Died: 11 Dec 1941 in Paris, France

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Emile Picard's father was the manager of a silk factory who died during the siege of Paris in 1870. The siege was a consequence of the Franco-German War which began on 19 July 1870. It went badly for France and on 19 September 1870 the Germans began a siege of Paris. This was a desperate time for the inhabitants of the town who killed their horses, cats and dogs for food. It was during this siege that Emile's father died. Paris surrendered on 28 January 1871 and The Treaty of Frankfurt, signed on 10 May 1871, was an humiliation for France. Picard's mother, the daughter of a medical doctor, was put in an extremely difficult position when her husband died. As well as Emile, she had a second young son, and in order to support them through their education she had to find employment. Only her determination to give her sons a good start, despite the tragedy, allowed Emile to receive the education which gave him the chance to achieve the highest international stading in mathematics. Picard's secondary education was at the Lycée Napoléon, later called the Lycée Henri IV. Strangely he was a brilliant pupil at almost all his subjects, particularly in translating Greek and Latin poetry, but he disliked mathematics. He himself wrote that he hated geometry but he:... learned it by heart in order to avoid being punished. It was only during the vacation after completing his secondary studies that Picard read an algebra book and suddenly he became fascinated in mathematics. He took the entrance examinations for Ecole Polytechnique and Ecole Normale Supérieure; he was placed second and first respectively in the two examinations. Hadamard wrote in [8]:As any young Frenchman of our time who was gifted in science, he was obliged to choose http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Picard_Emile.html (1 of 4) [2/16/2002 11:26:37 PM]

Picard_Emile

between the Ecole Polytechnique which, in principle, prepared one to be an engineer, and the Ecole Normale, with its pure scientific orientation. He was ranked first and chose the latter. It is said that he made this decision after an exciting visit to Pasteur, during which the father of bacteriology spoke about pure science in such lofty terms that the young man was completely persuaded. Picard received his agrégation in 1877, being placed first. He remained at the Ecole Normale Supérieure for a year where he was employed as an assistant. He was appointed lecturer at the University of Paris in 1878 and then professor at Toulouse in 1879. In 1881 he returned to Paris when appointed maître de conference in mechanics and astronomy at the Ecole Normale. In 1881 Picard was nominated for membership of the mathematics section of the Académie des Sciences. It says much of the extraordinary ability that he was showing at such a young age that he was nominated. He had already proved two important theorems which are both today known under Picard's name, yet it was still a little soon to gain admission to the prestigious academy and he would have to wait a few more years. In this year of his first nomination he married Hermite's daughter. Picard and his wife had three children, a daughter and two sons, who were all killed in World War I. His grandsons were wounded and captured in World War II. In 1885 Picard was appointed to the chair of differential calculus at the Sorbonne in Paris when the chair fell vacant on the death of Claude Bouquet. However a university regulation prevented anyone below the age of thirty holding a chair. The regulations were circumvented by making Picard his own suppléant until he reached the age of thirty which was in the following year. He requested exchanging his chair for that of analysis and higher algebra in 1897 so that he was able to train research students. Picard made his most important contributions in the fields of analysis, function theory, differential equations, and analytic geometry. He used methods of successive approximation to show the existence of solutions of ordinary differential equations solving the Cauchy problem for these differential equations. Starting in 1890, he extended properties of the Laplace equation to more general elliptic equations. Picard's solution was represented in the form of a convergent series. In 1879 he proved that an entire function which is not constant takes every value an infinite number of times, with one possible exception. Picard used the theory of Hermite's modular functions in the proof of this important result. Building on work by Abel and Riemann, Picard's study of the integrals attached to algebraic surfaces and related topological questions developed into an important part of algebraic geometry. On this topic he published, with Georges Simart, Théorie des fonctions algébriques de deux variables indépendantes which was a two volume work, the first volume appearing in 1897 and the second in 1906. Picard also discovered a group, now called the Picard group, which acts as a group of transformations on a linear differential equation. His three volume masterpiece Traité d'analyse was published between 1891 and 1896. The treatise [1]:... immediately became a classic and was revised with each subsequent edition. The work was accessible to many students through its range of subjects, clear exposition, and lucid style. Picard examined several specific cases before discussion his general theory. Picard also applied analysis to the study of elasticity, heat and electricity. He studied the transmission of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Picard_Emile.html (2 of 4) [2/16/2002 11:26:37 PM]

Picard_Emile

electrical pulses along wires finding a beautiful solution to the problem. As can be seen his contributions were both wide ranging and important. Among the honours given to Picard was his election to the Académie des Sciences in 1889, eight years after he was first unsuccessfully nominated. He later served the Academy as its permanent secretary from 1917 until his death in 1941. In this role [1]:... he wrote an annual notice on either a scientist or a subject of current interest. He also wrote many prefaces to mathematical books and participated in the publication of works of C Hermite and G H Halphen. Picard was awarded the Poncelet Prize in 1886 and the Grand Prix des Sciences Mathématiques in 1888. In addition to honorary doctorates from five universities and honorary membership of thirty-seven learned societies he received the Grande Croix de la Légion d'Honneur in 1932 and the Mittag-Leffler Gold Medal in 1937. He became a member of the Académie Française in 1924. Another honour was given to him was making him President of the International Congress of Mathematicians at Strasbourg in September 1920. Hadamard had this to say of Picard as a teacher when he addressed him in 1937:You were able to make [mechanics] almost interesting; I have always wondered how you went about this, because I was never able to do it when it was my turn. But you also escaped, you introduced us not only to hydrodynamics and turbulence, but to many other theories of mathematical physics and even of infinitesimal geometry; all this in lectures, the most masterly I have heard in my opinion, where there was not one word too many nor one word too little, and where the essence of the problem and the means used to overcome it appeared crystal clear, with all secondary details treated thoroughly and at the same time consigned to their right place. Hadamard wrote in [8]:A striking feature of Picard's scientific personality was the perfection of his teaching, one of the most marvellous, if not the most marvellous, that I have ever known. It is a remarkable fact that between 1894 and 1937 he trained over 10000 engineers who were studying at the Ecole Centrale des Arts et Manufactures. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) A Poster of Emile Picard

Mathematicians born in the same country

Honours awarded to Emile Picard (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1909

Other Web sites

Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Picard_Emile.html (3 of 4) [2/16/2002 11:26:37 PM]

Picard_Emile

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Picard_Emile.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Picard_Emile.html (4 of 4) [2/16/2002 11:26:37 PM]

Picard_Jean

Jean Picard Born: 21 July 1620 in La Flèche, France Died: 12 Oct 1682 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean Picard studied at the Jesuit college at La Flèche. He worked with Gassendi for some time and helped him with observations of a solar eclipse on 21 August 1645. He received an M.A. from the University of Paris is 1650. In 1655 Picard became professor of astronomy at the Collège de France in Paris. He became a member of the Académie Royale des Sciences in 1666, just after its foundation, and from this time on devoted himself to working for the Académie. Picard devised a micrometer to measure the diameters of celestial objects such as the Sun, Moon and planets. In 1667 Picard added a telescope to the quadrant making it much more useful in observations. He greatly increased the accuracy of measurements of the Earth, using Snell's method of triangulation. He measured the length of the arc of the meridian; the measurements appear in Mesure de la Terre (1671). This data was used by Newton in his gravitational theory. In the same year Picard went to Tycho Brahe's observatory at Hven Island in Sweden so that its location could be determined accurately and so Tycho observations could be directly compared with others. In 1673 Picard moved to the Paris Observatory where he collaborated with Cassini, Römer and, slightly later, with La Hire. Picard was one of the first to apply scientific methods to the making of maps. He produced a map of the Paris region, then went on to join a project to map France. Among his other skills were hydraulics, a topic on which he wrote but one where he put his skills into practice; he solved the problem of supplying the fountains at Versailles with water. Picard corresponded with many of the leading scientists of his time including Bartholin, Hevelius, Hudde and Huygens. Article by: J J O'Connor and E F Robertson List of References (5 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Picard_Jean.html (1 of 2) [2/16/2002 11:26:38 PM]

Picard_Jean

Mathematicians born in the same country Cross-references to History Topics

1. Longitude and the Académie Royale 2. English attack on the Longitude Problem

Honours awarded to Jean Picard (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Picard

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Picard_Jean.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Picard_Jean.html (2 of 2) [2/16/2002 11:26:38 PM]

Pieri

Mario Pieri Born: 22 June 1860 in Lucca, Italy Died: 1 March 1913 in S Andrea di Compito (near Lucca), Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mario Pieri entered the University of Bologna in 1880 and there his talents for mathematics were quickly spotted by Pincherle. Pieri was awarded a scholarship to study at the Scuola Normale Superiore in Pisa and he began his studies there in November 1881. His graduation was on 27 June 1884 and, after graduating, he began teaching at the Technical School in Pisa. After teaching only a short while in Pisa, Pieri moved to the Military Academy in Turin where he became professor of projective geometry. In 1888 he also was appointed an assistant in projective geometry at the University of Turin. In 1891 Pieri received his doctorate from the University of Turin and he taught projective geometry courses there for several years. In 1900 Pieri left Turin to take up an appointment at the University of Catania in eastern Sicily, after winning the competition for a chair. In Catania he taught projective geometry and descriptive geometry. After spending eight years in Sicily, Pieri moved to the north of Italy, taking up an appointment in Parma. Pieri's main area was projective geometry and he is an important member of the Italian School of Geometers. However, after he moved to Turin, Pieri became influenced by Peano at the University and Burali-Forti who was a colleague at the Military Academy. This influence led Pieri to study the foundations of geometry. In 1895 he set up an axiomatic system for projective geometry with three undefined terms, namely points, lines and segments. He improved on results of Pasch and Peano and then, in 1905, Pieri gave the first axiomatic definition of complex projective geometry which does not build on real projective geometry. In 1898 Pieri published the memoir The principles of the geometry of position through the Academy of Sciences of Turin. Russell was impressed by this memoir and wrote, in his Principia :This is, in my opinion, the best work on the present subject. Pieri attended both the Congress of Philosophy and the International Congress of Mathematicians in Paris in 1900. At the first of these conferences he lectured on Geometry considered as a purely logical system and certainly impressed Hans Freudenthal who wrote:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pieri.html (1 of 2) [2/16/2002 11:26:39 PM]

Pieri

In the field of the philosophy of sciences the Italian phalanx was supreme: Peano, Burali-Forti, Padoa, Pieri absolutely dominated the discussion. The first Lobachevsky Prize was awarded to Lie in 1897. Pieri submitted an entry for the Lobachevsky Prize on the third time the Prize was offered. He received an 'honourable mention', as did Barbarin, Lemoine and Study, while the Prize went to Hilbert for the 1903 edition of his Die Grundlagen der Geometrie. In 1911 Pieri became interested the the vector calculus through the work of Burali-Forti and Marcolongo. However, around this time his health began to fail and cancer was diagnosed. His mathematical work came to an abrupt end at a time when he was at the height of his creative powers. Peano in [8] writes:Pieri was totally dedicated to science and teaching. He was an untiring worker, honest, and of a singular modesty. When, some twenty years ago, the professors in Italy agitated for higher salaries, Pieri declared that their salaries were already above the work they did and their merit. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pieri.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pieri.html (2 of 2) [2/16/2002 11:26:39 PM]

Pillai

K C Sreedharan Pillai Born: 24 Feb 1920 in Kerala, India Died: 5 June 1985 in Lafayette, Indiana, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sree Pillai studied at the University of Travancore in Trivandrum. In 1937, just after Pillai began his studies there, the University of Travancore changed its name to the University of Kerala. He graduated in 1941 and obtained his Master's Degree in 1945. Pillai was appointed a lecturer at the University of Kerala in 1945 and worked there for six years until he went to the United States in 1951. After studying for one year at Princeton, Pillai went to the University of North Carolina where he was awarded a doctorate in statistics in 1954. His first post was as a statistician with the United Nations, a post he held from 1954 until 1962. Part of his duties in this post involved him founding the Statistical Center of the University of the Philippines. He was a visiting Professor and Advisor to the University over a number of years and supervised graduate students there. In 1962 Pillai was appointed Professor of Statistics and Mathematics at the University of Purdue. In [2] his contributions to Purdue as described as follows:In the 23 years he served Purdue, he directed the research of 15 Ph.D. students. He was also an active consultant on several projects both within and outside the University. He was a close friend of his students and maintained a correspondence with most of them, some of whom are in remote parts of the world. Pillai's research was in statistics, in particular in multivariate statistical analysis. In [2] his work is described:... he obtained the probability distributions of statistics relating to several multivariate http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pillai.html (1 of 2) [2/16/2002 11:26:41 PM]

Pillai

procedures. Perhaps his best known contribution is the widely used multivariate analysis of variance test which bears his name. Pillai was honoured by being elected a Fellow of the American Statistical Association and a Fellow of the Institute of Mathematical Statistics. He was an elected member of the International Statistics Institute. As well as his work at Purdue in developing the graduate programmes these Pillai was a keen golfer. This is described in [2]:His unique and unforgettable style charmed his playing companions and confused his opponents in the Purdue Staff League. His performances in the League matches were legendary. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pillai.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pillai.html (2 of 2) [2/16/2002 11:26:41 PM]

Pincherle

Salvatore Pincherle Born: 11 March 1853 in Trieste, Austria (now Italy) Died: 10 July 1936 in Bologna, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Salvatore Pincherle was born in Trieste (part of Austria at the time) into a Jewish family. His father was a business man and, after Pincherle had undertaken part of his education in Trieste, he moved to Marseilles taking his family with him. So Pincherle completed his school education in Marseilles. When he was at school in Trieste, Pincherle's interests were in the humanities, but the school in Marseilles specialised in science teaching and Pincherle soon became fascinated with mathematics through excellent teaching there. Leaving Marseilles when his schooling was complete, Pincherle entered the University of Pisa in 1869 to study mathematics. A student of Betti and taught by Dini, Pincherle was strongly influenced by both men. Graduating in 1874 he taught in a school in Pavia but won a scholarship to enable him to study abroad for a year. Pincherle spent his year abroad in Germany, studying at the University of Berlin. There he was strongly influenced by Weierstrass during 1877-78 and all his mathematical work from this time on shows the influence of the great mathematician. In 1880 Pincherle was appointed to the chair of infinitesimal calculus at the University of Palermo. It was a post he only held for a few months for he was offered the chair of mathematics at Bologna and accepted the post which he continued to hold until he retired in 1928. Pincherle worked on functional equations and functional analysis. Together with Volterra, he can claim to be one of the founders of functional analysis. Tricomi writes in [1]:Remaining faithful to the ideas of Weierstrass, he did not take the topological approach that later proved to be most successful, but tried to start from a series of powers of the D http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pincherle.html (1 of 2) [2/16/2002 11:26:43 PM]

Pincherle

derivation symbol. Although his efforts did not prove very fruitful, he was able to study in depth the Laplace transform, iteration problems, and series of generalised factors. Pincherle contributed to the development and dissemination of Weierstrass's development of a theory of analytic functions. He wrote an expository paper in 1880 which was published in the Giornale di Matematiche which was inspired by the lectures of Weierstrass. This work is important both in the development of analysis and in particular the progress of mathematics in Italy. The Italian Mathemtical Union was established in Bologna by Pincherle in 1922. He became its first President. He was also President of the Third International Congress of Mathematicians which was held in Bologna in 1928. It was through his efforts that German mathematicians were allowed to attend the Congress; they had been banned previously due to World War I.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Cross-references to History Topics

Abstract linear spaces

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pincherle.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pincherle.html (2 of 2) [2/16/2002 11:26:43 PM]

Pitiscus

Bartholomeo Pitiscus Born: 24 Aug 1561 in Grünberg, Silesia (now Zielona, Poland) Died: 2 July 1613 in Heidleberg, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bartholomeo Pitiscus studied theology, first at Zerbst, then at Heidelberg. He was a Calvinist, studying Calvinist theology. Pitiscus's future was much tied to Friedrich der Aufrichtige, known as Frederick IV, elector Palatine of the Rhine. Frederick's father died in 1583 and John Casimir, his uncle, became guardian to the ten year old Frederick. John Casimir was an ardent Calvinist and appointed Pitiscus to teach Frederick in 1584. Later Pitiscus was appointed court chaplain at Breslau and court preacher to Frederick. When John Casimir died in 1592 Frederick undertook the government of the Palatinate continuing his uncle's policies of hostility to the Catholic Church. Pitiscus strongly supported the Calvinist policies from a major position of influence. The word 'trigonometry' is due to Pitiscus in the title of a book Trigonometria published in Heidelberg in 1595. It consists of 5 books on plane and spherical trigonometry. It was translated into English in 1614 and into French in 1619. He also published Thesaurus mathematicus. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country Cross-references to History Topics

The trigonometric functions

Other references in MacTutor

Chronology: 1500 to 1600

Honours awarded to Bartholomeo Pitiscus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Pitiscus

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pitiscus.html (1 of 2) [2/16/2002 11:26:44 PM]

Pitiscus

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pitiscus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pitiscus.html (2 of 2) [2/16/2002 11:26:44 PM]

Plana

Giovanni Antonio Amedeo Plana Born: 6 Nov 1781 in Voghera, Italy Died: 20 Jan 1864 in Turin, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giovanni Plana's father was Antonio Maria Plana while his mother was Giovanna Giacoboni. Two of Giovanni's uncles lived in Grenoble and his father sent him there in 1796, at age about 15, to complete his schooling. In Grenoble he became a close friend of Henri Beyle, the famous French author better known by his pseudonym Stendhal. We should mention the political events which occurred in Europe around this time and had a large influence on the lives of many of the mathematicians we study in this archive. In April 1796 the French army under the command of Napoleon had invaded Italy. There followed a period when many intellectuals moved to a new way of thinking and enthusiastically took part in the democratic movement. Plana, although a young man at the time, was certainly one of those who embraced this new way. This aspect of his life between 1796 and 1799 is studied in [4]. In 1800 Plana entered the Ecole Polytechnique in Paris. There he was greatly influenced by Lagrange who was one of his teachers and, of course, also someone of Italian birth. Fourier, like Lagrange, was greatly impressed by Plana's abilities and he tried to arrange for him to be appointed to the chair of mathematics at the school of artillery at Grenoble. However, Fourier failed in his attempt on behalf of Plana, so he tried again, this time to have Plana appointed to the chair of mathematics at the school of artillery in the Piedmont which was located part in Turin and part in Alessandria. This attempt was successful and Plana returned to Italy in 1803 to take up this post. After further victories by Napoleon, the Treaty of Pressburg was signed by Austria and France on 26 December 1805. Austria was forced to give up much territory to France, and in particular Napoleon wanted Austrian influence completely removed from Italy. As a consequence Piedmont was given to France so Plana found himself in France again without making a move! In 1811 Lagrange recommended Plana for the chair of astronomy at the University of Turin, and Plana was appointed to the position. He would teach in Turin for the rest of his life, teaching both astronomy and mathematics, and teaching both at the university and at the school of artillery there. Topics Plana worked on, in addition to astronomy, were integrals, elliptic functions, heat, electrostatics and geodesy. In astronomy his most famous work relates to the motion of the moon. Plana had already worked with Francesco Carlini on geodesy, and the director of the observatory in Milan suggested to Plana that he might collaborate with Carlini on problems relating to the motion of the moon. This episode http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plana.html (1 of 3) [2/16/2002 11:26:45 PM]

Plana

is described in detail in [5] and we quote here part of the summary of that paper:In 1818 Laplace proposed that the Académie des Sciences in Paris set up a prize to be awarded to whoever succeeded in constructing lunar tables based solely on the law of universal gravity. In 1820 the prize was awarded to Carlini and Plana and to Damoiseau by a committee of which Laplace was a member. But Laplace strongly criticised the Carlini-Plana approach to lunar theory. A dispute ensued ... letters [were] exchanged between Carlini-Plana and Laplace, and ... papers published in Connaissance des temps and in Zach's Correspondance. After the exchanges, public and private, between Carlini-Plana and Laplace, the latter concluded that the results of the Italian astronomers and those arrived at by Damoiseau following the method of Laplace's Mecanique celeste were fairly close, and that the purpose of the Académie in establishing the prize had been reasonably fulfilled. In fact Plana fell out with Carlini and Carlini withdrew from the collaboration. Plana continued with the work on his own and published Théorie du mouvement de la lune in Turin in 1832. By this time Plana was astronomer royal, and he went on to became a hereditary baron in 1844 and a senator in 1848. When he was nearly 80 years old, in 1860, he was elected a member of the Académie des Sciences in Paris. Finally let us return to talk a little of Plana's association with the Turin Academy of Sciences. The article [3] describing the Academy of Sciences states that Plana's studies on lunar motion were the most important presented to the Academy at the beginning of the nineteenth century. An important event for mathematics in Turin occurred when Cauchy lived in Turin during the year 1832-33. He held the chair of mathematical physics at the University of Turin during a period when he was highly active in research, yet political events forced him to leave Paris. Cauchy and Plana certainly interacted during this time and their relationship is discussed in [6]. Another famous scientist who interacted with Plana was Babbage. He was elected to the Turin Academy of Sciences in 1841 and the article [7] examines his relations with the Academy and with some of its members, including in particular Plana. Perhaps the most important aspect of Plana's contribution is summed up by Tricomi in [1] as follows:Plana is generally considered one of the major Italian scientists of his age because, at a time when the quality of instruction at Italian universities had greatly deteriorated, his teaching was of the highest quality, quite comparable with that of the grandes écoles of Paris, at which he had studied. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plana.html (2 of 3) [2/16/2002 11:26:45 PM]

Plana

Honours awarded to Giovanni Plana (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1827

Royal Society Copley Medal

Awarded 1834

Lunar features

Crater Plana

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Plana.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plana.html (3 of 3) [2/16/2002 11:26:45 PM]

Planck

Max Karl Ernst Ludwig Planck Born: 23 April 1858 in Kiel, Schleswig-Holstein, Germany Died: 4 Oct 1947 in Göttingen, Germany

Click the picture above to see nine larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Max Planck came from an academic family, his father being professor of law at Kiel and both his grandfather and great-grandfather had been professors of theology at Göttingen. In 1867 Planck's family moved to Munich and he attended school there. He did well at school, but not brilliantly, usually coming somewhere between third and eighth in his class. In 1874, at the age of 16, he entered the University of Munich. Before he began his studies he discussed the prospects of research in physics with Philipp von Jolly, the professor of physics there, and was told that physics was essentially a complete science with little prospect of further developments. Fortunately Planck decided to study physics despite the bleak future for research that was presented to him. In [7] Planck describes why he chose physics:The outside world is something independent from man, something absolute, and the quest for the laws which apply to this absolute appeared to me as the most sublime scientific pursuit in life. Planck then studied at Berlin where his teachers included Helmholtz and Kirchhoff. He later wrote that he admired Kirchhoff greatly but found him dry and monotonous as a teacher. Planck returned to Munich and received his doctorate at the age of 21 with a thesis on the second law of thermodynamics. He was then appointed to a teaching post at the University of Munich in 1880 and he taught there until 1885. In 1885 Planck was appointed to a chair in Kiel and held this chair for four years. After the death of Kirchhoff in 1887, Planck succeeded him in the chair of theoretical physics at the University of Berlin in 1889. He was to hold the Berlin chair for 38 years until he retired in 1927. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Planck.html (1 of 3) [2/16/2002 11:26:47 PM]

Planck

While in Berlin Planck did his most brilliant work and delivered outstanding lectures. He studied thermodynamics in particular examining the distribution of energy according to wavelength. By combining the formulas of Wien and Rayleigh, Planck announced in 1900 a formula now known as Planck's radiation formula. In a letter written a year later Planck described proposing the formula saying:... the whole procedure was an act of despair because a theoretical interpretation had to be found at any price, no matter how high that might be. Within two months Planck made a complete theoretical deduction of his formula renouncing classical physics and introducing the quanta of energy. At first the theory met resistance but due to the successful work of Niels Bohr in 1913, calculating positions of spectral lines using the theory, it became generally accepted. Planck himself in [7] explains how despite having invented quantum theory he did not understand it himself at first:I tried immediately to weld the elementary quantum of action somehow in the framework of classical theory. But in the face of all such attempts this constant showed itself to be obdurate ... My futile attempts to put the elementary quantum of action into the classical theory continued for a number of years and they cost me a great deal of effort. Planck received the Nobel Prize for Physics in 1918. Planck took little part in the further development of quantum theory, this being left to Paul Dirac and others. Planck took on administrative duties such as Secretary of the Mathematics and Natural Science Section of the Prussian Academy of Sciences, a post he held from 1912 until 1943. He had been elected to the Academy in 1894. Planck was president of the Kaiser Wilhelm Gesellschaft, the main German research organisation, from 1930 until 1937. He remained in Germany during World War II through what must have been times of the deepest difficulty since his son Erwin was executed for plotting to assassinate Hitler. In [4] Heilbron describes the impact of wars on Planck and his family:He would remember, even in his old age, the sight of Prussian and Austrian troops marching into his native town when he was six years old. Throughout his life, war would cause him deep personal sorrow. He lost his eldest son during World War I. In World War II, his house in Berlin was burned down in an air raid. In 1945 his other son was executed when declared guilty of complicity in a plot to kill Hitler. After World War II he again became president of the Kaiser Wilhelm Gesellschaft in 1945-1946 for the second time defending German science through another period of exceptional difficulty. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles)

Some Quotations (3)

A Poster of Max Planck

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Planck.html (2 of 3) [2/16/2002 11:26:47 PM]

Planck

Cross-references to History Topics

1. The quantum age begins 2. General relativity 3. Pi through the ages 4. Special relativity

Other references in MacTutor

Chronology: 1900 to 1910

Honours awarded to Max Planck (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1918

Fellow of the Royal Society

Elected 1926

Royal Society Copley Medal

Awarded 1929

Lunar features

Crater Planck

Other Web sites

1. Nobel prizes site (A biography of Planck and his Nobel prize presentation speech) 2. University of Glasgow 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Planck.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Planck.html (3 of 3) [2/16/2002 11:26:47 PM]

Plateau

Joseph Antoine Ferdinand Plateau Born: 14 Oct 1801 in Brussels, Belgium Died: 15 Sept 1883 in Ghent, Belgium

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Plateau was a physicist who is best remembered in mathematics for Plateau Problems. He used a solution of soapy water and glycerine and dipped wire contours into it, noting that the surfaces formed were minimal surfaces. Plateau did not have the mathematical skills to investigate the problem theoretically but Weierstrass, Riemann and Schwarz worked on the problem which was finally solved by Douglas and Radó. Plateau wrote some mathematical work on number theory and wrote a joint article with Quetelet. He was blind for the last 40 years of his life after he experimented by staring at the Sun for 25 seconds. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to Famous Curves

Plateau Curves

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plateau.html (1 of 2) [2/16/2002 11:26:49 PM]

Plateau

Honours awarded to Joseph Plateau (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1870

Other Web sites

The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Plateau.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plateau.html (2 of 2) [2/16/2002 11:26:49 PM]

Plato

Plato Born: 427 BC in Athens, Greece Died: 347 BC in Athens, Greece

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Before giving details of Plato's life we will take a few moments to discuss how definite the details are which we give below. The details are mostly given by Plato himself in letters which seem, on the face of it, to make them certain. However, it is disputed whether Plato did indeed write the letters so there are three possible interpretations. Firstly that Plato wrote the letters and therefore the details are accurate. Secondly that although not written by Plato, the letters were written by someone who knew him or at least had access to accurate information on his life. The third possibility, which unfortunately cannot be ruled out, is that they were written by someone as pure fiction. Next we should comment on the name 'Plato'. In [13] Rowe writes:It was claimed that Plato's real name was Aristocles, and that 'Plato' was a nickname (roughly 'the broad') derived either from the width of his shoulders, the results of training for wrestling, or from the breadth of his style, or from the size of his forehead. Plato was the youngest son of Ariston and Perictione who both came from famous wealthy families who had lived in Athens for generations. While Plato was a young man his father died and his mother remarried, her second husband being Pyrilampes. It was mostly in Pyrilampes' house that Plato was brought up. Aristotle writes that when Plato was a young man he studied under Cratylus who was a student of Heracleitus, famed for his cosmology which is based on fire being the basic material of the universe. It almost certain that Plato became friends with Socrates when he was young, for Plato's mother's brother Charmides was a close friend of Socrates. The Peloponnesian War was fought between Athens and Sparta between 431 BC and 404 BC. Plato was

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plato.html (1 of 5) [2/16/2002 11:26:53 PM]

Plato

in military service from 409 BC to 404 BC but at this time he wanted a political career rather than a military one. At the end of the war he joined the oligarchy of the Thirty Tyrants in Athens set up in 404 BC, one of whose leaders being his mother's brother Charmides, but their violent acts meant that Plato quickly left. In 403 BC there was a restoration of democracy at Athens and Plato had great hopes that he would be able to enter politics again. However, the excesses of Athenian political life seem to have persuaded him to give up political ambitions. In particular, the execution of Socrates in 399 BC had a profound effect on him and he decided that he would have nothing further to do with politics in Athens. Plato left Athens after Socrates had been executed and travelled in Egypt, Sicily and Italy. In Egypt he learnt of a water clock and later introduced it into Greece. In Italy he learned of the work of Pythagoras and came to appreciate the value of mathematics. This was an event of great importance since from the ideas Plato gained from the disciples of Pythagoras, he formed his idea [6]:... that the reality which scientific thought is seeking must be expressible in mathematical terms, mathematics being the most precise and definite kind of thinking of which we are capable. The significance of this idea for the development of science from the first beginnings to the present day has been immense. Again there was a period of war and again Plato entered military service. It was claimed by later writers on Plato's life that he was decorated for bravery in battle during this period of his life. It is also thought that he began to write his dialogues at this time. On his return to Athens Plato founded, in about 387 BC, on land which had belonged to Academos, a school of learning which being situated in the grove of Academos was called the Academy. Plato presided over his Academy in Athens, an institution devoted to research and instruction in philosophy and the sciences, from 387 BC until his death. His reasons for setting up the Academy were connected with his earlier ventures into politics. He had been bitterly disappointed with the standards displayed by those in public office and he hoped to train, in his Academy, young men who would become statesmen. However, having given them the values that Plato believed in, Plato thought that these men would be able to improve the political leadership of the cities of Greece. Only two further episodes in Plato's life are recorded. He went to Syracuse in 367 BC following the death of Dionysius I who had ruled the city. Dion, the brother-in-law of Dionysius I, persuaded Plato to come to Syracuse to tutor Dionysius II, the new ruler. Plato did not expect the plan to succeed but because both Dion and Archytas of Tarentum believed in the plan then Plato agreed. Their plan was that if Dionysius II was trained in science and philosophy he would be able to prevent Carthage invading Sicily. However, Dionysius II was jealous of Dion who he forced out of Syracuse and the plan, as Plato had expected, fell apart. Plato returned to Athens, but visited Syracuse again in 361 BC hoping to be able to bring the rivals together. He remained in Syracuse for part of 360 BC but did not achieve a political solution to the rivalry. Dion attacked Syracuse in a coup in 357, gained control, but was murdered in 354. Field writes in [6] that Plato's life:... makes it clear that the popular conception of Plato as an aloof unworldly scholar, spinning theories in his study remote from practical life, is singularly wide of the mark. On

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plato.html (2 of 5) [2/16/2002 11:26:53 PM]

Plato

the contrary, he was a man of the world, an experienced soldier, widely travelled, with close contacts with many of the leading men of affairs, both in his own city and elsewhere. Plato's main contributions are in philosophy, mathematics and science. However, it is not as easy as one might expect to discover Plato's philosophical views. The reason for this is that Plato wrote no systematic treatise giving his views, rather he wrote a number of dialogues (about 30) which are written in the form of conversations. Firstly we should comment on what superb pieces of literature these dialogues are [6]:They show the mastery of language, the power of indicating character, the sense of a situation, and the keen eye for both its tragic and its comic aspects, which set Plato among the greatest writers of the world. He uses these gifts to the full in inculcating the lessons he wants to teach. In letters written by Plato he makes it clear that he understands that it will be difficult to work out his philosophical theory from the dialogues but he claims that the reader will only understand it after long thought, discussion and questioning. The dialogues do not contain Plato as a character so he does not declare that anything asserted in them are his own views. The characters are historic with Socrates usually the protagonist so it is not clear how much these characters express views with which they themselves would have put forward. It is thought that, at least in the early dialogues, the character of Socrates expresses views that Socrates actually held. Through these dialogues, Plato contributed to the theory of art, in particular dance, music, poetry, architecture, and drama. He discussed a whole range of philosophical topics including ethics, metaphysics where topics such as immortality, man, mind, and Realism are discussed. He discussed the philosophy of mathematics, political philosophy where topics such as censorship are discussed, and religious philosophy where topics such as atheism, dualism and pantheism are considered. In discussing epistemology he looked at ideas such as a priori knowledge and Rationalism. In his theory of Forms, Plato rejected the changeable, deceptive world that we are aware of through our senses proposing instead his world of ideas which were constant and true. Let us illustrate Plato's theory of Forms with one of his mathematical examples. Plato considers mathematical objects as perfect forms. For example a line is an object having length but no breadth. No matter how thin we make a line in the world of our senses, it will not be this perfect mathematical form, for it will always have breadth. In the Phaedo Plato talks of objects in the real world trying to be like their perfect forms. By this he is thinking of thinner and thinner lines which are tending in the limit to the mathematical concept of a line but, of course, never reaching it. Another example from the Phaedo is given in [6]:The instance taken there is the mathemtical relation of equality, and the contrast is drawn between the absolute equality we think of in mathematics and the rough, approximate equality which is what we have to be content with in dealing with objects with our senses. Again in the Republic Plato talks of geometrical diagrams as imperfect imitations of the perfect mathematical objects which they represent. Plato's contributions to the theories of education are shown by the way that he ran the Academy and his idea of what constitutes an educated person. He also contributed to logic and legal philosophy, including rhetoric. Although Plato made no important mathematical discoveries himself, his belief that mathematics http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plato.html (3 of 5) [2/16/2002 11:26:53 PM]

Plato

provides the finest training for the mind was extremely important in the development of the subject. Over the door of the Academy was written:Let no one unversed in geometry enter here. Plato concentrated on the idea of 'proof' and insisted on accurate definitions and clear hypotheses. This laid the foundations for Euclid's systematic approach to mathematics. In [2] his contributions to mathematics through his students are summarised:All of the most important mathematical work of the 4th century was done by friends or pupils of Plato. The first students of conic sections, and possibly Theaetetus, the creator of solid geometry, were members of the Academy. Eudoxus of Cnidus - author of the doctrine of proportion expounded in Euclid's "Elements", inventor of the method of finding the areas and volumes of curvilinear figures by exhaustion, and propounder of the astronomical scheme of concentric spheres adopted and altered by Aristotle - removed his school from Cyzicus to Athens for the purpose of cooperating with Plato; and during one of Plato's absences he seems to have acted as the head of the Academy. Archytas, the inventor of mechanical science, was a friend and correspondent of Plato. In mathematics Plato's name is attached to the Platonic solids. In the Timaeus there is a mathematical construction of the elements (earth, fire, air, and water), in which the cube, tetrahedron, octahedron, and icosahedron are given as the shapes of the atoms of earth, fire, air, and water. The fifth Platonic solid, the dodecahedron, is Plato's model for the whole universe. Plato's beliefs as regards the universe were that the stars, planets, Sun and Moon move round the Earth in crystalline spheres. The sphere of the Moon was closest to the Earth, then the sphere of the Sun, then Mercury, Venus, Mars, Jupiter, Saturn and furthest away was the sphere of the stars. He believed that the Moon shines by reflected sunlight. Perhaps the best overview of Plato's views can be gained from examining what he thought that a proper course of education should consist. Here is his course of study [2]:... the exact sciences - arithmetic, plane and solid geometry, astronomy, and harmonics would first be studied for ten years to familiarise the mind with relations that can only be apprehended by thought. Five years would then be given to the still severer study of 'dialectic'. Dialectic is the art of conversation, of question and answer; and according to Plato, dialectical skill is the ability to pose and answer questions about the essences of things. The dialectician replaces hypotheses with secure knowledge, and his aim is to ground all science, all knowledge, on some 'unhypothetical first principle'. Plato's Academy flourished until 529 AD when it was closed down by the Christian Emperor Justinian who claimed it was a pagan establishment. Having survived for 900 years it is the longest surviving university known. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (43 books/articles)

Some Quotations (16)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plato.html (4 of 5) [2/16/2002 11:26:53 PM]

Plato

A Poster of Plato

Mathematicians born in the same country

Cross-references to History Topics

1. Greek Astronomy 2. Doubling the cube 3. Trisecting an angle

Other references in MacTutor

1. Platonic solids 2. Chronology: 500BC to 1AD

Honours awarded to Plato (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Plato

Other Web sites

1. The Catholic Encyclopedia 2. Gutenberg Project (Some of Plato's works) 3. Columbia University (Texts by Plato) 4. Evansville 5. S M Cohen (Plato's Phaedo) 6. S M Cohen (Plato's Timaeus) 7. Kevin Brown (Platonic solids) 8. Mark Harden's Artchive (The School of Athens by Raphael) 9. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Plato.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plato.html (5 of 5) [2/16/2002 11:26:53 PM]

Playfair

John Playfair Born: 10 March 1748 in Benvie (near Dundee), Scotland Died: 20 July 1819 in Burntisland, Fife, Scotland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Playfair was the eldest son of the Reverend James Playfair, minister of Benvie, a small town near Dundee (then in Perthshire, now in Tayside), Scotland. He was educated by his father at home until the age of fourteen, when he was sent to the University of St Andrews to study for a general degree with the aim of entering the Church. Playfair was awarded a scholarship to the University in 1762, and there his aptitude and keenness to study gained him both the respect and friendship of his professors. His progress in the mathematical sciences was so rapid that the professor of natural philosophy (physics was still called natural philosophy in St Andrews when I [EFR] studied it in the 1960s), Professor Wilkie, when suffering from an illness, found him to be the person best qualified to deliver his lectures on natural philosophy. Playfair graduated from the University of St Andrews with an M.A. in 1765. In 1766, while still only eighteen, Playfair entered a contest for the Chair of Mathematics at Marischal College in Aberdeen. In this contest, which lasted eleven days, he distinguished himself and gained great recognition. The extent of mathematical knowledge required to be successful in such a contest was immense. Playfair was unsuccessful, however, finishing third out of the six candidates, behind the Reverend Dr Trail, who was appointed to the Chair, and Dr Hamilton, who succeeded him in the Chair. However Playfair, at a very young age, had proved his extraordinary talent combined with his comprehensive knowledge of mathematics. Going on to study divinity at the University of St Andrews, Playfair undertook his theological studies at St Mary's College, St Andrews. On completion of his studies in 1769, he left the University, and from then on spent much of his time until 1773 in Edinburgh. There he mixed with the luminaries of the Scottish Enlightenment (see [2]); which included such great scholars as Dugald Stewart the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Playfair.html (1 of 7) [2/16/2002 11:26:55 PM]

Playfair

mathematician (son of Matthew Stewart), Adam Smith the economist, Joseph Black the chemist, James Hutton the geologist, Robert Adam the architect and engineer, and Principal Robinson the historian. During the period between 1769 and 1773, Playfair had twice attempted to obtain an academic post. His first attempt was in 1769 but it was unsuccessful. He continued, however, in his vocation as a minister and was licensed to preach by Dundee Presbytery in 1770. In 1772 Playfair applied for the Chair of Natural Philosophy in the University of St Andrews, which was left vacant after the death of his friend Wilkie but again another candidate was appointed. Having failed to obtain an academic post Playfair returned to Edinburgh where he remained until his father's death in 1772. Playfair was nominated by Lord Gray to succeed his father as the Parish Minister of Liff and Benvie and he moved to Liff to supervise the education of his brothers and sisters. Almost a year had elapsed, however, before his nomination was confirmed, as Lord Gray's rights of presentation were disputed by the Crown of Lawyers. The case went before the Court of Session and, in August 1773, Playfair received confirmation by a resolution of the General Assembly of the Church. He was then ordained the Minister of Liff and Benvie in succession to his father. During this period Playfair did not neglect his own academic studies, and beside making occasional visits to Edinburgh, he made an excursion in 1774 to Schiehallion, Perthshire, to conduct experiments with Neville Maskelyne, the Astronomer Royal. They became lifelong friends and Maskelyne introduced him to the leading scientific men of the day. He persuaded Playfair to submit his first successful paper on mathematics to the Royal Society of London and this was published in the Philosophical Transactions in 1779. This first mathematical paper by Playfair On the Arithmetic of Impossible Quantities, has been described as exhibiting [11]:... a greater taste for purely analytical investigation than shown by any of the British mathematicians of that age. Playfair became Moderator of the Synod but soon after this he received, in 1782, a lucrative offer to resign his church position and to become the tutor to the two sons of Ferguson of Raith. He tutored Ronald Ferguson and his brother from 1782 until 1787. This involved moving closer to Edinburgh, and he was thus able to participate in the city's intellectual life. Playfair became involved in the establishment of the Royal Society of Edinburgh in 1783 and was one of the original Fellows of that Society. During a vacation he made his first visit to London, where Maskelyne introduced him to the scientific world. In 1785 Playfair was appointed Joint Professor of Mathematics in the University of Edinburgh, a position which he was to hold for twenty years. Two years later, after completing his tutoring duties for the Ferguson's, he moved to Edinburgh, joining his mother and sisters, who had for some years been resident in Edinburgh. From 1787 Playfair published on various topics in the Transactions of the Royal Society of Edinburgh and also contributed to the Edinburgh Review. In 1793 Playfair's brother James, who was established in London as an architect, died suddenly. Playfair interrupted his studies to make the family's arrangements. In the following year, he adopted James's eldest son, William Henry Playfair, then only six years of age. William would follow in his father's footsteps and also become an renowned architect. In the eighteenth century geometry was systematically studied from Euclid's Elements in the universities, while the schools were generally content to accept the theorems and constructions without proof. However, mathematicians began to demand more rigour with the growing interest in analytic http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Playfair.html (2 of 7) [2/16/2002 11:26:55 PM]

Playfair

investigation. In 1795 Playfair published an edition of the Elements which he intended for use by his students. The main innovation was Playfair's use of algebraic notation to abbreviate the proofs which he taught in his class. This was intended to avoid the "tediousness and circumlocution" of geometric theory. The difficulties encountered by those who studied the Elements in the eighteenth century centred around two problems. Firstly, there was the contentious "parallel" postulate. The second problem was Euclid's theory of proportion, derived from Eudoxus. Robert Simson of Glasgow University had, in his 1756 edition of the Elements, given a proof of the parallel axiom based on another assumption. Playfair solved this difficulty in 1795 with Playfair's Axiom, his alternative to Euclid's parallel axiom:Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line. This form of axiom was certainly not new as it had already been given in the fifth century by Proclus. It is curious that Playfair's name should be associated with this axiom, particularly since he clearly points out that he derived the axiom from Proclus. Playfair standardised the notation for points and sides of figures in the first six books of his edition of Euclid. To these books, which specifically deal with plane geometry, Playfair added three more books intended to supplement the preceding six; On the Quadrature of the Circle and the Geometry of Solids, Elements of Plane and Spherical Trigonometry and The Arithmetic of Sines. He also included a section of notes in the form of an appendix, which gave his reasons for the alterations made throughout the volumes, and an illuminating discussion on the difficult topic of parallel lines. The fact that it ran to six editions shows the popularity of Playfair's edition of Euclid. The author of [14] claims that:... Playfair's intervention saved Euclid for a hundred years from its inevitable fate! Playfair suffered a severe attack of rheumatism, during the early part of 1797. This did not prevent him writing however, and during this time he wrote An Analytical Treatise on the Conic Sections, and an Essay on the Accidental Discoveries Which Have Been Made By Men of Science, Whilst In Pursuit of Something Else, Or When They Had No Determinate Object in View. The death of his friend, James Hutton, moved Playfair to compose a biographical memoir, which gradually became a reply to the critics of Hutton's theories of geology. This in turn gave rise to Playfair's geological work Illustrations of the Huttonian Theory of the Earth. Playfair presented Hutton's theories in a different style from Hutton's original presentation. Hutton had a rather peculiar style of presentation which made his theory less intelligible and, as a result, he had received less acclaim than he deserved. It was a style which led to many erroneous misrepresentations and to attacks from the few who had read it. Playfair's simple and eloquent style consisted of a series of chapters clearly stating the Huttonian theory, giving the facts to support it, and the arguments given against it. The success of Playfair's presentation can be judged by the fame and credit which have since been given to Hutton, who is now regarded as the first great British geologist. The Illustrations [11]:... not only gave popularity to Hutton's theory, but help to create the modern science of geology. Playfair spent almost five years, from 1797 to 1802, writing the Illustrations. The majority of his spare time he spent travelling through Great Britain, in pursuit of his geological studies. Playfair had hoped to extend his researches to the Continent, but he was prevented in doing so by the war in Europe. He turned his attention to Ireland, making visits to Dublin and to the Giant's Causeway. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Playfair.html (3 of 7) [2/16/2002 11:26:55 PM]

Playfair

In 1803 Playfair published his biographical sketch of Hutton in the Transactions of the Royal Society of Edinburgh. This work was described over a century later as [11]:... a work for which luminous treatment and graceful direction, stands still without a rival in English Geological literature. Playfair was a successful teacher in his position as Professor of Mathematics at the University of Edinburgh, lecturing with a verve for the subject, doing his utmost to inspire his students with an enthusiasm for mathematical investigation, and rewarding those who succeeded by praising them in front of the class. He was described as a 'magnetic teacher' who [2]:... carried on with considerable aplomb after 1800 an established tradition of brilliant exposition and effective pedagogy associated since at least the seventeenth century with Scotland's solid Presbyterian schooling and its eloquent university facilities. Playfair was among the first in Britain to teach modern analysis. His course on this topic was attended by many who had long before completed their academic studies. To express their gratitude, class members presented Playfair with a precious astronomical circle, which was placed in the Observatory of the Astronomical Institution. However, despite his success as a mathematician, Playfair exchanged the Chair of Mathematics for the Chair of Natural Philosophy in 1805. Two years later he was elected a Fellow of the Royal Society of London. The Astronomical Institution of Edinburgh was founded in 1811, preceding the Royal Astronomical Society in England by nine years, making it the first British society devoted to astronomy. Playfair was its first president. The New Observatory on Calton Hill was built largely through Playfair's efforts in support of the project. In 1812 Playfair published the first of the volumes of his Outlines of Natural Philosophy, again intended primarily for the use of his students. The first volume covered dynamics, mechanics, hydrostatics, hydraulics, aerostatics, and pneumatics. The second volume was entirely devoted to astronomy, while a third volume, which was intended to complete the series and cover the subjects of optics, electricity, and magnetism, was never completed. In 1815 Playfair succeeded his friend and colleague, Professor Robison, as the Secretary of the Royal Society of Edinburgh. Playfair published many papers in the Transactions of the Society including a set of meteorological tables constructed from his own observations. Later in 1815 peace in Europe followed the defeat of Napoleon and Playfair began a 17 month, 4000 mile geological study of the Continent to gather material for the second edition of the Illustrations of the Huttonian Theory of the Earth. Although 68 years of age, Playfair set out on an arduous and extensive journey through France and Switzerland, continuing to the southern tip of Italy, examining the geological structure of the parts of the world he visited. He was accompanied for part of the time by his eldest nephew, James George Playfair, who assisted him by recording the details of their journeys. The second edition of the Illustrations was designed to be a much more major work than the first. It was intended to be more like scientific texts of today, beginning with the well authenticated facts, followed by general inferences that were deduced from these facts, with an examination of the various geological models that had been hypothesised. It was Playfair's aim to base the principles of geology on unquestionable assumptions and arguments. It would conclude with Playfair's model of geology and its applications. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Playfair.html (4 of 7) [2/16/2002 11:26:55 PM]

Playfair

However, this plan was interrupted when Playfair received a request he write an essay entitled Dissertation on the Progress of the Mathematical and Physical Science since the Revival of Letters in Europe for the supplement to Encyclopaedia Britannica. He moved to Burntisland in Fife in 1818 after seeing work begin on the New Observatory for the Astronomical Institution of Edinburgh, in order to complete this essay. While in Burntisland, he also wrote his Memoir on Naval Tactics, published posthumously in the Transactions of the Royal Society of Edinburgh.. Soon after completing the Dissertation Playfair suffered a severe attack of a disease of the bladder which prevented him from continuing his planned second edition of the Illustrations and interrupted his lectures. He regained his health sufficiently to finish the course of lectures in Edinburgh but, sadly, the second edition of the Illustrations was never completed. In June 1819 the bladder disease recurred with increased severity and Playfair returned to Burntisland. Although suffering very severe pain, he spent the last days of his life dictating corrections to the proof sheets of the Dissertation. After an illness lasting a month Playfair died. There were over 500 mourners at his burial in the Old Calton Burial Ground, overlooked by the Observatory which he helped to create. His grave is adjacent to that of David Hume, the famous philosopher, however it does not bear any indication whatever of who is interred there and, sadly, over the years has been neglected. Playfair earned for himself a high reputation in at least three branches of pure science, not primarily as a discoverer but rather as an expounder of theories. In the field of mathematics he introduced continental methods to Britain through his articles in scientific journals and encyclopaedias, and by his lecture courses. His nephew, James George Playfair, who edited The Works of John Playfair in 1822 wrote [10]:... we believe we hazard nothing in saying that he was one of the most learned mathematicians of his age, and among the first, if not the very first, who introduced the beautiful discoveries of the later continental geometers to the knowledge of his countrymen, and gave their just value and true place, in the scheme of European knowledge, to those important improvements by which the whole aspect of the abstract sciences has been renovated since the days of our illustrious Newton.... He possessed, indeed, in the highest degree, all characteristics both of a fine and powerful understanding, - at once penetrating and vigilant, - but more distinguished, perhaps, for the caution on sureness of its march, than for the brilliancy or rapidity of its movements, -and guided and adorned through all its progress by the most genuine enthusiasm for all that is grand, and the justest taste for all that is beautiful in the Truth or the Intellectual Energy with which he was habitually conversant. Playfair's character made him a popular personality. He possessed a [11]:... cordial combination of the two aristocracies of rank and of letters. Lord Henry Coburn wrote that Playfair was [5]:Admired by all men, and beloved by all women, of whose virtues and intellect he was always champion, society felt itself the happier and the more respectable from his presence. His nephew writes in [10]:... though the most social of human beings, and the most disposed to encourage and sympathise with the gaiety and joviality of others, his own spirits were in general rather cheerful than gay, or at least never rose to any turbulence or tumult of merriment... His own http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Playfair.html (5 of 7) [2/16/2002 11:26:55 PM]

Playfair

satisfaction might generally be traced in the slow and temperate smile, gradually mantling over his benevolent and intelligent features, and lighting up the countenance of the Sage with the expression of the mildest and most genuine philanthropy. In [15] Playfair's contributions are summed up as follows:The wide learning, the calm intellect and the clear thought, so apparent in all his writings, also marked his lectures. He was, according to one of his many illustrious pupils, 'a charming teacher, so simple, unaffected and sincere in manner, so chaste in style, so clear in demonstration'. By consolidating the learning of past generations and collating the discoveries and theories of his own time, he gave a comprehensive and unified presentation of the subjects he professed and thus laid the basis for future constructive researches in the fields of mathematics and natural philosophy. [This article was based on an honours project written by Mark Anderson at the University of St Andrews in 1999. We thank him for allowing us to reproduce this modified and shortened version.] Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles) A Poster of John Playfair

Mathematicians born in the same country

Cross-references to History Topics

1. Non-Euclidean geometry 2. Thomas Harriot's manuscripts

Honours awarded to John Playfair (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1807

Fellow of the Royal Society of Edinburgh Lunar features

Crater Playfair

Planetary features

Crater Playfair on Mars

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Playfair.html (6 of 7) [2/16/2002 11:26:55 PM]

Playfair

JOC/EFR August 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Playfair.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Playfair.html (7 of 7) [2/16/2002 11:26:55 PM]

Plessner

Abraham Ezechiel Plessner Born: 13 Feb 1900 in Lodz, Russian Empire (now Poland) Died: 18 April 1961 in Moscow, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Abraham Plessner attended secondary school (1909-1918) where up until 1916 he was taught in Russian, then for one year in German, then for his last year in Polish. In 1919 he entered the University of Giessen and studied under Schlesinger and Engel. In 1921 Plessner went to Göttingen where he took courses on Dirichlet series and Galois theory by Landau; algebraic number fields by Emmy Noether; and calculus of variations by Courant. Then in session 1921/22 he studied in Berlin where von Mises lectured on differential and integral equations, Bieberbach on differential geometry and Schur on algebra. Plessner obtained his doctorate from Giessen in 1922 for a thesis on conjugate trigonometrical series. Then he worked in Marburg with Hensel editing Kronecker's collected works. During his time in Marburg he published a paper containing what is today called Plessner's theorem, relating the boundary behaviour of functions meromorphic in the unit disk. In 1929 Plessner's Habilitationsschrift was submitted to the faculty at Giessen. Although it was an outstanding piece of work the Senate refused to give its approval since Plessner was a Russian citizen. He was told by the city officials that he was required to have 20 years continuous residence in Germany to obtain citizenship. Plessner moved to Berlin and the Senate of Giessen was again requested to confer the degree. They decided to postpone a decision indefinitely and so Plessner had no choice, he could not get a lectureship in Germany since he was a Russian citizen so he moved to Moscow. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plessner.html (1 of 2) [2/16/2002 11:26:57 PM]

Plessner

There he joined the research group of Luzin. His interests at this time moved to functional analysis and particularly spectral theory. It seems that his interest in functional analysis arose when he read Banach's book of 1932. He was greatly respected in Moscow: a colleague wrote Abrahm Ezechiel Plessner knew so much that it seemed he knew everything. He understood works in any field and every young mathematician tried to tell him of his new results. His students noted that his lectures were filled with comments like: this is false and this is trivial. They jokingly wrote:Paradise in the sense of Plessner is an abstract space in which all theorems are both false and trivial. Plessner was promoted to professor in 1939 and held posts both at Moscow University and at the Mathematical Institute of the Academy. However he was dismissed from both posts in 1949. His last years were ones of financial hardship and his health, which had never been good, became steadily worse. His role in mathematics is however a major one and must be considered as a founder of the Moscow school of functional analysis. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Plessner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plessner.html (2 of 2) [2/16/2002 11:26:57 PM]

Plucker

Julius Plücker Born: 16 June 1801 in Elberfeld (now Wuppertal), Duchy of Berg (now Germany) Died: 22 May 1868 in Bonn, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Julius Plücker was educated at Heidelberg, Berlin and Paris. He was appointed to Bonn in 1829, and became professor of mathematics at Halle in 1834, then at Bonn in 1836. He made important contributions to analytic geometry and physics. He initiated the investigation of geometrical configurations associated with line complexes. In this way of specifying coordinates a point has a linear equation, namely that of all lines through the point while a line has a pair of numbers namely the x and y coordinates of where it cuts the axes. His work on combinatorics considers Steiner type systems. He also introduced the notion of a ruled surface. In 1847 he turned to physics, accepting the chair of physics at Bonn working on magnetism, electronics and atomic physics. He anticipated Kirchhoff and Bunsen in indicating that spectral lines were characteristic for each chemical substance. In 1865 he returned to mathematics and Klein served as his assistant 1866-1868. Article by: J J O'Connor and E F Robertson List of References (8 books/articles) A Poster of Julius Plücker

Mathematicians born in the same country

Cross-references to History Topics

An overview of the history of mathematics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plucker.html (1 of 2) [2/16/2002 11:26:59 PM]

Plucker

Cross-references to Famous Curves

Trident of Newton

Other references in MacTutor

1. Chronology: 1820 to 1830 2. Chronology: 1860 to 1870

Honours awarded to Julius Plücker (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1855

Royal Society Copley Medal

Awarded 1866

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Plucker.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Plucker.html (2 of 2) [2/16/2002 11:26:59 PM]

Poincare

Jules Henri Poincaré Born: 29 April 1854 in Nancy, Lorraine, France Died: 17 July 1912 in Paris, France

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henri Poincaré's father was Léon Poincaré and his mother was Eugénie Launois. They were 26 and 24 years of age, respectively, at the time of Henri's birth. Henri was born in Nancy where his father was Professor of Medicine at the University. Léon Poincaré's family produced other men of great distinction during Henri's lifetime. Raymond Poincaré, who was prime minister of France several times and president of the French Republic during World War I, was the elder son of Léon Poincaré's brother Antoine Poincaré. The second of Antoine Poincaré's sons, Lucien Poincaré, achieved high rank in university administration. Henri was [2]:... ambidextrous and was nearsighted; during his childhood he had poor muscular coordination and was seriously ill for a time with diphtheria. He received special instruction from his gifted mother and excelled in written composition while still in elementary school. In 1862 Henri entered the Lycée in Nancy (now renamed the Lycée Henri Poincaré in his honour). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. Henri was described by his mathematics teacher as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France. Poincaré entered the Ecole Polytechnique in 1873, graduating in 1875. He was well ahead of all the other students in mathematics but, perhaps not surprisingly given his poor coordination, performed no better than average in physical exercise and in art. Music was another of his interests but, although he enjoyed http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html (1 of 7) [2/16/2002 11:27:01 PM]

Poincare

listening to it, his attempts to learn the piano while he was at the Ecole Polytechnique were not successful. Poincaré read widely, beginning with popular science writings and progressing to more advanced texts. His memory was remarkable and he retained much from all the texts he read but not in the manner of learning by rote, rather by linking the ideas he was assimilating particularly in a visual way. His ability to visualise what he heard proved particularly useful when he attended lectures since his eyesight was so poor that he could not see properly what his lecturers were writing on the blackboard. After graduating from the Ecole Polytechnique Poincaré continued his studies at the Ecole des Mines. His [20]:... meticulous notes taken on field trips while a student there exhibit a deep knowledge of the scientific and commercial methods of the mining industry; a subject that interested him throughout his life. After completing his studies at the Ecole des Mines Poincaré spent a short while as a mining engineer at Vesoul while completing his doctoral work. As a student of Charles Hermite, Poincaré received his doctorate in mathematics from the University of Paris in 1879. His thesis was on differential equations and the examiners were somewhat critical of the work. They praised the results near the beginning of the work but then reported that the (see for example [20]):... remainder of the thesis is a little confused and shows that the author was still unable to express his ideas in a clear and simple manner. Nevertheless, considering the great difficulty of the subject and the talent demonstrated, the faculty recommends that M Poincaré be granted the degree of Doctor with all privileges. Immediately after receiving his doctorate, Poincaré was appointed to teach mathematical analysis at the University of Caen. Reports of his teaching at Caen were not wholly complimentary, referring to his sometimes disorganised lecturing style. He was to remain there for only two years before being appointed to a chair in the Faculty of Science in Paris in 1881. In 1886 Poincaré was nominated for the chair of mathematical physics and probability at the Sorbonne. The intervention and the support of Hermite was to ensure that Poincaré was appointed to the chair and he also was appointed to a chair at the Ecole Polytechnique. In his lecture courses to students in Paris [2]:... changing his lectures every year, he would review optics, electricity, the equilibrium of fluid masses, the mathematics of electricity, astronomy, thermodynamics, light, and probability. Poincaré held these chairs in Paris until his death at the early age of 58. Before looking briefly at the many contributions that Poincaré made to mathematics and to other sciences, we should say a little about his way of thinking and working. He is considered as one of the great geniuses of all time and there are two very significant sources which study his thought processes. One is a lecture which Poincaré gave to l'Institut Général Psychologique in Paris in 1908 entitled Mathematical invention in which he looked at his own thought processes which led to his major mathematical discoveries. The other is the book [29] by Toulouse who was the director of the Psychology Laboratory of l'Ecole des Hautes Etudes in Paris. Although published in 1910 the book recounts conversations with Poincaré and tests on him which Toulouse carried out in 1897. In [29] Toulouse explains that Poincaré kept very precise working hours. He undertook mathematical research for four hours a day, between 10 am and noon then again from 5 pm to 7 pm. He would read articles in journals later in the evening. An interesting aspect of Poincaré's work is that he tended to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html (2 of 7) [2/16/2002 11:27:01 PM]

Poincare

develop his results from first principles. For many mathematicians there is a building process with more and more being built on top of the previous work. This was not the way that Poincaré worked and not only his research, but also his lectures and books, were all developed carefully from basics. Perhaps most remarkable of all is the description by Toulouse in [29] of how Poincaré went about writing a paper. Poincaré:... does not make an overall plan when he writes a paper. He will normally start without knowing where it will end. ... Starting is usually easy. Then the work seems to lead him on without him making a wilful effort. At that stage it is difficult to distract him. When he searches, he often writes a formula automatically to awaken some association of ideas. If beginning is painful, Poincaré does not persist but abandons the work. Toulouse then goes on to describe how Poincaré expected the crucial ideas to come to him when he stopped concentrating on the problem:Poincaré proceeds by sudden blows, taking up and abandoning a subject. During intervals he assumes ... that his unconscious continues the work of reflection. Stopping the work is difficult if there is not a sufficiently strong distraction, especially when he judges that it is not complete ... For this reason Poincaré never does any important work in the evening in order not to trouble his sleep. As Miller notes in [20]:Incredibly, he could work through page after page of detailed calculations, be it of the most abstract mathematical sort or pure number calculations, as he often did in physics, hardly ever crossing anything out. Let us examine some of the discoveries that Poincaré made with this method of working. Poincaré was a scientist preoccupied by many aspects of mathematics, physics and philosophy, and he is often described as the last universalist in mathematics. He made contributions to numerous branches of mathematics, celestial mechanics, fluid mechanics, the special theory of relativity and the philosophy of science. Much of his research involved interactions between different mathematical topics and his broad understanding of the whole spectrum of knowledge allowed him to attack problems from many different angles. Before the age of 30 he developed the concept of automorphic functions which are functions of one complex variable invariant under a group of transformations characterised algebraically by ratios of linear terms. The idea was to come in an indirect way from the work of his doctoral thesis on differential equations. His results applied only to restricted classes of functions and Poincaré wanted to generalise these results but, as a route towards this, he looked for a class of functions where solutions did not exist. This led him to functions he named Fuchsian functions after Lazarus Fuchs but were later named automorphic functions. The crucial idea came to him as he was about to get onto a bus, as he relates in Science and Method (1908):At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformation that I had used to define the Fuchsian functions were identical with those of non-euclidean geometry. In a correspondence between Klein and Poincaré many deep ideas were exchanged and the development of the theory of automorphic functions greatly benefited. However, the two great mathematicians did not remain on good terms, Klein seeming to become upset by Poincaré's high opinions of Fuchs' work. Rowe examines this correspondence in [149].

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html (3 of 7) [2/16/2002 11:27:01 PM]

Poincare

Poincaré's Analysis situs, published in 1895, is an early systematic treatment of topology. He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology. For 40 years after Poincaré published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work. Even today the Poincaré conjecture remains as one of the most baffling and challenging unsolved problems in algebraic topology. Homotopy theory reduces topological questions to algebra by associating with topological spaces various groups which are algebraic invariants. Poincaré introduced the fundamental group (or first homotopy group) in his paper of 1894 to distinguish different categories of 2-dimensional surfaces. He was able to show that any 2-dimensional surface having the same fundamental group as the 2-dimensional surface of a sphere is topologically equivalent to a sphere. He conjectured that this result held for 3-dimensional manifolds and this was later extended to higher dimensions. Surprisingly proofs are known for the equivalent of Poincaré's conjecture for all dimensions strictly greater than three. No complete classification scheme for 3-manifolds is known so there is no list of possible manifolds that can be checked to verify that they all have different homotopy groups. Poincaré is also considered the originator of the theory of analytic functions of several complex variables. He began his contributions to this topic in 1883 with a paper in which he used the Dirichlet principle to prove that a meromorphic function of two complex variables is a quotient of two entire functions. He also worked in algebraic geometry making fundamental contributions in papers written in 1910-11. He examined algebraic curves on an algebraic surface F(x, y, z) = 0 and developed methods which enabled him to give easy proofs of deep results due to Emile Picard and Severi. He gave the first correct proof of a result stated by Castelnuovo, Enriques and Severi, these authors having suggested a false method of proof. His first major contribution to number theory was made in 1901 with work on [1]:... the Diophantine problem of finding the points with rational coordinates on a curve f(x, y) = 0, where the coefficients of f are rational numbers. In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology. In the field of celestial mechanics he studied the three-body-problem, and the theories of light and of electromagnetic waves. He is acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity. We should describe in a little more detail Poincaré's important work on the 3-body problem. Oscar II, King of Sweden and Norway, initiated a mathematical competition in 1887 to celebrate his sixtieth birthday in 1889. Poincaré was awarded the prize for a memoir he submitted on the 3-body problem in celestial mechanics. In this memoir Poincaré gave the first description of homoclinic points, gave the first mathematical description of chaotic motion, and was the first to make major use of the idea of invariant integrals. However, when the memoir was about to be published in Acta Mathematica, Phragmen, who was editing the memoir for publication, found an error. Poincaré realised that indeed he had made an error and Mittag-Leffler made strenuous efforts to prevent the publication of the incorrect version of the memoir. Between March 1887 and July 1890 Poincaré and Mittag-Leffler exchanged fifty letters mainly relating to the Birthday Competition, the first of these by Poincaré telling Mittag-Leffler

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html (4 of 7) [2/16/2002 11:27:01 PM]

Poincare

that he intended to submit an entry, and of course the later of the 50 letters discuss the problem concerning the error. It is interesting that this error is now regarded as marking the birth of chaos theory. A revised version of Poincaré's memoir appeared in 1890. Poincaré's other major works on celestial mechanics include Les Méthodes nouvelles de la méchanique celeste in three volumes published between 1892 and 1899 and Leçons de mecanique celeste (1905). In the first of these he aimed to completely characterise all motions of mechanical systems, invoking an analogy with fluid flow. He also showed that series expansions previously used in studying the 3-body problem were convergent, but not in general uniformly convergent, so putting in doubt the stability proofs of Lagrange and Laplace. He also wrote many popular scientific articles at a time when science was not a popular topic with the general public in France. As Whitrow writes in [2]:After Poincaré achieved prominence as a mathematician, he turned his superb literary gifts to the challenge of describing for the general public the meaning and importance of science and mathematics. Poincaré's popular works include Science and Hypothesis (1901), The Value of Science (1905), and Science and Method (1908). A quote from these writings is particularly relevant to this archive on the history of mathematics. In 1908 he wrote:The true method of foreseeing the future of mathematics is to study its history and its actual state. Finally we look at Poincaré's contributions to the philosophy of mathematics and science. The first point to make is the way that Poincaré saw logic and intuition as playing a part in mathematical discovery. He wrote in Mathematical definitions in education (1904):It is by logic we prove, it is by intuition that we invent. In a later article Poincaré emphasised the point again in the following way:Logic, therefore, remains barren unless fertilised by intuition. McLarty [119] gives examples to show that Poincaré did not take the trouble to be rigorous. The success of his approach to mathematics lay in his passionate intuition. However intuition for Poincaré was not something he used when he could not find a logical proof. Rather he believed that formal arguments may reveal the mistakes of intuition and logical argument is the only means to confirm insights. Poincaré believed that formal proof alone cannot lead to knowledge. This will only follow from mathematical reasoning containing content and not just formal argument. It is reasonable to ask what Poincaré meant by "intuition". This is not straightforward, since he saw it as something rather different in his work in physics to his work in mathematics. In physics he saw intuition as encapsulating mathematically what his senses told him of the world. But to explain what "intuition" was in mathematics, Poincaré fell back on saying it was the part which did not follow by logic:... to make geometry ... something other than pure logic is necessary. To describe this "something" we have no word other than intuition. The same point is made again by Poincaré when he wrote a review of Hilbert's Foundations of geometry (1902):The logical point of view alone appears to interest [Hilbert]. Being given a sequence of propositions, he finds that all follow logically from the first. With the foundations of this http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html (5 of 7) [2/16/2002 11:27:01 PM]

Poincare

first proposition, with its psychological origin, he does not concern himself. We should not give the impression that the review was negative, however, for Poincaré was very positive about this work by Hilbert. In [180] Stump explores the meaning of intuition for Poincaré and the difference between its mathematically acceptable and unacceptable forms. Poincaré believed that one could choose either euclidean or non-euclidean geometry as the geometry of physical space. He believed that because the two geometries were topologically equivalent then one could translate properties of one to the other, so neither is correct or false. for this reason he argued that euclidean geometry would always be preferred by physicists. This, however, has not proved to be correct and experimental evidence now shows clearly that physical space is not euclidean. Poincaré was absolutely correct, however, in his criticism of those like Russell who wished to axiomatise mathematics were doomed to failure. The principle of mathematical induction, claimed Poincaré, cannot be logically deduced. He also claimed that arithmetic could never be proved consistent if one defined arithmetic by a system of axioms as Hilbert had done. These claims of Poincaré were eventually shown to be correct. We should note that, despite his great influence on the mathematics of his time, Poincaré never founded his own school since he did not have any students. Although his contemporaries used his results they seldom adopted his techniques. Poincaré achieved the highest honours for his contributions of true genius. He was elected to the Académie des Sciences in 1887 and in 1906 was elected President of the Academy. The breadth of his research led to him being the only member elected to every one of the five sections of the Academy, namely the geometry, mechanics, physics, geography and navigation sections. In 1908 he was elected to the Académie Francaise and was elected director in the year of his death. He was also made chevalier of the Légion d'Honneur and was honoured by a large number of learned societies around the world. He won numerous prizes, medals and awards.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (199 books/articles)

Some Quotations (27)

A Poster of Henri Poincaré

Mathematicians born in the same country

Cross-references to History Topics

1. Topology enters mathematics 2. Orbits and gravitation 3. General relativity 4. Special relativity

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html (6 of 7) [2/16/2002 11:27:01 PM]

Poincare

Other references in MacTutor

1. Chronology: 1880 to 1890 2. Chronology: 1890 to 1900 3. Chronology: 1900 to 1910

Honours awarded to Henri Poincaré (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1894

Royal Society Sylvester Medal

Awarded 1901

ASP Bruce Medallist

1911

Lunar features

Crater Poincare

Paris street names

Rue Henri Poincaré (20th Arrondissement)

Other Web sites

1. Arnaud Riess 2. Internet Encyclopedia of Philosophy 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Poincare.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html (7 of 7) [2/16/2002 11:27:01 PM]

Poinsot

Louis Poinsot Born: 3 Jan 1777 in Paris, France Died: 5 Dec 1859 in Paris, France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Louis Poinsot attended the school of Louis-le-Grand in Paris. He sat the entrance examination of the Ecole Polytechnique in October 1794 but he performed poorly in the algebra assessment, which he failed. Despite this poor performance in one part of the entrance examinations, Poinsot was accepted for admission to the Ecole Polytechnique. From 1794 until 1797 Poinsot was a student at the Ecole Polytechnique but then he left and went to study at the Ecole des Ponts et Chaussée with the intention of becoming an engineer. Poinsot seems to have somewhat lacked vision at this stage in his career as to what he wanted for the future. Once on the course to become an engineer, he reacted by failing to study the practical subjects and became fully engrossed in abstract mathematics. Having made rather a mess of his education, Poinsot decided to give up the idea of becoming an engineer and left the Ecole des Ponts et Chaussée to become a mathematics teacher. From 1804 until 1809 Poinsot was a mathematics teacher at the Lycée Boneparte in Paris. Then he was appointed as inspector general of the Imperial University. Another famous mathematician held a post with this university at this time for Delambre had been appointed treasurer to the Imperial University in 1808, holding the position until 1815. Poinsot took on another appointment, in addition to the one with the Imperial University, when he accepted the position of assistant professor of analysis and mechanics at the Ecole Polytechnique on 1 November 1809. We should say a little about how someone like Poinsot, who had not really made a great success of his education, had moved from school teaching into higher education in the way that he did. The reason is that Poinsot had been very active in mathematics even while he was supposed to be studying for an engineering qualification. He had published a number of works on geometry, mechanics and statics http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poinsot.html (1 of 3) [2/16/2002 11:27:04 PM]

Poinsot

beginning with Eléments de statique in 1803 and following this with [1]:... memoirs that dealt with the composition of moments and the composition of areas (1806), the general theory of equilibrium and of movements in systems (1806), and polygons and polyhedra (1809). Certainly Poinsot had been very busy during his years as a school teacher of mathematics and by 1809 he had made for himself a fine reputation. In 1812 Poinsot arranged for Reynaud to substitute for him in lecturing at the Ecole Polytechnique and, from this time on, he did no further teaching there asking Cauchy to substitute for him after Reynaud. In September 1816 the Ecole Polytechnique was reorganised and Poinsot lost the post which he had held in name only for the previous four years. By 1816 Poinsot had been elected to the Académie des Science for, on 31 May 1813, he had replaced Lagrange in the mathematics section of the Académie. The 1816 reorganisation of the Ecole Polytechnique saw Poinsot become admissions examiner and he held this post for ten years. From 1839 until his death he worked at the Bureau des Longitudes. Poinsot had high principles when it came to mathematics research and the publication of it [1]:Poinsot was determined to publish only fully developed results and to present them with clarity and elegance. Consequently he left a rather limited body of work ... His research in geometry, statics and dynamics is important. He was the inventor of geometrical mechanics, investigating how a system of forces acting on a rigid body could be resolved into a single force and a couple. He wrote an important work on polyhedra in 1809 (already mentioned above), discovering four new regular polyhedra, two of which appear in Kepler's work of 1619 but Poinsot was unaware of this. In 1810 Cauchy proved that, with this definition of regular, the enumeration of regular polyhedra is complete. A mistake was discovered in Poinsot's (and hence Cauchy's) definition in 1990 when an internal inconsistency became apparent. In addition Poinsot worked on number theory and on this topic he studied Diophantine equations, how to express numbers as the difference of two squares and primitive roots. However he is best known for his dedication to geometry and, together with Monge, he contributed to the topic regaining its leading role in mathematical research in France in the eighteenth century. As well as his research in geometry, Poinsot contributed to its increasing importance by creating a chair of advanced geometry at the Sorbonne in 1846. Poinsot intended the chair for Chasles and indeed he was appointed to the new chair which he occupied until his death in 1880. Poinsot did, as many academics at this time in France, become involved in politics. He was involved both in the politics of higher education and also with the politics of France. In the former capacity he served on the Conseil de Perfectionnement of the Ecole Polytechnique on a number of occasions after 1830. In 1840 he was appointed to the Conseil Royal de l'Instruction Publique which dealt with higher education. He was [1]:Moderately liberal in his political opinions [and] he protested against the clericalism of the Restoration but later accepted nomination to the Chambre des Paris (1846) and the Senate (1852). Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poinsot.html (2 of 3) [2/16/2002 11:27:04 PM]

Poinsot

List of References (2 books/articles)

A Quotation

A Poster of Louis Poinsot

Mathematicians born in the same country

Cross-references to History Topics

1. A comment from Thomas Hirst's diary 2. The fundamental theorem of algebra

Other references in MacTutor

Chronology: 1800 to 1810

Honours awarded to Louis Poinsot (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1858

Lunar features

Crater Poinsot

Paris street names

Rue Poinsot (14th Arrondissement)

Commemorated on the Eiffel Tower

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Poinsot.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poinsot.html (3 of 3) [2/16/2002 11:27:04 PM]

Poisson

Siméon Denis Poisson Born: 21 June 1781 in Pithiviers, France Died: 25 April 1840 in Sceaux (near Paris), France

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Originally forced to study medicine, Siméon Poisson began to study mathematics in 1798 at the Ecole Polytechnique. His teachers Laplace and Lagrange were to become friends for life. A memoir on finite differences, written when Poisson was 18, attracted the attention of Legendre. Poisson taught at Ecole Polytechnique from 1802 until 1808 when he became an astronomer at Bureau des Longitudes. In 1809 he was appointed to the chair of pure mathematics in the newly opened Faculté des Sciences. His most important works were a series of papers on definite integrals and his advances in Fourier series. This work was the foundation of later work in this area by Dirichlet and Riemann. In Recherchés sur la probabilité des jugements..., an important work on probability published in 1837, the Poisson distribution first appeared. The Poisson distribution describes the probability that a random event will occur in a time or space interval under the conditions that the probability of the event occurring is very small, but the number of trials is very large so that the event actually occurs a few times. He published between 300 and 400 mathematical works including applications to electricity and magnetism, and astronomy. His Traité de mécanique published in 1811 and again in 1833 was the standard work on mechanics for many years. His name is attached to a wide area of ideas, for example:- Poisson's integral, Poisson's equation in potential theory, Poisson brackets in differential equations, Poisson's ratio in elasticity, and Poisson's http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poisson.html (1 of 3) [2/16/2002 11:27:06 PM]

Poisson

constant in electricity. Libri said of him: His only passion has been science: he lived and is dead for it. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (22 books/articles)

A Quotation

A Poster of Siméon Poisson

Mathematicians born in the same country

Some pages from publications

The title page of Récherches sur la probabilité des jugements (1837).

Cross-references to History Topics

1. General relativity

Other references in MacTutor

1. Chronology: 1810 to 1820 2. Chronology: 1830 to 1840

Honours awarded to Siméon Poisson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1818

Royal Society Copley Medal

Awarded 1832

Lunar features

Crater Poisson

Paris street names

Rue Denis Poisson (17th Arrondissement)

Commemorated on the Eiffel Tower Other Web sites

1. Rouse Ball 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poisson.html (2 of 3) [2/16/2002 11:27:06 PM]

Poisson

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Poisson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poisson.html (3 of 3) [2/16/2002 11:27:06 PM]

Poleni

Giovanni Poleni Born: 23 Aug 1683 in Venice, Italy Died: 15 Nov 1761 in Padua, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giovanni Poleni studied first philosophy and then theology in Venice. He began a judicial career but, after being introduced to mathematics and science by his father, he went on to accept the chair of astronomy at the University of Padua in 1709. In 1715 he became, in addition to his astronomy chair, professor of physics. Poleni was invited to investigate the problem of hydraulics relating to the irrigation of Lower Lombardy. He was appointed to the chair of mathematics at Padua in 1719 which had been vacated by Nicolaus(II) Bernoulli. In 1738 he acquired a laboratory to conduct physics experiments and began to lecture on this topic. Poleni worked on hydraulics, physics, astronomy and archaeology. He conducted meteorological observations. His work was honoured by the award of a number of prizes, for calculating the distance travelled by a ship, for a study of ship's anchors, and for a study of cranes and windlasses. Article by: J J O'Connor and E F Robertson List of References (9 books/articles) Mathematicians born in the same country Honours awarded to Giovanni Poleni (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1710

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poleni.html (1 of 2) [2/16/2002 11:27:06 PM]

Poleni

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Poleni.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poleni.html (2 of 2) [2/16/2002 11:27:06 PM]

Polozii

Georgii Nikolaevich Polozii Born: 23 April 1914 in Transbaikal, Russia Died: 26 Nov 1968 in Kiev, Ukraine Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georgii Polozii studied at Saratov University which had been founded in 1919. He graduated in 1937 and then was appointed to the teaching staff of the university. In 1949 Polozii was appointed to the University of Kiev and he remained at Kiev until his death in 1968. Polozii's major pure mathematical contributions were to the theory of functions of a complex variable, approximation theory, and numerical analysis. He also made major contributions to mathematical physics and applied mathematics in particular working on the theory of elasticity. Between 1962 and 1966 Polozii developed the theory for a new class of (p,q) analytic functions. Petryshyn, writing in [2], summarises Polozii's work:Original results in the theory of functions of a complex variable were obtained in the 1950s and 1960s by G Polozii of Kiev, who introduced a new notion of p-analytic functions, defined the notion of derivative and integral for these functions, developed their calculus, obtained a generalised Cauchy formula, and devised a new approximation method for solution of problems in elasticity and filtration. His results were further developed by his students ... In approximation theory Polozii worked mainly with the aim of developing effective methods to solve boundary value problems which arise in mathematical physics. He work here produced the method of summary representation. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Polozii.html (1 of 2) [2/16/2002 11:27:07 PM]

Polozii

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Polozii.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Polozii.html (2 of 2) [2/16/2002 11:27:07 PM]

Polya

George Pólya Born: 13 Dec 1887 in Budapest, Hungary Died: 7 Sept 1985 in Palo Alto, California, USA

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Pólya enrolled at the University of Budapest to study law but found it boring. He then studied languages and literature. In order to understand philosophy he studied mathematics. He was awarded a doctorate from Budapest in mathematics in 1912. He spent 1913 in Göttingen where he met Hilbert, Weyl and others. In 1914 he went to Zürich to an appointment arranged by Hurwitz. He spent 1924 in England working with Hardy and Littlewood and their joint work Inequalities was published in 1934. While in Zürich his output of mathematics was very large and wide ranging. In 1918 he published papers on series, number theory, combinatorics and voting systems. The following year in addition to papers in these topics he published on astronomy, probability. While he was doing this wide range of work he was working on some of his deepest results in the study of integral functions. The political situation in Europe forced Pólya to move to the USA where after working at Brown University for two years he took up an appointment at Stanford. Before going to the USA Pólya had a draft of a book How to solve it written in German. Pólya had to try four publishers before finding one to publish the English version in the USA. It sold over one million copies over the years. Pólya gave wise advice If you cannot solve a problem, then there is an easier problem you cannot solve: find it.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Polya.html (1 of 2) [2/16/2002 11:27:09 PM]

Polya

In probability Pólya looked at the Fourier transform of a distribution and proved a celebrated theorem on random walks. Geometric symmetry and the enumeration of symmetry classes of objects was a major area of interest over many years. His main contribution to combinatorics is his enumeration theorem, published in 1937. Pólya's interest in complex analysis, conformal mappings and potential theory led him to study boundary value problems for partial differential equations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles)

Some Quotations (28)

A Poster of George Pólya

Mathematicians born in the same country

Other Web sites

Pólya's Random Walk Constants

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Polya.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Polya.html (2 of 2) [2/16/2002 11:27:09 PM]

Poncelet

Jean Victor Poncelet Born: 1 July 1788 in Metz, Lorraine, France Died: 22 Dec 1867 in Paris, France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean-Victor Poncelet was a pupil of Monge. His development of the pole and polar lines associated with conics led to the principle of duality. Poncelet took part in Napoleon's 1812 Russian campaign as an engineer. He was left for dead at Krasnoy and imprisoned until 1814 when he returned to France. During his imprisonment he studied projective geometry. He also wrote a treatise on analytic geometry Applications d'analyse et de géométrie based on what he had learnt at the Ecole Polytechnique but it was only published 50 years later. From 1815 to 1825 he was a military engineer at Metz and from 1825 to 1835 professor of mechanics there. He applied mechanics to improve turbines and waterwheels more than doubling the efficiency of the waterwheel. Poncelet was one of the founders of modern projective geometry simultaneously discovered by Joseph Gergonne and Poncelet. His development of the pole and polar lines associated with conics led to the principle of duality. He also discovered circular points at infinity. He published Traité des propriétés projectives des figures in 1822 which is a study of those properties which remain invariant under projection. This work contains fundamental ideas of projective geometry such as the cross-ratio, perspective, involution and the circular points at infinity. While writing this book he consulted with Servois. Poncelet published Applications d'analyse et de géométrie in two volumes: 1862 and 1864.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poncelet.html (1 of 2) [2/16/2002 11:27:11 PM]

Poncelet

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) A Poster of Jean-Victor Poncelet

Mathematicians born in the same country

Some pages from publications

Part of the Introduction of Traitè des propriés projectives de figures (1822)

Cross-references to History Topics

1. Abstract linear spaces 2. A history of group theory

Other references in MacTutor

1. Chronology: 1820 to 1830 2. Chronology: 1830 to 1840

Honours awarded to Jean-Victor Poncelet (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1842

Lunar features

Crater Poncelet

Paris street names

Passage Poncelet and Rue Poncelet (17th Arrondissement)

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Poncelet.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poncelet.html (2 of 2) [2/16/2002 11:27:11 PM]

Pontryagin

Lev Semenovich Pontryagin Born: 3 Sept 1908 in Moscow, Russia Died: 3 May 1988

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lev Semenovich Pontryagin's father, Semen Akimovich Pontryagin was a civil servant. Pontryagin's mother, Tat'yana Andreevna Pontryagina, was 29 years old when he was born and she was a remarkable woman who played a crucial role in his path to becoming a mathematician. Perhaps the description of 'civil servant', although accurate, gives the wrong impression that the family were reasonably well off. In fact Semen Akimovich's job left the family without enough money to allow them to give their son a good education and Tat'yana Andreevna worked using her sewing skills to help out the family finances. Pontryagin attended the town school where the standard of education was well below that of the better schools but the family's poor circumstances put these well out of reach financially. At the age of 14 years Pontryagin suffered an accident and an explosion left him blind. This might have meant an end to his education and career but his mother had other ideas and devoted herself to help him succeed despite the almost impossible difficulties of being blind. The help that she gave Pontryagin is described in [1] and [2]:From this moment Tat'yana Andreevna assumed complete responsibility for ministering to the needs of her son in all aspects of his life. In spite of the great difficulties with which she had to contend, she was so successful in her self-appointed task that she truly deserves the gratitude ... of science throughout the world. For many years she worked, in effect, as Pontryagin's secretary, reading scientific works aloud to him, writing in the formulas in his manuscripts, correcting his work and so on. In order to do this she had, in particular, to learn to read foreign languages. Tat'yana Andreevna helped Pontryagin in all other respects, seeing to his needs and taking very great care of him.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pontryagin.html (1 of 4) [2/16/2002 11:27:13 PM]

Pontryagin

It is not unreasonable to pause for a moment and think about how Tat'yana Andreevna, with no mathematical training or knowledge, made by her determination and extreme efforts a major contribution to mathematics by allowing Pontryagin to become a mathematician against all the odds. There must be many other non-mathematicians, perhaps many of whom are unrecorded by history, who have also by their unselfish acts allowed mathematics to flourish. As we try to show in this archive, the development of mathematics depends on a wide number of influences other than the talents of the mathematicians themselves: political influences, economic influences, social influences, and the acts of non-mathematicians like Tat'yana Andreevna. But how does one read a mathematics paper without knowing any mathematics? Of course it is full of mysterious symbols and Tat'yana Andreevna, not knowing their mathematical meaning or name, could only describe them by their appearance. For example an intersection sign became a 'tails down' while a union symbol became a 'tails up'. If she read 'A tails right B' then Pontryagin knew that A was a subset of B! Pontryagin entered the University of Moscow in 1925 and it quickly became apparent to his lecturers that he was an exceptional student. Of course that a blind student who could not make notes yet was able to remember the most complicated manipulations with symbols was in itself truly remarkable. Even more remarkable was the fact that Pontryagin could 'see' (if you will excuse the bad pun) far more clearly than any of his fellow students the depth of meaning in the topics presented to him. Of the advanced courses he took, Pontryagin felt less happy with Khinchin's analysis course but he took a special liking to Aleksandrov's courses. Pontryagin was strongly influenced by Aleksandrov and the direction of Aleksandrov's research was to determine the area of Pontryagin's work for many years. However this was as much to do with Aleksandrov himself as with his mathematics ([1] and [2]):Aleksandrov's personal charm, his attention and helpfulness influenced the formation of Pontryagin's scientific interests to a remarkable extent, as much in fact as the personal abilities and inclinations of the young scholar himself. The year 1927 was the year of the death of Pontryagin's father. By 1927, although he was still only 19 years old, Pontryagin had begun to produce important results on the Alexander duality theorem. His main tool was to use link numbers which had been introduced by Brouwer and, by 1932, he had produced the most significant of these duality results when he proved the duality between the homology groups of bounded closed sets in Euclidean space and the homology groups in the complement of the space. Pontryagin graduated from the University of Moscow in 1929 and was appointed to the Mechanics and Mathematics Faculty. In 1934 he became a member of the Steklov Institute and in 1935 he became head of the Department of Topology and Functional Analysis at the Institute. Pontryagin worked on problems in topology and algebra. In fact his own description of this area that he worked on was:... problems where these two domains of mathematics come together. The significance of this work of Pontryagin on duality ([1] and [2]):... lies not merely in its effect on the further development of topology; of equal significance is the fact that his theorem enabled him to construct a general theory of characters for commutative topological groups. This theory, historically the first really exceptional

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pontryagin.html (2 of 4) [2/16/2002 11:27:13 PM]

Pontryagin

achievement in a new branch of mathematics, that of topological algebra, was one of the most fundamental advances in the whole of mathematics during the present century... One of the 23 problems posed by Hilbert in 1900 was to prove his conjecture that any locally Euclidean topological group can be given the structure of an analytic manifold so as to become a Lie group. This became known as Hilbert's Fifth Problem. In 1929 von Neumann, using integration on general compact groups which he had introduced, was able to solve Hilbert's Fifth Problem for compact groups. In 1934 Pontryagin was able to prove Hilbert's Fifth Problem for abelian groups using the theory of characters on locally compact abelian groups which he had introduced. Among Pontryagin's most important books on the above topics is topological groups (1938). The authors of [1] and [2] rightly assert:This book belongs to that rare category of mathematical works that can truly be called classical - book which retain their significance for decades and exert a formative influence on the scientific outlook of whole generations of mathematicians. In 1934 Cartan visited Moscow and lectured in the Mechanics and Mathematics Faculty. Pontryagin attended Cartan's lecture which was in French but Pontryagin did not understand French so he listened to a whispered translation by Nina Bari who sat beside him. Cartan's lecture was based around the problem of calculating the homology groups of the classical compact Lie groups. Cartan had some ideas how this might be achieved and he explained these in the lecture but, the following year, Pontryagin was able to solve the problem completely using a totally different approach to the one suggested by Cartan. In fact Pontryagin used ideas introduced by Morse on equipotential surfaces. Pontryagin's name is attached to many mathematical concepts. The essential tool of cobordism theory is the Pontryagin-Thom construction. A fundamental theorem concerning characteristic classes of a manifold deals with special classes called the Pontryagin characteristic class of the manifold. One of the main problems of characteristic classes was not solved until Sergi Novikov proved their topological invariance. In 1952 Pontryagin changed the direction of his research completely. He began to study applied mathematics problems, in particular studying differential equations and control theory. In fact this change of direction was not quite as sudden as it appeared. From the 1930s Pontryagin had been friendly with the physicist A A Andronov and had regularly discussed with him problems in the theory of oscillations and the theory of automatic control on which Andronov was working. He published a paper with Andronov on dynamical systems in 1932 but the big shift in Pontryagin's work in 1952 occurred around the time of Andronov's death. In 1961 he published The Mathematical Theory of Optimal Processes with his students V G Boltyanskii, R V Gamrelidze and E F Mishchenko. The following year an English translation appeared and, also in 1962, Pontryagin received the Lenin prize for his book. He then produced a series of papers on differential games which extends his work on control theory. Pontryagin's work in control theory is discussed in the historical survey [3]. Another book by Pontryagin Ordinary differential equations appeared in English translation, also in 1962.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pontryagin.html (3 of 4) [2/16/2002 11:27:13 PM]

Pontryagin

Pontryagin received many honours for his work. He was elected to the Academy of Sciences in 1939, becoming a full member in 1959. In 1941 he was of one the first recipients of the Stalin prizes (later called the State Prizes). He was honoured in 1970 by being elected Vice-President of the International Union.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Pontryagin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pontryagin.html (4 of 4) [2/16/2002 11:27:13 PM]

Poretsky

Platon Sergeevich Poretsky Born: 15 Oct 1846 in Elisavetgrad (now Kirovograd), Russia Died: 22 Aug 1907 in Joved, Chernigov guberniya, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Platon Poretsky worked as an astronomer and observer first at Kharkov Observatory and then at Kazan University. He became interested in logic through Vasiliev just before he retired in 1889 through ill health. Poretsky worked on mathematical logic for the rest of his life, extending and augmenting results of Boole, Jevons and Schröder. He applied his logic calculus to the theory of probability. Recently further material on Poretsky had come to light and this is described in [3]. V A Bazhanov, the author states in his summary:In February 1992, T Ivanovic, working with the author in the archives at Kazan University, discovered new material relating to the life and work of the logician P S Poretsky (1846-1907). These include: various documents related to Poretsky's lectures on mathematical logic for mathematics department students at Kazan University which were intended to be given for three semesters in the autumn of 1887 and all of 1888 but were delivered only during the 1888 Spring semester, a complete mathematical logic program compiled by Poretsky, materials related to Poretsky's father and family, Poretsky's Magister's (master's) dissertation and the decision of the physics-mathematics faculty council to award him the doctorate in astronomy rather than the Magister, a complete list of the sources he used (including Boole, Jevons, Schröder, and Peano), biographical data and materials regarding his illness and subsequent dismissal from Kazan University. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poretsky.html (1 of 2) [2/16/2002 11:27:15 PM]

Poretsky

Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Poretsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Poretsky.html (2 of 2) [2/16/2002 11:27:15 PM]

Porphyry

Porphyry Malchus Born: 233 in Tyre (now Sur, Lebanon) Died: 309 in Rome

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Porphyry's father was called Malkhos or Malchus, which means 'king'. Both Porphyry's parents were Syrian and he would only get the nickname Porphyry later in his life as we shall explain below. Porphyry was named after his father so for many years he was known as Malchus. As a young man Porphyry tried to gain as broad a knowledge as he possibly could by studying many languages and religions. At that time Athens was the main centre for learning, so it was natural that someone with a thirst for knowledge as Porphyry had should travel there to continue his studies. In Athens Porphyry became a student of Longinus who [8]:... was a 'living library and walking museum' and the academic's critical attention to detail, clarity of style and erudition left their permanent mark on the keen student. It was Longinus who gave Porphyry that nickname. In fact it was a clever pun since 'Porphyry' means 'purple' in Greek and he was given this name since he came from Tyre which was famous for the production of the royal purple dye and his name 'Malchus' meant 'king' = 'royal' = 'purple'. In about 263 Porphyry left Athens and went to Rome where he worked with Plotinus, the founder of Neoplatonism. Plotinus taught that there is an ultimate reality which is beyond the reach of thought or language. The object of life was to aim at this ultimate reality which could never be precisely described. Plotinus stressed that people did not have the mental capacity to fully understand both the ultimate reality itself or the consequences of its existence. Porphyry had mixed feelings when he heard the teachings of Plotinus. On the one hand he came to appreciate the point of view that Plotinus was putting forward, although it was somewhat different from the views of Longinus. However Porphyry was very disappointed in the way that Plotinus expressed himself and he found Plotinus's lectures poorly structured and his arguments rather woolly. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.html (1 of 3) [2/16/2002 11:27:17 PM]

Porphyry

Porphyry spent five years in Rome with Plotinus and during this time he [2]:... tried to rouse in the master the ambition to organise his doctrines. It is not entirely clear why Porphyry left Rome but there does seem to have been disputes over doctrine between the philosophers who formed Plotinus's circle. Porphyry went to Sicily where he wrote a text bringing the philosophies of Aristotle and Plato together. Porphyry had a strong respect for the views of Aristotle and further work by Porphyry at this time led to a revival in studies of the works of Aristotle. In particular his commentary on Aristotle's Categories led to the later developments of logic. It would appear that Porphyry's views on Aristotle had something to do with the disputes in Plotinus's school which had led to the break-up. Also while Porphyry was in Sicily, he wrote a work on vegetarianism and a critical work against Christian doctrines. Porphyry did not break his links with Plotinus, however, and he continued to correspond with him on presenting his views in the most coherent fashion. According to Heath [5] Porphyry was:The disciple of Plotinus and the reviser and editor of his works. Despite the fact that Porphyry's views were not completely at one with those of Plotinus, this description by Heath is a fair one. Porphyry certainly did go on to edit the works of Plotinus, for he returned to Rome in about 282 (which was about 12 years after Plotinus died). There he continued to write commentaries on Plato and other philosophers but mainly he continued the major work of his life which was to edit and publish Enneads on the teachings of Plotinus which he had completed about 301. Also while in Rome Porphyry taught Iamblichus who was another important developer of Neoplatonism. However, Iamblichus developed his views away from those of Plotinus and soon found himself disagreeing with Porphyry. We should make some comments as to the importance of Porphyry in the history of mathematics. He wrote a commentary on Euclid's Elements which was used by Pappus when he wrote his own commentary. Proclus appears to have Porphyry's original comments to hand when he wrote his own commentary. This is a point on which it is impossible to be certain for there is a slight possibility that all Proclus knew about Porphyry's commentary was what Pappus had written. Another important contribution made by Porphyry was in writing his Life of Pythagoras. Certain important fragments of other mathematician's writings have also been preserved in the works of Porphyry including Nicomachus and Eudemus. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.html (2 of 3) [2/16/2002 11:27:17 PM]

Porphyry

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Porphyry.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Porphyry.html (3 of 3) [2/16/2002 11:27:17 PM]

Porta

Giambattista Della Porta Born: 1 Nov 1535 in Vico Equense (near Naples), Italy Died: 4 Feb 1615 in Naples, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giambattista della Porta was educated at home where discussions on scientific topics frequently took place. His father, from 1541, was in the service of Emperor Charles V and della Porta was well educated by private tutors. Charles V was Holy Roman emperor and king of Spain at this time and his empire extended across Europe to the Netherlands, Austria and the Kingdom of Naples. Della Porta travelled widely in Italy, France and Spain always returning to his estate near Naples where he was able to study in peace. He never needed to earn a living as the wealth of the family seems to have been sufficient to allow della Porta to devote himself to study. In 1579 della Porta moved to Rome and entered the service of Luigi, cardinal d'Este, and frequented the court of Duke Alfonso II d'Este at Ferrara. He also lived in Venice while working for the Cardinal. In fact he was one of a number of dramatists who worked for the Cardinal, like Torquato Tasso, the greatest Italian poet of the late Renaissance. Della Porta, however, also undertook scientific work for the Cardinal, making optical instruments for him while in Venice. Della Porta's work was wide ranging and, having studied refraction in De refractione, optices parte (1593), he claimed to be the inventor of the telescope although he does not appear to have constructed one before Galileo. Other topics he wrote on include cryptography in De furtivis literarum (1563), mechanics and squaring the circle. He was the first to propose adding a convex lens to the camera obscura. He described a steam engine in De' spiritali (1606) and he was the first to recognise the heating effect of light rays.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Porta.html (1 of 2) [2/16/2002 11:27:19 PM]

Porta

Della Porta formed a society, Accademia dei Segreti dedicated to discussing and studying nature, which had regular meetings at his home. This Society was closed down by the Inquisition about 1578 after they examined della Porta. In 1585 he joined the Jesuit Order but his support of the Roman Catholic Church did not prevent the Inquisition from banning publication of his work between 1594 and 1598. Della Porta's major work is Magia naturalis (1558), in which he examines the natural world claiming it can be manipulated by the natural philosopher through theoretical and practical experiment. The work discusses many subjects including demonology, magnetism and the camera obscura. Della Porta also published Villae (1583-92), an agricultural encyclopaedia and De distillatione (1609), describing his work in chemistry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Giambattista della Porta

Mathematicians born in the same country

Cross-references to History Topics

Squaring the circle

Other Web sites

1. The Galileo Project 2. Silverdale (Magia naturalis) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Porta.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Porta.html (2 of 2) [2/16/2002 11:27:19 PM]

Posidonius

Posidonius of Rhodes Born: 135 BC in Apameia, Syria Died: 51 BC in Rhodes Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Posidonius of Rhodes is also known as Posidonius of Apameia. The first of these names refers to where he taught while the second refers to the town of his birth, Apameia on the Orontes. One must not think of these two as different people. Although he was born in Apameia in Syria, Posidonius was from a Greek family and he was brought up in the Greek tradition. He went to Athens to complete his education, and there he studied under the Stoic philosopher Panaetius of Rhodes. Posidonius travelled widely in the western Mediterranean region and he made many scientific studies on his travels relating to astronomy, geography and geology. Some time not long after 100 BC Posidonius became the head of the Stoic School in Rhodes. While in this position he also held political office in Rhodes. It was in a political position, as ambassador of Rhodes, that he travelled to Rome in 87-86 BC. There he met a number of men who he had known and taught earlier including Cicero. In Rome Posidonius visited Gaius Marius, the Roman general and politician who was consul seven times. Marius died on 13 January 86 BC while Posidonius was still in Rome. While there Posidonius became friends with Pompey the Great who had been educated in the Greek tradition. Pompey the Great kept up his friendship with Posidonius and visited him in Rhodes on a number of later occasions when on his military campaigns. None of the writing of Posidonius has survived but much has been written about his achievements and much work has been undertaken trying to reconstruct his views from the fragments of his writings which are preserved in quotations by later authors. Posidonius made some minor contributions to pure mathematics where he is [2]:... quoted as the author of certain definitions, or for views on technical terms. e.g. 'theorem' and 'problem', and subjects belonging to elementary geometry. ... he wrote a separate work in refutation of the Epicurean Zeno of Sidon, who had objected to the very beginnings of the "Elements" on the ground that they contained unproved assumptions. His work on astronomy is fairly well known to us through the treatise by Cleomedes On the Circular Motions of the Celestial Bodies. The work is in two volumes and as Heath comments [2]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Posidonius.html (1 of 3) [2/16/2002 11:27:21 PM]

Posidonius

... the very long first chapter of Book II (nearly half of the Book) ... seems for the most part to be copied bodily from Posidonius. Cleomedes explains in his work the method used by Posidonius to calculate the length of the circumference of the earth. His method is based on observations of the star Canopus at Rhodes and Alexandria. At Rhodes he observes that Canopus touches the horizon while at Alexandria it reaches an altitude of 7 30'. Using a distance of 5000 stadia between Rhodes and Alexandria this gave Posidonius a value of 240000 stadia for the circumference of the earth. This a very accurate value, but it is produced because of two compensating errors. Both figures used by Posidonius in the above calculation are inaccurate. The 7 30' should be really 5 15' while the figure of 5000 stadia for the Rhodes to Alexandria distance is also incorrect. Later Ptolemy informs us via the writings of Cleomedes, Posidonius used the more accurate 3750 stadia for the Rhodes to Alexandria distance but kept his very inaccurate 7 30' thus obtaining the figure of 180000 stadia for the circumference which is far too small. We should note, however, that Taisbak in [11] attempts to prove that attributing this far too small value of 180000 stadia to Posidonius is unfounded. Eratosthenes had given a much more accurate value of 252000 stadia 150 years before Posidonius. Posidonius also made calculations of the size and distance to the moon, and the size and distance to the sun. His measurements of the moon are inaccurate partly because he assumes a cylindrical rather than conical shadow. As to his calculations of the sun, Neugebauer writes [3]:Posidonius's attempts (according to Cleomedes) to determine the size of the sun are rather naive and make it difficult to understand that his astronomy was not ridiculed by authors like Cicero and Pliny who pretend to know the work of Hipparchus. As to Posidonius's views on knowledge he believed that [1]:... fundamental principles depended on philosophers and individual problems on scientists; and he believed that, among early men, the philosophically wise managed everything and discovered all crafts and industry. ... For true judgement the standard is right reasoning; but precepts, persuasion, consolation, and exhortation are necessary; and enquiry into causes as opposed to matter is important. Posidonius wrote on meteorology, a topic where he closely followed the teachings of Aristotle. He gave theories to explain clouds, mist, wind and rain. He also gave opinions on frost, hail and rainbows. Lightning and earthquakes interested him and he tried to approach all these topics in a scientific manner although he had little chance of coming up with explanations which were anywhere close to being correct. In moral philosophy he followed the Stoic teachings and gave opinions on virtue, evil, the soul, and emotions. He wrote historical works covering the period from about 146 BC to about 63 BC. These works give an account of the Roman civil wars and the contacts by the Greeks and the Romans with other peoples such as the Celts, Germans, and peoples of Spain and Gaul. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Posidonius.html (2 of 3) [2/16/2002 11:27:21 PM]

Posidonius

List of References (11 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Posidonius.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Posidonius.html (3 of 3) [2/16/2002 11:27:21 PM]

Post

Emil Leon Post Born: 11 Feb 1897 in Augustów, Poland Died: 21 April 1954 in New York, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Emil Post's father was Arnold Post and his mother was Pearl Post. Arnold and Pearl were Polish Jews and their son Emil was born in Poland and spent the first seven years of his life there. The family emigrated to the United States in May 1904 looking for a better life, and set up home in New York. Emil was an extraordinarily bright child but his life was one of great tragedy. When he was a child he lost an arm in an accident but this handicap was one which he handled well. He had to face mental problems in his adult life which had a devastating effect on him, making the physical problem of having lost an arm seem rather trivial in comparison. There was free secondary schooling available for specially gifted children in New York. This was at the Townsend Harris High School which was situated on the same site as the College of the City of New York. After graduating from the High School Post remained on the same campus as he continued his studies at the City College. We now think of Post as a mathematical logician but the first subject which attracted him was astronomy. While studying at the College of the City of New York he studied mathematics but there is little sign that at this stage he was particularly attracted towards logic. While an undergraduate at the College he wrote his first paper which was on generalised differentiation. The question he asked was a fascinating one: what does the differential operator Dn mean when n is not an integer? Although written while he was an undergraduate, Post did not submit the paper to the American Mathematical Society until 1923 and it was not finally published until 1930. It does contain a really important idea, for in the paper Post proves an important result about inverting the Laplace transform. This publication appeared long after Post's

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Post.html (1 of 5) [2/16/2002 11:27:23 PM]

Post

graduation with his first degree which was his B.S. awarded by the City College in 1917. After graduating with his first degree, Post began postgraduate research at Columbia University. Now the significant event for Post's career had been the publication of Russell and Whitehead's Principia Mathematica. The first volume of Principia Mathematica was published in 1910, the second in 1912, and the third in 1913. When Post began his graduate studies it was an exciting new development and Post participated in Cassius J Keyser's seminar at Columbia which studied the Principia Mathematica. Post was awarded the degree of A.M. in 1918 and of Ph.D. in 1920. His Ph.D. thesis was on mathematical logic, and we shall discuss it further in a moment, but first let us note that Post wrote a second paper as a postgraduate, which was published before his first paper, and this was a short work on the functional equation of the gamma function. We now turn to Post's Ph.D. thesis, in which he proved the completeness and consistency of the propositional calculus described in the Principia Mathematica by introducing the truth table method. He then generalised his truth table method, which was based on the two values "true" and "false", to a method which had an arbitrary finite number of truth values. The final, and perhaps the most remarkable, new idea which Post introduced in his thesis was to give a framework for systems of logic as inference systems based on a finite process of manipulation of symbols. Such a system of logic that Post proposed produces, in today's terminology, a recursively enumerable set of words on a finite alphabet. It would be fair to say that Post's thesis marks the beginning of proof theory. After receiving his doctorate, Post went to Princeton University for a year as Proctor Fellow. He returned to Columbia University and, shortly after this, he had his first bout of an illness which was to recur throughout his career and limit what he might have achieved. As Davis writes in [4]:He suffered all his adult life from crippling manic-depressive disease at a time when no drug therapy was available for this malady. In 1924 Post went to Cornell but again became ill. He resumed work as a high school teacher in New York in 1927. He married Gertrude Singer in 1929 and they had one child, a daughter Phyllis. Then in 1932 he was appointed to the City College. He left after a short spell, again struggling with his mental illness, but returned three years later and spent the rest of his life there. At the City College his teaching load was 16 contact hours per week which made finding time for research very difficult. Also members of staff has no offices of their own but were all put in a single room with one large table in the middle. Post chose to work at home, but with a young child this put a strain on the family. Post's daughter Phyllis explained later in her life how Gertrude Post had struggled to give her husband the opportunity to devote time to research:My father was a genius; my mother was a saint ... Besides typing letters of recommendation, my mother also typed my father's manuscripts and correspondence ... My mother was also the one who handled all financial matters ... she was the buffer in daily life that permitted my father to devote his attention to mathematics (as well as his varied interests in contemporary world affairs). Would he have accomplished so much without her? I for one, don't think so. Post's early death at the age of 57 was almost certainly a direct consequence of the treatment he received for his mental illness. At that time such manic-depressive illnesses were treated with electric shock treatment. It was an horrific treatment for an horrific illness and one which caused great distress. It was based on nothing better than the fact that after patients received this treatment many had periods of more normal mental states. Post received the electric shock treatment on a number of occasions and it was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Post.html (2 of 5) [2/16/2002 11:27:23 PM]

Post

while he was in a mental institution, shortly after receiving electric shocks, that he suffered a heart attack and died. Post is best known for his work on polyadic groups, recursively enumerable sets, and degrees of unsolvability, as well as for his contribution to the unsolvability of problems in combinatorial mathematics. He introduced the concepts of completeness and consistency in a paper on truth-table methods which developed from the work of his doctoral thesis. He attributed these methods to his teacher at Columbia, C J Keyser, rather than to Charles Peirce and E Schröder as had been done previously. In the 1920's Post proved results similar to those which Gödel, Church and Turing discovered later, but he did not publish them. He reason he did not publish was because he felt that a 'complete analysis' was necessary to gain acceptance. He wrote:The correctness of this result is clearly entirely dependent on the trustworthiness of the analysis leading to the above generalisation... it is fundamentally weak in its reliance on the logic of Principia Mathematica ... for full generality a complete analysis would have to be given of all possible ways in which the human mind could set up finite processes for generating sequences. He also made a mathematical study of Lukasiewicz's three-valued logic. At around this time he wrote in his diary:I study Mathematics as a product of the human mind not as absolute. When Gödel published his Incompleteness Theorems in 1931, Post realised that he had waited too long to publish what he had proved and that now the whole credit would go to Gödel. In a postcard written to Gödel in 1938, just after they had met for the first time, Post wrote:... for fifteen years I carried around the thought of astounding the mathematical world with my unorthodox ideas, and meeting the man chiefly responsible for the vanishing of that dream rather carried me away. Since you seemed interested in my way of arriving at these new developments perhaps Church can show you a long letter I wrote to him about them. As for any claims I might make perhaps the best I can say is that I would have proved Gödel's Theorem in 1921 - had I been Gödel. In a follow-up letter written the day after he writes:... after all it is not ideas but the execution of ideas that constitute a mark of greatness. In 1936 he proposed what is now known as a Post machine, a kind of automaton which predates the notion of a program which von Neumann studied in 1946. In 1941 he wrote:... mathematical thinking is, and must be, essentially creative... but he said there are limitations and symbolic logic is:... the indisputable means for revealing and developing these limitations. Post showed that the word problem for semigroups was recursively insoluble in 1947, giving the solution to a problem which had been posed by Thue in 1914. Quine, in a letter written in 1954 after Post's death, said:Modern proof theory, and likewise the modern theory of machine computation, hinge on the concept of the recursive function. This important number theoretic concept ... was

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Post.html (3 of 5) [2/16/2002 11:27:23 PM]

Post

discovered independently ... by four mathematicians, and one of these was Post. Subsequent work by Post was instrumental to the further progress of the theory of recursive functions. Quine added in 1972:The theory of recursive functions of which Post was cofounder is now nearly twice as old as when I wrote that letter. What a fertile field it has proved to be. The way that Post conducted his classes at the City College was, to say the least, unusual. Davis attended such classes at the College of the City of New York during the late 1940s, and in [4] he gives us a clear picture:Post's classes were tautly organised affairs. Each period would begin with student recitations covering problems and proofs of theorems from the day's assignment. These were handed out apparently at random and had to be put on the blackboard without the aid of textbooks or notes. Woe betide the hapless student who was unprepared. He (or rarely she) would have to face Post's "more in sorrow than anger look". In turn, the students would recite on their work. Afterwards, Post would get out his 3 by 5 cards and explain various fine points. The class would be a success if he completed his last card just as the bell rang. Questions from the class were discouraged: there was no time. Surprisingly, these inelastic pedagogic methods were extremely successful, and Post was a very popular teacher. Paul Chessin recalls being taught by Post in New York in about 1943:I recall that he was a short stocky fellow who invariably dressed in a three piece suit, empty sleeve carefully tucked into the side suitcoat pocket. He would stride steadily up and down before the blackboard, speaking clearly, vigorous in his motions. He would frequently, suddenly whirl around to face the board, chalk in hand, to write. This motion always tended to loosen that sleeve from its anchor until finally (to the relief of the class) it flapped loosely about as a cape might. That freedom of motion seemed to us to liberate his thinking as he lectured. Finally we give this very fine tribute to Post from Davis:Post's significance transcends his scientific contributions, important as those were. He remains an inspiration as well, for the manner in which he overcame his potentially crippling mental disability, for his distinctive voice, and for his continued devotion to science and his students. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Post.html (4 of 5) [2/16/2002 11:27:23 PM]

Post

History Topics Time lines Glossary index

Societies, honours, etc.

Famous curves

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2001 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Post.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Post.html (5 of 5) [2/16/2002 11:27:23 PM]

Potapov

Vladimir Petrovich Potapov Born: 24 Jan 1914 in Odessa, Ukraine Died: 21 Dec 1980 in Kharkov, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vladimir Potapov did not attend school but was taught, from the age of six, by his father who was a lecturer in Old Russian Literature at the University of Odessa. Potapov's father taught him mathematics, history, literature and languages. However he had a special music teacher, as music was at this stage his first love, and his mother also assisted in his education also teaching him literature. Potapov entered Odessa conservatory to study music and for three years he followed the course. At this time, see [3]:He was well educated and, although only seventeen, very serious. Whatever he did, he did in a very serious fashion, he walked and spoke slowly, and had his own opinion about everything. Even his teachers at the conservatory respected his profundity and thoroughness and called him 'professor'. And, as time proved, they were not mistaken. However he did not complete the four year course, transferring instead to the Faculty of Physics and Mathematics at Odessa University. At Odessa University, Potapov was a student of Livsic and he was also taught by Livsic's teacher Krein. After graduating in 1939 he began to work for his doctorate on the problem of divisors of almost periodic polynomials. Europe was disrupted by World War II soon after he began his doctoral work but due to poor eyesight Potapov did not have to undertake military duties. He was sent to the Odessa Institute of Marine Engineer, however, to undertake mathematical work in keeping with the war effort. In 1941 Potapov was evacuated from Odessa which is in south-western Ukraine on the Black Sea coast. He taught at a school in Fari but was allowed to return to Odessa in 1944 and, much delayed by the war, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Potapov.html (1 of 3) [2/16/2002 11:27:25 PM]

Potapov

submitted his doctoral thesis to Odessa University in 1945. When the war was over people of Jewish descent were victimised. In particular Jewish lecturers at Odessa such as Krein and Livsic were dismissed from their posts. Potapov was not Jewish but he wrote an article attacking the policy of dismissing Jewish lecturers:The leaders of the university have to correct their mistakes and to revive immediately the famous traditions of the Odessa School of Mathematics. The Faculty of Physics and Mathematics has to become active again, as a really creative centre of scientific and mathematical thought in our city .... He did not stop at an article as is explained in [3]:During the forties and fifties V P Potapov made many unsuccessful attempts to bring M G Krein and his students back to the university. The leaders of the Faculty of Physics and Mathematics could not forgive him for this. In fact they prohibited the use of any subject connected with Potapov's investigations as a theme for a diploma. In 1948 Potapov was invited to the Pedagogical Institute at Odessa. He soon became Head of Mathematics and, from 1952, Dean of the Faculty of Physics and Mathematics. He used his position to invite Livsic and others to the Institute. During the 1950s Potapov worked on the theory of J-contractive matrix functions and the analysis of matrix functions became his main work. He published papers on the multiplicative theory of analytic matrix functions in the years 1950-55 which contain work from his doctoral thesis. He also worked on interpolation problems. From 1974 Potapov lectured at Odessa Institute of National Economy, then he went to Kharkov to head the Department of Applied Mathematics at the Institute for low temperature physics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Potapov.html (2 of 3) [2/16/2002 11:27:25 PM]

Potapov

JOC/EFR June 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Potapov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Potapov.html (3 of 3) [2/16/2002 11:27:25 PM]

Pratt

John Henry Pratt Born: 4 June 1809 in London, England Died: 28 Dec 1871 in Ghazipur, India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Pratt attended Gonville and Caius College, Cambridge receiving his B.A. in 1833. He was third Wrangler, an excellent result, and he was elected to a Fellowship. In 1836 he received his M.A. from Christ's and Sidney Sussex College. John's father was secretary of the Church Missionary Society and John left university with two strong drives inside him, one coming from his exceptional scientific ability, the other from the missionary zeal coming through his father. Pratt obtained a chaplaincy appointment with the East India Company in 1838. He remained in India for the rest of his life, becoming chaplain to the Bishop of Calcutta in 1844. Then in 1850 Pratt was appointed Archdeacon of Calcutta. His first work, published in 1836, was The mathematical principles of mechanical philosophy. He revised the work in 1842, then expanded and republished it under the title On attractions, Laplace's functions and the figure of the Earth in 1860. This work of Pratt's was on the shape of the Earth. He assumed that the Earth behaved as a fluid and showed, as Newton had done, that the resulting shape would be an oblate spheroid. He gave 26.9 miles as the difference between the equatorial and polar axes, quite a good result. In 1855 he postulated density differences in the Earth's crust, lower densities under mountains, higher densities in lowlands, to explain the (too nearly constant) values obtained for gravity at a given latitude. This is included in the rewrite of his work in 1860. In the same year, 1855, Airy offered a different explanation of the gravitational data. Both Pratt's and Airy's proposals have their merits but are oversimplifications of the actual situation. Pratt also published religious works; Scripture and science not at variance (1856) was a popular work which ran to many editions. He also edited his father's Notes of discussion on religious topics at meetings of the Eclectic Society, London, during 1798 - 1814. He was elected a Fellow of the Royal Society in 1866. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pratt.html (1 of 2) [2/16/2002 11:27:26 PM]

Pratt

Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to John Pratt (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1866

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pratt.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pratt.html (2 of 2) [2/16/2002 11:27:26 PM]

Pringsheim

Alfred Pringsheim Born: 2 Sept 1850 in Ohlau (now Olawa), Lower Silesia (now Poland) Died: 25 June 1941 in Zurich, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Having studied in Berlin Alfred Pringsheim became a follower of Weierstrass's mathematics. He taught in Munich and worked on real and complex functions. He gave a very simple proof of Cauchy's integral theorem. He also has important results on the singularities of power series with positive coefficients. From 1933 to 1939 his life was made impossible as a non-Aryan. His house was taken away and eventually he moved to Zurich. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pringsheim.html (1 of 2) [2/16/2002 11:27:28 PM]

Pringsheim

Glossary index

Quotations index

Mathematicians of the day JOC/EFR December 1996

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pringsheim.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pringsheim.html (2 of 2) [2/16/2002 11:27:28 PM]

Privalov

Ivan Ivanovich Privalov Born: 11 Feb 1891 in Nizhniy Lomov, Penza guberniya (now oblast), Russia Died: 13 July 1941 in Moscow, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ivan Privalov attended school in Novgorod, then entered the University of Moscow in 1909. He graduated in 1913 studying under Egorov and Luzin. His Master's Degree was awarded in 1916 and after this he taught at the University of Moscow. In 1917 Privalov became professor at Saratov University where he remained for five years, then returned to Moscow. He was appointed professor of the theory of functions and complex variable there and he held this post for the rest of his life. Privalov studied analytic functions in the vicinity of singular points by means of measure theory and Lebesgue integrals. He also obtained important results on conformal mappings showing that angles were preserved on the boundary almost everywhere. Privalov wrote Cauchy Integral (1918) which built on work by Fatou. He also worked on many problems jointly with Luzin. In 1934 he studied subharmonic functions, building on the work of Riesz. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Privalov.html (1 of 2) [2/16/2002 11:27:30 PM]

Privalov

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Privalov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Privalov.html (2 of 2) [2/16/2002 11:27:30 PM]

Privat_de_Molieres

Joseph Privat de Molières Born: 1677 in Tarascon, Bouches-du-Rhône, France Died: 12 May 1742 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Privat de Molières studied at Oratorian schools in Aix, Marseilles, Arles and Angers. At the last of these schools he studied under Charles-René Reyneau from 1698 until 1699. He chose, against his parents wishes, an ecclesiastical life and entered the Congregation of the Oratory in 1699. His first posts were teaching in schools belonging to the Congregation of the Oratory, first at Saumur, then at Juilly and finally at Soissons. In 1704 he went to Paris to take up a more active scientific career. He studied mathematics and physics with Malebranche until 1715. In 1723 he was appointed to a chair at the Collège Royal to succeed Varignon. Privat de Molières was elected to the Académie Royal des Sciences in 1721, and became a Fellow of the Royal Society of London in 1729. He argued against Newton and for Descartes' view of physics although he knew Newton's to be the more precise. He attempted to bring Newton's calculations into the vortex theory of matter of Malebranche. Although his arguments were very effective, eventually Newtonian physics came to the fore in France. Privat de Molières published Leçons de mathematiques (1726), a work on the principles of algebra and calculus. His Leçons de physique (1734-1739), was a four volume work based on his lectures at the Collège Royale. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Privat de Molières (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Privat_de_Molieres.html (1 of 2) [2/16/2002 11:27:31 PM]

Privat_de_Molieres

Fellow of the Royal Society

Elected 1729

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Privat_de_Molieres.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Privat_de_Molieres.html (2 of 2) [2/16/2002 11:27:31 PM]

Proclus

Proclus Diadochus Born: 8 Feb 411 in Constantinople (now Istanbul), Byzantium (now Turkey) Died: 17 April 485 in Athens, Greece Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Proclus's father, Particius, and his mother, Marcella, were citizens of high social position in Lycia. Particius was a senior law official in the courts at Byzantium. Proclus was brought up at Xanthus, on the south coast of Lycia, where he attended school. It was intended that Proclus should follow his father and enter the legal profession. With this aim in mind he was sent to Alexandria but, while in the middle of his studies, he visited Byzantium and he became convinced that his calling in life was the study of philosophy. He returned to Alexandria where now he studied philosophy under Olympiodorus the Elder, in particular making a deep study of the works of Aristotle. He also learnt mathematics in Alexandria and in this subject his teacher was Heron (not the famous mathematician, Heron was a common name at this time). Proclus was not entirely satisfied with the education he was receiving in philosophy in Alexandria so, while still a teenager, he moved from Alexandria to Athens where he studied at Plato's Academy under the philosophers Plutarch and Syrianus (a pupil of Plutarch). He progressed from being a student at the Academy to teaching there then, on the death of Syrianus, Proclus became head of the Academy. The title Diadochus was given to him at this time, the meaning of the word being successor. At the Academy Proclus appears to have been well off and to have helped his friends and relations financially. He never married and lived a life which was, in certain respects, not unlike that proposed by Pythagoras. He did not eat meat and tried to live a religious life, composing hymns to the gods. His hymns were clearly highly thought of since seven of them have been preserved and are seen today as having considerable literary merit. Proclus was to remain as head of the Academy until his death. A man of great learning, Proclus was regarded with great veneration by his contemporaries. He followed the neoplatonist philosophy which Plotinus founded, and Porphyry and Iamblichus developed around 300 AD. Other developers of these ideas were Plutarch and Syrianus, the teachers of Proclus. Heath writes [4]:He was an acute dialectician and pre-eminent among his contemporaries in the range of his learning; he was a competent mathematician; he was even a poet. At the same time he was a believer in all sorts of myths and mysteries, and a devout worshipper of divinities both http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.html (1 of 4) [2/16/2002 11:27:33 PM]

Proclus

Greek and Oriental. He was much more a philosopher than a mathematician. Of course, as one might expect, his belief in many religious sayings meant that he was highly biased in his views on many issues of science. For example he mentions the hypothesis that the sun is at the centre of the planets as proposed by Hipparchus but rejects it immediately since it contradicted the views of a Chaldean whom he says that it is unlawful not to believe. Proclus wrote Commentary on Euclid which is our principal source about the early history of Greek geometry. The book is certainly the product of his teaching at the Academy. This work is not coloured by his religious beliefs and Martin, writing in the middle of the 19th century, says (see for example [4]):... for Proclus the "Elements of Euclid" had the good fortune not to be contradicted either by the Chaldean Oracles or by the speculations of Pythagoreans old and new. Proclus had access to books which are now lost and others, already lost in Proclus's time, were described based on extracts in other books available to Proclus. In particular he certainly used the History of Geometry by Eudemus, which is now lost, as is the works of Geminus which he also used. Heath, describing Proclus's Commentary on Euclid writes [4]:Proclus deals historically and critically with all the definitions, postulates and axioms in order. The notes on the postulates and axioms are preceded by a general discussion of the principles of geometry, hypotheses, postulates and axioms, and their relation to one another; here as usual Proclus quotes the opinions of all the important authorities. Another interesting part of Proclus's commentary is his discussion of the critics of geometry. He writes:... it is against [the principles of geometry] that most critics of geometry have raised objections, endeavouring to show that these parts are not firmly established. Of those in this group whose arguments have become notorious some, such as the Sceptics, would do away with all knowledge ... whereas others, like the Epicureans, propose only to discredit the principles of geometry. Another group of critics, however, admit the principles but deny that the propositions coming after the principles can be demonstrated unless they grant something that is not contained in the principles. This method of controversy was followed by Zeno of Sidon, who belonged to the school of Epicurus and against whom Posidonius has written a whole book and shown that his views are thoroughly unsound. Morrow in [1] confirms the great debt that we owe to Proclus, and in particular his Commentary on Euclid when he writes in [1]:Proclus was not a creative mathematician; but he was an acute expositor and critic, with a thorough grasp of mathematical method and a detailed knowledge of the thousand years of Greek mathematics from Thales to his own time. The recent book [7] gives a good description of the writings of Proclus found in his commentary on Book I of Euclid's Elements. The book [7] is an important contribution to the study of the philosophy of Proclus and in particular his philosophy of mathematics. Proclus also wrote Hypotyposis, an introduction to the astronomical theories of Hipparchus and Ptolemy in which he described the mathematical theory of the planets based on epicycles and on eccentrics. He combined his geometrical skills and his knowledge of astronomy to give a geometrical proof that the epicycle theory for the planets is equivalent to the eccentric theory. In the epicycle theory the Earth is in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.html (2 of 4) [2/16/2002 11:27:33 PM]

Proclus

the centre of a circle which has smaller circles rotating round its circumference. In the eccentric theory the planets move round in circles whose centres do not coincide with the Earth. Nothing here is original and Proclus is proving results first given by Hipparchus and Ptolemy. However, although Proclus believed that this theory should be studied by his students at the Academy, he was not uncritical, suggesting that the theory was overly complicated and also that it was an ad hoc theory with no reason to explain its various parts. In his astronomical writings, Proclus described how the water clock invented by Heron could be used to measure the apparent diameter of the Sun. Proclus's method can be used at the equinox. Water is collected from the clock in a container while the sun rises. As soon as the Sun has risen the water is collected in another container and this measurement continues until sunrise the following day. Then the ratio of the weights of water in the two containers gives the apparent diameter of the Sun. Among Proclus's many works are Liber de causis (Book of Causes), Institutio theologica (Elements of Theology), a concise exposition of metaphysics, Elements of Physics, largely giving Aristotle's views, and In Platonis theologiam (Platonic Theology) giving Plato's metaphysics. His contribution is well summarised in [1] as follows:Proclus deserves to be remembered ... for the qualities he possessed that are exceedingly rare in any age and were almost unique in his: the logical clarity and firmness of his thought, the acuteness of his analyses, his eagerness to understand and readiness to present the views of his predecessors on controversial issues, the sustained coherence of his lengthy expositions, and the large horizon, as broad as the whole of being, within which his thinking moved. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles)

Some Quotations (5)

Mathematicians born in the same country Cross-references to History Topics

1. Non Euclidean geometry 2. Squaring the circle 3. How do we know about Greek mathematicians?

Other references in MacTutor

1. A quotation from Proclus 2. Chronology: 1AD to 500

Honours awarded to Proclus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Proclus

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.html (3 of 4) [2/16/2002 11:27:33 PM]

Proclus

Other Web sites

1. Math Forum 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Proclus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Proclus.html (4 of 4) [2/16/2002 11:27:33 PM]

Prthudakasvami

Prthudakasvami Born: about 830 in India Died: about 890 in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Prthudakasvami is best known for his work on solving equations. The solution of a first-degree indeterminate equation by a method called kuttaka (or "pulveriser") was given by Aryabhata I. This method of finding integer solutions resembles the continued fraction process and can also be seen as a use of the Euclidean algorithm. Brahmagupta seems to have used a method involving continued fractions to find integer solutions of an indeterminate equation of the type ax + c = by. Prthudakasvami's commentary on Brahmagupta's work is helpful in showing how "algebra", that is the method of calculating with the unknown, was developing in India. Prthudakasvami discussed the kuttaka method which he renamed as "bijagnita" which means the method of calculating with unknown elements. To see just how this new idea of algebra was developing in India, we look at the notation which was being used by Prthudakasvami in his commentary on Brahmagupta's Brahma Sputa Siddhanta. In this commentary Prthudakasvami writes the equation 10x + 8 = x2 + 1 as: yava 0 ya 10 ru 8 yava 1 ya 0 ru 1 Here yava is an abbreviation for yavat avad varga which means the "square of the unknown quantity", ya is an abbreviation for yavat havat which means the "unknown quantity", and ru is an abbreviation for rupa which means "constant term". Hence the top row reads 0x2 + 10x + 8 while the second row reads x2 + 0x + 1 The whole equation is therefore 0x2 + 10x + 8 = x2 + 0x + 1 or 10x + 8 = x2 + 1. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Prthudakasvami.html (1 of 2) [2/16/2002 11:27:34 PM]

Prthudakasvami

List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Prthudakasvami.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Prthudakasvami.html (2 of 2) [2/16/2002 11:27:34 PM]

Prufer

Ernst Paul Heinz Prüfer Born: 10 Nov 1896 in Germany Died: 7 April 1934 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Heinz Prüfer worked on abelian groups, algebraic numbers and knot theory. He also published papers on Sturm-Liouville theory. After his death, four years after being appointed to a chair at Münster, his papers on projective geometry were collected into a book. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Prufer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Prufer.html [2/16/2002 11:27:35 PM]

Ptolemy

Claudius Ptolemy Born: about 85 in Egypt Died: about 165 in Alexandria, Egypt

Click the picture above to see five larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

One of the most influential Greek astronomers and geographers of his time, Ptolemy propounded the geocentric theory in a form that prevailed for 1400 years. However, of all the ancient Greek mathematicians, it is fair to say that his work has generated more discussion and argument than any other. We shall discuss the arguments below for, depending on which are correct, they portray Ptolemy in very different lights. The arguments of some historians show that Ptolemy was a mathematician of the very top rank, arguments of others show that he was no more than a superb expositor, but far worse, some even claim that he committed a crime against his fellow scientists by betraying the ethics and integrity of his profession. We know very little of Ptolemy's life. He made astronomical observations from Alexandria in Egypt during the years AD 127-41. In fact the first observation which we can date exactly was made by Ptolemy on 26 March 127 while the last was made on 2 February 141. It was claimed by Theodore Meliteniotes in around 1360 that Ptolemy was born in Hermiou (which is in Upper Egypt rather than Lower Egypt where Alexandria is situated) but since this claim first appears more than one thousand years after Ptolemy lived, it must be treated as relatively unlikely to be true. In fact there is no evidence that Ptolemy was ever anywhere other than Alexandria. His name, Claudius Ptolemy, is of course a mixture of the Greek Egyptian 'Ptolemy' and the Roman 'Claudius'. This would indicate that he was descended from a Greek family living in Egypt and that he was a citizen of Rome, which would be as a result of a Roman emperor giving that 'reward' to one of Ptolemy's ancestors. We do know that Ptolemy used observations made by 'Theon the mathematician', and this was almost certainly Theon of Smyrna who almost certainly was his teacher. Certainly this would make sense since

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html (1 of 8) [2/16/2002 11:27:37 PM]

Ptolemy

Theon of Smyrna was both an observer and a mathematician who had written on astronomical topics such as conjunctions, eclipses, occultations and transits. Most of Ptolemy's early works are dedicated to Syrus who may have also been one of his teachers in Alexandria, but nothing is known of Syrus. If these facts about Ptolemy's teachers are correct then certainly in Theon of Smyrna he did not have a great scholar, for Theon of Smyrna seems not to have understood in any depth the astronomical work he describes. On the other hand Alexandria had a tradition for scholarship which would mean that even if Ptolemy did not have access to the best teachers, he would have access to the libraries where he would have found the valuable reference material of which he made good use. Ptolemy's major works have survived and we shall discuss them in this article. The most important, however, is the Almagest which is a treatise in thirteen books. We should say straight away that, although the work is now almost always known as the Almagest that was not its original name. Its original Greek title translates as The Mathematical Compilation but this title was soon replaced by another Greek title which means The Greatest Compilation. This was translated into Arabic as "al-majisti" and from this the title Almagest was given to the work when it was translated from Arabic to Latin. The Almagest is the earliest of Ptolemy's works and gives in detail the mathematical theory of the motions of the Sun, Moon, and planets. Ptolemy made his most original contribution by presenting details for the motions of each of the planets. The Almagest was not superseded until a century after Copernicus presented his heliocentric theory in the De revolutionibus of 1543. Grasshoff writes in [8]:Ptolemy's "Almagest" shares with Euclid's "Elements" the glory of being the scientific text longest in use. From its conception in the second century up to the late Renaissance, this work determined astronomy as a science. During this time the "Almagest" was not only a work on astronomy; the subject was defined as what is described in the "Almagest". Ptolemy describes himself very clearly what he is attempting to do in writing the work (see for example [15]):We shall try to note down everything which we think we have discovered up to the present time; we shall do this as concisely as possible and in a manner which can be followed by those who have already made some progress in the field. For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients. However, those topics which have not been dealt with by our predecessors at all, or not as usefully as they might have been, will be discussed at length to the best of our ability. Ptolemy first of all justifies his description of the universe based on the earth-centred system described by Aristotle. It is a view of the world based on a fixed earth around which the sphere of the fixed stars rotates every day, this carrying with it the spheres of the sun, moon, and planets. Ptolemy used geometric models to predict the positions of the sun, moon, and planets, using combinations of circular motion known as epicycles. Having set up this model, Ptolemy then goes on to describe the mathematics which he needs in the rest of the work. In particular he introduces trigonometrical methods based on the chord function Crd (which is related to the sine function by sin a = (Crd 2a)/120). Ptolemy devised new geometrical proofs and theorems. He obtained, using chords of a circle and an inscribed 360-gon, the approximation http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html (2 of 8) [2/16/2002 11:27:37 PM]

Ptolemy

= 3 17/120 = 3.14166 and, using 3 = chord 60 , 3 = 1.73205. He used formulas for the Crd function which are analogous to our formulas for sin(a + b), sin(a - b) and sin a/2 to create a table of the Crd function at intervals of 1/2 a degree. This occupies the first two of the 13 books of the Almagest and then, quoting again from the introduction, we give Ptolemy's own description of how he intended to develop the rest of the mathematical astronomy in the work (see for example [15]):[After introducing the mathematical concepts] we have to go through the motions of the sun and of the moon, and the phenomena accompanying these motions; for it would be impossible to examine the theory of the stars thoroughly without first having a grasp of these matters. Our final task in this way of approach is the theory of the stars. Here too it would be appropriate to deal first with the sphere of the so-called 'fixed stars', and follow that by treating the five 'planets', as they are called. In examining the theory of the sun, Ptolemy compares his own observations of equinoxes with those of Hipparchus and the earlier observations Meton in 432 BC. He confirmed the length of the tropical year as 1/300 of a day less than 365 1/4 days, the precise value obtained by Hipparchus. Since, as Ptolemy himself knew, the accuracy of the rest of his data depended heavily on this value, the fact that the true value is 1/28 of a day less than 365 1/4 days did produce errors in the rest of the work. We shall discuss below in more detail the accusations which have been made against Ptolemy, but this illustrates clearly the grounds for these accusations since Ptolemy had to have an error of 28 hours in his observation of the equinox to produce this error, and even given the accuracy that could be expected with ancient instruments and methods, it is essentially unbelievable that he could have made an error of this magnitude. A good discussion of this strange error is contained in the excellent article [19]. Based on his observations of solstices and equinoxes, Ptolemy found the lengths of the seasons and, based on these, he proposed a simple model for the sun which was a circular motion of uniform angular velocity, but the earth was not at the centre of the circle but at a distance called the eccentricity from this centre. This theory of the sun forms the subject of Book 3 of the Almagest. In Books 4 and 5 Ptolemy gives his theory of the moon. Here he follows Hipparchus who had studied three different periods which one could associate with the motion of the moon. There is the time taken for the moon to return to the same longitude, the time taken for it to return to the same velocity (the anomaly) and the time taken for it to return to the same latitude. Ptolemy also discusses, as Hipparchus had done, the synodic month, that is the time between successive oppositions of the sun and moon. In Book 4 Ptolemy gives Hipparchus's epicycle model for the motion of the moon but he notes, as in fact Hipparchus had done himself, that there are small discrepancies between the model and the observed parameters. Although noting the discrepancies, Hipparchus seems not to have worked out a better model, but Ptolemy does this in Book 5 where the model he gives improves markedly on the one proposed by Hipparchus. An interesting discussion of Ptolemy's theory of the moon is given in [24]. Having given a theory for the motion of the sun and of the moon, Ptolemy was in a position to apply

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html (3 of 8) [2/16/2002 11:27:37 PM]

Ptolemy

these to obtain a theory of eclipses which he does in Book 6. The next two books deal with the fixed stars and in Book 7 Ptolemy uses his own observations together with those of Hipparchus to justify his belief that the fixed stars always maintain the same positions relative to each other. He wrote (see for example [15]):If one were to match the above alignments against the diagrams forming the constellations on Hipparchus's celestial globe, he would find that the positions of the relevant stars on the globe resulting from the observations made at the time of Hipparchus, according to what he recorded, are very nearly the same as at present. In these two book Ptolemy also discusses precession, the discovery of which he attributes to Hipparchus, but his figure is somewhat in error mainly because of the error in the length of the tropical year which he used. Much of Books 7 and 8 are taken up with Ptolemy's star catalogue containing over one thousand stars. The final five books of the Almagest discuss planetary theory. This must be Ptolemy's greatest achievement in terms of an original contribution, since there does not appear to have been any satisfactory theoretical model to explain the rather complicated motions of the five planets before the Almagest. Ptolemy combined the epicycle and eccentric methods to give his model for the motions of the planets. The path of a planet P therefore consisted of circular motion on an epicycle, the centre C of the epicycle moving round a circle whose centre was offset from the earth. Ptolemy's really clever innovation here was to make the motion of C uniform not about the centre of the circle around which it moves, but around a point called the equant which is symmetrically placed on the opposite side of the centre from the earth. The planetary theory which Ptolemy developed here is a masterpiece. He created a sophisticated mathematical model to fit observational data which before Ptolemy's time was scarce, and the model he produced, although complicated, represents the motions of the planets fairly well. Toomer sums up the Almagest in [1] as follows:As a didactic work the "Almagest" is a masterpiece of clarity and method, superior to any ancient scientific textbook and with few peers from any period. But it is much more than that. Far from being a mere 'systemisation' of earlier Greek astronomy, as it is sometimes described, it is in many respects an original work. We will return to discuss some of the accusations made against Ptolemy after commenting briefly on his other works. He published the tables which are scattered throughout the Almagest separately under the title Handy Tables. These were not merely lifted from the Almagest however but Ptolemy made numerous improvements in their presentation, ease of use and he even made improvements in the basic parameters to give greater accuracy. We only know details of the Handy Tables through the commentary by Theon of Alexandria but in [77] the author shows that care is required since Theon was not fully aware of Ptolemy's procedures. Ptolemy also did what many writers of deep scientific works have done, and still do, in writing a popular account of his results under the title Planetary Hypothesis. This work, in two books, again follows the familiar route of reducing the mathematical skills needed by a reader. Ptolemy does this rather cleverly by replacing the abstract geometrical theories by mechanical ones. Ptolemy also wrote a work on astrology. It may seem strange to the modern reader that someone who wrote such excellent scientific

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html (4 of 8) [2/16/2002 11:27:37 PM]

Ptolemy

books should write on astrology. However, Ptolemy sees it rather differently for he claims that the Almagest allows one to find the positions of the heavenly bodies, while his astrology book he sees as a companion work describing the effects of the heavenly bodies on people's lives. In a book entitled Analemma he discussed methods of finding the angles need to construct a sundial which involves the projection of points on the celestial sphere. In Planisphaerium he is concerned with stereographic projection of the celestial sphere onto a plane. This is discussed in [49] where it is stated:In the stereographic projection treated by Ptolemy in the "Planisphaerium" the celestial sphere is mapped onto the plane of the equator by projection from the south pole. Ptolemy does not prove the important property that circles on the sphere become circles on the plane. Ptolemy's major work Geography, in eight books, attempts to map the known world giving coordinates of the major places in terms of latitude and longitude. It is not surprising that the maps given by Ptolemy were quite inaccurate in many places for he could not be expected to do more than use the available data and this was of very poor quality for anything outside the Roman Empire, and even parts of the Roman Empire are severely distorted. In [19] Ptolemy is described as:... a man working [on map-construction] without the support of a developed theory but within a mathematical tradition and guided by his sense of what is appropriate to the problem. Another work on Optics is in five books and in it Ptolemy studies colour, reflection, refraction, and mirrors of various shapes. Toomer comments in [1]:The establishment of theory by experiment, frequently by constructing special apparatus, is the most striking feature of Ptolemy's "Optics". Whether the subject matter is largely derived or original, "The Optics" is an impressive example of the development of a mathematical science with due regard to physical data, and is worthy of the author of the "Almagest". An English translation, attempting to remove the inaccuracies introduced in the poor Arabic translation which is our only source of the Optics is given in [14]. The first to make accusations against Ptolemy was Tycho Brahe. He discovered that there was a systematic error of one degree in the longitudes of the stars in the star catalogue, and he claimed that, despite Ptolemy saying that it represented his own observations, it was merely a conversion of a catalogue due to Hipparchus corrected for precession to Ptolemy's date. There is of course definite problems comparing two star catalogues, one of which we have a copy of while the other is lost. After comments by Laplace and Lalande, the next to attack Ptolemy vigorously was Delambre. He suggested that perhaps the errors came from Hipparchus and that Ptolemy might have done nothing more serious than to have failed to correct Hipparchus's data for the time between the equinoxes and solstices. However Delambre then goes on to say (see [8]):One could explain everything in a less favourable but all the simpler manner by denying Ptolemy the observation of the stars and equinoxes, and by claiming that he assimilated everything from Hipparchus, using the minimal value of the latter for the precession motion. However, Ptolemy was not without his supporters by any means and further analysis led to a belief that

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html (5 of 8) [2/16/2002 11:27:37 PM]

Ptolemy

the accusations made against Ptolemy by Delambre were false. Boll writing in 1894 says [4]:To all appearances, one will have to credit Ptolemy with giving an essentially richer picture of the Greek firmament after his eminent predecessors. Vogt showed clearly in his important paper [78] that by considering Hipparchus's Commentary on Aratus and Eudoxus and making the reasonable assumption that the data given there agreed with Hipparchus's star catalogue, then Ptolemy's star catalogue cannot have been produced from the positions of the stars as given by Hipparchus, except for a small number of stars where Ptolemy does appear to have taken the data from Hipparchus. Vogt writes:This allows us to consider the fixed star catalogue as of his own making, just as Ptolemy himself vigorously states. The most recent accusations of forgery made against Ptolemy came from Newton in [12]. He begins this book by stating clearly his views:This is the story of a scientific crime. ... I mean a crime committed by a scientist against fellow scientists and scholars, a betrayal of the ethics and integrity of his profession that has forever deprived mankind of fundamental information about an important area of astronomy and history. Towards the end Newton, having claimed to prove every observation claimed by Ptolemy in the Almagest was fabricated, writes [12]:[Ptolemy] developed certain astronomical theories and discovered that they were not consistent with observation. Instead of abandoning the theories, he deliberately fabricated observations from the theories so that he could claim that the observations prove the validity of his theories. In every scientific or scholarly setting known, this practice is called fraud, and it is a crime against science and scholarship. Although the evidence produced by Brahe, Delambre, Newton and others certainly do show that Ptolemy's errors are not random, this last quote from [12] is, I [EFR] believe, a crime against Ptolemy (to use Newton's own words). The book [8] is written to study validity of these accusations and it is a work which I strongly believe gives the correct interpretation. Grasshoff writes:... one has to assume that a substantial proportion of the Ptolemaic star catalogue is grounded on those Hipparchan observations which Hipparchus already used for the compilation of the second part of his "Commentary on Aratus". Although it cannot be ruled out that coordinates resulting from genuine Ptolemaic observations are included in the catalogue, they could not amount to more than half the catalogue. ... the assimilation of Hipparchan observations can no longer be discussed under the aspect of plagiarism. Ptolemy, whose intention was to develop a comprehensive theory of celestial phenomena, had no access to the methods of data evaluation using arithmetical means with which modern astronomers can derive from a set of varying measurement results, the one representative value needed to test a hypothesis. For methodological reason, then, Ptolemy was forced to choose from a set of measurements the one value corresponding best to what he had to consider as the most reliable data. When an intuitive selection among the data was no longer possible ... Ptolemy had to consider those values as 'observed' which could be confirmed by theoretical predictions. As a final comment we quote the epigram which is accepted by many scholars to have been written by http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html (6 of 8) [2/16/2002 11:27:37 PM]

Ptolemy

Ptolemy himself, and it appears in Book 1 of the Almagest, following the list of contents (see for example [11]):Well do I know that I am mortal, a creature of one day. But if my mind follows the winding paths of the stars Then my feet no longer rest on earth, but standing by Zeus himself I take my fill of ambrosia, the divine dish. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (81 books/articles)

A Quotation

A Poster of Ptolemy

Mathematicians born in the same country

Cross-references to History Topics

1. Pi through the ages 2. A chronology of pi 3. Longitude and the Académie Royale 4. Greek Astronomy 5. Arabic mathematics : forgotten brilliance? 6. A brief history of cosmology 7. The trigonometric functions 8. An overview of Indian mathematics 9. A history of Zero

Other references in MacTutor

Chronology: 1AD to 500

Honours awarded to Ptolemy (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Ptolemaeus

Planetary features

Crater Ptolemaeus on Mars

Other Web sites

1. Science Museum, Florence 2. Vatican exhibition 3. High Altitude Observatory 4. Linda Hall Library (Star Atlas) 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html (7 of 8) [2/16/2002 11:27:37 PM]

Ptolemy

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ptolemy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ptolemy.html (8 of 8) [2/16/2002 11:27:37 PM]

Puiseux

Victor Alexandre Puiseux Born: 16 April 1820 in Argenteuil, Val-d'Oise, France Died: 9 Sept 1883 in Frontenay, Jura, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Victor Puiseux's family moved to Lorraine when he was only three years old and he was brought up in that region of France. He attended the Collège de Pont-à-Mousson then, in 1834, he entered the Collège Rollin in Paris where he attended lecture courses by Charles Sturm. Victor received prizes in physics in 1836 then a mathematics prize in 1837, the year he graduated and entered Ecole Normale Supérieure. Here he became friends with Briot and Bouquet and, in 1840, he was placed first in his final examination. The following year he was awarded his doctorate for a thesis on astronomy and mechanics in which he studied planetary orbits. His thesis is a competent piece of work without showing much in the way of originality. From 1841 until 1844 Puiseux was professor of mathematics at the Collège Royal in Rennes then, until 1849, he was professor of mathematics in the Faculty of Science in Besançon. During this period he published a series of papers in Liouville's Journal. He wrote on geometry, where he discovered new properties of evolutes and involutes and mechanics where he studied the conical pendulum, the tautochrone and similar topics. After this Puiseux held a number of posts. He worked at the Ecole Normale Supérieure during the years 1849 to 1855 and again between 1862 and 1868. From 1855 to 1859 he worked at the Paris Observatory. In 1857 he was appointed professor of mathematical astronomy at the Faculty of Science, after teaching courses for Le Verrier, where he succeeded to Cauchy's position. From 1868 to 1872 Puiseux held a post at the Bureau de Longitudes.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Puiseux.html (1 of 2) [2/16/2002 11:27:39 PM]

Puiseux

Puiseux had attended courses by Cauchy early in his career and he soon became interested in research in topics Cauchy was studying. He further developed Cauchy's work on functions of a complex variable, being the first to distinguish poles, essential points and branch points. He examined series expansions and looked at series with fractional powers. Although his work in this area was exceptionally good it became rather redundant after Riemann introduced the concept of a Riemann surface. Laplace's theory of the Moon, presented in 1787, had been shown to be inadequate by Adams in 1853. Puiseux contributed to the problem of the acceleration of the mean motion of the Moon and his work was used by Hill in his more precise understanding of the lunar motion in 1877. Puiseux also worked on elliptic functions and studied computational methods which were used to reduce astronomical data. He was elected to the mathematics section of the Academy of Sciences in 1871 where he succeeded Lamé. He was a keen mountaineer and was the first to scale an Alpine peak which is now named after him. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Victor Puiseux

Mathematicians born in the same country

Honours awarded to Victor Puiseux (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Puiseux

Other Web sites

The Catholic Encyclopedia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Puiseux.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Puiseux.html (2 of 2) [2/16/2002 11:27:39 PM]

Puissant

Louis Puissant Born: 22 Sept 1769 in France Died: 10 Jan 1843 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Louis Puissant directed the Ecole de Géographes in Paris from 1809 to 1833. He is best remembered for his invention of a new map projection for a new map of France and he was involved in the production of the map. The map was produced with considerable detail, the projection used spherical trigonometry, truncated power series and differential geometry. He wrote on geodesy, the shape of the Earth and spherical trigonometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Puissant.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Puissant.html [2/16/2002 11:27:40 PM]

Pythagoras

Pythagoras of Samos Born: about 569 BC in Samos, Ionia Died: about 475 BC

Click the picture above to see ten larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pythagoras of Samos is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure. We do have details of Pythagoras's life from early biographies which use important original sources yet are written by authors who attribute divine powers to him, and whose aim was to present him as a god-like figure. What we present below is an attempt to collect together the most reliable sources to reconstruct an account of Pythagoras's life. There is fairly good agreement on the main events of his life but most of the dates are disputed with different scholars giving dates which differ by 20 years. Some historians treat all this information as merely legends but, even if the reader treats it in this way, being such an early record it is of historical importance. Pythagoras's father was Mnesarchus ([12] and [13]), while his mother was Pythais [8] and she was a native of Samos. Mnesarchus was a merchant who came from Tyre, and there is a story ([12] and [13]) that he brought corn to Samos at a time of famine and was granted citizenship of Samos as a mark of gratitude. As a child Pythagoras spent his early years in Samos but travelled widely with his father. There are accounts of Mnesarchus returning to Tyre with Pythagoras and that he was taught there by the Chaldaeans and the learned men of Syria. It seems that he also visited Italy with his father. Little is known of Pythagoras's childhood. All accounts of his physical appearance are likely to be fictitious except the description of a striking birthmark which Pythagoras had on his thigh. It is probable http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html (1 of 7) [2/16/2002 11:27:42 PM]

Pythagoras

that he had two brothers although some sources say that he had three. Certainly he was well educated, learning to play the lyre, learning poetry and to recite Homer. There were, among his teachers, three philosophers who were to influence Pythagoras while he was a young man. One of the most important was Pherekydes who many describe as the teacher of Pythagoras. The other two philosophers who were to influence Pythagoras, and to introduce him to mathematical ideas, were Thales and his pupil Anaximander who both lived on Miletus. In [8] it is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old. By this time Thales was an old man and, although he created a strong impression on Pythagoras, he probably did not teach him a great deal. However he did contribute to Pythagoras's interest in mathematics and astronomy, and advised him to travel to Egypt to learn more of these subjects. Thales's pupil, Anaximander, lectured on Miletus and Pythagoras attended these lectures. Anaximander certainly was interested in geometry and cosmology and many of his ideas would influence Pythagoras's own views. In about 535 BC Pythagoras went to Egypt. This happened a few years after the tyrant Polycrates seized control of the city of Samos. There is some evidence to suggest that Pythagoras and Polycrates were friendly at first and it is claimed [5] that Pythagoras went to Egypt with a letter of introduction written by Polycrates. In fact Polycrates had an alliance with Egypt and there were therefore strong links between Samos and Egypt at this time. The accounts of Pythagoras's time in Egypt suggest that he visited many of the temples and took part in many discussions with the priests. According to Porphyry ([12] and [13]) Pythagoras was refused admission to all the temples except the one at Diospolis where he was accepted into the priesthood after completing the rites necessary for admission. It is not difficult to relate many of Pythagoras's beliefs, ones he would later impose on the society that he set up in Italy, to the customs that he came across in Egypt. For example the secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths made from animal skins, and their striving for purity were all customs that Pythagoras would later adopt. Porphyry in [12] and [13] says that Pythagoras learnt geometry from the Egyptians but it is likely that he was already acquainted with geometry, certainly after teachings from Thales and Anaximander. In 525 BC Cambyses II, the king of Persia, invaded Egypt. Polycrates abandoned his alliance with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. After Cambyses had won the Battle of Pelusium in the Nile Delta and had captured Heliopolis and Memphis, Egyptian resistance collapsed. Pythagoras was taken prisoner and taken to Babylon. Iamblichus writes that Pythagoras (see [8]):... was transported by the followers of Cambyses as a prisoner of war. Whilst he was there he gladly associated with the Magoi ... and was instructed in their sacred rites and learnt about a very mystical worship of the gods. He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians... In about 520 BC Pythagoras left Babylon and returned to Samos. Polycrates had been killed in about 522 BC and Cambyses died in the summer of 522 BC, either by committing suicide or as the result of an accident. The deaths of these rulers may have been a factor in Pythagoras's return to Samos but it is nowhere explained how Pythagoras obtained his freedom. Darius of Persia had taken control of Samos after Polycrates' death and he would have controlled the island on Pythagoras's return. This conflicts with the accounts of Porphyry and Diogenes Laertius who state that Polycrates was still in control of Samos when Pythagoras returned there.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html (2 of 7) [2/16/2002 11:27:42 PM]

Pythagoras

Pythagoras made a journey to Crete shortly after his return to Samos to study the system of laws there. Back in Samos he founded a school which was called the semicircle. Iamblichus [8] writes in the third century AD that:... he formed a school in the city [of Samos], the 'semicircle' of Pythagoras, which is known by that name even today, in which the Samians hold political meetings. They do this because they think one should discuss questions about goodness, justice and expediency in this place which was founded by the man who made all these subjects his business. Outside the city he made a cave the private site of his own philosophical teaching, spending most of the night and daytime there and doing research into the uses of mathematics... Pythagoras left Samos and went to southern Italy in about 518 BC (some say much earlier). Iamblichus [8] gives some reasons for him leaving. First he comments on the Samian response to his teaching methods:... he tried to use his symbolic method of teaching which was similar in all respects to the lessons he had learnt in Egypt. The Samians were not very keen on this method and treated him in a rude and improper manner. This was, according to Iamblichus, used in part as an excuse for Pythagoras to leave Samos:... Pythagoras was dragged into all sorts of diplomatic missions by his fellow citizens and forced to participate in public affairs. ... He knew that all the philosophers before him had ended their days on foreign soil so he decided to escape all political responsibility, alleging as his excuse, according to some sources, the contempt the Samians had for his teaching method. Pythagoras founded a philosophical and religious school in Croton (now Crotone, on the east of the heal of southern Italy) that had many followers. Pythagoras was the head of the society with an inner circle of followers known as mathematikoi. The mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras held were [2]:(1) that at its deepest level, reality is mathematical in nature, (2) that philosophy can be used for spiritual purification, (3) that the soul can rise to union with the divine, (4) that certain symbols have a mystical significance, and (5) that all brothers of the order should observe strict loyalty and secrecy. Both men and women were permitted to become members of the Society, in fact several later women Pythagoreans became famous philosophers. The outer circle of the Society were known as the akousmatics and they lived in their own houses, only coming to the Society during the day. They were allowed their own possessions and were not required to be vegetarians. Of Pythagoras's actual work nothing is known. His school practised secrecy and communalism making it hard to distinguish between the work of Pythagoras and that of his followers. Certainly his school made outstanding contributions to mathematics, and it is possible to be fairly certain about some of Pythagoras's mathematical contributions. First we should be clear in what sense Pythagoras and the mathematikoi were studying mathematics. They were not acting as a mathematics research group does in a modern university or other institution. There were no 'open problems' for them to solve, and they were not in any sense interested in trying to formulate or solve mathematical problems.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html (3 of 7) [2/16/2002 11:27:42 PM]

Pythagoras

Rather Pythagoras was interested in the principles of mathematics, the concept of number, the concept of a triangle or other mathematical figure and the abstract idea of a proof. As Brumbaugh writes in [3]:It is hard for us today, familiar as we are with pure mathematical abstraction and with the mental act of generalisation, to appreciate the originality of this Pythagorean contribution. In fact today we have become so mathematically sophisticated that we fail even to recognise 2 as an abstract quantity. There is a remarkable step from 2 ships + 2 ships = 4 ships, to the abstract result 2 + 2 = 4, which applies not only to ships but to pens, people, houses etc. There is another step to see that the abstract notion of 2 is itself a thing, in some sense every bit as real as a ship or a house. Pythagoras believed that all relations could be reduced to number relations. As Aristotle wrote:The Pythagorean ... having been brought up in the study of mathematics, thought that things are numbers ... and that the whole cosmos is a scale and a number. This generalisation stemmed from Pythagoras's observations in music, mathematics and astronomy. Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments. In fact Pythagoras made remarkable contributions to the mathematical theory of music. He was a fine musician, playing the lyre, and he used music as a means to help those who were ill. Pythagoras studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers, perfect numbers etc. However to Pythagoras numbers had personalities which we hardly recognise as mathematics today [3]:Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. This feeling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry. Ten was the very best number: it contained in itself the first four integers - one, two, three, and four [1 + 2 + 3 + 4 = 10] - and these written in dot notation formed a perfect triangle. Of course today we particularly remember Pythagoras for his famous geometry theorem. Although the theorem, now known as Pythagoras's theorem, was known to the Babylonians 1000 years earlier he may have been the first to prove it. Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [7]):After [Thales, etc.] Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational and the construction of the cosmic figures. Again Proclus, writing of geometry, said:I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life. Heath [7] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans. (i) The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans knew the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html (4 of 7) [2/16/2002 11:27:42 PM]

Pythagoras

generalisation which states that a polygon with n sides has sum of interior angles 2n - 4 right angles and sum of exterior angles equal to four right angles. (ii) The theorem of Pythagoras - for a right angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. We should note here that to Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical square constructed on the side. To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square. (iii) Constructing figures of a given area and geometrical algebra. For example they solved equations such as a (a - x) = x2 by geometrical means. (iv) The discovery of irrationals. This is certainly attributed to the Pythagoreans but it does seem unlikely to have been due to Pythagoras himself. This went against Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers. However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number. (v) The five regular solids. It is thought that Pythagoras himself knew how to construct the first three but it is unlikely that he would have known how to construct the other two. (vi) In astronomy Pythagoras taught that the Earth was a sphere at the centre of the Universe. He also recognised that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realise that Venus as an evening star was the same planet as Venus as a morning star. Primarily, however, Pythagoras was a philosopher. In addition to his beliefs about numbers, geometry and astronomy described above, he held [2]:... the following philosophical and ethical teachings: ... the dependence of the dynamics of world structure on the interaction of contraries, or pairs of opposites; the viewing of the soul as a self-moving number experiencing a form of metempsychosis, or successive reincarnation in different species until its eventual purification (particularly through the intellectual life of the ethically rigorous Pythagoreans); and the understanding ...that all existing objects were fundamentally composed of form and not of material substance. Further Pythagorean doctrine ... identified the brain as the locus of the soul; and prescribed certain secret cultic practices. In [3] their practical ethics are also described:In their ethical practices, the Pythagorean were famous for their mutual friendship, unselfishness, and honesty. Pythagoras's Society at Croton was not unaffected by political events despite his desire to stay out of politics. Pythagoras went to Delos in 513 BC to nurse his old teacher Pherekydes who was dying. He remained there for a few months until the death of his friend and teacher and then returned to Croton. In 510 BC Croton attacked and defeated its neighbour Sybaris and there is certainly some suggestions that Pythagoras became involved in the dispute. Then in around 508 BC the Pythagorean Society at Croton was attacked by Cylon, a noble from Croton itself. Pythagoras escaped to Metapontium and the most authors say he died there, some claiming that he committed suicide because of the attack on his Society. Iamblichus in [8] quotes one version of events:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html (5 of 7) [2/16/2002 11:27:42 PM]

Pythagoras

Cylon, a Crotoniate and leading citizen by birth, fame and riches, but otherwise a difficult, violent, disturbing and tyrannically disposed man, eagerly desired to participate in the Pythagorean way of life. He approached Pythagoras, then an old man, but was rejected because of the character defects just described. When this happened Cylon and his friends vowed to make a strong attack on Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon and his followers to persecute the Pythagoreans to the very last man. Because of this Pythagoras left for Metapontium and there is said to have ended his days. This seems accepted by most but Iamblichus himself does not accept this version and argues that the attack by Cylon was a minor affair and that Pythagoras returned to Croton. Certainly the Pythagorean Society thrived for many years after this and spread from Croton to many other Italian cities. Gorman [6] argues that this is a strong reason to believe that Pythagoras returned to Croton and quotes other evidence such as the widely reported age of Pythagoras as around 100 at the time of his death and the fact that many sources say that Pythagoras taught Empedokles to claim that he must have lived well after 480 BC. The evidence is unclear as to when and where the death of Pythagoras occurred. Certainly the Pythagorean Society expanded rapidly after 500 BC, became political in nature and also spilt into a number of factions. In 460 BC the Society [2]:... was violently suppressed. Its meeting houses were everywhere sacked and burned; mention is made in particular of "the house of Milo" in Croton, where 50 or 60 Pythagoreans were surprised and slain. Those who survived took refuge at Thebes and other places. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (27 books/articles)

Some Quotations (3)

A Poster of Pythagoras

Mathematicians born in the same country

Some pages from publications

Pythagorus in competition with Boethius in Margaista Philosophica (1504)

Cross-references to History Topics

1. Greek Astronomy 2. Perfect numbers 3. Prime numbers 4. The Indian Sulbasutras

Other references in MacTutor

1. Pythagoras's theorem 2. Chronology: 30000BC to 500BC

Honours awarded to Pythagoras (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html (6 of 7) [2/16/2002 11:27:42 PM]

Pythagoras

Lunar features

Crater Pythagoras

Other Web sites

1. Theosophy Online (Pythagoras and his School) 2. Yahoo (More links) 3. G Don Allen 4. Internet Encyclopedia of Philosophy 5. Mathgym, Australia 6. Simon Fraser University 7. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pythagoras.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html (7 of 7) [2/16/2002 11:27:42 PM]

Qadi_Zada

Qadi Zada al-Rumi Born: 1364 in Bursa, Turkey Died: 1436 in Samarkand, Uzbekistan Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Qadi Zada means "son of the judge" and we must assume that indeed Qadi Zada's father was the judge. Qadi Zada, however, is not his proper name which was Salah al-Din Musa Pasha. While we are commenting on his name we should also note that it is often written as Qadizada. Dilgan points out in [1] that certain historians have made errors regarding Qadi Zada's name. For example Montucla say that he was a Greek convert to Islam which Dilgan suggests may come from a misunderstanding of the name al-Rumi [1]:... for the peoples who lived in Asia Minor were called Rum, meaning Roman (not Greek), because Asia Minor was once Roman. It was in his home town of Bursa that Qadi Zada was brought up. He completed his standard education in Basra and then studied geometry and astronomy with al-Fanari. His teacher al-Fanari realised that Qadi Zada was a young man with great abilities in mathematics and astronomy and he advised him to visit the cultural centres of the empire, Khorasan or Transoxania, where he could benefit from coming in contact with the top mathematicians of his time. When Qadi Zada was a young man, Timur, who is often known as Tamerlane, ruled the empire stretching across present day Iran, Iraq, and eastern Turkey. After Timur's death in 1405, his empire was disputed among his sons. Shah Rukh was the fourth son of Timur and, by 1407, he had gained overall control of most of the empire, including Iran and Turkistan, regaining control of Samarkand. The cultural centres where Qadi Zada would be advised to visit would include Herat in Khorasan (today in western Afghanistan) and Bukhara and Samarkand in Transoxania. It was not until around 1407 that Qadi Zada set off to visit these cities. It is unclear why he waited so long for by this time he was over forty years of age, not really a young man setting off to begin a career. He had already gained a good reputation as a mathematician and a treatise on arithmetic, which he composed in Bursa in 1383, has survived. It is a work which covers arithmetic, algebra and mensuration. After visiting a number of cities, Qadi Zada reached Samarkand in about 1410. The previous year Shah Rukh, having gained control of his father Timur's empire, had decided to make Herat in Khorasan his new capital and put his own son Ulugh Beg in control of Samarkand. Ulugh Beg was only 17 years old when Qadi Zada met him at Samarkand in 1410. He was far more interested in science and culture than in politics or military conquest but he was, nevertheless, deputy ruler of the whole empire and, in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Qadi_Zada.html (1 of 3) [2/16/2002 11:27:44 PM]

Qadi_Zada

particular, sole ruler of the Mawaraunnahr region. Meeting Ulugh Beg was certainly a turning point for Qadi Zada, for he would spend the rest of his life working in Samarkand. He married in that city and his son Shams al-Din Muhammad was born there. Qadi Zada wrote a number of commentaries on works on mathematics and astronomy during his first years in Samarkand. These seem to have been written for Ulugh Beg and it would appear that Qadi Zada was producing material as a teacher of the brilliant young mathematician. One commentary on the compendium of the astronomer al-Jaghmini was written by Qadi Zada in 1412-13, while a second commentary was on a work by al-Samarqandi. This second commentary is on al-Samarqandi's famous short work of only 20 pages in which he discusses thirty-five of Euclid's propositions. Qadi Zada wrote the work in 1412. In 1417, perhaps encouraged by Qadi Zada, Ulugh Beg began building a madrasah which is a centre for higher education. The madrasah, fronting the Rigestan Square in Samarkand, was completed in 1420 and Ulugh Beg then began to appoint the best scientists he could find to teaching positions in his university. In addition to Qadi Zada, Ulugh Beg invited al-Kashi to join his madrasah, as well as around sixty other scientists. There is little doubt that al-Kashi, Qadi Zada and Ulugh Beg himself, were the leading astronomers and mathematicians at this prestigious establishment in Samarkand. Construction of an observatory in Samarkand began in 1424 and, while the observatory was under construction, al-Kashi wrote to his father, who lived in Kashan, about the scientific life in Samarkand. In the letters al-Kashi praises the mathematical abilities of Ulugh Beg and Qadi Zada but considers the other scientist second rate compared with them. Scientific meetings were led by Ulugh Beg and in these sessions problems in astronomy were freely discussed. Usually these problems were too difficult for all except al-Kashi and Qadi Zada. Qadi Zada's most original work was a computation of sin 1 with remarkable accuracy. He published his methods in his Treatise in the sine and, although al-Kashi also produced a method for solving this problem, the two methods are different and show that two remarkable scientists were both working on the same problems at Samarkand. Qadi Zada computed sin 1 to an accuracy of 10-12 (if expressed in decimals), as did al-Kashi. The method is fully described in [1] (see also [5]). The major work undertaken at the Observatory in Samarkand was the production of the Catalogue of the stars, the first comprehensive stellar catalogue since that of Ptolemy. This star catalogue, the Zij-i Sultani, set the standard for such works up to the seventeenth century. Published in 1437, in the year following Qadi Zada's death, it gives the positions of 992 stars. The catalogue was a collaborative effort by a number of scientists working at the Observatory but the principal contributors were certainly Ulugh Beg, al-Kashi, and Qadi Zada. As well as tables of observations made at the Observatory, the work contained calendar calculations and results in trigonometry. A commentary by Qadi Zada which is incomplete, was on the astronomical treatise of Nasir ad-Din al-Tusi. The contents of this surviving work are described in [3]. A treatise by Qadi Zada on the problem of facing Mecca, an important problem which many Muslim astronomers and mathematicians discussed, has also survived. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Qadi_Zada.html (2 of 3) [2/16/2002 11:27:44 PM]

Qadi_Zada

List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Qadi_Zada.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Qadi_Zada.html (3 of 3) [2/16/2002 11:27:44 PM]

Qalasadi

Qalasadi This biography is now under Al-Qalasadi. You should be automatically forwarded to the correct page. If your browser leaves you on this page, then press HERE. JOC/EFR September 1999

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Qalasadi.html [2/16/2002 11:27:45 PM]

Quetelet

Lambert Adolphe Jacques Quetelet Born: 22 Feb 1796 in Ghent, Flanders, Belgium Died: 17 Feb 1874 in Brussels, Belgium

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adolphe Quetelet received his first doctorate in 1819 from Ghent for a dissertation on the theory of conic sections. After receiving this doctorate he taught mathematics in Brussels, then, in 1823, he went to Paris to study astronomy at the Observatory there. He learnt astronomy from Arago and Bouvard and the theory of probability under Joseph Fourier and Pierre Laplace. Influenced by Laplace and Fourier, Quetelet was the first to use the normal curve other than as an error law. His studies of the numerical consistency of crimes stimulated wide discussion of free will versus social determinism. For his government he collected and analysed statistics on crime, mortality etc. and devised improvements in census taking. His work produced great controversy among social scientists of the 19th century. At an observatory in Brussels that he established in 1833 at the request of the Belgian government, he worked on statistical, geophysical, and meteorological data, studied meteor showers and established methods for the comparison and evaluation of the data. In Sur l'homme et le developpement de ses facultés, essai d'une physique sociale (1835) Quetelet presented his conception of the average man as the central value about which measurements of a human trait are grouped according to the normal curve. Quetelet organised the first international statistics conference in 1853. The internationally used measue of obesity is the Quetelet index, sometimes also called the Body mass index (BMI). This is: http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Quetelet.html (1 of 2) [2/16/2002 11:27:46 PM]

Quetelet

QI = (weight in kilograms)/(height in metres)2 703/(height in inches)2.

In non-metric measurements, QI = (weight in pounds) If QI > 30 then a person is officially obese.

The portrait above is taken from a French stamp issued in his honour. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles)

Some Quotations (2)

A Poster of Adolphe Quetelet

Mathematicians born in the same country

Other references in MacTutor

1. Chronology: 1830 to 1840 2. Chronology: 1840 to 1850

Honours awarded to Adolphe Quetelet (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1839

Lunar features

Crater Quetelet

Other Web sites

1. Louvain, Belgium 2. University of Minnesota 3. James Cook University, Australia 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR December 1996 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Quetelet.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Quetelet.html (2 of 2) [2/16/2002 11:27:46 PM]

Quillen

Daniel Grey Quillen Born: 27 June 1940 in Orange, New Jersey, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dan Quillen's father trained as a chemical engineer but made his career as a physics teacher. Daniel attended Newark Academy and, from there, he entered Harvard University. He received his B.A. in 1961 and then began research at Harvard under R Bott's supervision. Quillen was awarded his Ph.D. for a thesis on partial differential equations in 1964 entitled Formal Properties of Over-Determined Systems of Linear Partial Differential Equations. After receiving his doctorate, Quillen was appointed to the faculty of Massachusetts Institute of Technology. He spent a number of years undertaking research at other universities which were to prove important in setting the direction of his research. He was a Sloan Fellow in Paris during academic year 1968-69 when he was greatly influenced by Grothendieck, a visiting member of the Institute for Advanced Study at Princeton during 1969-70 when he was strongly influenced by Atiyah, and a Guggenheim Fellow again in France during 1973-74. Quillen at present works at the University of Oxford in England. In the 1960's, Quillen described how to define the homology of simplical objects over many different categories, including sets, algebras over a ring, and unstable algebras over the Steenrod algebra. Frank Adams had formulated a conjecture in homotopy theory which Quillen work on. Quillen approached the Adams conjecture with two quite distinct approaches, namely using techniques from algebraic geometry and also using techniques from the modular representation theory of groups . Both approaches proved successful, the proof in the first approach being completed by one of Quillen's students, the second approach leading to a proof by Quillen.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Quillen.html (1 of 3) [2/16/2002 11:27:48 PM]

Quillen

The techniques using modular representation theory of groups were used by Quillen to great effect in later work on cohomology of groups and algebraic K-theory. The work on cohomology led to Quillen giving a structure theorem for mod p cohomology rings of finite groups, this structure theorem solving a number of open questions in the area. Quillen received a Fields Medal at the International Congress of Mathematicians held in Helsinki in 1978. He received the award as the principal architect of the higher algebraic K-theory in 1972, a new tool that successfully used geometric and topological methods and ideas to formulate and solve major problems in algebra, particularly ring theory and module theory. Algebraic K-theory is an extension of ideas of Grothendieck to commutative rings. Grothendieck's ideas were used by Atiyah and Hirzebruch when they created topological K-theory. Clearly Quillen's year spent in Paris under Grothendieck's influence and Princeton working with Atiyah were important factors in Quillen's development of algebraic K-theory. Bass describes in [2] how Quillen resolved the problem that the higher algebraic K-groups, Kn for n 3, being constructed in an essentially different way from the Grothendieck construction presented great difficulties:... he borrowed techniques from homotopy theory, and in a completely novel way. The paper in which this so-called Q-construction occurs is essentially without mathematical precursors. Reading it for the first time is like landing on a new and friendly mathematical planet. One meets there not only new theorems and new methods, but new mathematical creatures and a complete paradigm of gestures for dealing with them. Higher algebraic K-theory is effectively built there from first principles and, in 63 pages, reaches a state of maturity that one normally expects from the efforts of several mathematicians over several years. As to his character, this is shown in [2]:When Quillen received his Ph.D. at the age of 24, he and his wife Jean, a violinist, were already caring for two of their five children. His precocity as a mathematician and as a father perhaps influenced the early greying of his hair, but it has not altered his boyish look or his easy and modest manner. He has a somewhat retiring life-style, appearing rarely in public, and then almost invariably with some extraordinary new theorem or idea in hand. In [2] Hyman Bass sums up Quillen's contribution leading up to the award of the Fields Medal in 1978 as follows:Mathematical talent tends to express itself either in problem solving or in theory building. It is with rare cases like Quillen that one has the satisfaction of seeing hard, concrete problems solved with general ideas of great force and scope and by the unification of methods from diverse fields of mathematics. Quillen has had a deep impact on the perceptions and the very thinking habits of a whole generation of young algebraists and topologists. One studies his work not only to be informed, but to be edified. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Quillen.html (2 of 3) [2/16/2002 11:27:48 PM]

Quillen

List of References (5 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1970 to 1980

Honours awarded to Dan Quillen (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1978

AMS Cole Prize winner

1975

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Quillen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Quillen.html (3 of 3) [2/16/2002 11:27:48 PM]

Quine

Willard Van Orman Quine Born: 25 June 1908 in Akron, Ohio, USA Died: 25 Dec 2000 in Boston, Massachusetts, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Willard Van Quine's father was Cloyd Robert Quine, an engineer who founded the Akron Equipment Company. His mother, Harriet Van Orman, was a teacher. Willard (or Van as he became known to his friends) was the youngest son of the family. His interests at school were mainly scientific but from a young age he began to interest himself with philosophical questions. For example, the concepts of heaven and hell worried him when he was only nine years old. His older brother William gave him James's Pragmatism before he had left school and the book fascinated him. After leaving school Quine studied at Oberlin College, Oberlin, Ohio, where his brother William had also studied. A fellow student suggested that he would find reading Bertrand Russell's works interesting and indeed reading Russell and Whitehead's Principia Mathematica quickly convinced Quine that he should study mathematics as his major subject with the philosophy of mathematics as a secondary topic. O'Grady writes in [5]:Quine's years at Oberlin were idyllic. His rooming house, full of kindred spirits was "an ideal setting in which to wax articulate". His appetite for cosmic understanding was sharpened by reading Russell. He graduated from Oberlin College in 1930 and then won a scholarship to study for his doctorate at Harvard University. He married Naomi Clayton, who he had known at Oberlin College, soon after arriving in Harvard. Quine completed his doctorate in two years supervised by Alfred North Whitehead. It was Whitehead who introduced him to Russell, who was visiting Harvard to lectured there, and from that time Quine began a correspondence with Russell. Awarded a Ph.D. in philosophy from Harvard in 1932, Quine described the following year, perhaps the most important for his future research, in a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Quine.html (1 of 5) [2/16/2002 11:27:50 PM]

Quine

somewhat matter of fact way:In 1932 - I already had my PhD and was married to my first wife - I had a travelling fellowship. That was a great year. We used up our resources very accurately - I had $7 when we got back to America. Then I came back to Harvard as a Junior Fellow in 1933. Indeed it was "a great year" for Quine who made an exciting visit to Europe financed by a Sheldon Traveling Fellowship. In Vienna he met Philip Frank, Moritz Schlick and other members of the Vienna Circle of Logical Positivists. He also met the British philosopher A J Ayer and Kurt Gödel. Quine spent six weeks in Warsaw with Tarski before going on to study at Prague under Rudolf Carnap. Quine became, in his own words, "an ardent disciple" of Carnap discovering:... what it was to be intellectually fired by a living teacher rather than by a dead book. After Quine returned to Harvard in 1933 to take up the Junior Fellowship he published his first book A System of Logistics which was the published version of his doctoral thesis. From this time up to the break in his career for war service Quine's research was mainly on logic although always with a philosophical motivation. One of the most important papers he published during this period was New Foundations for Mathematical Logic in the American Mathematical Monthly in 1937. In this paper he presented his invention of the heterodox system of set theory, which known as NF set theory after the title of the paper. Quine presented no model for this theory, nor did he prove that the system was consistent. This posed an awkward problem for mathematical logicians and led to quite a bit of interest. Quine was appointed onto the staff at Harvard in 1936 as an Instructor in Philosophy. He later wrote about his teaching at this stage in his career:What I enjoyed most was more the mathematical end than the philosophical, because of it being less a matter of opinion. Clarifying, not defending. Resting on proof. I taught first in both departments, but my appointment was in the Philosophy Department. I taught Mathematical Logic and Set Theory as well as a general course of Logic in Philosophy. Harvard was good about letting me teach my own interests. I gave a course in Philosophy of my own choosing, my own ideas, for concentrators. The year 1940 was an exciting one for Quine at Harvard for in that year both Carnap and Tarski visited Harvard and the three debated logical positivism. Although Quine was a firm friend of Carnap the two took somewhat different stands on many philosophical issues which led to lively discussions and challenges. Quine was promoted to Associate Professor in 1941 but shortly after, because of World War II, he undertook war service. From 1942 he spent four years in the United States Navy Intelligence, first as a Lieutenant then as a Lieutenant Commander, decrypting messages from German submarines off the coast. He later wrote of this work decrypting messages:The Germans had a replica Enigma breaking complicated ciphers. Each day they had a different setting on the machine. We had to get it the hard way, by intercepting a message from a submarine that gave direction finders. We would know, say, from the preceding day's message he had been sent on a refuelling rendezvous, so a good guess was that some word would be 'refuelling.' Then if our men could fit the word, they could get the setting for the whole. In 1945 Quine and his wife, having had two daughters, separated and they were divorced two years later.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Quine.html (2 of 5) [2/16/2002 11:27:50 PM]

Quine

He returned to Harvard where he was promoted to Professor in 1948. He married again in 1948, his second wife being Marjorie Boynton who he met while serving in the Navy. Quine and his second wife had two children, a daughter and a son. The son, Douglas B Quine, has written an obituary of his father. During 1953-54 Quine was Eastman Visiting Professor at Oxford and during that time he published a book From a Logical Point of View which was a collection of his earlier articles. One article in this work Two Dogmas of Empiricism, which he had first published in 1951, was the one which made his reputation as a leading philosopher. This article [5]:... challenged received notions of knowledge, meaning and truth, and exceeded even the extreme empiricism of logical positivism by arguing that logic and mathematics, like factual statements, are open to revision in the light of experience. In the article Quine argued that it was:... folly to seek a boundary between synthetic statements, which hold contingently on experience, and analytic statements, which hold come what may. He argued against the dogma of reductionism, or:... the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience. Quine claimed:Any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system [of our beliefs]. The totality of our so-called knowledge or beliefs, from the most casual matters of geography and history to the profoundest laws of atomic physics or even of pure mathematics and logic, is a man-made fabric which impinges on experience only along the edges. Quine became Edgar Pierce Professor of Philosophy at Harvard in 1956, a post he retained until he retired in 1978. He continued as Edgar Pierce Professor Emeritus at Harvard, commuting daily to his corner office in Emerson Hall, for many years after he retired. His passion for travelling continued after his retirement and throughout his life he visited 118 countries. We have commented above about Quine's work in mathematical logic. Symbolic logic represented for him the framework for the language of science. He modestly said:I do not do anything with computers, although one of my little results in mathematical logic has become a tool of the computer theory, the Quine McCluskey principle. And corresponds to terminals in series, or to those in parallel, so that if you simplify mathematical logical steps, you have simplified your wiring. I arrived at it not from an interest in computers, but as a pedagogical device, a slick way of introducing that way of teaching mathematical logic. Quine developed a new type of philosophy, which he called naturalized epistemology. He claimed that epistemology's only legitimate role is to describe the way knowledge is actually obtained so, according to Quine, its function is to describe how present science arrives at the beliefs accepted by the scientific community. Among Quine's publications are works on logic, metaphysics, the philosophy of language and the philosophy of logic. His 22 books include A System of Logistic (1934), Mathematical Logic (1940), Elementary Logic (1941), On What There Is (1948), From a Logical Point of View (1953), Word and Object (1960), Set Theory and Its Logic (1963), Philosophy of Logic (1970), The Time of My Life: an autobiography (1985), Quiddities (1990), and From Stimulus to Science (1995). http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Quine.html (3 of 5) [2/16/2002 11:27:50 PM]

Quine

Quine won many prizes and medals for his outstanding contributions. These included the Murray Butler gold medal (1965), the F Polacky gold medal in Prague (1991), the Charles University gold medal in Prague (1993), the Rolf Schock Prize in Stockholm (1993), and the Kyoto Prize in Tokyo (1996). The Kyoto Prize for Creative Arts and Moral Sciences focused on the field of philosophy and made the award to Quine as one of America's pre-eminent 20th century philosophers. His achievements were summarised as:We may therefore say that the many theses and arguments of Dr. Willard Van Orman Quine have become the centre of debate for modern epistemology and ontology as well as philosophy of language and philosophy of science in general. He has created a profound, powerful influence without which it would not be possible to understand the current state of philosophy. The University of Lille, Oxford University, Cambridge University, Uppsala University, the University of Bern, and Harvard University were among the eighteen universities awarding him an honorary degree. He was elected to fellowships of many learned societies including the American Academy of Arts and Sciences (1949), the British Academy (1959), the Instituto Brasileiro de Filosophia (1963), the National Academy of Sciences (1977), the Institut de France (1978), and the Norwegian Academy of Sciences (1979). A little of his character is shown by some entertaining episodes. All his books were typed on the 1927 Remington typewriter, on which he wrote his doctoral thesis, which he had modified by including some mathematical symbols instead of characters such as !, ?, and 1. When once he was asked how he managed without a question mark he replied:Well, you see, I deal in certainties. He gave this picture of himself in his book The Time of My Life :I am orderly and I am frugal. For the most part my only emotion is impatience, I am deeply moved by occasional passages of poetry, and so, characteristically, I read little of it. Outside philosophy and mathematics Quine loved music, especially Dixieland jazz, Mexican folksongs, and Gilbert and Sullivan. He enjoyed playing the piano and he played banjo in several jazz groups. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles)

Some Quotations (5)

Mathematicians born in the same country Other Web sites

1. The Guardian, UK (Obituary) 2. The Times, UK (Obituary) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Quine.html (4 of 5) [2/16/2002 11:27:50 PM]

Quine

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Quine.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Quine.html (5 of 5) [2/16/2002 11:27:50 PM]

Rademacher

Hans Rademacher Born: 3 April 1892 in Wandsbeck (part of Hamburg), Schleswig-Holstein, Germany Died: 7 Feb 1969 in Haverford, Pennsylvania, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hans Rademacher studied at Göttingen and was persuaded to study mathematics by Courant. His initial interests were in the theory of real functions which he was taught by Carathéodory who also taught him the calculus of variations. At Göttingen he also studied number theory with Landau. Continuing his interest in the theory of real functions he was awarded his doctorate in 1916 for a dissertation on single-valued mappings and mensurability. He taught at a school in Thuringia for a short while before being appointed to the University of Berlin as a Privatdozent in December 1916. He was a colleague of Schmidt and Schur and was certainly influenced by them. He changed his area of mathematical interest from the theory of real functions to number theory in 1922 when he accepted the position of extraordinary professor at Hamburg. He was led towards number theory by Hecke who had been appointed to Hamburg three years before Rademacher. At Easter 1925 Rademacher left Hamburg to become an ordinary professor at Breslau. It was a difficult decision for Rademacher, particularly since Hecke was so keen for him to stay in Hamburg. Had Hecke succeeded in his attempt to get Hamburg to offer Rademacher an ordinary professorship then he would almost certainly have remained there, but the university would not make the offer that Hecke requested and, after much thought, Rademacher went to Breslau. In different political circumstances one would have expected Rademacher to remain at Breslau for the rest of his career. However when Hitler came to power in 1933 normal expectations were completely http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rademacher.html (1 of 2) [2/16/2002 11:27:52 PM]

Rademacher

overturned for most people. Rademacher was not Jewish, and he was certainly racially acceptable to the Nazi regime, but his views were not acceptable to the Nazis. He was forced out of his professorship in 1933 because of his pacifist views and he left Germany in 1934. He spent the rest of his life in the United States, first at Swarthmore College, and later at the University of Pennsylvania. Rademacher's early arithmetical work dealt with applications of Brun's sieve method and with the Goldbach problem in algebraic number fields. About 1928 he began research on the topics for which he is best known among mathematicians today, namely his work in connection with questions concerning modular forms and analytic number theory. Perhaps his most famous result, obtained in 1936 when he was in the United States, is his proof of the asymptotic formula for the growth of the partition function (the number of representations of a number as a sum of natural numbers). This answered questions of Leibniz and Euler and followed results obtained by Hardy and Ramanujan. Rademacher also wrote important papers on Dedekind sums and investigated many problems relating to algebraic number fields. In addition to the significant contributions to real analysis and measure theory which we have briefly mentioned above, complex analysis, geometry, and numerical analysis. P T Bateman, reviewing Rademacher's collected works, wrote that they:... serve not only as a fitting memorial to a great mathematician and human being, but also provide excellent examples of how mathematics should be presented, and serve as leisurely but authentic introductions to some fascinating parts of analysis and number theory. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rademacher.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rademacher.html (2 of 2) [2/16/2002 11:27:52 PM]

Rado

Tibor Radó Born: 2 June 1895 in Budapest, Hungary Died: 12 Dec 1965 in New Smyrna Beach, Florida, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Tibor Radó enlisted in 1915 and fought on the Russian front before being captured and imprisoned for four years. While he was in prison, the only books he had to read were on mathematics. After his release he went to Szeged University, taking a Ph.D. under Riesz. In 1929 he went to the USA and held posts at Harvard and Rice before being appointed to a chair at Ohio State University. Radó worked on conformal mappings, real analysis, calculus of variations, partial differential equations, integration theory and topology. He worked on the Plateau Problem and found the solution independently of Douglas. He also worked on surface measure continuing the work of Lebesgue and Riesz. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Honours awarded to Tibor Radó (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1945

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rado.html (1 of 2) [2/16/2002 11:27:54 PM]

Rado

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rado.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rado.html (2 of 2) [2/16/2002 11:27:54 PM]

Rado_Richard

Richard Rado Born: 28 April 1906 in Berlin, Germany Died: 23 Dec 1989 in Reading, Berkshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Richard Rado made a decision while at school to choose between being a concert pianist or a mathematician. He chose mathematics and entered the University of Berlin to study mathematics. He also spent some time at Göttingen but returned to the University of Berlin to study for his doctorate under Schur. At this stage he was also strongly influenced by Schmidt. Rado's thesis, entitled Studies on combinatorics, earned him a doctorate in 1933. Rado's family were Jewish so, when the Nazis came to power in 1933, the family left for England. Rado entered Fitzwilliam House, University of Cambridge and completed a second PhD under Hardy's supervision on Linear transformations of sequences. While at Cambridge Rado was influenced by many mathematicians working there at the time including Hardy, Littlewood, Hall, Besicovitch and B H Neumann. Rado also worked with Heilbronn and Davenport and, at around this time, Rado also began to correspond with Erdös. They met in 1934 and began a fruitful collaboration which resulted in a number of joint papers. This collaboration was described by Erdös in [1]. In 1936 Rado was appointed to Sheffield where, after Mirsky was appointed in 1942, the two became close friends. In 1947 Rado moved to King's College London, moving seven years later to a chair at the University of Reading. He remained at Reading until he retired in 1971. Rado's work covered a wide range of mathematics but his most important work was in combinatorics. Some of his more minor work was in topics such as the convergence of sequences and series. He studied

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rado_Richard.html (1 of 2) [2/16/2002 11:27:56 PM]

Rado_Richard

inequalities and geometry and measure theory, particularly working in this area with Besicovitch. In graph theory he worked on infinite graphs and hypergraphs. His important combinatorial results were in the area of Hall's theorem, Ramsey's theorem and partitions. Rado's approach to mathematics is described by Erdös in [1] where he comments:I was good at discovering perhaps difficult and interesting special cases and Richard was good at generalising them and putting them in their proper perspective. In [5] Rado is described by saying:Richard was fascinated by mathematical beauty and sought after it. He always tried to formulate his results at their natural level of generality, so that their full power was exhibited, without their content being obscured by over-elaboration. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Honours awarded to Richard Rado (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1978

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rado_Richard.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rado_Richard.html (2 of 2) [2/16/2002 11:27:56 PM]

Radon

Johann Radon Born: 16 Dec 1887 in Tetschen, Bohemia (now Decin, Czech Republic) Died: 25 May 1956 in Vienna, Austria

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Radon attended school in Leitmeritz (now Litomerice) in Bohemia between 1897 and 1905. He then entered the University of Vienna where he was awarded a doctorate in 1910 for a dissertation on the calculus of variations. The year 1911 he spent in Göttingen, became assistant professor at the University of Brünn (now Brno) for a year and then moved to the Technische Hochschule in Vienna. In 1919 Radon became assistant professor at Hamburg becoming a full professor in Greifswald in 1922. He taught in Erlangen in 1925, then from 1928 until 1945 he worked at the University of Breslau. He was appointed to the University of Vienna in 1947 and he remained there for the rest of his life. Radon applied the calculus of variations to differential geometry which led to applications in number theory. He discovered curves whch are now named after him. His best known results involve combining the integration theories of Lebesgue and Stieltjes. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Johann Radon

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Radon.html (1 of 2) [2/16/2002 11:27:58 PM]

Radon

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Radon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Radon.html (2 of 2) [2/16/2002 11:27:58 PM]

Rahn

Johann Heinrich Rahn Born: 10 March 1622 in Zurich, Switzerland Died: 1676 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Rahn was the first to use the symbol for division in his algebra book published in 1659. Rahn's book was written in German. Pell helped Rahn with this book which contains an example of Pell's equation. The book is important for the innovation in algebraic symbolism that it contains but recent work has suggested that credit for this may be due more to Pell than to Rahn. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Other references in MacTutor

1. Pell's equation 2. Chronology: 1650 to 1675

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rahn.html (1 of 2) [2/16/2002 11:27:59 PM]

Rahn

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rahn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rahn.html (2 of 2) [2/16/2002 11:27:59 PM]

Rajagopal

Cadambathur Tiruvenkatacharlu Rajagopal Born: 8 Sept 1903 in Triplicane, Madras, India Died: 25 April 1978 in Madras, India

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Rajagopal was educated in Madras, India. He graduated in 1925 from the Madras Presidency College with Honours in mathematics. He spent a short while in the clerical service, another short while teaching in Annamalai University then, from 1931 to 1951, he taught in the Madras Christian College. Here he gained an outstanding reputation as a teacher of classical analysis. In 1951 Rajagopal was persuaded to join the Ramanujan Institute of Mathematics then, four years later, he became head of the Institute. Under his leadership the Institute became the major Indian mathematics research centre. Rajagopal studied sequences, series, summability. He published 89 papers in this area generalising and unifying Tauberian theorems. He also studied functions of a complex variable giving an analogue of a theorem of Landau on partial sums of Fourier series. In several papers he studied the relation between the growth of the mean values of an entire function and that of its Dirichlet series. A final topic to interest him was the history of medieval Indian mathematics. He showed that the series for tan-1 x discovered by Gregory and those for sin x and cos x discovered by Newton were known to the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rajagopal.html (1 of 2) [2/16/2002 11:28:01 PM]

Rajagopal

Hindus 150 years earlier. He identified the Hindu mathematician Madhava as the first discoverer of these series. Rajagopal is described in [1] as follows:Rajagopal was a teacher par excellence and a reliable and inspiring research guide. No words can adequately describe his modesty. Rational thinking and interest in psychic studies were two attributes which he imbibed with pride from his teacher Ananda Rau. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rajagopal.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rajagopal.html (2 of 2) [2/16/2002 11:28:01 PM]

Ramanujam

Chidambaram Padmanabhan Ramanujam Born: 9 Jan 1938 in Madras, India Died: 27 Oct 1974 in Bangalore, India

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Before we look at the life and work of Chidambaram Padmanabhan Ramanujam we must warn the reader that this article is on Ramanujam, NOT Ramanujan the number theorist who worked with G H Hardy (there is only a difference of one letter in their names!). Ramanujam's father was Shri C Ramanujam who was an advocate working in Madras, India, at the High Court. C P Ramanujam was educated in Madras, first at Ewart's School, where he had his primary and the first part of his secondary education, and then at the Sir M Ct Muthiah Chetty High School at Vepery, Madras. His interests on the academic side were in mathematics and chemistry while on the sporting side he was an enthusiastic tennis player. Chemistry experiments were particularly fascinating to him and he made a chemistry laboratory in a room in his home. There he would spend happy times carrying out experiments with one of his friends. In 1952, while still only 14 years old, he passed his final High School examinations and entered Loyola College in Madras. Ramanujam's achievements at High School had been outstanding and he had shown that he was extraordinarily gifted, so he entered Loyola College with great expectations. He continued his interest in chemistry but it was mathematics that he specialised in, taking Mathematics Honours after obtaining his Intermediate qualification. He was awarded a B.A. with Honours in Mathematics in 1957 but, strangely for such an outstanding student, he only obtained a second class degree. This may have been a result of starting his university education at so young an age before he was really ready, for the second class degree no way reflected his remarkable mathematical abilities. On the other hand it may have resulted from a lack of belief in himself which haunted Ramanujam throughout his life.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujam.html (1 of 4) [2/16/2002 11:28:03 PM]

Ramanujam

He had been taught mathematics by Father C Racine in his final honours years at Loyola College and he encouraged Ramanujam to apply for entry to the School of Mathematics at the Tata Institute in Bombay. In his letter of recommendation Father Racine wrote:He has certainly originality of mind and the type of curiosity which is likely to suggest that he will develop into a good research worker if given sufficient opportunity. In Madras there was another prestigious Institute, the Ramanujan Institute of Mathematics. In 1957 Ramanujam learnt deep results in analytic number theory from the former director of this Institute (who had retired three years earlier) in the months before he left Madras for Bombay to begin his studies at the Tata Institute. At the Institute, Ramanujam quickly became an expert in many different mathematical areas. His wide expertise made him a natural person to write up lecture notes from courses given by visitors to the Institute and in 1958-59 Max Deuring gave a course on the theory of algebraic functions of one variable which was expertly written up by Ramanujam. He seemed able to soak up huge amounts of deep and difficult mathematics and he gave many talks showing what a deep understand he had of many topics. What he was not doing was producing original mathematical advances while some of his less able colleagues were being much more successful. Ramanujam felt that he did not have what it takes to solve the big problems of mathematics, and he had no wish to solve small routine problems. Again, as in his undergraduate course, it would appear to be a psychological problem rather than a mathematical one but for Ramanujam it was a very real problem and he became more and more frustrated. He decided that his strengths were in teaching mathematics rather than producing original mathematics, and consequently he began applying to a variety of universities and colleges for a teaching position. His applications failed so reluctantly Ramanujam remained at the Tata Institute. At this stage K G Ramanathan, the author of [4], began working with Ramanujam. He directed Ramanujam to work on some generalisations of the Waring problem to algebraic number fields. On this topic Ramanujam produced some outstanding results, generalising methods due to Davenport to attack certain questions which had been posed by Carl Siegel. For his deep results in number theory he was promoted to Associate Professor at the Tata Institute. It was not a position he easily accepted, arguing strongly that he was not worthy of such a post. However his friends and colleagues persuaded him to accept. There is a fine line between whether someone behaves in a certain way because they have an illness or whether it is just their personality which determines their behaviour. Up to 1964 Ramanujam's lack of belief in his own abilities could have been described as part of his personality, but in 1964 he was struck with an illness which was diagnosed as severe depression and schizophrenia. Again feeling totally inadequate as a research mathematician he applied for university teaching posts. During 1964-65 I R Shafarevich visited the Tata Institute and lectured on minimal models and birational transformations of two dimensional schemes. Ramanujam took notes at the lectures for publication and, as he had done previously he showed his deep understanding of mathematics in doing this task. On seeing the notes which Ramanujam had written, Shafarevich wrote to the Institute:I want to thank [Ramanujam] for the splendid job he has done. He not only corrected several mistakes but also complemented proofs of many results that were only stated in oral exposition. To mention some of them, he has written proofs of the Castelnuovo theorem... of the chain conditions ..., the example of Nagata of a non-projective surface ... and the proof http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujam.html (2 of 4) [2/16/2002 11:28:03 PM]

Ramanujam

of Zariski's theorem ... In July 1965 Ramanujam was offered a Professorship at the Punjab University in Chandigarh. He accepted and began teaching there. However his depression returned and [4]:... amidst tragic circumstances he had to cut short his stay there after about eight months. Back in the Tata Institute, Ramanujam received an invitation to spend six months at the Institut des Hautes Etudes Scientifique in Paris. Again his illness forced him to return from Paris before the end of the six months. However his ability to do mathematics seemed as remarkable as ever outside his periods of illness. In 1967-68 David Mumford visited the Tata Institute and again Ramanujam wrote up his lectures for publication. In the Introduction to Abelian Varieties Mumford wrote:... these lectures were subsequently written up, and improved in many ways, by C P Ramanujam. The present text is a joint effort. ... C P Ramanujam continuing my lectures at the Tata Institute lectured on and wrote up notes on Tate's theorem on homomorphisms between abelian varieties over finite fields. Severe depression struck Ramanujam frequently. On one occasion he tried to take his life with barbiturates but was quickly treated and recovered. In February 1970, while again suffering depression, he resigned from the Tata Institute. The Director refused his resignation but later in the year he again resigned and went to the 1970-71 Algebraic geometry year at the University of Warwick in England. Mumford was also at the meeting and writes in [2]:... we were together in Warwick where he ran seminars on étale cohomology and on classification of surfaces. His excitement and enthusiasm was one of the main factors that made "Algebraic geometry year" a success. We discussed many topics involving topology and algebraic geometry at that time, and especially Kodaira's Vanishing Theorem. My wife and I spent many evenings together with him, talking about life, religion and customs both in India and the West and we looked forward to a warm and continuing friendship. As a result of his work with Shafarevich and Mumford, Ramanujam went on to make contributions to algebraic geometry which Mumford describes in [2]. These include a characterization of C2, a version of the Kodaira vanishing theorem, a study of the automorphism group of a variety, a study of the purity of the discriminant locus, a proof that the invariance of the Milnor number implies the invariance of the topological type, and a geometric interpretation of multiplicity. The work on the Milnor number was done in collaboration with Le Dung Trang. Back in India after his year at the University of Warwick, Ramanujam asked for a Professorship at the Tata Institute but be based in Bangalore where a new branch dealing with applications of mathematics was being set up. This was agreed and he taught analysis in Bangalore but, again in the depths of depression caused by his illness, he tried again to leave the Institute and obtain a university teaching post. While waiting for an offer of such a post from the Indian Institute at Simla he took his life with an overdose of barbiturates. In [3] Ramanan pays this tribute to Ramanujam:For sheer elegance and economy, I have come across few mathematicians who were C P Ramanujam's equal. he made so many remarks which clarified and threw light on different branches of mathematics that personally I derived immense pleasure from his company. Mumford writes in [2]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujam.html (3 of 4) [2/16/2002 11:28:03 PM]

Ramanujam

It was a stimulating experience to know and collaborate with C P Ramanujam. He loved mathematics and he was always ready to take up a new thread or pursue an old one with infectious enthusiasm. He was equally ready to discuss a problem with a first year student or a colleague, to work through an elementary point or puzzle over a deep problem. On the other hand he had high standards. He felt the spirit of mathematics demanded of him not merely routine developments but the right theorem an any given topic. He was sometimes tormented by these high standards, but, in retrospect, it is clear to us how often he succeeded in adding to our knowledge, results both new, beautiful and with a genuine original stamp. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ramanujam.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujam.html (4 of 4) [2/16/2002 11:28:03 PM]

Ramanujan

Srinivasa Aiyangar Ramanujan Born: 22 Dec 1887 in Erode, Tamil Nadu state, India Died: 26 April 1920 in Kumbakonam, Tamil Nadu state, India

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He made substantial contributions to the analytical theory of numbers and worked on elliptic functions, continued fractions, and infinite series. Ramanujan was born in his grandmother's house in Erode, a small village about 400 km southwest of Madras. When Ramanujan was a year old his mother took him to the town of Kumbakonam, about 160 km nearer Madras. His father worked in Kumbakonam as a clerk in a cloth merchant's shop. In December 1889 he contracted smallpox. When he was nearly five years old, Ramanujan entered the primary school in Kumbakonam although he would attend several different primary schools before entering the Town High School in Kumbakonam in January 1898. At the Town High School, Ramanujan was to do well in all his school subjects and showed himself an able all round scholar. In 1900 he began to work on his own on mathematics summing geometric and arithmetic series. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried (and of course failed) to solve the quintic. It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary results in pure mathematics. This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujan.html (1 of 7) [2/16/2002 11:28:05 PM]

Ramanujan

he had of written mathematical arguments. The book contained theorems, formulas and short proofs. It also contained an index to papers on pure mathematics which had been published in the European Journals of Learned Societies during the first half of the 19th century. The book, published in 1856, was of course well out of date by the time Ramanujan used it. By 1904 Ramanujan had begun to undertake deep research. He investigated the series (1/n) and calculated Euler's constant to 15 decimal places. He began to study the Bernoulli numbers, although this was entirely his own independent discovery. Ramanujan, on the strength of his good school work, was given a scholarship to the Government College in Kumbakonam which he entered in 1904. However the following year his scholarship was not renewed because Ramanujan devoted more and more of his time to mathematics and neglected his other subjects. Without money he was soon in difficulties and, without telling his parents, he ran away to the town of Vizagapatnam about 650 km north of Madras. He continued his mathematical work, however, and at this time he worked on hypergeometric series and investigated relations between integrals and series. He was to discover later that he had been studying elliptic functions. In 1906 Ramanujan went to Madras where he entered Pachaiyappa's College. His aim was to pass the First Arts examination which would allow him to be admitted to the University of Madras. He attended lectures at Pachaiyappa's College but became ill after three months study. He took the First Arts examination after having left the course. He passed in mathematics but failed all his other subjects and therefore failed the examination. This meant that he could not enter the University of Madras. In the following years he worked on mathematics developing his own ideas without any help and without any real idea of the then current research topics other than that provided by Carr's book. Continuing his mathematical work Ramanujan studied continued fractions and divergent series in 1908. At this stage he became seriously ill again and underwent an operation in April 1909 after which he took him some considerable time to recover. He married on 14 July 1909 when his mother arranged for him to marry a nine year old girl S Janaki Ammal. Ramanujan did not live with his wife, however, until she was twelve years old. Ramanujan continued to develop his mathematical ideas and began to pose problems and solve problems in the Journal of the Indian Mathematical Society. He devoloped relations between elliptic modular equations in 1910. After publication of a brilliant research paper on Bernoulli numbers in 1911 in the Journal of the Indian Mathematical Society he gained recognition for his work. Despite his lack of a university education, he was becoming well known in the Madras area as a mathematical genius. In 1911 Ramanujan approached the founder of the Indian Mathematical Society for advice on a job. After this he was appointed to his first job, a temporary post in the Accountant General's Office in Madras. It was then suggested that he approach Ramachandra Rao who was a Collector at Nellore. Ramachandra Rao was a founder member of the Indian Mathematical Society who had helped start the mathematics library. He writes in [30]:A short uncouth figure, stout, unshaven, not over clean, with one conspicuous feature-shining eyes- walked in with a frayed notebook under his arm. He was miserably poor. ... He opened his book and began to explain some of his discoveries. I saw quite at once that there was something out of the way; but my knowledge did not permit me to judge whether he talked sense or nonsense. ... I asked him what he wanted. He said he wanted a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujan.html (2 of 7) [2/16/2002 11:28:05 PM]

Ramanujan

pittance to live on so that he might pursue his researches. Ramachandra Rao told him to return to Madras and he tried, unsuccessfully, to arrange a scholarship for Ramanujan. In 1912 Ramanujan applied for the post of clerk in the accounts section of the Madras Port Trust. In his letter of application he wrote [3]:I have passed the Matriculation Examination and studied up to the First Arts but was prevented from pursuing my studies further owing to several untoward circumstances. I have, however, been devoting all my time to Mathematics and developing the subject. Despite the fact that he had no university education, Ramanujan was clearly well known to the university mathematicians in Madras for, with his letter of application, Ramanujan included a reference from E W Middlemast who was the Professor of Mathematics at The Presidency College in Madras. Middlemast, a graduate of St John's College, Cambridge, wrote [3]:I can strongly recommend the applicant. He is a young man of quite exceptional capacity in mathematics and especially in work relating to numbers. He has a natural aptitude for computation and is very quick at figure work. On the strength of the recommendation Ramanujan was appointed to the post of clerk and began his duties on 1 March 1912. Ramanujan was quite lucky to have a number of people working round him with a training in mathematics. In fact the Chief Accountant for the Madras Port Trust, S N Aiyar, was trained as a mathematician and published a paper On the distribution of primes in 1913 on Ramanujan's work. The professor of civil engineering at the Madras Engineering College C L T Griffith was also interested in Ramanujan's abilities and, having been educated at University College London, knew the professor of mathematics there, namely M J M Hill. He wrote to Hill on 12 November 1912 sending some of Ramanujan's work and a copy of his 1911 paper on Bernoulli numbers. Hill replied in a fairly encouraging way but showed that he had failed to understand Ramanujan's results on divergent series. The recommendation to Ramanujan that he read Bromwich's Theory of infinite series did not please Ramanujan much. Ramanujan wrote to E W Hobson and H F Baker trying to interest them in his results but neither replied. In January 1913 Ramanujan wrote to G H Hardy having seen a copy of his 1910 book Orders of infinity. In Ramanujan's letter to Hardy he introduced himself and his work [10]:I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'. Hardy, together with Littlewood, studied the long list of unproved theorems which Ramanujan enclosed with his letter. On 8 February he replied to Ramanujan [3], the letter beginning:I was exceedingly interested by your letter and by the theorems which you state. You will however understand that, before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions. Your results seem to me to fall into roughly three classes: (1) there are a number of results that are already known, or easily deducible from known theorems;

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujan.html (3 of 7) [2/16/2002 11:28:05 PM]

Ramanujan

(2) there are results which, so far as I know, are new and interesting, but interesting rather from their curiosity and apparent difficulty than their importance; (3) there are results which appear to be new and important... Ramanujan was delighted with Hardy's reply and when he wrote again he said [8]:I have found a friend in you who views my labours sympathetically. ... I am already a half starving man. To preserve my brains I want food and this is my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the university of from the government. Indeed the University of Madras did give Ramanujan a scholarship in May 1913 for two years and, in 1914, Hardy brought Ramanujan to Trinity College, Cambridge, to begin an extraordinary collaboration. Setting this up was not an easy matter. Ramanujan was an orthodox Brahmin and so was a strict vegetarian. His religion should have prevented him from travelling but this difficulty was overcome, partly by the work of E H Neville who was a colleague of Hardy's at Trinity College and who met with Ramanujan while lecturing in India. Ramanujan sailed from India on 17 March 1914. It was a calm voyage except for three days on which Ramanujan was seasick. He arrived in London on 14 April 1914 and was met by Neville. After four days in London they went to Cambridge and Ramanujan spent a couple of weeks in Neville's home before moving into rooms in Trinity College on 30th April. Right from the beginning, however, he had problems with his diet. The outbreak of World War I made obtaining special items of food harder and it was not long before Ramanujan had health problems. Right from the start Ramanujan's collaboration with Hardy led to important results. Hardy was, however, unsure how to approach the problem of Ramanujan's lack of formal education. He wrote [1]:What was to be done in the way of teaching him modern mathematics? The limitations of his knowledge were as startling as its profundity. Littlewood was asked to help teach Ramanujan rigorous mathematical methods. However he said ([31]):... that it was extremely difficult because every time some matter, which it was thought that Ramanujan needed to know, was mentioned, Ramanujan's response was an avalanche of original ideas which made it almost impossible for Littlewood to persist in his original intention. The war soon took Littlewood away on war duty but Hardy remained in Cambridge to work with Ramanujan. Even in his first winter in England, Ramanujan was ill and he wrote in March 1915 that he had been ill due to the winter weather and had not been able to publish anything for five months. What he did publish was the work he did in England, the decision having been made that the results he had obtained while in India, many of which he had communicated to Hardy in his letters, would not be published until the war had ended. On 16 March 1916 Ramanujan graduated from Cambridge with a Bachelor of Science by Research (the degree was called a Ph.D. from 1920). He had been allowed to enrol in June 1914 despite not having the proper qualifications. Ramanujan's dissertation was on Highly composite numbers and consisted of seven of his papers published in England. Ramanujan fell seriously ill in 1917 and his doctors feared that he would die. He did improve a little by

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujan.html (4 of 7) [2/16/2002 11:28:05 PM]

Ramanujan

September but spent most of his time in various nursing homes. In February 1918 Hardy wrote (see [3]):Batty Shaw found out, what other doctors did not know, that he had undergone an operation about four years ago. His worst theory was that this had really been for the removal of a malignant growth, wrongly diagnosed. In view of the fact that Ramanujan is no worse than six months ago, he has now abandoned this theory - the other doctors never gave it any support. Tubercle has been the provisionally accepted theory, apart from this, since the original idea of gastric ulcer was given up. ... Like all Indians he is fatalistic, and it is terribly hard to get him to take care of himself. On 18 February 1918 Ramanujan was elected a fellow of the Cambridge Philosophical Society and then three days later, the greatest honour that he would receive, his name appeared on the list for election as a fellow of the Royal Society of London. He had been proposed by an impressive list of mathematicians, namely Hardy, MacMahon, Grace, Larmor, Bromwich, Hobson, Baker, Littlewood, Nicholson, Young, Whittaker, Forsyth and Whitehead. His election as a fellow of the Royal Society was confirmed on 2 May 1918, then on 10 October 1918 he was elected a Fellow of Trinity College Cambridge, the fellowship to run for six years. The honours which were bestowed on Ramanujan seemed to help his health improve a little and he renewed his effors at producing mathematics. By the end of November 1918 Ramanujan's health had greatly improved. Hardy wrote in a letter [3]:I think we may now hope that he has turned to corner, and is on the road to a real recovery. His temperature has ceased to be irregular, and he has gained nearly a stone in weight. ... There has never been any sign of any diminuation in his extraordinary mathematical talents. He has produced less, naturally, during his illness but the quality has been the same. .... He will return to India with a scientific standing and reputation such as no Indian has enjoyed before, and I am confident that India will regard him as the treasure he is. His natural simplicity and modesty has never been affected in the least by success - indeed all that is wanted is to get him to realise that he really is a success. Ramanujan sailed to India on 27 February 1919 arriving on 13 March. However his health was very poor and, despite medical treatment, he died there the following year. The letters Ramanujan wrote to Hardy in 1913 had contained many fascinating results. Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function. On the other hand he had only a vague idea of what constitutes a mathematical proof. Despite many brilliant results, some of his theorems on prime numbers were completely wrong. Ramanujan independently discovered results of Gauss, Kummer and others on hypergeometric series. Ramanujan's own work on partial sums and products of hypergeometric series have led to major development in the topic. Perhaps his most famous work was on the number p(n) of partitions of an integer n into summands. MacMahon had produced tables of the value of p(n) for small numbers n, and Ramanujan used this numerical data to conjecture some remarkable properties some of which he proved using elliptic functions. Other were only proved after Ramanujan's death. In a joint paper with Hardy, Ramanujan gave an asymptotic formula for p(n). It had the remarkable property that it appeared to give the correct value of p(n), and this was later proved by Rademacher.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujan.html (5 of 7) [2/16/2002 11:28:05 PM]

Ramanujan

Ramanujan left a number of unpublished notebooks filled with theorems that mathematicians have continued to study. G N Watson, Mason Professor of Pure Mathematics at Birmingham from 1918 to 1951 published 14 papers under the general title Theorems stated by Ramanujan and in all he published nearly 30 papers which were inspired by Ramanujan's work. Hardy passed on to Watson the large number of manuscripts of Ramanujan that he had, both written before 1914 and some written in Ramanujan's last year in India before his death. The picture above is taken from a stamp issued by the Indian Post Office to celebrate the 75th anniversary of his birth. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (32 books/articles) A Poster of Srinivasa Ramanujan

Mathematicians born in the same country

Cross-references to History Topics

Squaring the circle

Other references in MacTutor

Chronology: 1910 to 1920

Honours awarded to Srinivasa Ramanujan (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1918

Other Web sites

1. Chennai, India 2. Landau-Ramanujan Constant 3. Nielsen-Ramanujan Constants 4. Kevin Brown (Something else about 1729) (See also the quotations of G H Hardy) 5. Kuala Lumpur, Malaysia (His influence on Chandrasekhar) 6. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujan.html (6 of 7) [2/16/2002 11:28:05 PM]

Ramanujan

JOC/EFR June 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ramanujan.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramanujan.html (7 of 7) [2/16/2002 11:28:05 PM]

Ramsden

Jesse Ramsden Born: 15 Oct 1735 in Halifax, Yorkshire, England Died: 5 Nov 1800 in Brighton, Sussex, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jesse Ramsden was apprenticed to a cloth maker, but at the age of 23 he chose to become an apprentice to a mathematical instrument maker. By the age of 27 he had his own business in London where he became acknowledged as the most skilful designer of mathematical, astronomical, surveying and navigational instruments in the 18th Century. The French scientist N Cassegrain proposed a design of a reflecting telescope in 1672. It was Ramsden, however, 100 years later who found that this design reduces blurring of the image caused by the sphericity of the lenses or mirrors. He was elected Fellow of the Royal Society (1786) and received the Copley Medal in 1795. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Jesse Ramsden (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Ramsden

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramsden.html (1 of 2) [2/16/2002 11:28:07 PM]

Ramsden

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ramsden.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramsden.html (2 of 2) [2/16/2002 11:28:07 PM]

Ramsey

Frank Plumpton Ramsey Born: 22 Feb 1903 in Cambridge, Cambridgeshire, England Died: 19 Jan 1930 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Frank Ramsey's parents were Arthur Stanley Ramsey and Agnes Mary Wilson. Arthur Ramsey was President of Magdalene College, Cambridge and a tutor in mathematics there. Frank was the oldest of his parents four children. He had one brother and two sisters and his brother Michael Ramsey went on to become Archbishop of Canterbury. Ramsey entered Winchester College in 1915 and from there he won a scholarship to Trinity College, Cambridge. He completed his secondary school education at Winchester in 1920 and he entered Trinity College, Cambridge to study mathematics. At Cambridge, Ramsey became a senior scholar in 1921 and graduated as a Wrangler in the Mathematical Tripos of 1923. After graduating, Ramsey went to Vienna for a short while, returning to Cambridge where he was elected a Fellow of King's College Cambridge in 1924. It is worth noting that this was a most unusual occurrence, and in fact Ramsey was only the second person ever to be elected to a Fellowship at King's College, not having previously studied at King's. In 1925 Ramsey married Lettice C Baker and they had two daughters. In 1926 he was appointed as a university lecturer in mathematics and he later became a Director of Studies in Mathematics at King's College. It was a short career, for sadly Ramsey died at the beginning of 1930. However, in the short time during which he lectured at Cambridge he had already established himself as an outstanding lecturer. Broadbent writes in [1]:His lectures on the foundations of mathematics impressed young students by their remarkable clarity and enthusiasm ...

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramsey.html (1 of 4) [2/16/2002 11:28:09 PM]

Ramsey

Although Ramsey was a lecturer in mathematics, he produced work in a remarkable range of topics over a short period. As well as starting up the new area of mathematics now vcalled 'Ramsey theory', which we say more about below, he wrote on the foundations of mathematics, economics and philosophy. He published his first major work The Foundations of Mathematics in 1925. In this work he accepted the claim by Russell and Whitehead made in the Principia Mathematica that mathematics is a part of logic. Ramsey's aim in this paper, however, was to improve on the Principia Mathematica and he does so in two ways. Firstly he proposed dropping the axiom of reducibility which, he writes, is:... certainly not self-evident and there is no reason to suppose it true; and if it were true, this would be a happy accident and not a logical necessity, for it is not a tautology. His second simplification is to suggest simplifying Russell's theory of types by regarding certain semantic paradoxes as linguistic. He accepted Russell's solution to remove the logical paradoxes of set theory arising from, for example, "the set of all sets which are not members of themselves". However, the semantic paradoxes such as "this is a lie" are, Ramsey claims, quite different and depend on the meaning of the word "lie". These he removed with his reinterpretation that removed the axiom of reducibility. Ramsey published Mathematical Logic in the Mathematical Gazette in 1926. In this he attacks the:... Bolshevik menace of Brouwer and Weyl ... for denying that propositions are either true of false. He writes:Brouwer would refuse to agree that either it was raining or it was not raining, unless he had looked to see. He also criticises Hilbert in Mathematical Logic saying that he had attempted to reduce mathematics to:... a meaningless game with marks on paper. His second paper on mathematics On a problem of formal logic was read to the London Mathematical Society on 13 December 1928 and published in the Proceedings of the London Mathematical Society in 1930. This examines methods for determining the consistency of a logical formula and it includes some theorems on combinatorics which have led to the study of a whole new area of mathematics called Ramsey theory. Harary describes this birth of Ramsey theory in [7] where he writes the following:The celebrated paper of Ramsey [in 1930] has stimulated an enormous study in both graph theory ..., and in other branches of mathematics .... Most certainly 'Ramsey theory' is now an established and growing branch of combinatorics. Its results are often easy to state (after they have been found) and difficult to prove; they are beautiful when exact, and colourful. Unsolved problems abound, and additional interesting open questions arise faster than solutions to the existing problems. The combinatorics was introduced by Ramsey to solve a special case of the decision problem for the first-order predicate calculus. However, as Mellor points out in [8], it is now known that there is a more direct proof than that given by Ramsey, while the general case of the decision problem cannot be solved. So Mellor points out that [8]:Ramsey's enduring fame in mathematics ... rests on a theorem he didn't need, proved in the course of trying to do something we now know can't be done! Ramsey made a systematic attempt to base the mathematical theory of probability on the notion of partial belief. This work on probability, and also important work on economics, came about mainly because http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramsey.html (2 of 4) [2/16/2002 11:28:09 PM]

Ramsey

Ramsey was a close friend of Keynes. Being a friend of Keynes certainly did not stop Ramsey attacking Keynes' work, however, and in Truth and probability, which Ramsey published in 1926, he argues against Keynes' ideas of an a priori inductive logic. Ramsey's arguments convinced Keynes who then abandoned his own ideas. Ramsey, proposing a probability measure based on strength of belief, [10]:... derives measures both of desires (subjective utilities) and of beliefs (subjective probabilities), thereby founding the now standard use of these concepts. In economics, Ramsey wrote two papers A contribution to the theory of taxation and A mathematical theory of saving. These would lead to important new areas in the subject. It was philosophy, however, that was Ramsey's real love. He wrote a number of works such as Universals (1925), Facts and propositions (1927), Universals of law an of fact (1928), Knowledge (1929), Theories (1929), and General propositions and causality (1929). Braithwaite writes in [5]:... in general philosophy took more and more of his attention. For profitable thought in this most difficult field Ramsey was superbly equipped, and there is no doubt that his early death has deprived the world of one of its most promising philosophers. One would have to say, however, that Ramsey's work in philosophy had been somewhat overshadowed by that of Wittgenstein. Recently, however, Ramsey's work in philosophy seems to be receiving more attention. Several of the articles cited in the references paint vivid pictures of Ramsey's character. For example Braithwaite writes in [5]:As a person, no less than as a thinker, Ramsey was an ornament to Cambridge. From his undergraduate days he had been recognised as an authority on any abstract subject, and his directness of approach and candour were an inspiration to his associates. His enormous physical size fitted well the range of his intellect, and his devastating laugh suited his power of humorously discarding irrelevancies, which power enabled him to be both subtle and profound in the highest degree. Mellor, in [9], paints a similar picture:He was a quiet, modest man, easy going and uninhibited, with a loud infectious laugh, his tolerance and good humour enabling him to disagree strongly without giving or taking offence; as with his brother Michael ... whose ordination ... Frank, as a militant atheist, regretted. He was tall (six feet three inches) and bulky, short-sighted, wore steel-rimmed spectacles and appeared clumsy but was in fact a good tennis player. He produced his remarkable output in four hours a day - he found it too exacting to do more - in the mornings, with afternoons and evenings often spent walking or listening to records. He listened a lot to classical music, both live and recorded, and was a keen hill-walker. Ramsey suffered an attack of jaundice and was taken to Guy's Hospital in London for an operation. He died following the operation. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramsey.html (3 of 4) [2/16/2002 11:28:09 PM]

Ramsey

List of References (13 books/articles)

A Quotation

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ramsey.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramsey.html (4 of 4) [2/16/2002 11:28:09 PM]

Ramus

Peter Ramus Born: 1515 in Cuts (near Noyon), Vermandois, France Died: 26 Aug 1572 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Peter Ramus is also known as Petrus Ramus and as Pierre de la Ramée. This latter name is the one he was given at birth. Peter Ramus's father Jacques de la Ramée was a labourer and his mother was Jeanne Charpentier. It might sound a little strange that Jacques de la Ramée was titled, yet worked as a labourer. This came about because the family had lost their money in 1468 when Liège was destroyed, yet they kept their title. Ramus was educated at home until, in 1527 at the age of twelve years, he entered the Collège de Navarre in Paris. He graduated with a Master's Degree in 1536, defending a thesis on Aristotle. After this Ramus taught, first at the Collège de Mans, then at the Collège de l'Ave Maria. His teaching was aimed at attacking Aristotle and in particular Aristotle's logic. He published his views in three works including Aristotelicae animadversiones in 1543, and following this he was forbidden to teach or publish philosophy by Francis I. It was mathematics and the plague which came to Ramus's rescue. Mathematics since when he was forbidden to teach and publish philosophy he turned to mathematics, and the plague for it created staff shortages which resulted in Ramus being reinstated as a teacher. In 1547 Cardinal Charles de Lorraine appealed to Henry II to have the ban against Ramus lifted and indeed this happened. He was then appointed to the Collège de Presles, and he soon became head of the College. Not only was Ramus back as a teacher, but he also returned to publishing texts. Of course those opposed to his views still strongly attacked him and these attacks became even stronger after he was appointed as Regius Professor of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramus.html (1 of 4) [2/16/2002 11:28:11 PM]

Ramus

Philosophy and Eloquence at the Collège de France in 1551. In 1562 Ramus, whose teaching were becoming more involved with political and religious issues, abandoned the Catholic Church and became a convert to Calvinism. In this year he proposed major reforms in the teaching and structure of the University of Paris. Convinced that mathematics was a subject of fundamental importance to all of learning, he proposed a chair of mathematics at the University. Later he would endow this chair with his own money. Other changes which Ramus proposed was the abolition of student fees (which 450 years later is again a topic of vigorous debate in Britain!). He also proposed changes to the arts syllabus which included a large component of physics and other sciences. Political events were to intervene, however, as the French Wars of Religion began. The Duc de Guise, a Catholic supporter, with his armed forces took control of the royal family in Paris. There were uprisings by the Huguenots around France. Conspirators were ruthlessly dealt with by the Duc de Guise. Near the end of 1562, Ramus was forced to flee Paris for fear of his life as the Calvinists were ordered out of the city. He went to Fontainebleau. The two sides in the War of Religion fought the Battle of Dreux in December 1562 and then looked for a peaceful settlement. Despite the assassination of the Duke de Guise by a Protestant fanatic, the Peace of Amboise was signed in March 1563. It granted certain rights of conscience to the Huguenots and Ramus saw it as sufficient to allow him to return to Paris. For a while he tried to keep a low profile but when one of his opponents was appointed to the Regius Chair of Mathematics, Ramus opposed the appointment, but lost. With tensions rising again in the religious wars, Ramus fled Paris for a second time in 1567. The situation for the Calvinists deteriorated and a third religious war broke out in 1568. Ramus returned briefly to Paris, found his library destroyed, and requested permission from the King to visit Germany. This Ramus did from 1568 to 1570, but in that year another treaty, the Peace of Saint-Germain, was signed in August. Feeling that it was again safe to return to Paris, Ramus gained the promise of protection from the King although he was again banned from teaching. In 1572 three thousand Huguenots assembled in Paris to celebrate the marriage of Marguerite de Valois to Henry III of Navarre. They were massacred on the eve of the feast St Bartholomew and, despite his royal protection, Ramus was murdered by hired assassins. As Mack writes in [6]:That he died a Protestant martyr had considerable consequences for his later reputation. His works were massively reprinted and became very influential in Protestant parts of Germany, in Britain and in New England well into the seventeenth century. The changes which Ramus proposed to the Arts courses taught at universities at that time was a return to the seven classical liberal arts, but with the syllabus more based on applied topics. He developed "method" as a pedagogical concept taking theory towards that required for practical problems. In a text that he wrote in 1546 Ramus describes his concept of methods as:... the organisation of different things in such a way that the whole subject may be more easily perceived and taught. Using this philosophy he proposed to reorganise the seven liberal arts using the following three "laws of method":(i) only things which are true and necessary may be included; (ii) all and only things which belong to the art in question must be included; (iii) general things must be dealt with in a general way, particular things in a particular http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramus.html (2 of 4) [2/16/2002 11:28:11 PM]

Ramus

way. Using this approach Ramus worked on many topics and wrote a whole series of textbooks on logic and rhetoric, grammar, mathematics, astronomy, and optics. It is reasonable to ask how important Ramus is for mathematics. There do not seem to be any theorems named after him, and indeed he is not considered to have been an original mathematician discovering new facts. This does not prevent him from being important, however, and recent work has suggested that he had a more major influence on the development of mathematics than had been once thought. The book [4] is devoted to considering Ramus's contribution to mathematics. Ramus believed that learning in mathematics had declined, and this was due in large part to Plato because of his refusal to consider applications of mathematics. Given these views it is not surprising that his 1569 textbook on geometry contained strong criticisms of Euclid's Elements. Having identified the problems, Ramus aimed to improve mathematical instruction. In order to achieve this he planned to prepare editions of classical mathematical texts. He wrote textbooks on arithmetic, algebra and geometry with the aim of including only theorems which could be applied to the crafts. Rigorous proof was of little importance to Ramus who preferred a "natural method". It was not that he did not believe in theoretical mathematics, but he only saw that it was of importance when it was placed in conjunction with applications. He studied the methods of the tradesmen and craftsmen in Paris in order to choose the directly applicable material. One of the topics which Ramus believed that mathematics should be applied to was astronomy. He seems to have been an early believer of the heliocentric theory of the solar system. He did not favour using theoretical hypotheses to decide between theories, but advocated basing theories on observational evidence. As Mahoney writes in [1]:By emphasising the central importance of mathematics and by insisting on the application of scientific theory to practical problem solving, Ramus helped to formulate the quest for operational knowledge of nature that marks the Scientific Revolution. Article by: J J O'Connor and E F Robertson List of References (7 books/articles) Mathematicians born in the same country Other Web sites

1. The Catholic Encyclopedia 2. The Galileo Project 3. Chris Dawson 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramus.html (3 of 4) [2/16/2002 11:28:11 PM]

Ramus

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ramus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ramus.html (4 of 4) [2/16/2002 11:28:11 PM]

Rankin

Robert Alexander Rankin Born: 27 Oct 1915 in Garlieston, Wigtownshire, Scotland Died: 27 Jan 2001 in Glasgow, Scotland

Click the picture above to see three larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Rankin's father was the Rev Oliver Shaw Rankin who was the parish minister of Sorbie, Wigtownshire at the time of Robert's birth. Robert was named after his grandfather, Robert Rankin, who was Minister of Lamington, Lanarkshire. Robert attended Garlieston School and from there went to Fettes College, an independent school in Edinburgh. He obtained a scholarship to Clare College, Cambridge which he entered in 1934. There he was particularly influenced by Littlewood and Ingham whose lectures he attended while taking Part III of the Mathematical Tripos. Rankin graduated in 1937 and in the same year his father became Professor of Old Testament Language, Literature and Theology at the University of Edinburgh. At Cambridge Rankin began to undertake research in number theory under Ingham's supervision. Dalyell recalls in [2] that Ingham told him:Robert was the most serious of all my gifted pupils. The research which Rankin undertook at this time, on the difference between two successive primes, won him the Rayleigh Prize in 1939. He published four papers on The difference between consecutive prime numbers between this time and 1950. Rankin was elected a Fellow of Clare College in 1939. In the same year he began to work with G H Hardy on the results of Ramanujan. Although Ramanujan had died nearly twenty years earlier, he had left a number of unpublished notebooks filled with theorems that Hardy and other mathematicians continued to study. Rankin did not work long on Ramanujan at this time however before World War II meant that he had to devote himself to war work. He did return to study Ramanujan's work and in many ways it can be seen as a continuing theme throughout his life. He wanted to join the army and become http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rankin.html (1 of 4) [2/16/2002 11:28:13 PM]

Rankin

involved in the fighting but he was ordered to work for the Ministry of Supply at Fort Halstead in Kent. A mathematician of his calibre was seen to have a much more important role to play in the war effort. At Fort Halstead Rankin worked on the development of rockets. He developed a theory to allow the trajectory of the rocket to be calculated from the initial conditions. The British Government, however, paid little attention to the work of Rankin and his team. He was transferred from Fort Halstead to Wales where he continued to work until the end of the war. Of course during the war his work on rockets was classified information, but once the war was over the information was declassified and Rankin was released early from his war service on the condition that he wrote up the theoretical work which he had done on rockets. Indeed he did write the work up as The mathematical theory of the motion of rotated and unrotated rockets and it was published in the Philosophical Transactions of the Royal Society in a paper which was longer than any previously published in that journal. Kelley, in a review of the paper, writes:The author makes a thorough and comprehensive study of the motion of a rocket during burning. ... The problem is to devise a mathematical theory which will, after experimental measurement of suitable constants, predict position, velocity, angular position, and angular velocity of a rocket at the end of burning from a knowledge of these and other physical data at the beginning of burning. Of necessity, since the work is intended for direct application to ballistic calculations, considerable detail is given, and this, in turn, requires a truly formidable list of notations. During the war Rankin had married Mary Llewellyn in 1942; they would have four children, one son and three daughters. Rankin returned to Cambridge with his wife in 1945 where he became a Faculty Assistant Lecturer. In 1947 he became an Assistant Tutor and then in the following year he was promoted from Assistant Lecturer to Lecturer. In 1949 he became a Praelector at Cambridge. A colleague who worked with him at Cambridge at this time recalled (see [2]):He was a conscientious teacher and had a wide interest in mathematics. Those who took the trouble to ask him serious questions were rewarded with precise and very serious answers. In 1951 Rankin left Cambridge when he was appointed Mason Professor of Pure Mathematics at Birmingham University. He was not to spend long in Birmingham for, in 1954, T M MacRobert retired from the Chair of Mathematics at the University of Glasgow and the Principal of that university tempted Rankin to move back to Scotland to fill the vacancy. Martin writes [3]:Thus began a period of 28 years during which Robert's powerful intellect, exceptionally accurate memory and tremendous energy, along with his absolute integrity and unstinted devotion to duty, enabled him to render signal service to the university. Rankin wrote over 100 research papers, mostly on the theory of numbers and the theory of functions. He wrote The modular group and its subgroups published in 1969 and Modular forms and functions which was published in 1977. The former of these is described by Rankin himself in the Preface:This short course of lectures was given at the Ramanujan Institute for Advanced Study in Mathematics, in the University of Madras, in September 1968. The object of the course was to study the modular group and some of its subgroups, with help of algebraic rather than analytic or topological methods. Of course the connection with Ramanujan, the continuing theme we mentioned earlier, was not only that the lectures were given in the Ramanujan Institute but that The Ramanujan Institute in Madras published the book. A major historical work published was Ramanujan : Letters and commentary a joint work with http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rankin.html (2 of 4) [2/16/2002 11:28:13 PM]

Rankin

B C Berndt which Rankin published in 1995; it has been instantly recognised as a exceptional contribution to the history of mathematics. Earlier he had published the papers Ramanujan as a patient in 1984, and Srinivasa Ramanujan (1887-1920) in 1987. At the conference Ramanujan revisited in UrbanaChampaign, Illinois in 1987 he presented a paper on Ramanujan's t-function and its generalizations. The paper appeared in the conference proceedings of which Rankin was himself an editor. Marvin Knopp writes:This is an informed (and informative) account of Ramanujan's function t(n) and many of the important later developments grounded in that work. With his usual fine sense of history, [Rankin] begins the discussion, not with Ramanujan himself, but rather with the older English mathematician J W L Glaisher (born in 1848), who initiated the study of multiplicative properties of the Fourier coefficients of modular forms in his series of papers, published in 1907, dealing with ... the number of representations of n as a sum of s squares. Taking up this work where severe complications had forced Glaisher to abandon it (18 was the largest value of s he treated), Ramanujan was led to his seminal work on t(n) in investigating [the number of representations of n as a sum of 24 squares]. We should note that Rankin himself had made a number of contributions to studying the number of representations of an integer as a sum of squares. Although the above comments on his publications concentrate on his historical contributions, we have only given it this slant since this aspect is more easily described. We should emphasise that his remarkable contributions to the theory of numbers have played a major part in the modern developemnt of the topic. One characteristic of Rankin was the care with which he undertook all things in his life. Not only was his research articles written with great care but he applied the same attention to detail in running the Department of Mathematics at Glasgow and also in his teaching. Directly coming out of his teaching was the undergraduate text An introduction to mathematical analysis. He was still publishing papers, and giving lectures, up to the time of his death. He published The books studied by Ramanujan in India in 2000 (again collaborating with B C Berndt) and travelled to London to lecture there shortly before his death although his doctors had strongly advised him not to go. Rankin received many honours for his outstanding contributions to mathematics. Elected a fellow of the Royal Society of Edinburgh in 1955, he received the Society's Keith Prize for his publications in 1961-63. The London Mathematical Society awarded him their Senior Whitehead Prize in 1987 and the De Morgan Medal in 1998. Mathematics was certainly not Rankin's only interest. He was very musical, took a scholarly approach to his interest in Scottish Gaelic being President of the Glasgow Gaelic Society from 1957 until his death, and he enjoyed hill walking. He used his knowledge of Gaelic in a professional capacity when he was external examiner at University College, Galway in Ireland when he examined mathematics papers written in Irish Gaelic. May I [EFR] finish this biography on a personal note. I knew Rankin from the late 1960s. He always made me feel very welcome whenever I visited Glasgow University and treated me with great kindness. Latterly, with his deep interest in the history of mathematics, Rankin was extremely encouraging to me in developing this archive. He contributed an excellent article on Robert Simson to the archive based on a lecture he gave at the Royal Society of Edinburgh; one which I was fortunate enough to hear. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rankin.html (3 of 4) [2/16/2002 11:28:13 PM]

Rankin

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other Web sites

The Scotsman (Obituary)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rankin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rankin.html (4 of 4) [2/16/2002 11:28:13 PM]

Rankine

William John Macquorn Rankine Born: 2 July 1820 in Edinburgh, Scotland Died: 24 Dec 1872 in Glasgow, Scotland

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Trained as a civil engineer, William Rankine was appointed to the chair of civil engineering and mechanics at Glasgow in 1855. He developed methods to solve the force distribution in frame structures. He worked on heat, and attempted to derive Sadi Carnot's law from his own hypothesis. His work was extended by Maxwell. Rankine also wrote on fatigue in the metal of railway axles, on Earth pressures in soil mechanics and the stability of walls. He was elected a Fellow of the Royal Society in 1853. Among his most important works are Manual of Applied Mechanics (1858), Manual of the Steam Engine and Other Prime Movers (1859) and On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance. Article by: J J O'Connor and E F Robertson List of References (10 books/articles) Mathematicians born in the same country Honours awarded to William Rankine (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1853

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rankine.html (1 of 2) [2/16/2002 11:28:15 PM]

Rankine

Fellow of the Royal Society of Edinburgh Lunar features

Crater Rankine

Other Web sites

1. iGEM 2. Maritime History Archive (Obituary) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rankine.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rankine.html (2 of 2) [2/16/2002 11:28:15 PM]

Raphson

Joseph Raphson Born: 1648 in Middlesex, England Died: 1715 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Raphson's life can only be deduced from a number of pointers. No obituary of Raphson seems to have been written and we can now only piece together details about his life from records which exist such at University of Cambridge records and records of the Royal Society. It is through the University of Cambridge records that we know that Raphson attended Jesus College Cambridge and graduated with an M.A. in 1692. Rather remarkably Raphson was made a member of the Royal Society in 1691, the year before he graduated. His election to that Society was on the strength of his book Analysis aequationum universalis which was published in 1690 contained the Newton method for approximating the roots of an equation. In Method of Fluxions Newton describes the same method and, as an example, finds the root of x3 - 2x 5 = 0 lying between 2 and 3. Although written in 1671 it was not published until 1736, so Raphson published the result nearly 50 years before Newton. Raphson's relation to Newton is important but not particularly well understood. In [2] Copenhaver writes:Raphson was one of the few people whom Newton allowed to see his mathematical papers. As early as 1691, he and Edmund Halley were involved in plans to publish Newton's work of the early 1670's on quadrature of curves, a project fulfilled only in 1704, and then in a much different form. In 1711, Roger Cotes and Willian Jones arranged for Raphson to see some of Newton's papers '... pertinent to his design of writing an History of the Method of Fluxions'. Raphson did indeed write his History of Fluxions which did not appear until 1715 after Raphson had died. It is unclear how pleased Newton was with this work despite its clear position in favour of Newton's claims over those of Leibniz. Certain letters which had passed between Newton and Leibniz appeared as an appendix to a reprint of Raphson's book in 1716-1718. Immediately a row broke out and Johann Bernoulli showed his anger. An attempt was made by Newton to calm things down when he wrote to Johann Bernoulli saying:I stopt [Raphson's History of Fluxions] coming abroad for three or four years. However, Newton admitted in a letter to Varignon that he was responsible for the letter being added to

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Raphson.html (1 of 3) [2/16/2002 11:28:17 PM]

Raphson

Raphson's book:When I heard that Mr Leibnitz was dead I caused what had passed between him and me to be printed at the end of Raphson's book because copies thereof had been dispersed by Mr Leibnitz. This was not Raphson's only publication relating to Newton's work. He translated Newton's algebraic work from Latin to English. Newton's Arithmetica universalis was translated by Raphson and appeared as Universal arithmetick in 1720 after Raphson's death. Early in his career Raphson published a mathematical dictionary. In 1691, the year Raphson was elected to the Royal Society, Ozanam published Dictionnaire mathématique. Raphson produced his shorter version A mathematical dictionary in 1702 which is:A mathematical dictionary or a compendious explication of all mathematical terms, abridg'd from Monsieur Ozanam and others ... written by J Raphson FRS. Raphson published a second edition of his analysis book and, at the same time, De spatio reali which is an application of mathematical reasoning to theological issues. Raphson wrote a second theological work Demonstratio de deo in 1710. De spatio reali discusses space and in it Raphson talks of 'real space' which he thinks of as being independent of the mind that perceives it. He discusses the infinite, distinguishing between the potentially infinite and the actual infinite. In discussing motion he argues that space is infinite but the collection of moving objects in it is finite. Raphson's ideas of space and philosophy were based on Cabalist ideas. The Cabala was a Jewish mysticism which was influential from the 12th century on. It was an oral tradition and initiation into its doctrines and practices was passed on. Cabala developed several basic doctrines which were strong influences on Raphson's philosophical thinking. The doctrines included the withdrawal of the divine light, thereby creating primordial space, the sinking of luminous particles into matter and a "cosmic restoration" that is achieved by Jews through living a mystical life. In these two works by Raphson De spatio reali and Demonstratio de deo, cosmology, natural philosophy, mathematics and his Cabalist beliefs combine. Of course his religious beliefs greatly influenced all his thinking. Newton's views of space were strongly influenced by Christian beliefs, and possible just slightly by his interaction with Raphson. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Newton-Raphson method of solving equations

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Raphson.html (2 of 3) [2/16/2002 11:28:17 PM]

Raphson

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Raphson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Raphson.html (3 of 3) [2/16/2002 11:28:17 PM]

Rasiowa

Helena Rasiowa Born: 20 June 1917 in Vienna, Austria Died: 9 Aug 1994 in Warsaw, Poland

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Although Helena Rasiowa was born in Vienna, her parents were Polish. In 1918 Poland regained its status as an independent nation and Rasiowa's parents moved to Warsaw. She was educated there, obtaining a good secondary school education with music lessons taken at a special music school. After completing her school studies she took a course in business management before entering university. Rasiowa entered the University of Warsaw in 1938 but, after the German invasion of Poland in 1939, the university closed. Rasiowa and her parents moved to Lvov but the Poles were trapped between the Soviets and the Germans and Lvov came under Soviet control. Life there seemed even more difficult than under German occupation, so after a year the family returned to Warsaw. There was an impressive collection of mathematicians at the University of Warsaw at this time including Borsuk, Lukasiewicz, Mazurkiewicz, Sierpinski, Mostowski and others. They had organised an underground version of the university which was strongly opposed by the Nazi authorities. Borsuk, for example, was imprisoned after the authorities found that he was helping to run the underground university. In this dangerous situation Rasiowa learnt mathematics, knowing that the penalties for being discovered were extreme. Yet in this environment Rasiowa studied for her Master's Degree under Lukasiewicz's supervision. When the Soviet forces came close to Warsaw in 1944, the Warsaw Resistance rose up against the weakened German garrison. However German reinforcements arrived and put down resistance. Around 160,000 people died in the Warsaw Uprising of 1944 and the city was left in a state of almost total http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rasiowa.html (1 of 3) [2/16/2002 11:28:19 PM]

Rasiowa

devastation. Rasiowa's time during the Uprising is described in [1]:In 1944 the Warsaw Uprising broke out and in consequence Warsaw was almost completely destroyed, not only because of warfare but also because of the systematic destruction which followed the uprising after it had been squashed down. Rasiowa's thesis burned together with the whole house. She herself survived with her mother in a cellar covered by ruins of the demolished building. After the war Rasiowa taught in a secondary school while her supervisor Lukasiewicz left Poland after the terrible suffering he had gone through. Mostowski however remembered Rasiowa's impressive work and persuaded her to return to the University of Warsaw to complete a second Master's Thesis under his supervision. In 1946, having obtained her Master's degree, she was appointed as an assistant at the University of Warsaw and continued to work for her doctorate under Mostowski's supervision. Her thesis, presented in 1950, was on algebra and logic Algebraic treatment of the functional calculus of Lewis and Heyting and these topics would be the main areas of her research throughout her life. Rasiowa was promoted steadily, reaching the rank of Professor in 1957 and Full Professor in 1967. She led the Foundations of Mathematics Section from 1964 and the Mathematical Logic Section after its creation in 1970. Her main research was in algebraic logic and the mathematical foundations of computer science. In algebraic logic she continued work by Post, Stone, Tarski and Lukasiewicz [1]:... aimed at finding a precise description for the mathematical structure of formalised logical systems. Of course Rasiowa's work on algebraic logic was in precisely the right area to make her a natural contributor to theoretical computer science. However it is one thing to be in the right area and yet another to have the ability to see the importance of a new subject such as computer science. Her contributions are described in [1]:Her contribution to theoretical computer science stems from her conviction that there are deep relations between methods of algebra and logic on the one side and essential problems of foundations of computer science on the other. Among these problems she clearly distinguished inference methods characteristic of computer science and its applications. This conviction of hers had been supported by her results on many-valued and non-classical logics, especially on applications of various generalisations of Post algebras to logics of programs and approximation logics. In fact in 1984 Rasiowa introduced an important concept of inference where the basic information was incomplete. This led to approximate reasoning and approximate logics which are now central to the study of artificial intelligence. Rasiowa wrote over 100 papers, books and monographs. She also supervised the doctoral dissertations of more than 20 students. However her contributions were not restricted to research. She helped set up the journal Fundamenta Informaticae which she was editor-in-chief from its setting up in 1977 until her death. In addition to these editorial duties she also was Collecting Editor of Studia Logica from 1974 and, from 1986, an associate editor of the Journal of Approximate Reasoning.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rasiowa.html (2 of 3) [2/16/2002 11:28:19 PM]

Rasiowa

She also played a major role in the mathematical life of Poland. A member of the Polish Mathematical Society, she was its secretary in 1955-57 and its vice-president in 1958/59. She served on the Committee on Mathematics of the Polish Academy of Sciences and chaired various committees of the Polish Ministry of Science and Higher Education. It was partly through her endeavours that the Polish Society for Logic and the Philosophy of Science was set up. Rasiowa remained active right up to her death, having completed eight chapters of a new monograph Algebraic analysis of non-classical first order logics before entering hospital with her final illness. Article by: J J O'Connor and E F Robertson A Reference (One book/article) A Poster of Helena Rasiowa

Mathematicians born in the same country

Other Web sites

Warsaw, Poland

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rasiowa.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rasiowa.html (3 of 3) [2/16/2002 11:28:19 PM]

Rayleigh

John William Strutt Lord Rayleigh Born: 12 Nov 1842 in Langford Grove (near Maldon), Essex, England Died: 30 June 1919 in Terling Place, Witham, Essex, England

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John William Strutt's father was the second Baron Rayleigh of Terling Place, Witham in the county of Essex. Certainly it was a family with little previous interests in science for they were mostly landowners with interests in the countryside. One exception was Robert Boyle, who was a distant relation. Let us say at the beginning of this article that we shall refer to Strutt as Rayleigh throughout this article although he did not succeed to the title until he was 30 years old. As a boy Rayleigh suffered from poor health and his schooling at both Eton and Harrow was disrupted. He had to leave both schools after a short period due to health problems. Four years spent at the Reverend Warner's boarding school, prepared Rayleigh for university and at this stage he did begin to show signs of mathematical ability. During these four years he had a private tutor but overall he showed little sign of being anything other than an average child of average ability. He entered Trinity College, Cambridge in October 1861 where he took the mathematical Tripos. His coach at Cambridge was Edward Routh who, in addition to being the most famous of the Cambridge coaches at that time (perhaps of all time), was himself a very fine applied mathematician making important contributions to dynamics. There is no doubt that the grounding in mathematical techniques which Rayleigh had from Routh was an important factor in his outstanding scientific career. It was more than just the mathematics which he learnt that was important to him, for in addition he learnt from Routh how to come up with the most appropriate mathematical methods to tackle each problem. There was another important influence on Rayleigh during his undergraduate years at Cambridge, namely that of Stokes who was the Lucasian professor of mathematics at the time. Stokes inspired

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rayleigh.html (1 of 7) [2/16/2002 11:28:21 PM]

Rayleigh

Rayleigh with his lectures which combined theory and practice in a novel way with many physical experiments being carried out during the lectures. Students did not have the opportunity to undertake physics experiments themselves, so seeing Stokes perform experiments in his course on light was Rayleigh's only exposure to the experimental side of science. Rayleigh himself later spoke of how important a role Stokes had played in his development as a scientist. However Stokes does not appear to have directly encouraged Rayleigh to undertake a scientific career. If Rayleigh had been an average school pupil he was far from an average student. He was awarded an astronomy scholarship in 1864, then in the Tripos examinations of 1865 he was Senior Wrangler (the top First Class student) and in the same year he was the first Smith's prizeman. One has to understand that Rayleigh was now faced with a difficult decision. For someone in his position, knowing that he would succeed to a title and become the third Baron Rayleigh, taking up a scientific career was not really acceptable, and certainly various members of his family felt exactly that way. By this time, however, Rayleigh was determined to devote his life to science so he was certain in his own mind that his social obligations must not stand in his way. His first paper was inspired by reading Maxwell's 1865 paper on electromagnetic theory. It was through reading widely the current scientific literature that Rayleigh tried to work out which were the important problems on which he should undertake research. The other scientist whose works he studied deeply was Helmholtz, in particular reading Helmholtz' 1860 results on the acoustic resonator. In 1866 Rayleigh was elected a Fellow of Trinity College, Cambridge and he was poised to make his mark in science. The usual course of action for young British men of social standing at this time was to take a European tour - the grand tour as it was called. Rayleigh, surprisingly, made a very different, and for that time unusual, tour for he set out on a trip to the United States. One advantage of Rayleigh's privileged social position was that he did not need an academic post to earn his living. Rather when he returned from the United States he purchased equipment for undertaking scientific experiments and set it up on the family estate at Terling. He did experiments on the galvanometer and presented his results to the British Association meeting in Norwich in 1868. Rayleigh's theory of scattering, published in 1871, was the first correct explanation of why the sky is blue. In the same year he married Evelyn Balfour, the sister of Arthur James Balfour who was to be a leading member of the Conservative Party for 50 years and Prime Minister of Britain 30 years later. Rayleigh had been a student at Cambridge with Arthur James Balfour and through him had met Evelyn. Shortly after their marriage Rayleigh had an attack of rheumatic fever which nearly brought his scientific activities to a premature end. He was advised to travel to Egypt and indeed he did just this with his wife. They sailed down the Nile during the last months of 1872 and early 1873, returning to England in the spring of 1873. It was a trip during which Rayleigh recovered his health but it was also a very profitable trip from a scientific point of view. Rather remarkably he began writing a major text The Theory of Sound while on the trip. It was five years after beginning this great classic before it appeared in print. The first volume, on the mechanics of a vibrating medium which produces sound, was published in 1877, while the second volume on acoustic wave propagation was published the following year. Shortly after returning from his trip down the Nile, Rayleigh's father died and Strutt, as he had been up to that time, succeeded to the title becoming the third Baron Rayleigh. He continued working at Terling

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rayleigh.html (2 of 7) [2/16/2002 11:28:21 PM]

Rayleigh

where he now took up residence. The laboratory which he had set up there was one where he made impressive discoveries but one should not think that this was because the rich Rayleigh was able to have better equipment than anyone else. On the contrary he obtained impressive experimental results with cheap equipment. Rayleigh was always one to economise and make do with unsophisticated equipment. Also he was not as well off as might have been expected, for the 1870s were a time of economic problems for farming in England and as a consequence his income was far less than might otherwise have been the case. From 1879 to 1884 Rayleigh was the second Cavendish professor of experimental physics at Cambridge. The laboratory had been opened five years earlier and Maxwell had been the first Cavendish professor. On the academic side Rayleigh was an obvious choice to succeed to Maxwell's chair, yet in other times he might have been content to work at Terling. The agricultural depression however swung the balance making the income from the post look attractive. There was no suggestion, however, that Rayleigh was just there for the money. On the contrary he took his duties very seriously making very substantial improvements to the teaching of physics at Cambridge. Heathcote, in [6], writes:To him fell the task of organising the laboratory as a centre of instruction and research, a task which he accomplished with outstanding success. We mentioned above the lack of experimental physics when Rayleigh himself was an undergraduate and although changes were being made, still a great deal was required to be done. With the same energy with which he approached everything, Rayleigh developed laboratory courses in heat, electricity and magnetism, properties of matter, optics, and acoustics. One of the important pieces of experimental work he carried during his time as Cavendish professor was a standardisation of the ohm. Maxwell and Chrystal had carried out experiments in Cambridge earlier and the apparatus was still available for Rayleigh. However the old equipment did not prove good enough to allow Rayleigh to obtain the accuracy he required and he had new apparatus built. In his Presidential Address to the British Association in Montreal in 1884 he explained the results. He introduced the topic by saying:During the last few years much interest has been felt in the reduction to an absolute standard of measurements of electromotive force, current, resistance, etc. and to this end many laborious investigations have been undertaken. The subject is one which has engaged a good deal of my own attention ... Then in 1884 he resigned his Chair at Cambridge to return to his research on his own estate at Terling. His financial position had improved and what he loved was scientific research, without the time-consuming responsibilities of a university post. There were many colleagues who tried to get him to reconsider his action and continue to hold the chair but Rayleigh knew exactly what he wanted from life. It was not a solitary scientific existence for him in Terling since he made frequent visits to London where he had duties to perform for many learned and scientific societies. Let us look briefly at some of his activities in this area. Rayleigh had been elected as a Fellow of the Royal Society in 1873. He received the Royal Medal from the Society in 1882, and became secretary of the Society in 1885, being awarded the Society's Copley Medal in 1899. He gave the Society's Bakerian Lecture in 1902 and he was elected President of the Society in 1905, holding the position until 1908. Rayleigh served as President of the London Mathematical Society in 1876-78 and he was awarded the Society's De Morgan Medal in 1890. He also http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rayleigh.html (3 of 7) [2/16/2002 11:28:21 PM]

Rayleigh

had connections with the Royal Institution, becoming professor of natural philosophy there in 1887. He became chancellor of Cambridge University in 1908. Other activities which deserve mention involve the work he put in helping towards establishing the National Physical Laboratory which was set up at Teddington in Middlesex in 1900. He was appointed scientific advisor to Trinity House, the association of English seamen, in 1896. Connected with the political scene through his wife, he was much involved in advisory roles such as serving on a committee on aeronautics. Clearly this level of activity meant that he did not lack contact with fellow scientists, and he also corresponded with many of the leading scientists. We should now turn to examine briefly some of the scientific work which he undertook. First, however, we note that [6] contains a complete list of Rayleigh's publications and remarkably there are 446 items in the list. They cover an incredible range of topics in applied mathematics and physics. Among the publications devoted to mathematics, rather than to its applications, are papers on Bessel functions, the relationship between Laplace functions and Bessel functions, and Legendre functions. In addition to the more usual topics of applied mathematics and physics which we say a little on below, he wrote on more unusual topics such as Insects and the colour of flowers (1874), On the irregular flight of a tennis ball (1877), The soaring of birds (1883), The sailing flight of the albatross (1889), and The problem of the Whispering Gallery (1910). We have mentioned above his work on electromagnetic phenomena, his major treatise on sound, the determination of the ohm, and his important paper of scattering of light which explained why the sky is blue. In addition [6]:... he applied the wave theory of light to the mathematical investigation of the resolving power of prisms and diffraction gratings; thus he showed that the resolving power of a grating is determined by the total number of lines in the grating multiplied by the order of the spectrum, and not by the closeness of the lines. ... In 1887 he published a paper in which he suggested the method of reproducing colours by photography later adopted in principle by Lippmann. Rayleigh is perhaps most famous for his discovery the inert gas argon in 1895, work which earned him a Nobel Prize in 1904. In his address on the occasion of receiving the Nobel Prize Rayleigh explained how he made his famous discovery (see for example [6]):The subject of the densities of gases has engaged a large part of my attention for over 20 years. ... Turning my attention to nitrogen, I made a series of determinations ... Air bubbled through liquid ammonia is passed through a tube containing copper at a red heat where the oxygen of the air is consumed by the hydrogen of the ammonia, the excess of the ammonia being subsequently removed with sulphuric acid. ... Having obtained a series of concordant observations on gas thus prepared I was at first disposed to consider the work on nitrogen as finished. ... Afterwards, however, ... I fell back upon the more orthodox procedure according to which, ammonia being dispensed with, air passes directly over red hot copper. Again a good agreement with itself resulted, but to my surprise and disgust the densities of the two methods differed by a thousandth part - a difference small in itself but entirely beyond experimental errors. ... It is a good rule in experimental work to seek to magnify a discrepancy when it first appears rather than to follow the natural instinct to trying to get quit of it. What was the difference between the two kinds of nitrogen? The one was wholly derived from air; the other partially, to the extent of about one-fifth part, from ammonia. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rayleigh.html (4 of 7) [2/16/2002 11:28:21 PM]

Rayleigh

The most promising course for magnifying the discrepancy appeared to be the substitution of oxygen for air in the ammonia method so that all the nitrogen should in that case be derived from ammonia. Success was at once attained, the nitrogen from the ammonia being now 1/200 part lighter than that from air. ... Among the explanations which suggested themselves are the presence of a gas heavier than nitrogen in air ... Rayleigh of course was correct and succeeded, with considerable difficulty, in isolating the gas. Since it refused to make chemical combinations it was called argon from the Greek word for inactive. In 1879 Rayleigh wrote a paper on travelling waves, this theory has now developed into the theory of solitons. The preface of [3] explains why Rayleigh-wave theory, introduced by him in 1885 in a paper in the Proceedings of the London Mathematical Society, has proved so important:There is no respect for mere age in science or technology. Yet the centenary of the discovery, by the third Lord Rayleigh, that elastic waves can be guided by a surface, is memorable for the contradictions which it encompasses: Rayleigh's assessment of his classic 1885 paper as a rather minor mathematical development with a potential value only in seismology on the one hand; on the other the rediscovery of the subject in a totally different field - that of electronic signal processing - which has led to its explosive growth over the last twenty years. In fact in his paper 1885 paper On waves propagated along the plane surface of an elastic solid Rayleigh writes:It is proposed to investigate the behaviour of waves upon the plane surface of an infinite homogeneous isotropic elastic solid, their character being such that the disturbance is confined to a superficial region, of thickness comparable with the wavelength. .... It is not improbable that the surface waves here investigated play an important part in earthquakes, and in the collision of elastic solids. Diverging in two dimensions only, they must acquire at a great distance from the source a continually increasing preponderance. Rott [11] looks at Rayleigh's contributions to hydrodynamics, in particular to hydrodynamic similarity:[There were] two domains in fluid mechanics in which Lord Rayleigh made explicit use of hydrodynamic similarity: the theory of aerodynamic drag and the treatment of the Aeolian tones. [There was a] great impact of Rayleigh's ideas on the development of hydrodynamic similarity theory and applications during his lifetime and beyond. Of course Rayleigh received many honours for his scientific work. In 1902, at the coronation of King Edward VII, he received the Order of Merit. In addition to the Nobel Prize he received thirteen honorary degrees, five government awards, and honorary membership of five learned societies world-wide. Rayleigh was a modest and generous man. He donated the proceeds of his Nobel Prize to the University of Cambridge to build an extension to the Cavendish laboratories. On receiving the Order of Merit in 1902 he said:... the only merit of which I personally am conscious was that of having pleased myself by my studies, and any results that may be due to my researches were owing to the fact that it has been a pleasure for me to become a physicist. We end this brief biography of Rayleigh by quoting from his Presidential Address to the British Association in Montreal in 1884:Without encroaching upon grounds appertaining to the theologian and the philosopher, the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rayleigh.html (5 of 7) [2/16/2002 11:28:21 PM]

Rayleigh

domain of natural sciences is surely broad enough to satisfy the wildest ambition of its devotees. In other departments of human life and interest, true progress is rather an article of faith than a rational belief; but in science a retrograde movements is, from the nature of the case, almost impossible. Increasing knowledge brings with it increasing power, and great as are the triumphs of the present century, we may well believe that they are but a foretaste of what discovery and invention have yet in store for mankind. ... The work may be hard, and the discipline severe; but the interest never fails, and great is the privilege of achievement. Article by: J J O'Connor and E F Robertson List of References (11 books/articles)

Some Quotations (2)

A Poster of John William Strutt

Mathematicians born in the same country

Honours awarded to John William Strutt (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1904

Fellow of the Royal Society

Elected 1873

Royal Society Copley Medal

Awarded 1899

Royal Society Royal Medal

Awarded 1882

Royal Society Bakerian lecturer

1902

London Maths Society President

1876 - 1878

LMS De Morgan Medal

Awarded 1900

Lunar features

Crater Rayleigh

Planetary features

Crater Rayleigh on Mars

Other Web sites

1. Nobel prizes site (A biography of Rayleigh and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rayleigh.html (6 of 7) [2/16/2002 11:28:21 PM]

Rayleigh

JOC/EFR May 2001

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rayleigh.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rayleigh.html (7 of 7) [2/16/2002 11:28:21 PM]

Razmadze

Andrei Mikhailovich Razmadze Born: 11 Aug 1889 in Chkhenisi, Russia (now Samtredia, Georgia) Died: 2 Oct 1929 in Tbilisi, USSR Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Andrei Razmadze's father was Mikhail Gavrilovich Razmadze, who was employed on the railways, while his mother was Nino Georgievna Nodia. Andrei attended secondary school in Kutaisi, completing his studies there in 1906, and in the same year he entered Moscow University. Graduating with a degree in mathematics in 1910, Razmadze taught mathematics in secondary schools while he continued to work for his Master's degree. He was awarded this degree in 1917, taught at Moscow University for a few months, then returned to Georgia near the end of the year 1917. Razmadze was one of the founders of Tbilisi University and he taught at this university from the time that it opened in 1918. He held a chair in the Physics and Mathematics Faculty in Tbilisi for the rest of his life. Chrelashvili describes some of his work and achievements there in [3]:As one of the founders of Tbilisi University, he (jointly with N I Muskhelishvili) directed the training of specialists in mathematics at the university. As a result, within a short period of time all the higher educational establishments of Georgia were supplied with teaching and scientific personnel in mathematics. Mathematics teachers for most secondary schools of Georgia were also trained here. Razmadze wrote the first textbooks in Georgian on analysis and integral calculus. In [2] Razmadze's Introduction to differential calculus a little-known textbook published in Russian in 1923 is described. His work was on the calculus of variations, continuing work by Weierstrass and Hilbert. The fundamental lemma of the calculus of variations is named after him. He also did important work on discontinuous solutions. Yushkevich writes [1]:[Razmadze] presented a report on his research to the International Congress of Mathematicians at Toronto in 1924, and for that paper he received the doctorate in mathematics from the Sorbonne. Chrelashvili also notes in [3] how Razmadze's work had been recognised by the international mathematical community. He writes that:Following Razmadze's death ..., the outstanding French mathematician Jacques Hadamard sent a telegram of condolence to Tbilisi University, saying that he, together with all the mathematicians of France and the world, was profoundly grieved at the death of Razmadze. This is undoubtedly another expression of the international recognition of Razmadze's scientific contribution and his talent. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Razmadze.html (1 of 2) [2/16/2002 11:28:22 PM]

Razmadze

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Razmadze.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Razmadze.html (2 of 2) [2/16/2002 11:28:22 PM]

Recorde

Robert Recorde Born: 1510 in Tenby, Wales Died: 1558 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Recorde is best known for inventing the 'equals' symbol '=' which appears in his book The Whetstone of Witte (1557). Recorde was educated at Oxford and Cambridge. He became physician to King Edward VI and Queen Mary. He served for a time in Ireland as 'Comptroller of Mines and Monies'. Recorde virtually established the English school of mathematics and first introduced algebra into England. He wrote many textbooks, for example The Grounde of Artes in 1540 was a very successful commercial arithmetic book teaching the perfect work and practice of Arithmeticke etc in Recorde's own words. The book discusses operations with arabic numerals, computation with counters, proportion, the 'rule of three', and fractions. In 1551 Recorde wrote Pathwaie to Knowledge which some consider an abridged version of Euclid's Elements. It is the only one of his books not written in the form of a dialogue between a master and scholar. The 'equals' symbol '=' appears in Recorde's book The Whetstone of Witte published in 1557. He justifies using two parallel line segments bicause noe 2 thynges can be moare equalle . The symbol = was not immediately popular. The symbol || was used by some and ae ( or oe ), from the word aequalis meaning equal, was widely used into the 1700's.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Recorde.html (1 of 2) [2/16/2002 11:28:24 PM]

Recorde

Recorde died in King's Bench prison in Southwark, where he was committed for debt. Although no official record remains of other crimes, some historians think he was guilty of much more serious offences. The picture above is the only portrait known of him and was thought to have been made from life in 1556 although more recent research suggests that it is a 17th Century work. Article by: J J O'Connor and E F Robertson List of References (6 books/articles)

Some Quotations (2)

A Poster of Robert Recorde

Mathematicians born in the same country

Some pages from publications

A page from The Whetsone of Witte (1557)showing the use of the = sign

Cross-references to History Topics

1. Mathematical games and recreations 2. An overview of the history of mathematics

Other references in MacTutor

Chronology: 1500 to 1600

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Recorde.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Recorde.html (2 of 2) [2/16/2002 11:28:24 PM]

Rees

Mina Spiegel Rees Born: 2 Aug 1902 in Cleveland, Ohio, USA Died: 25 Oct 1997 in New York City, USA

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mina Rees grew up in New York City and attended public school there. She attended Hunter College High School with which she was to have a long association. She graduated with distinction in 1923 and was appointed as an assistant teacher at the school while she continued her studies at Columbia University. In an interview, see [1], she spoke of the attitude that she encountered there:When I had taken four of their six-credit graduate courses in mathematics and was beginning to think about a thesis, the word was conveyed to me - no official ever told me this but I learned - that the Columbia mathematics department was really not interested in having women candidates for Ph.D's. This was a very unpleasant shock. ... I decided to switch to Teacher's College and take the remaining courses necessary for an M.A. there. After receiving her M.A. in 1925 she returned to Hunter College where she was appointed to the post of instructor. Determined not to allow the attitude of Columbia University to prevent her from completing her doctorate, she enrolled at the University of Chicago in 1929 after obtaining leave of absence from Hunter College. At Chicago her doctorate was supervised by Dickson who agreed to a topic in associative algebra despite his own interests having moved to number theory by this time. In 1931 Rees graduated with her doctorate for a thesis entitled Division algebras associated with an equation whose group has four generators. Saunders MacLane writes in [4]:Now for a missed opportunity for mathematics in connection with Mina. Mina's thesis was finished in the spring of 1931. Professor Dickson might then have recommended her for an NRC postdoctoral fellowship. If she had won one, and if she had heard of Noether, she could have gone to Göttingen, where she would surely have attended Noether's lectures on

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rees.html (1 of 3) [2/16/2002 11:28:26 PM]

Rees

hypercomplex systems and factor sets. From Mina's subsequent accomplishments we know that she would have understood these notions and made use of them to simplify her earlier proof. We know that Emmy Noether took care of her associates and students, both men and women. We also know about Mina and so can imagine many splendid research results. The opportunity was missed, however, and Rees retuned to Hunter College where she was promoted to assistant professor in 1932 and then to associate professor in 1940. However, in 1943 she took leave so that she could contribute to the war effort. She worked as a technical aide and executive assistant with the Applied Mathematics Panel in the Office of Scientific Research and Development. In this job she had to take problems submitted to Panel, find the underlying mathematics behind the problems, and then find the right university mathematician to solve it. For this work, Rees was received the President's Certificate of Merit at the end of the war. She also received the King's Medal for Service in the Cause of Freedom from the British government. In 1946 the US Navy invited Rees to become Head of the mathematics branch of the Office of Naval Research to support scientific and mathematical research. In 1949 she became Director of the Mathematical Sciences Division and then, in 1952, Deputy Science Director. At the December 1953 meeting of the American Mathematical Society, Rees's achievements in these important roles was recognised with the following resolution (see Bull. Amer. Math. Soc. 60 (1954), 134):Under her guidance, basic research in general, and especially in mathematics, received the most intelligent and wholehearted support. No greater wisdom and foresight could have been displayed and the whole post-war development of mathematical research in the United States owes an immeasurable debt to the pioneer work of the Office of Naval Research and to the alert, vigorous and farsighted policy conducted by Miss Rees. In 1953 Rees returned to Hunter College that she had left on extended leave 10 years previously to undertake war work. She was appointed professor of mathematics and dean of the faculty, positions she held until 1961. However, during these eight years back at Hunter College she served on numerous committees bodies which included the National Research Council, the National Bureau of Standards and the National Science Foundation. Rees also acted as a consultant on the machine handling of data for the 1960 census. Rees left Hunter College in 1961, taking up the post of dean of graduate studies in the newly established City University of New York. Graduate studies at CUNY were very much directed by Rees during her 11 years there as she was appointed provost of the graduate division for 1968-1969 and then president of the Graduate School and University Center from 1969 until she retired in 1972. While dean of graduate studies at CUNY she wrote in 1965 (see [4]):It may be because the Graduate Dean is a woman, or it may be for completely objective reasons, that ours is proving an ideal university to draw into advanced graduate work the most obvious source of unused talent in a society that desperately needs additional numbers of persons with training through the doctorate, namely women. Rees received many awards for her outstanding contributions. In 1962 Rees received the first Award for Distinguished Service to Mathematics from the Mathematical Association of America:... for outstanding service to mathematics, other than mathematical research ... [and for] contributions [that] influence significantly the field of mathematics or mathematical education on a national scale. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rees.html (2 of 3) [2/16/2002 11:28:26 PM]

Rees

In 1970 she became president elect of the American Association for the Advancement of Science and in 1971 she became the first woman president of the Association. In 1983 Rees was awarded the National Academy of Sciences Public Welfare Medal:... in recognition of distinguished contributions in the application of science to the public welfare. In addition Rees was awarded honorary degrees from around twenty universities and colleges. In [4] Uta Merzbach describes Rees in these terms:Mina Rees was eminently rational. Her devotion to reason helped her formulate goals clearly and allocate resourses judiciously in accordance with these goals. ... Mina Rees was eminently intelligent. She comprehended quickly, communicated effectively, and thought creatively. Her ability to attach raelisable pieces of basic research to mission-oriented applications of mathematics did much to develop a broadened base of support for mathematicians' work. Mina Rees was eminently civilised. Her diplomatic skills were considerable; her conversational technique bespoke her broad knowledge base as well as her wide interest in mathematical and non-mathematical topics. Experience and reflection led her to a balanced outlook on teaching and research, the arts and sciences, long-range and short-range planning and obligations of the professional and the private life. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rees.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rees.html (3 of 3) [2/16/2002 11:28:26 PM]

Regiomontanus

Johann Müller Regiomontanus Born: 6 June 1436 in Königsberg, Archbishopric of Mainz (now Germany) Died: 8 July 1476 in Rome, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Regiomontanus was born Johann Müller of Königsberg but he used the Latin version of his name (Königsberg = "King's mountain"). Regiomontanus was a pupil of Peurbach. In 1461 he was appointed professor of astronomy at the University of Vienna, filling the chair vacated by Peurbach. In 1468 Regiomontanus was appointed astronomer to King Matthias Corvinus of Hungary. A translator and publisher, Regiomontanus made important contributions to trigonometry and astronomy. His book De triangulis omnimodis (1464) is a systematic account of methods for solving triangles. Regiomontanus built an observatory in Nuremberg in 1471 with funds from his patron and fellow scientist Bernard Walther. He also built a workshop to construct instruments at Nuremberg. He wrote Scipta giving details of his instruments. In January 1472 he made observations of a comet which were accurate enough to allow it to be identified with Halley's comet 210 years (and three returns of the 70 year period comet) later. He also observed several eclipses of the Moon, a total eclipse on 3 September 1457, a partial eclipse on 3 July 1460 and another total eclipse on 22 June 1461. Regiomontanus's interest in the motion of the Moon led him to make the important observation that the method of lunar distances could be used to determine longitude at sea. It was many years, however, before the position of the Moon could be predicted with sufficient accuracy to make the method practical http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Regiomontanus.html (1 of 2) [2/16/2002 11:28:28 PM]

Regiomontanus

and instruments constructed to give the lunar position with the high degree of accuracy necessary for the method. Regiomontanus describes how the position of the Moon can be used to determine longitude in the ephemerides for the years 1474-1506 which he published. In common with many others he wrote on calendar reform, for example he wrote Kalendarium and De Reformatione Kalendarii. Pope Sixus IV summoned Regiomontanus to Rome in 1475 to advise on calendar reform and to become bishop of Regensburg. However he died before he could take office. Some accounts say he was poisoned by his enemies, other accounts say he died of the plague. Article by: J J O'Connor and E F Robertson List of References (45 books/articles)

A Quotation

A Poster of Johann Regiomontanus

Mathematicians born in the same country

Cross-references to History Topics

1. Squaring the circle 2. Perfect numbers 3. The trigonometric functions

Other references in MacTutor

Chronology: 1300 to 1500

Honours awarded to Johann Regiomontanus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Regiomontanus

Other Web sites

1. Vatican exhibition 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Regiomontanus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Regiomontanus.html (2 of 2) [2/16/2002 11:28:28 PM]

Reichenbach

Hans Reichenbach Born: 26 Sept 1891 in Hamburg, Germany Died: 9 April 1953 in Los Angeles, California, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hans Reichenbach studied mathematics, physics and philosophy at Berlin, Munich and Göttingen. Planck, Sommerfeld, Hilbert and Cassirer were among his teachers. He contracted a severe illness on the Russian front during World War I. Only 5 people attended Einstein's first course on relativity and Reichenbach was one of them. Appointed to a chair in Berlin in 1926 Reichenbach founded the Berlin school of logical positivism. he When Hitler came to power in 1933 he fled to Turkey and taught at Istanbul from 1933 to 1938. He emigrated to the USA and worked at the University of California. Reichenbach wrote on quantum mechanics, time, induction, probability and the philosophy of science. Among his works are Elements of Symbolic Logic (1947) and The Rise of Scientific Philosophy (1951). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

Some Quotations (2)

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reichenbach.html (1 of 2) [2/16/2002 11:28:29 PM]

Reichenbach

Other Web sites

1. Internet Encyclopedia of Philosophy 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Reichenbach.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reichenbach.html (2 of 2) [2/16/2002 11:28:29 PM]

Reidemeister

Kurt Werner Friedrick Reidemeister Born: 13 Oct 1893 in Brunswick, Germany Died: 8 July 1971 in Göttingen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kurt Reidemeister was examined as a student by Landau and became an assistant of Hecke. His doctoral thesis was on algebraic number theory, the particular problem having been suggested by Hecke, and the resulting publication Relativklassenzahl gewisser relativ- quadratischer Zahlköper appeared in 1921. In [4] it is noted that:Of all of the 71 papers listed in Reidemeister's obituary by Artzy ([2]), this is the only one which deals with number theory. It rarely happens that a highly productive mathematician deserts the field of his PhD thesis so consistently later on. Immediately he had written his doctoral thesis, Reidemeister became interested in differential geometry. It was Blaschke who came up with the particular problems in differential geometry on which Reidemeister began to work. On Hahn's recommendation, Reidemeister was appointed as associate professor of geometry at the University of Vienna in 1923. Here he became a colleague of Wirtinger who interested Reidemeister in knot theory. In particular Wirtinger showed Reidemeister how to compute the fundamental group of a knot from its projection. This method, originally due to Wirtinger, appears in work of Artin which was published in 1925. While in Vienna, Reidemeister came across the Tractatus by Wittgenstein. Led by Reidemeister, the group of mathematicians at Vienna spent a year studying the deep ideas on logic and mathematics in this work.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reidemeister.html (1 of 3) [2/16/2002 11:28:31 PM]

Reidemeister

In 1927 Reidemeister was offered a chair in Königsberg which he accepted. In 1930 the German Mathematical Congress met in Königsberg and Reidemeister organised the first international conference on the philosophy of mathematics to be a part of the larger Congress. He was forced to leave his chair in Königsberg in 1933 by the Nazis, who he strongly opposed, who classed him as 'politically unsound'. After being temporarily suspended from his chair he was later appointed to Hensel's chair in what was considered a smaller and less prestigious university. Reidemeister worked on the foundations of geometry and he wrote an important book on knot theory Knoten und gruppen (1926). He established a geometry and topology based on group theory without the concept of a limit. In particular he wrote an important book Einführung in die kombinatorische Topologie (1932) on combinatorial topology. As is remarked in [4]:Although Reidemeister ... was, above all, a geometer, his book on 'combinatorial topology' contains hardly any drawings. Abstraction and rigor were very much in fashion. Reidemeister had an important influence on group theory, partly through his work on knots and groups, partly through his influence on Schreier. Talking of this influence on group theory, Chandler and Magnus write in [4]:Reidemeister was ... essentially a geometer. His influence on combinatorial group theory is largely that of a pioneer. His ideas were stimulating and had, at least in some cases, a long-lasting effect. Reidemeister's other interests included the philosophy and the foundations of mathematics as described above. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Kurt Reidemeister

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1920 to 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reidemeister.html (2 of 3) [2/16/2002 11:28:31 PM]

Reidemeister

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Reidemeister.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reidemeister.html (3 of 3) [2/16/2002 11:28:31 PM]

Reiner

Irving Reiner Born: 8 Feb 1924 in Brooklyn, New York, USA Died: 28 Oct 1986 in Urbana, Illinois, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Irving Reiner attended school in Brooklyn then, in 1940, he entered Brooklyn College. His first paper on formulas for primes was published while he was still an undergraduate. Reiner graduated in 1944 and became a graduate student at Cornell. He was awarded a Master's Degree for a thesis on binary quadratic forms in 1945 and his doctorate in 1949 for a thesis on a generalisation of Meyer's theorem. His doctoral supervisor was Burton Jones. After completing work on his doctoral thesis Reiner went to the Institute for Advanced Study at Princeton. After a year at Princeton he was appointed to the University of Illinois. Although he was to remain at the University of Illinois throughout his career, he held various visiting appointments in particular in London and at the University of Warwick. G J Janusz wrote in [1]:[Reiner lived] a life dedicated to mathematics. He gave encouragement to everyone in whom he saw some mathematical talent and, in return, he was stimulated by the success of other mathematicians with whom he had contact. ... He would often work in streaks, putting in long hours on many consecutive days and nights. In order to refresh himself after such periods of intensive work, he would like to relax by attending a concert. ... During the summers, the vacation of his choice would be a couple of weeks in the mountains. After his doctoral thesis Reiner worked on classical subgroups of GL(n,Z). He solved problems concerning minimal generating sets. He then found generators for the automorphism group of GL(n,Z) and of related groups. In 1955 he wrote his first paper on representations of groups. His most famous book, Representation theory of finite groups and associative algebras was published in 1962. It was written jointly with C Curtis who was based at the University of Wisconsin, Madison and Reiner claimed it was the only mathematics book written in a museum and a hotel lobby. Since Chicago was approximately half way between Madison and Urbana, Curtis and Reiner would make day trips there, about once a month, to discuss the progress of the book, meeting first at the Art Institute (which was between their train stations) where the day's work was planned during a stroll through the galleries. When it was time to work on their manuscript... they usually found a congenial place in some out-of-the-way corner of the lobby of the Palmer House to spread out their papers. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reiner.html (1 of 2) [2/16/2002 11:28:33 PM]

Reiner

Curtis and Reiner produced a completely rewritten book on representation theory in two volumes published in 1981 and 1987. Reiner wrote over 100 papers and books in a highly productive career. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to Irving Reiner (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Reiner

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Reiner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reiner.html (2 of 2) [2/16/2002 11:28:33 PM]

Remak

Robert Remak Born: 1888 in Germany Died: 1942 in Auschwitz, Poland

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Remak was the grandson of the first Jew in Prussia to be given an habilitation without giving up the Jewish faith. Also called Robert Remak, he was awarded his habilitation from the University of Berlin in 1847 and, with support from Alexander von Humboldt, he went on be made a professor in the medical faculty. His grandson Robert Remak the mathematician, studied at the University of Berlin and was supervised by Frobenius for his doctoral work. His doctoral dissertation Uber die Zerlegung der enlichen Gruppen in indirekte unzerlegbare Faktoren was submitted in 1911 with Schwarz as the other expert involved. This important work considered the decomposition of finite groups into a direct product of irreducible factors, a result that his name is often now attached to along with those of Wedderburn, Schmidt and Krull. Although his doctorate was awarded in 1911, it was a long and difficult road for Remak to be awarded his habilitation. He had submitted a thesis for his habilitation several times, and each time it had been rejected. However he persevered: as Schappacher writes in [2]:The Remak family had something of a tradition in slowly overcoming administrative hurdles at the University of Berlin. In 1929 Remak eventually received his habilitation and the right to teach at the university, but there was no post for him. He did lecture on groups at the University of Berlin during the summer semester 1929 and these lectures were attended by Bernhard Neumann. The same year 1929 was one in which Remak published an essay on applications of mathematics to economics. He had broad interests, working on mathematical economics as well as group theory and the geometry of numbers.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Remak.html (1 of 3) [2/16/2002 11:28:35 PM]

Remak

In his 1929 essay Can economics became an exact science Remak writes:I emphasise ... that I have not made any politico-economic statements, but only stated problems and indicated some calculational schemes, ... that it is still open as to if the outcome of the computation favours capitalism, socialism, or communism. ... these equations are very awkward to handle mathematically. There is, however, work in progress concerning the numerical solution of linear equations with several unknowns using electrical circuits. In this work Remak is far sighted in seeing the applications that computers would have in the subject, but these were areas into which is was unwise for someone like Remak to be venturing in Germany at this time. Remak made important contibutions to algebraic number theory. In a publication in 1932 he gave a lower bound for the regulator of the units of an algebraic number field which depends only on the number of real conjugates and the number of pairs of complex conjugates. He went on to produce further extensions of this work which continued to be published ten years after his death. Further papers by Remak on finite algebraic number fields with unit defect appeared in 1952 and 1954. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Under this law, Remak lost the right to teach at the university in September 1933. Remak did not leave Berlin at this time, however, and he continued to live in the city and continued with his mathematical research. He was particularly interested in the exciting new mathematical developments which were written up in van der Waerden's two volume Algebra published in 1930 which contained the new developments in ring theory by Emmy Noether, Hilbert, Dedekind and Artin. Remak was married to a German woman, who did satisfy the Aryan condition, and this certainly made him wish to continue living in Germany. He might also have expected that it would also give him a certain protection against the Nazi policies. On 10 July 1936 Schur wrote a report on Remak:I consider Dr Robert Remak to be an outstanding researcher, who is distinguished by his versatility, originality, strength and brilliance. ... He may, without doubt, be called a leading scholar in the splendid and important field of geometry of numbers. On the Kristallnacht (so called because of the broken glass in the streets on the following morning), the 9-10 November 1938, Remak was arrested. On that night 91 Jews were murdered, hundreds were seriously injured, and thousands were subjected to horrifying experiences. Thousands of Jewish businesses were burnt down together with over 150 synagogues. The Gestapo arrested 30,000 well-off Jews and a condition of their release was that they emigrate. Remak was put into the Sachsenhausen concentration camp near Berlin and his wife made strenuous attempts to obtain an affidavit which would allow them to emigrate to the United States. Having failed to obtain permission to emigrate to the United States, Remak was released after over eight weeks in the concentration camp after his wife organised that he go to Amsterdam. This Remak did in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Remak.html (2 of 3) [2/16/2002 11:28:35 PM]

Remak

April 1939 but his wife did not go to Amsterdam with him. Hans Freudenthal wrote to Hopf on 7 March 1940 describing the problems that Remak was causing:My main problem is Remak, who is not satisfied with visiting my lectures but who also gives us no end of trouble. It mostly concerns conflicts with his landlords, who then immediately run to the alien registration office. ... also Remak's expression "then I'd rather go to a concentration camp" has infuriated enough people already. The matter is extremely serious; it is doubtful how long we can still prevent him from being expelled to Germany. ... It is understandable that his wife didn't want to come here, but demonstrates a lack of loyalty to him. In fact Remak's wife divorced him which almost certainly made his position impossible. He was arrested by the German occupying forces in Amsterdam in 1942 and taken to the concentration camp in Auschwitz. He died there on some unknown day in 1942. Merzbach writes in [1] that Remak's:... refusal - in mathematics and everyday affairs - to compromise, or to be 'realistic', swept him out of the mainstream of mathematics and cost him his life. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Remak.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Remak.html (3 of 3) [2/16/2002 11:28:35 PM]

Remez

Evgeny Yakovlevich Remez Born: 17 Feb 1896 in Mstsislau, Mahiliou gubernia, Belarus Died: 31 Aug 1975 in Kiev, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Evgeny Remez was educated in Kiev University graduating in 1924. He taught at various institutions in Kiev for most of his life. He worked at the Pedagogical Institute from 1933 to 1955 and he became a professor at Kiev University in 1935. He also worked at the Institute of Mathematics of the Ukranian Academy of Sciences in Kiev from 1935 until his death in 1975. His main work was on the constructive theory of functions and approximation theory. In the mid 1930s, he developed general computational methods of Chebyshev approximation and the Remez algorithm which allows uniform approximation. It constructs, with prescribed degree of exactness, a polynomial of the best Chebyshev approximation for a given continuous function. A similar algorithm was later developed which allowed rational approximation of continuous functions defined on an interval. Remez generalised Chebyshev-Markov characterisation theory and used it to obtain approximate solutions of differential equations. He proved results about bounded polynomials and created general operator methods of sequence approximation. He also worked on approximate solutions of differential equations and the history of mathematics. Remez received many honours for his achievements including election to the Academy of Sciences of the Ukraine in 1939. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Remez.html (1 of 2) [2/16/2002 11:28:37 PM]

Remez

List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Remez.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Remez.html (2 of 2) [2/16/2002 11:28:37 PM]

Renyi

Alfréd Rényi Born: 30 March 1921 in Budapest, Hungary Died: 1 Feb 1970 in Budapest, Hungary

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alfréd Rényi received a literary, rather than scientific, schooling. In 1944 he was forced to a Fascist Labour Camp but somehow managed to escape. He obtained false papers and hid for six months avoiding capture. During this time his parents were held prisoners in the Budapest ghetto. Alfréd rescued them with an extreme act of bravery:Alfréd got hold of a soldier's uniform, walked into the ghetto, and marched his parents out. ... It requires familiarity with the circumstances to appreciate the skill and courage needed to perform these feats. At the end of World War II, Rényi obtained a Ph.D. at Szeged under F Riesz for work on Cauchy-Fourier series. He was taught by Fejér at Budapest, then he went to Russia and worked with Linnik on the theory of numbers, in particular working on the Goldbach conjecture. He discovered methods described by Turán as at present one of the strongest methods of analytical number theory. After returning to Hungary he worked on probability which was to be his main research topic throughout his life. He published joint work with Erdös on random graphs and also considered random space filling curves. Known by the nickname of Buba, he is best remembered for proving that every even integer is the sum of a prime and an almost prime number (one with only two prime factors), he is also remembered as the author of the anecdote a mathematician is a machine for converting coffee into theorems http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Renyi.html (1 of 2) [2/16/2002 11:28:39 PM]

Renyi

Turán developed the anecdote by describing weak coffee as fit only for lemmas. Rényi was the founder, and for 20 years the director, of the Mathematical Institute of the Hungarian Academy of Sciences. He was a famous raconteur remembered for many performances of his dialogue, which he spoke with his daughter, on the nature of mathematics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (16 books/articles)

A Quotation

Mathematicians born in the same country Other Web sites

Rényi's Parking Constants

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Renyi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Renyi.html (2 of 2) [2/16/2002 11:28:39 PM]

Rey_Pastor

Julio Rey Pastor Born: 16 Aug 1888 in Logrono, Spain Died: 21 Feb 1962 in Buenos Aires, Argentina

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Julio Rey Pastor studied science at the University of Zaragoza and published his first mathematics book at the age of 17. He had broad interests in his youth, however, and before he became completely occupied with mathematics, he had written poetry. He was appointed professor of mathematical analysis at Oviedo University in 1911. The university, in Oviedo in northern Spain, was an ancient university founded in 1608. Rey Pastor was accused of being unpatriotic in describing the deplorable state of Spanish science under the Hapsburgs. This happened as the result of his inaugural address given for the session 1913-14 on Spanish mathematicians of the 16th century. A series of visits by Rey Pastor to Germany resulted in two major publications on geometry in 1912 and 1916. The 1916 monograph was on synthetic geometry in n-dimensions and introduced [1]:... concepts of great generality (for example the definition of the curve) and developing them in all their consequences. Rey Pastor moved to a chair in Madrid in 1915. However he was not one to remain fixed in one place for a long time and went to Barcelona in 1915 to give a series of lectures at the Institut d'Estudia. His lectures there on n-dimensional geometry and conformal mappings, developing the work of Schwarz, was written up by Esteban Terrades who attended the lectures, and the course was published in Catalan. It was not only a Spanish visit to lecture that Rey Pastor made, however, for he made a visit to Argentina in 1917 and lectured the University of Buenos Aires. Although he was still a young man only 29 years old, he was asked to help promote mathematics in Argentina and a way was found to enable him to do http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rey_Pastor.html (1 of 2) [2/16/2002 11:28:41 PM]

Rey_Pastor

this. A contract was proposed which enabled him to spend six months each year in Argentina and six months in Spain. Rey Pastor was pleased to sign the contract:... to direct the advanced study of the exact sciences in Argentina. A famous monograph which Rey Pastor published in 1917 contained most of his own mathematical discoveries. The history of mathematics had always interested Rey Pastor and late in his career his interests in historical topics extended to cartography. Of course Spain has a reputation for remarkable cartography so his monograph (written jointly with E Garcia Camarero in 1960) on the history of Spanish cartography was a particularly useful addition to knowledge of the topic. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rey_Pastor.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rey_Pastor.html (2 of 2) [2/16/2002 11:28:41 PM]

Reye

Theodor Reye Born: 20 June 1838 in Ritzebüttel (near Cuxhaven), Germany Died: 2 July 1919 in Würzburg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Theodor Reye attended school in Hamburg. He then studied mechanical engineering and mathematical physics at Hannover, Zurich and Göttingen. He received a doctorate from Göttingen in 1861 for a thesis on gas dynamics. Reye presented his habilitation thesis to the University of Zurich and then became a lecturer there in 1863. He lectured there on mathematical physics until 1870. From 1870 to 1872 Reye worked at Aachen, then he was appointed to a chair of geometry at Strasbourg. After he retired in 1909 he remained at Strasbourg until the start of World War I when he moved to Würzburg. Reye's early interest in mathematical physics and meteorology turned to an interest in geometry even while he held the lectureship in mathematical physics at Zurich. The reason for his change of area was that he was led towards geometry by his work in mechanics. This led him to graphical statics and von Staudt's work on geometry. He published a two volume work on synthetics geometry Geometrie der Lage in 1866 and 1868. His work in geometry included a study of conics, quadrics and projective geometry. A configuration of 12 points, 12 planes and 16 lines which he published in 1878 is named after him. W Burau writes in [1]:Reye treated in detail the theory of conics and quadrics and of their linear systems... He was one of the leading geometers of his time, and he published a great deal on synthetic geometry. It is not unreasonable to ask why Reye's work is not better known today. Part of the reason must be that some of Reye's work was later interpreted in the geometry set up by Corrado Segre, in particular it was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reye.html (1 of 2) [2/16/2002 11:28:43 PM]

Reye

interpreted in the setting of Segre manifolds. Reye's work on linear manifolds of projective plane pencils and of bundles on spheres went into the Segre manifold setting. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Theodor Reye

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Reye.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reye.html (2 of 2) [2/16/2002 11:28:43 PM]

Reynaud

Antoine-André-Louis Reynaud Born: 12 Sept 1771 in Paris, France Died: 24 Feb 1844 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Antoine-André-Louis Reynaud's father was a lawyer in the parliament in Paris. Reynaud was attracted to literature when he was young and he wrote a number of plays. Reynaud lived through a time of great political upheaval in France since he was 17 years of age when the storming of the Bastille took place in 1789. He was attracted to the principles of the Revolution and, in 1790, he was made a Capitaine au Régiment d'Elèves. He wanted to make a career for himself in the military, and his joining the National Guard, a strongly republican branch of the military, in 1792 shows that at this stage he was on track with his aims. The direction of Reynaud's career was changed, however, through pressure by his family who persuaded him to give up his military aspirations in favour of a career as an accountant. For four years, until 1796, he worked as an accountant but his heart was never in the occupation and he would spend his evenings studying mathematics. In 1796 Reynaud gave up his accountant job and entered the Ecole Polytechnique in Paris. He studied mathematics there and graduated in 1796 as the top student of his year. As many other mathematicians of this period he was assigned to the Corps des Ponts et Chaussés but allowed to study mathematics for a third year at the Ecole Polytechnique. In 1800 a school was founded to prepare pupils for entry to the Ecole Polytechnique and Reynaud began teaching mathematics at the school, although he received no salary for this post. He also taught at a Lycée and, in 1804, he was appointed to a teaching post in the Ecole Polytechnique. Reynaud was to hold a variety of different posts. He headed the land survey of France in 1806, while he was appointed admissions examiner at the Ecole Polytechnique in 1809. Between 1808 and 1811 he assisted de Prony with the mechanics course and, from 1812 to 1814 he replaced Poinsot on the analysis course. Cauchy began to teach this course from 1815. Although strongly republican in views in his youth, Reynaud became more liberal in his views and from 1814 he supported Louis XVIII. Reynaud published a number of extremely influential textbooks. He published a mathematics manual for surveyors as well as Traité d'algèbre, Trigonométrie rectiligne et sphérique, Théorèmes et problèmes de

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reynaud.html (1 of 2) [2/16/2002 11:28:45 PM]

Reynaud

géométrie and Traité de statistique. His best known texts, however, were his editions of Bezout's Traité d'arithmetique which appeared in at least 26 versions containing much original work by Reynaud. It appears that Reynaud became interested in algorithms when he was working with de Prony. At this time de Prony was very much involved in trying to get his logarithmic and trigonometric tables published and it seems to have made Reynaud think about analysing algorithms. Certainly Reynaud, although his results in this area were rather trivial, must get the credit for being one of the first people to give an explicit analysis of an algorithm, an area of mathematics which is of major importance today. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Reynaud.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reynaud.html (2 of 2) [2/16/2002 11:28:45 PM]

Reyneau

Charles René Reyneau Born: 11 June 1656 in Brissac, Maine-et-Loire, France Died: 24 Feb 1728 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles René Reyneau's father was a surgeon. He studied at the Oratorian College in Angers. In October 1676 he entered the Maison d'Institution in Paris where he met Malebranche. In 1679 he moved to the Collège de Toulon and, in 1681 he was ordained a priest there. In 1682 Reyneau was appointed professor of mathematics at the University of Angers. However, in 1705, he had to give up teaching as be became deaf. He had been going deaf for a number of years but he managed to keep his post by having former students give his lectures for him. After giving up the struggle to continue his job in these difficult circumstances, Reyneau went to Paris and lived at the Oratorian house there for the rest of his life. This was a time when there were major new mathematical ideas coming through the work of Johann Bernoulli and being brought into France through de L'Hôpital and others. For many years Reyneau was not really abreast of these new developments, even when Johann Bernoulli visited Paris in 1692 and Reyneau did not rush to keep up to date with the important new ideas. Malebranche asked Reyneau to undertake some editorial duties in 1694 but then, in 1698, he persuaded Reyneau to write a new textbook to provide instruction in the new mathematics. Reyneau struggled to assimilate the differential and integral calculus participating in debates provoked by Rolle on these topics. He worked with other mathematicians but, mainly due to his having to learn revolutionary new ideas as he went along, the book took a long time to complete. The two volume work Analyse démontrée was published in 1708 and a second enlarged edition was produced which was the text from which d'Alembert learnt mathematics. In 1705 Reyneau obtained a copy of Leçons prepared by Johann Bernoulli for de L'Hôpital. Reyneau lent some documents to Montmort who lost them. Reyneau wrote a second work in 1714 which [1]:... attempted to preserve the central conceptions of the Oratorian mathematics of the end of the preceding century, [but] was less successful than the first. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reyneau.html (1 of 2) [2/16/2002 11:28:46 PM]

Reyneau

Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Reyneau.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reyneau.html (2 of 2) [2/16/2002 11:28:46 PM]

Reynolds

Osborne Reynolds Born: 23 Aug 1842 in Belfast, Ireland Died: 21 Feb 1912 in Watchet, Somerset, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Osborne Reynolds's father was a priest in the Anglican church but he had an academic background having graduated from Cambridge in 1837, being elected to a fellowship at Queens' College, and being headmaster of first Belfast Collegiate School and then Dedham School in Essex. In fact it was a family with a tradition of the Church and three generations of Osborne's father's family had been the rector of Debach-with-Boulge. Osborne was born in Belfast when his father was Principal of the Collegiate School there but began his schooling at Dedham when his father was headmaster of the school in that Essex town. After that he received private tutoring to complete his secondary education. He did not go straight to university after his secondary education, however, but rather he took an apprenticeship with the engineering firm of Edward Hayes in 1861. Reynolds wrote (actually in his application for the chair in Manchester in 1868) of his father's influence on him while he was growing up (see for example Lamb's article [7]):In my boyhood I had the advantage of the constant guidance of my father, also a lover of mechanics, and a man of no mean attainments in mathematics and its application to physics. Reynolds, after gaining experience in the engineering firm, studied mathematics at Cambridge, graduating in 1867. As an undergraduate Reynolds had attended some of the same classes as Rayleigh who was one year ahead of him. As his father had before him, Reynolds was elected to a scholarship at Queens' College. He again took up a post with an engineering firm, this time the civil engineers John Lawson of London, spending a year as a practicing civil engineer. In 1868 Reynolds became the first professor of engineering in Manchester (and the second in England). Kargon writes in [1]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reynolds.html (1 of 3) [2/16/2002 11:28:48 PM]

Reynolds

... a newly created professorship of engineering was advertised at Owens College, Manchester, at £500 per annum. Reynolds applied for the position and, despite his youth and inexperience, was awarded the post. We should note in passing that Owens College would later become the University of Manchester. In his application for the post Reynolds wrote (see for example [7]):From my earliest recollection I have had an irresistible liking for mechanics and the physical laws on which mechanics as a science is based. Reynolds held this post until he retired in 1905. His early work was on magnetism and electricity but he soon concentrated on hydraulics and hydrodynamics. He also worked on electromagnetic properties of the sun and of comets, and considered tidal motions in rivers. After 1873 Reynolds concentrated mainly on fluid dynamics and it was in this area that his contributions were of world leading importance. We summarise these contributions. He studied the change in a flow along a pipe when it goes from laminar flow to turbulent flow. In 1886 he formulated a theory of lubrication. Three years later he produced an important theoretical model for turbulent flow and it has become the standard mathematical framework used in the study of turbulence. The author of [2] notes that:His studies of condensation and heat transfer between solids and fluids brought radical revision in boiler and condenser design, while his work on turbine pumps permitted their rapid development. An account of Reynolds' work on hydrodynamic stability published in 1883 and 1895 is looked at in [8]. The 1883 paper is called An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels. The 'Reynolds number' (as it is now called) used in modelling fluid flow which is named after him appears in this work. Reynolds became a Fellow of the Royal Society in 1877 and, 11 years later, won their Royal Medal. In 1884 he was awarded an honorary degree by the University of Glasgow. By the beginning of the 1900s Reynolds health began to fail and he retired in 1905. Not only did he deteriorate physically but also mentally, which was sad to see in so brilliant a man who was hardly 60 years old. Not only is Reynolds important in terms of his research, but he is also important for the applied mathematics course he set up at Manchester. Anderson writes in [3]:Reynolds was a scholarly man with high standards. Engineering education was new to English universities at that time, and Reynolds had definite ideas about its proper form. He believed that all engineering students, no matter what their speciality, should have a common background based in mathematics, physics, and particularly the fundamentals of classical mechanics. ... Despite his intense interest in education, he was not a great lecturer. His lectures were difficult to follow, and he frequently wandered among topics with little or no connection. In [7] Lamb, who knew Reynolds well both as a man and as a fellow worker in fluid dynamics, wrote:The character of Reynolds was like his writings, strongly individual. He was conscious of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reynolds.html (2 of 3) [2/16/2002 11:28:48 PM]

Reynolds

the value of his work, but was content to leave it to the mature judgement of the scientific world. For advertisement he had no taste, and undue pretension on the part of others only elicited a tolerant smile. To his pupils he was most generous in the opportunities for valuable work which he put in their way, and in the share of cooperation. Somewhat reserved in serious or personal matters and occasionally combative and tenacious in debate, he was in the ordinary relations of life the most kindly and genial of companions. Article by: J J O'Connor and E F Robertson List of References (8 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1880 to 1890

Honours awarded to Osborne Reynolds (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1877

Royal Society Royal Medal

Awarded 1888

Royal Society Bakerian lecturer

1897

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Reynolds.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Reynolds.html (3 of 3) [2/16/2002 11:28:48 PM]

Rheticus

Georg Joachim von Lauchen Rheticus Born: 16 Feb 1514 in Feldkirch, Austria Died: 4 Dec 1574 in Kassa, Hungary (now Kosice) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Joachim von Lauchen Rheticus's father, Georg Iserin, was the town doctor in Feldkirch and also a government official. Rheticus was, therefore, born Georg Joachim Iserin. His mother, Thomasina de Porris, was Italian. He was taught by his father for the first 14 years of his life but, in 1528, his father was tried on a charge of sorcery, convicted and beheaded. One of the legal requirements of such an execution was that his name could no longer be used, so Rheticus's mother reverted to her maiden name and Rheticus's became Georg Joachim de Porris. 'De porris' means 'of the leeks' in Italian and, since Rheticus did not consider himself Italian he translated it into German 'von Lauchen' and called himself Georg Joachim von Lauchen. He later took the additional name of Rheticus after the Roman province of Rhaetia in which he had been born. Achilles Gasser took over the medical practice in Feldkirch after Rheticus's father was executed. He helped Rheticus continue his studies and was a strong support to him. After his father's execution, Rheticus studied at the Latin school in Feldkirch, then went to Zurich where he studied at the Frauenmuensterschule from 1528 to 1531. In 1533 he entered the University of Wittenberg receiving his M.A. from that university three years later on 27 April 1536. Philipp Melanchthon, Martin Luther's "right hand man", was a theologian, and educator who reorganised the whole educational system of Germany, founding and reforming several of its universities. Melanchthon played a major role in getting Rheticus an appointment to teach mathematics and astronomy at the University of Wittenberg in 1536. This appointment, which involved teaching arithmetic and geometry, gave Rheticus a salary of 100 gulden. Two years later Melanchthon again used his influence to arrange leave for Rheticus to study with some of the leading astronomers of the day, but his main reason was to visit Copernicus. Leaving Wittenberg in October 1538 he travelled to Nuremberg and there visited Johann Schöner who was publishing books, including those that Regiomontanus had intended to publish 60 years earlier. In Nuremberg Rheticus also visited the printer Petreius. He then visited Peter Apianus in Ingolstadt, next Joachim Camerarius in Tübingen and then he fitted in a visit to his home town of Feldkirch to visit Achilles Gasser whom he presented with a copy of Sacrobosco. In May 1539 Rheticus arrived at Frauenberg in Ermland where he spent about two years with Copernicus. Rheticus wrote ([7], [13]):http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rheticus.html (1 of 4) [2/16/2002 11:28:50 PM]

Rheticus

I heard of the fame of Master Nicolaus Copernicus in the northern lands, and although the University of Wittenberg had made me a Public Professor in those arts, nonetheless, I did not think that I should be content until I had learned something more through the instruction of that man. And I also say that I regret neither the financial expenses nor the long journey nor the remaining hardships. Yet, it seems to me that there came a great reward for these troubles, namely. that I, a rather daring young man compelled this venerable man to share his ideas sooner in this discipline with the whole world. In September 1539 Rheticus went to Danzig, visiting the mayor of Danzig, who gave Rheticus some financial assistance to help publish the Narratio Prima or, to give it its full title First report to Johann Schöner on the Books of the Revolutions of the learned gentleman and distinguished mathematician, the Reverend Doctor Nicolaus Copernicus of Torun, Canon of Warmia, by a certain youth devoted to mathematics. Swerdlow writes in [12]:Copernicus could not have asked for a more erudite, elegant, and enthusiastic introduction of his new astronomy to the world of good letters; indeed to this day the Narratio Prima remains the best introduction to Copernicus's work. Of course Rheticus sent a copy to Schöner and also to Petreius, who found it splendid. In August 1541 Rheticus presented a copy of his work on a map of Prussia to Duke Albert of Prussia and the following day he sent him an instrument he had made to determine the length of the day. Rheticus knew that Duke Albert had tried unsuccessfully to find out how to compute the time of sunrise. Having put himself in good favour with the Duke he asked for the favour he wanted: would the Duke permit the publication of Copernicus's De Revolutionibus. Duke Albert replied quickly giving permission for the publication and, at the same time requesting that Rheticus retain his chair. In October 1541 Rheticus returned to the University of Wittenberg and there he was elected dean of the Faculty of Arts. In early 1541 Rheticus published the trigonometrical sections of Copernicus's De Revolutionibus adding tables of his own giving tables of sines and cosines (although he did not call them by these names). This was the first published table of cosines and, see [1], Rheticus's:... place in the history of mathematics is due precisely to his computation of innovative and monumental trigonometrical tables. Joachim Camerarius, who was head of the University of Tübingen, working with Melanchthon, arranged for Rheticus to be offered a post at the University of Leipzig. In 1542 Rheticus was appointed professor of higher mathematics at Leipzig. Initially he was offered the same salary as he had received from the University of Wittenberg but he soon negotiated a 40% rise. He left Wittenberg in May 1542, travelling to Nuremberg where he supervised the printing of De Revolutionibus but before the work was finished he had to go to Leipzig to begin teaching in October 1542. Rheticus remained at Leipzig until 1545 when he again arranged leave to allow him to study abroad. After initially returning to his home town of Feldkirch he spent some time in Italy, where he visited Cardan in Milan. Rheticus continued on his travels until, at Lindau, a city in Bavaria on an island in Lake Constance his health broke down and he had severe mental problems during the first half of 1547. His health recovered sufficiently to allow him to teach mathematics at Constance for three months in late 1547 then he studied medicine in Zurich before he returned to Leipzig in February 1548. With Melanchthon's influence Rheticus was made a member of the theological faculty at Leipzig. A man of many talents, Rheticus published a calendar and ephemeris of 1550 and also an ephemeris and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rheticus.html (2 of 4) [2/16/2002 11:28:50 PM]

Rheticus

calendar of 1551. However a scandal forced him to leave Leipzig in April 1551; he was accused of having a homosexual affair with one of his students. He had to flee and this he did rapidly, spending some time at Chemnitz and a further period in Prague. He was tried in his absence and his friends, such as Melanchthon, stopped supporting him: they probably had little option if they were to retain their own positions. Although he was not present to defend himself, Rheticus was sentenced to 101 years in exile. In 1551-52 he studied medicine at the University of Prague but his interest in medicine only ever seemed to be used to treat patients and never to undertake scholarly research so he never seems to have produced innovations in medicine in the way he did in mathematics. In 1553 he was offered an appointment as professor of mathematics at Vienna. He went to Vienna but never took up the appointment. He moved to Krakóv in 1554 where he remained for 20 years as a practising doctor. He certainly did not give up his mathematical interest while in Krakóv, for he worked on his famous trigonometric tables as well as making instruments, carrying out astronomical observations and alchemy experiments. In fact he did rather well for himself at this stage employing six research assistants and he was funded by Emperor Maximilian II for his work on trigonometric tables which went a long way to providing rather good salaries for his assistants. Rheticus's important work on trigonometry Opus Palatinum de triangulis uses all six trigonometric functions. He gave tables of all these six functions in this major work which was completed and published in 1596 by Valentine Otho many years after Rheticus's death. Otho had studied at Wittenberg and then set out to visit Rheticus in a similar way to that in which Rheticus himself had visited Copernicus. Otho writes (see [13]):We had hardly exchanged a few words on this and that when, on learning the cause of my visit, he burst forth with these words: "You come to see me at the same age as I was myself when I visited Copernicus. If I had not visited him, none of his works would have seen the light." Other works by Rheticus include ones on map making (he published a map of Prussia), and works on navigational instruments, Chorographia tewsch. He designed many instruments such as sea compasses and the instrument to show the length of the day throughout the year which he gave to Duke Albert as we mentioned above. Given the dramatic and eventful life that Rheticus led it is interesting to think about his personality. Westman writes in [13]:If ... one were to point to the single most prominent trait in Rheticus's personality, based upon the tone of his writings, the testimonies of his contemporaries, and his own life activities, one would have to seize upon his great energy and intensity - whether in the vitality of his work, in his widespread travels, or in his evident pursuit to lay to rest something inside himself. Article by: J J O'Connor and E F Robertson List of References (13 books/articles)

A Quotation

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rheticus.html (3 of 4) [2/16/2002 11:28:50 PM]

Rheticus

Mathematicians born in the same country Cross-references to History Topics

The trigonometric functions

Other references in MacTutor

Chronology: 1500 to 1600

Honours awarded to Georg Joachim Rheticus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Rhaeticus

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rheticus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rheticus.html (4 of 4) [2/16/2002 11:28:50 PM]

Riccati

Jacopo Francesco Riccati Born: 28 May 1676 in Venice, Venetian Republic (now Italy) Died: 15 April 1754 in Treviso, Venetian Republic (now Italy)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Originally entering Padua to read law, Jacopo Riccati became friends with Angeli who encouraged him to study mathematics. He soon attained fame and turned down offers from Peter the Great to become President of the St Petersburg Acanemy of Science, and other offers, to remain in Italy. His work in hydraulics was useful to the city of Venice and he helped construct dikes along the canals. In the study of differential equations his methods of lowering the order of an equation and separating variables were important. He considered many general classes of differential equations and found methods of solution which were widely adopted. He is chiefly known for the Riccati differential equation of which he made elaborate study and gave solutions for certain special cases. The equation had already been studied by Jacob Bernoulli, and was discussed by Riccati in a paper of 1724. He corresponded with a large number of mathematicians throughout Europe and had a wide influence on Daniel Bernoulli, Euler. He also worked on cycloidal pendulums, the laws of resistance in a fluid and differential geometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riccati.html (1 of 2) [2/16/2002 11:28:52 PM]

Riccati

A Poster of Jacopo Riccati

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1720 to 1740

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Riccati.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riccati.html (2 of 2) [2/16/2002 11:28:52 PM]

Riccati_Vincenzo

Vincenzo Riccati Born: 11 Jan 1707 in Castelfranco Veneto (near Treviso), Italy Died: 17 Jan 1775 in Treviso, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vincenzo Riccati was the second son of Jacopo Riccati and his early education was at home and from the Jesuits. He entered the Jesuit order in 1726 going to the Jesuit College in Piacenza in 1728 to teach literature. In 1729 he moved to Padua, then to Parma in 1734, studied theology in Rome for a while, then returned to Bologna in 1739 where he taught mathematics in the College of San Francesco Saverio for 30 years. Vincenzo continued his father's work on integration and differential equations. He was skilled in hydraulic engineering and carried out flood control projects which saved the Venetian and Bolognian region from flooding. Vincenzo studied hyperbolic functions and used them to obtain solutions of cubics. He found the standard addition formulas for hyperbolic functions, their derivatives and their relation to the exponential function. Lambert is often cited as the first to introduce the hyperbolic fuctions but he did not do so until 1770 while Vincenzo Riccati's work (some joint with Saladini) was published between 1757 and 1767. Vincenzo Riccati and Saladini worked on the 'rose curves' introduced by Grandi. They also worked on Alhazen (al-Haytham)'s problem:Given points A and B find the point C on a circular mirror so that a ray of light from A is reflected at C to B. Al-Haytham had a less than satisfactory solution, Huygens found a good solution which Vincenzo Riccati and Saladini simplified and improved.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riccati_Vincenzo.html (1 of 2) [2/16/2002 11:28:54 PM]

Riccati_Vincenzo

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Cross-references to Famous Curves

Rhodenea curves

Other Web sites

Jesuit Scientists

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Riccati_Vincenzo.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riccati_Vincenzo.html (2 of 2) [2/16/2002 11:28:54 PM]

Ricci-Curbastro

Gregorio Ricci-Curbastro Born: 12 Jan 1853 in Lugo, Papal States (now Italy) Died: 6 Aug 1925 in Bologna, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gregorio Ricci-Curbastro's father was Antonio Ricci-Curbastro and his mother was Livia Vecchi. It was a family of high status known throughout the province of Ravenna. Antonio Ricci-Curbastro, although certainly never achieving anything close to the fame achieved by his son Gregorio, nevertheless was himself well known as an engineer. Neither Gregorio nor his brother Domenico attended school. All their education prior to entering university was carried out at home where their parents employed private tutors. In 1869 Ricci-Curbastro entered the University of Rome with the intention of studying mathematics and philosophy. He was only sixteen years old at the time and, although he had not attended school, he was well prepared academically. Political events, however, conspired to make Rome a somewhat unfortunate choice, although a very natural one given his place of birth. When Ricci-Curbastro began his studies in Rome, although the Kingdom of Italy had been created a few years earlier, Rome was not part of that Kingdom being part of the Papal States in which Ricci was born and brought up. Rome had been attacked by Italian troops in 1867 but France had defended the city and employed its troops against the attack. In 1870, however, Italian troops captured Rome and it became the capital of the Kingdom of Italy. Ricci-Curbastro studied at Rome for one year from 1869 to 1870 and then returned to his parents home where he remained for two years before beginning a second university career. This time he went, not to Rome but to the University of Bologna. He studied there during the years 1872-73, then moved to Pisa where he attended the Scuola Normale Superiore which, under Betti's leadership, was becoming the leading Italian centre for mathematical research and mathematical education. As well as attending lectures by Betti in Pisa, Ricci-Curbastro also attended lectures by Dini.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ricci-Curbastro.html (1 of 4) [2/16/2002 11:28:56 PM]

Ricci-Curbastro

In 1875 Ricci-Curbastro was awarded a doctorate for his thesis On Fuchs's research concerning linear differential equations. He remained at Pisa working on a paper which he presented the following year to fulfil the requirements necessary to teach. The paper was On a generalisation of Riemann's problem concerning hypergeometric functions. Neither this paper, nor his doctoral thesis, have been published. A perceptive reader will have noticed that both of these first two works by Ricci-Curbastro were based on works by German, rather than by Italian, mathematicians. The next pieces of work which he undertook were, likewise, not based on ideas by Italian mathematicians. The first was a series of articles on Maxwell's theory of electrodynamics and the work of Clausius which Betti asked him to write. Three of these articles appeared in Nuovo Cimento in 1877 and, in the same year, an article appeared in Giornale di matematiche di Battaglini which Dini had asked him to write on Lagrange's problem on a system of linear differential equations. Ricci-Curbastro now competed for a scholarship and he won one which allowed him to spend the year 1877-78 abroad. That he chose to go to Germany should be no surprise and in fact he chose to study at the Technische Hochschule in Munich where Klein had been appointed to the chair two years earlier. As well as Klein, Brill worked at the Technische Hochschule in Munich and Ricci-Curbastro attended lectures by both these famous mathematicians. As Speziali writes in [1]:Ricci greatly admired Klein, and his esteem was soon reciprocated; nevertheless, Ricci does not seem to have been decisively influenced by Klein's teaching. It was, rather, Riemann, Christoffel, and Lipschitz who inspired his future research. Indeed, their influence on him was even greater than that of his Italian teachers. Returning to Pisa in 1879, Ricci-Curbastro became Dini's assistant. Then, from 1880 until his death in 1925 he was professor of mathematical physics at the University of Padua. He did not only teach mathematical physics, however, for from 1891 he also taught courses on advanced algebra at Padua. It was only after he was appointed to the chair at Padua that he had the security that would allow him to marry and, in 1884, he married Bianca Bianchi Azzarani. They had three children; two sons and a daughter. Ricci-Curbastro's early work was in mathematical physics, particularly on the laws of electric circuits and differential equations. He changed area somewhat to undertake research in differential geometry and was the inventor of the absolute differential calculus between 1884 and 1894. The initial contributions had been made by Gauss, then the ideas had been developed in Riemann's 1854 Probevorlesung and in an 1861 paper which he wrote for a prize contest of the Paris Académie des Sciences. However, it was a paper of Christoffel, published in Crelle's Journal in 1868, which was the main influence on Ricci-Curbastro to begin his investigations in 1884 on quadratic differential forms. He first systematically presented the important ideas in 1888 in a paper written for the 800th anniversary of the University of Bologna. Speziali writes in [1]:The method he used to demonstrate [the invariance of the quadratics] led him to the technique of absolute differential calculus, which he discussed in its entirety in four publications written between 1888 and 1892. Much of Ricci-Curbastro's work after 1900 was done jointly with his student Levi-Civita. In a fundamental joint paper that year Méthodes de calcul différentiel absolu et leurs applications he used (for the only time) the name Ricci instead of his full name. This paper had been requested five years http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ricci-Curbastro.html (2 of 4) [2/16/2002 11:28:56 PM]

Ricci-Curbastro

earlier by Klein. The authors state their aims in the preface to their important seventy-seven page paper:The algorithm of absolute differential calculus, the instrument matériel of the methods ... can be found complete in a remark due to Christoffel. But the methods themselves and the advantages they offer have their raison d'être and their source in the intimate relationships that join them to the notion of an n-dimensional variety, which we owe to the brilliant minds of Gauss and Riemann. ... Being thus associated in an essential way with Vn, it is the natural instrument of all those studies that have as their subject, such a variety, or in which one encounters as a characteristic element a positive quadratic form of the differentials of n variables or of their derivatives. In the paper, applications are given by Ricci-Curbastro and Levi-Civita to the classification of the quadratic forms of differentials and there are other analytic applications; they give applications to geometry including the theory of surfaces and groups of motions; and mechanical applications including dynamics and solutions to Lagrange's equations. The main ideas of this paper are discussed in [1]. Ricci-Curbastro's absolute differential calculus became the foundation of tensor analysis and was used by Einstein in his theory of general relativity. The paper [7], written by Ricci-Curbastro's student Levi-Civita, lists sixty-one of his publications. However, he found time to also contribute to local government as did many of the Italian mathematicians of his time. He served as a councillor for his home town of Lugo and in this capacity was involved in many projects relating to the supply of water and to swamp drainage (an activity which many Italian mathematicians became involved with over several centuries). Later he served as a councillor for Padua and there his interests included school education and finance. Offered the position of mayor of Padua, however, he declined. Ricci-Curbastro received many honours for his outstanding contributions, although one would have to say that the importance of his work was not fully understood at the time when he produced it, but rather it was realised some time later. He was honoured with membership of several academies such as the Istituto Veneto which he was admitted to in 1892 and which he served as president in 1916-18. He also was a member of the Reale Accademia dei Lincei from 1899, the Accademia dei Padua from 1905, the Reale Accademia dei Turin from 1918, the Società dei Quaranta from 1921, the Reale Accademia dei Bologna from 1922 and the Accademia Pontifica from 1925. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. The quantum age begins 2. General relativity

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ricci-Curbastro.html (3 of 4) [2/16/2002 11:28:56 PM]

Ricci-Curbastro

Other references in MacTutor

1. Chronology: 1880 to 1890 2. Chronology: 1900 to 1910

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ricci-Curbastro.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ricci-Curbastro.html (4 of 4) [2/16/2002 11:28:56 PM]

Ricci

Michelangelo Ricci Born: 30 Jan 1619 in Rome, Italy Died: 12 May 1682 in Rome, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Michelangelo Ricci was a friend of Torricelli; in fact both were taught by Benedetti Castelli. He studied theology and law in Rome and at this time he became friends with René de Sluze. It is clear that Sluze, Torricelli and Ricci had a considerable influence on each other in the mathematics which they studied. Ricci made his career in the Church. His income came from the Church, certainly from 1650 he received such funds, but perhaps surprisingly he was never ordained. Ricci served the Pope in several different roles before being made a cardinal by Pope Innocent XI in 1681. Ricci's main work was Exercitatio geometrica, De maximis et minimis (1666) which was later reprinted as an appendix to Nicolaus Mercator's Logarithmo-technia (1668). It only consisted of 19 pages, remarkable that his high reputation rests solely on such a short publication. In this work Ricci finds the maximum of xm(a - x)n and the tangents to ym = kxn. The methods are early examples of induction. He also studied spirals (1644), generalised cycloids (1674) and states explicitly that finding tangents and finding areas are inverse operations (1668). In his own time Ricci's fame as a mathematician rested more on the many letters he wrote on mathematical topics, rather than on his published work. He corresponded with many mathematicians across Europe including Clavius, Viviani and de Sluze. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ricci.html (1 of 2) [2/16/2002 11:28:57 PM]

Ricci

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ricci.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ricci.html (2 of 2) [2/16/2002 11:28:57 PM]

Ricci_Matteo

Matteo Ricci Born: 6 Oct 1552 in Macerata, Papal States (now Italy) Died: 11 May 1610 in Peking, China

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

After being educated at home by his parents, Matteo Ricci entered the Jesuit School in Macerata in 1561. He went to Rome in 1568 to study law but he was attracted to the Jesuit religious order which he joined in 1571. He then continued his studies in Rome, studying mathematics and astronomy under Clavius. Ricci set out on his sea voyages in 1577. He arrived first in Portugal where he studied at the University of Coimbra for a while. Then, in 1578, he sailed to the Portuguese city of Goa on the west coast of India. In Goa Ricci studied for the priesthood, and he was ordained in 1580. Two years later he sailed to China. Ricci arrived at Macau on the east coast of China in 1582. He settled in Chao-ch'ing, Kwangtung Province and began his study of Chinese. He also worked at acquiring understanding of Chinese culture. While there Ricci produced the first edition of his map of the world Great Map of Ten Thousand Countries which is a remarkable achievement showing China's geographical position in the world. In 1589 Ricci moved to Shao-chou and began to teach Chinese scholars the mathematical ideas that he had learnt from his teacher Clavius. This is perhaps the first time that European mathematics and Chinese mathematics had interacted and it must be seen as an important event. Ricci attempted to visit Peking in 1595 but found the city closed to foreigners. He went instead to Nanking where he lived from 1599, working on mathematics, astronomy and geography. Ricci was well received in Nanking and this encouraged him to try again to visit Peking which he did in 1601. This time he was allowed to live in the city and he made this his home from that time until his death nine years later. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ricci_Matteo.html (1 of 3) [2/16/2002 11:28:59 PM]

Ricci_Matteo

There was at that time a problem with the European's understanding of whether the country which Marco Polo had visited by an overland route, and called Cathay, was the same country as China which had been visited by sea. Marco Polo, also an Italian, had travelled from Europe through Asia beginning his journey in 1271 and living in Cathay for 17 years before returning to Italy. Ricci was convinced that these countries were the same but, until another overland journey was made, this could not be confirmed. Ricci's hypothesis was proved by another Jesuit by the name of De Goes, who set out from India in 1602, and although he died in 1607 before reaching Peking, he had by that time made contact by letter with Ricci and proved that Marco Polo's Cathay was China. By the time he was living in Peking, Ricci's skill at Chinese was sufficient to allow him to publish several books in Chinese. He wrote The Secure Treatise on God (1603), The Twenty-five Words (1605), The First Six Books of Euclid (1607), and The Ten Paradoxes (1608). The First Six Books of Euclid was based on Clavius's Latin version of Euclid's Elements which Ricci had studied under Clavius's guidance while in Rome. The Chinese reaction to Ricci's book, which showed them the logical construction in Euclid's Elements for the first time, is discussed in [10]. Certainly the style of Euclid was far from the style of Chinese mathematics and this mixing of mathematical cultures must have been a cultural shock to both sides. Ricci of course had to dress in the style of a Chinese scholar and be known under a Chinese name, he used 'Li Matou', to become accepted by the Chinese. However he became famous in China for more than his mathematical skills, becoming known for his extraordinary memory and for his knowledge of astronomy. He even became known as a painter and a painting of a landscape around Peking has recently been attributed to him. Article by: J J O'Connor and E F Robertson List of References (10 books/articles) Mathematicians born in the same country Honours awarded to Matteo Ricci (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Riccius

Other Web sites

1. Jesuit Scientists 2. The Catholic Encyclopedia 3. The Galileo Project 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ricci_Matteo.html (2 of 3) [2/16/2002 11:28:59 PM]

Ricci_Matteo

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ricci_Matteo.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ricci_Matteo.html (3 of 3) [2/16/2002 11:28:59 PM]

Richard_Jules

Jules Antoine Richard Born: 12 Aug 1862 in Blet, Cher, France Died: 14 Oct 1956 in Châteauroux, Indre, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jules Richard taught at Lycées in Tours, Dijon and Châteauroux. He obtained a doctorate on Fresnel waves from Paris in 1901. Jules Richard wrote on the philosophy of mathematics in 1903 and 1908 wrote Sur la nature des axiomes de la géométrie in which he looked critically at four different approaches to geometry:i) Geometry is founded on arbitrarily chosen axioms - there are infinitely many equally true geometries. ii) Experience provides the axioms of geometry, the basis is experimental, the development deductive. iii) The axioms of geometry are definitions. iv) Axioms are neither experimental nor arbitrary, they force themselves on us. Jules Richard is remembered for Richard's paradox involving the set of real numbers which can be defined in a finite number of words. The paradox first appeared in a letter from Richard to Louis Olivier. Article by: J J O'Connor and E F Robertson A Reference (One book/article) A Poster of Jules Richard

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richard_Jules.html (1 of 2) [2/16/2002 11:29:01 PM]

Richard_Jules

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Richard_Jules.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richard_Jules.html (2 of 2) [2/16/2002 11:29:01 PM]

Richard_Louis

Louis Paul Émile Richard Born: 31 March 1795 in Rennes, France Died: 11 March 1849 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Louis Richard was left with a physical impediment as a result of a childhood accident so he was unable to persue a military career as he had wished. He taught at several lycées, then in 1815 he was appointed professor at College de Pontivy. He held posts at the Collège Saint-Louis and then, from 1820, at Collège Louis-le-Grand where he attained his greatest fame as the teacher of Galois. Richard was appointed to a special chair at Collège Louis-le-Grand in 1822 and held this post until his death. Richard also taught Le Verrier, Serret and Hermite. He was an outstanding teacher of mathematics but, despite being encouraged by his friends to do so, he published nothing. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richard_Louis.html (1 of 2) [2/16/2002 11:29:02 PM]

Richard_Louis

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Richard_Louis.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richard_Louis.html (2 of 2) [2/16/2002 11:29:02 PM]

Richardson

Lewis Fry Richardson Born: 11 Oct 1881 in Newcastle upon Tyne, Northumberland, England Died: 30 Sept 1953 in Kilmun, Argyll, Scotland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lewis Richardson attended Newcastle Preparatory School where he his favourite subject was the study of Euclid. He then went to school in York from 1894 which he left in 1898 to spend two years in Newcastle attending the Durham College of Science. There he studied mathematics, physics, chemistry, botany and zoology. His education was completed at King's college Cambridge, graduating with a First Class degree in Natural Science in 1903. Richardson held a large number of posts. He worked in the National Physical Laboratory (1903-04, 1907-09) and the Meteorological Office (1913-16), and he held university posts at University College Aberystwyth (1905-06) and Manchester College of Technology (1912-13). In addition he was a chemist with National Peat Industries (1906-07) and in charge of the physical and chemical laboratory of the Sunbeam Lamp Company (1909-12). From 1920 to 1929 Richardson was head of the Physics Department at Westminster Training College, then from 1929 to 1940, he was Principal of Paisley College of Technology in Scotland. Richardson was a scientist who was the first to apply mathematics, in particular the method of finite differences, to predicting the weather in Weather Prediction by Numerical Process (1922). In this work he uses data from work by Vilhelm Bjerknes published in Dynamical meteorology and hydrography. Another application of mathematics by Richardson was in his study of the causes of war in Generalized Foreign Politics (1939), Arms and Insecurity (1949), and Statistics of Deadly Quarrels (1950). http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richardson.html (1 of 2) [2/16/2002 11:29:04 PM]

Richardson

He made contributions to the calculus and to the theory of diffusion, in particular eddy-diffusion in the atmosphere. The 'Richardson number', a fundamental quantity involving gradients of temperature and wind velocity is named after him. Article by: J J O'Connor and E F Robertson List of References (5 books/articles)

Some Quotations (3)

Mathematicians born in the same country Other references in MacTutor

Chronology: 1920 to 1930

Honours awarded to Lewis Richardson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1926

Planetary features

Crater Richardson on Mars

Other Web sites

1. L F Richardson home page 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Richardson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richardson.html (2 of 2) [2/16/2002 11:29:04 PM]

Richer

Jean Richer Born: 1630 in France Died: 1696 in Paris, France Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nothing is known of Jean Richer's education. He became a member of the Académie Royal des Sciences in 1666 with the title of 'astronomer'. By 1670, however, he had been given the title 'mathematician' by the Académie. He spent most of his life after this time undertaking work for the Académie. In 1670 Richer was sent by the Académie to La Rochelle to measure the heights of the tides there at both the spring and vernal equinoxes. Also in 1670 he set out on a voyage to Canada (France controlled parts of the country). On the voyage he had the task of testing two clocks made by Huygens. Accurate clocks were important in determining longitude. However there was a storm and Huygens's clocks stopped. On his return Richer reported the failure of the clocks to Huygens and to the Académie. Huygens accused Richer of incompetence but this was certainly untrue. Richer had made many important observations on the voyage and the problem with Huygens's clocks was certainly not his fault. In 1671 Richer was sent on an expedition to Cayenne, French Guyana by the French Government. His first task there was to measure the parallax of Mars and the observations were to be compared with that taken at other sites to compute the distance to the planet. This data enabled the scale of the solar system to be computed, the first reasonably accurate results to be found. Richer's second important work was to examine the periods of pendulums at different points on the Earth. He examined the period of a pendulum while on the expedition to Cayenne, French Guyana and found that the pendulum beat more slowly than in Paris. From this Richer deduced that gravity was weaker at Cayenne, so it was further from the centre of the Earth than was Paris. Richer published his observations in his only written work Observations astronomiques et physiques faites en l'isle de Caienne. Newton and Huygens used Richer's gravity data to show that the Earth is an oblate sphere. In 1673 Richer returned to Paris where he was given the title of 'royal engineer' and undertook work on fortifications. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richer.html (1 of 2) [2/16/2002 11:29:05 PM]

Richer

Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Richer.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richer.html (2 of 2) [2/16/2002 11:29:05 PM]

Richmond

Herbert William Richmond Born: 17 July 1863 in Tottenham, Middlesex, England Died: 22 April 1948 in Cambridge, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Herbert Richmond entered Merchant Taylors' School in London in 1875 and studied there until he graduated in 1882. In the same year he entered King's College Cambridge to read mathematics. He went to Cambridge having won both an open Eton Scholarship and a scholarship from his own school. At King's College, Richmond was coached by the famous coach Dr Routh. He was highly successful, and he was placed third in the Tripos examinations (Love was one place above him). He graduated with a BA in 1885. However the rigours of the Tripos had taken its toll. According to Richmond the intense work needed to succeed at Cambridge made him turn away from mathematics after his graduation and he took up music for a while. However his love for mathematics soon returned and, after writing a dissertation on algebraic geometry, he was awarded a Fellowship to King's College, Cambridge in 1880. Richmond was a college lecturer from 1891 to 1927 and, in addition, a university lecturer from 1901 until he retired in 1923. His main work was in algebraic geometry. Writing in February 1946 he set the scene for his research work in the following, rather delightful, way:In 1885 Cayley and Salmon had carried forward the investigations of the earlier German geometers, Hesse, Steiner, Plücker and others; and Salmon had expounded the subject in treatises which for clarity of style are still unrivalled. Further, the Italians, Corrado Segre and Castelnuovo were opening the way into a vast unexplored field, geometry of more than three dimensions. Rarely has a branch of science offered more inviting prospects for a novice hoping to undertake research. Yet I must now admit that my devotion to this one branch of mathematics has been in some http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richmond.html (1 of 3) [2/16/2002 11:29:07 PM]

Richmond

degree unfortunate; for time has shown (what might perhaps have been foreseen) that the elementary algebraic methods, so effective and so admirable when applied to the simpler curves and surfaces, fail to produce results when applied to loci of higher order. Often, no doubt, no relations of an elementary nature exist, and the search for them is bound to end in disappointment. Nevertheless, methods of elementary algebra may still be employed with success both in geometry proper and in applications such as arithmetical properties of rational functions. It is true that the scope of these methods is restricted, but there is compensation in the fact that when geometry is successful in solving a problem the solution is almost invariably both simple and beautiful. The last sentence explains why so many of my published papers are very short. A result already known is obtained in a simple manner. Admittedly I have spent much time advocating old-fashioned methods which have fallen into undeserved neglect (in my opinion). This does not imply lack of appreciation of modern methods... Richmond was honoured by being elected a Fellow of the Royal Society in 1911. He was an active member of the London Mathematical Society for many years and served as its President from 1920 to 1922. He was also honoured by the University of St Andrews in the award of an honorary degree in 1923. Special mention should be made of the work Richmond undertook during World War I. He was attached to the Ministry of Munitions and he undertook research into ballistics at Portsmouth. He continued his work on spinning shells after the war ended and published the two volume work Text book of AA (anti-aircraft) Gunnery in 1924. His work in this area strongly influenced later work on ballistics undertaken during World War II. Unfortunately this work at Portsmouth involved experimental work with explosives which left him partially deaf. In 1923 he resigned from the committees he was serving on since he felt that his deafness prevented him following the business and playing a full role in the work of the committee. Outside mathematics his other interests included music, an interest in flowers, birds and flora. He enjoyed trips to the Orkneys and Shetlands to photograph birds which he did with photographs of professional quality. A E Milne, writing in [1], says:But shining through his special interests was his genius for friendships with men of all academic generations. He endeared himself to all of them by his utter sincerity and unpretentiousness, his extreme modesty, his engaging humour..., his faithfulness... . He was a lovely person to know. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richmond.html (2 of 3) [2/16/2002 11:29:07 PM]

Richmond

Honours awarded to Herbert Richmond (Click a link below for the full list of mathematicians honoured in this way) London Maths Society President

1920 - 1922

Honorary Fellow of the Edinburgh Maths Society

Elected 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Richmond.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Richmond.html (3 of 3) [2/16/2002 11:29:07 PM]

Riemann

Georg Friedrich Bernhard Riemann Born: 17 Sept 1826 in Breselenz, Hanover (now Germany) Died: 20 July 1866 in Selasca, Italy

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bernhard Riemann's father, Friedrich Bernhard Riemann, was a Lutheran minister. Friedrich Riemann married Charlotte Ebell when he was in his middle age. Bernhard was the second of their six children, two boys and four girls. Friedrich Riemann acted as teacher to his children and he taught Bernhard until he was ten years old. At this time a teacher from a local school named Schulz assisted in Bernhard's education. In 1840 Bernhard entered directly into the third class at the Lyceum in Hannover. While at the Lyceum he lived with his grandmother but, in 1842, his grandmother died and Bernhard moved to the Johanneum Gymnasium in Lüneburg. Bernhard seems to have been a good, but not outstanding, pupil who worked hard at the classical subjects such as Hebrew and theology. He showed a particular interest in mathematics and the director of the Gymnasium allowed Bernhard to study mathematics texts from his own library. On one occasion he lent Bernhard Legendre's book on the theory of numbers and Bernhard read the 900 page book in six days. In the spring of 1846 Riemann enrolled at the University of Göttingen. His father had encouraged him to study theology and so he entered the theology faculty. However he attended some mathematics lectures and asked his father if he could transfer to the faculty of philosophy so that he could study mathematics. Riemann was always very close to his family and he would never have changed courses without his father's permission. This was granted, however, and Riemann then took courses in mathematics from Moritz Stern and Gauss. It may be thought that Riemann was in just the right place to study mathematics at Göttingen, but at this

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.html (1 of 7) [2/16/2002 11:29:10 PM]

Riemann

time the University of Göttingen was a rather poor place for mathematics. Gauss did lecture to Riemann but he was only giving elementary courses and there is no evidence that at this time he recognised Riemann's genius. Stern, however, certainly did realise that he had a remarkable student and later described Riemann at this time saying that he:... already sang like a canary. Riemann moved from Göttingen to Berlin University in the spring of 1847 to study under Steiner, Jacobi, Dirichlet and Eisenstein. This was an important time for Riemann. He learnt much from Eisenstein and discussed using complex variables in elliptic function theory. The main person to influence Riemann at this time, however, was Dirichlet. Klein writes in [4]:Riemann was bound to Dirichlet by the strong inner sympathy of a like mode of thought. Dirichlet loved to make things clear to himself in an intuitive substrate; along with this he would give acute, logical analyses of foundational questions and would avoid long computations as much as possible. His manner suited Riemann, who adopted it and worked according to Dirichlet's methods. Riemann's work always was based on intuitive reasoning which fell a little below the rigour required to make the conclusions watertight. However, the brilliant ideas which his works contain are so much clearer because his work is not overly filled with lengthy computations. It was during his time at the University of Berlin that Riemann worked out his general theory of complex variables that formed the basis of some of his most important work. In 1849 he returned to Göttingen and his Ph.D. thesis, supervised by Gauss, was submitted in 1851. However it was not only Gauss who strongly influenced Riemann at this time. Weber had returned to a chair of physics at Göttingen from Leipzig during the time that Riemann was in Berlin, and Riemann was his assistant for 18 months. Also Listing had been appointed as a professor of physics in Göttingen in 1849. Through Weber and Listing, Riemann gained a strong background in theoretical physics and, from Listing, important ideas in topology which were to influence his ground breaking research. Riemann's thesis studied the theory of complex variables and, in particular, what we now call Riemann surfaces. It therefore introduced topological methods into complex function theory. The work builds on Cauchy's foundations of the theory of complex variables built up over many years and also on Puiseux's ideas of branch points. However, Riemann's thesis is a strikingly original piece of work which examined geometric properties of analytic functions, conformal mappings and the connectivity of surfaces. In proving some of the results in his thesis Riemann used a variational principle which he was later to call the Dirichlet Principle since he had learnt it from Dirichlet's lectures in Berlin. The Dirichlet Principle did not originate with Dirichlet, however, as Gauss, Green and Thomson had all made use if it. Riemann's thesis, one of the most remarkable pieces of original work to appear in a doctoral thesis, was examined on 16 December 1851. In his report on the thesis Gauss described Riemann as having:... a gloriously fertile originality. On Gauss's recommendation Riemann was appointed to a post in Göttingen and he worked for his Habilitation, the degree which would allow him to become a lecturer. He spent thirty months working on his Habilitation dissertation which was on the representability of functions by trigonometric series. He gave the conditions of a function to have an integral, what we now call the condition of Riemann http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.html (2 of 7) [2/16/2002 11:29:10 PM]

Riemann

integrability. In the second part of the dissertation he examined the problem which he described in these words:While preceding papers have shown that if a function possesses such and such a property, then it can be represented by a Fourier series, we pose the reverse question: if a function can be represented by a trigonometric series, what can one say about its behaviour. To complete his Habilitation Riemann had to give a lecture. He prepared three lectures, two on electricity and one on geometry. Gauss had to choose one of the three for Riemann to deliver and, against Riemann's expectations, Gauss chose the lecture on geometry. Riemann's lecture Über die Hypothesen welche der Geometrie zu Grunde liegen (On the hypotheses that lie at the foundations of geometry), delivered on 10 June 1854, became a classic of mathematics. There were two parts to Riemann's lecture. In the first part he posed the problem of how to define an n-dimensional space and ended up giving a definition of what today we call a Riemannian space. Freudenthal writes in [1]:It possesses shortest lines, now called geodesics, which resemble ordinary straight lines. In fact, at first approximation in a geodesic coordinate system such a metric is flat Euclidean, in the same way that a curved surface up to higher-order terms looks like its tangent plane. Beings living on the surface may discover the curvature of their world and compute it at any point as a consequence of observed deviations from Pythagoras' theorem. In fact the main point of this part of Riemann's lecture was the definition of the curvature tensor. The second part of Riemann's lecture posed deep questions about the relationship of geometry to the world we live in. He asked what the dimension of real space was and what geometry described real space. The lecture was too far ahead of its time to be appreciated by most scientists of that time. Monastyrsky writes in [6]:Among Riemann's audience, only Gauss was able to appreciate the depth of Riemann's thoughts. ... The lecture exceeded all his expectations and greatly surprised him. Returning to the faculty meeting, he spoke with the greatest praise and rare enthusiasm to Wilhelm Weber about the depth of the thoughts that Riemann had presented. It was not fully understood until sixty years later. Freudenthal writes in [1]:The general theory of relativity splendidly justified his work. In the mathematical apparatus developed from Riemann's address, Einstein found the frame to fit his physical ideas, his cosmology, and cosmogony: and the spirit of Riemann's address was just what physics needed: the metric structure determined by data. So this brilliant work entitled Riemann to begin to lecture. However [6]:Not long before, in September, he read a report "On the Laws of the Distribution of Static Electricity" at a session of the Göttingen Society of Scientific researchers and Physicians. In a letter to his father, Riemann recalled, among other things, "the fact that I spoke at a scientific meeting was useful for my lectures". In October he set to work on his lectures on partial differential equations. Riemann's letters to his dearly-loved father were full of recollections about the difficulties he encountered. Although only eight students attended the lectures, Riemann was completely happy. Gradually he overcame his natural shyness and established a rapport with his audience. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.html (3 of 7) [2/16/2002 11:29:10 PM]

Riemann

Gauss's chair at Göttingen was filled by Dirichlet in 1855. At this time there was an attempt to get Riemann a personal chair but this failed. Two years later, however, he was appointed as professor and in the same year, 1857, another of his masterpieces was published. The paper Theory of abelian functions was the result of work carried out over several years and contained in a lecture course he gave to three people in 1855-56. One of the three was Dedekind who was able to make the beauty of Riemann's lectures available by publishing the material after Riemann's early death. The abelian functions paper continued where his doctoral dissertation had left off and developed further the idea of Riemann surfaces and their topological properties. He examined multi-valued functions as single valued over a special Riemann surface and solved general inversion problems which had been solved for elliptic integrals by Abel and Jacobi. However Riemann was not the only mathematician working on such ideas. Klein writes in [4]:... when Weierstrass submitted a first treatment of general abelian functions to the Berlin Academy in 1857, Riemann's paper on the same theme appeared in Crelle's Journal, Volume 54. It contained so many unexpected, new concepts that Weierstrass withdrew his paper and in fact published no more. The Dirichlet Principle which Riemann had used in his doctoral thesis was used by him again for the results of this 1857 paper. Weierstrass, however, showed that there was a problem with the Dirichlet Principle. Klein writes [4]:The majority of mathematicians turned away from Riemann ... Riemann had quite a different opinion. He fully recognised the justice and correctness of Weierstrass's critique, but he said, as Weierstrass once told me, that he appealed to Dirichlet's Principle only as a convenient tool that was right at hand, and that his existence theorems are still correct. We return at the end of this article to indicate how the problem of the use of Dirichlet's Principle in Riemann's work was sorted out. In 1858 Betti, Casorati and Brioschi visited Göttingen and Riemann discussed with them his ideas in topology. This gave Riemann particular pleasure and perhaps Betti in particular profited from his contacts with Riemann. These contacts were renewed when Riemann visited Betti in Italy in 1863. In [16] two letter from Betti, showing the topological ideas that he learnt from Riemann, are reproduced. In 1859 Dirichlet died and Riemann was appointed to the chair of mathematics at Göttingen on 30 July. A few days later he was elected to the Berlin Academy of Sciences. He had been proposed by three of the Berlin mathematicians, Kummer, Borchardt and Weierstrass. Their proposal read [6]:Prior to the appearance of his most recent work [Theory of abelian functions], Riemann was almost unknown to mathematicians. This circumstance excuses somewhat the necessity of a more detailed examination of his works as a basis of our presentation. We considered it our duty to turn the attention of the Academy to our colleague whom we recommend not as a young talent which gives great hope, but rather as a fully mature and independent investigator in our area of science, whose progress he in significant measure has promoted. A newly elected member of the Berlin Academy of Sciences had to report on their most recent research and Riemann sent a report on On the number of primes less than a given magnitude another of his great masterpieces which were to change the direction of mathematical research in a most significant way. In it http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.html (4 of 7) [2/16/2002 11:29:10 PM]

Riemann

Riemann examined the zeta function (s) = (1/ns) = (1 - p-s)-1 which had already been considered by Euler. Here the sum is over all natural numbers n while the product is over all prime numbers. Riemann considered a very different question to the one Euler had considered, for he looked at the zeta function as a complex function rather than a real one. Except for a few trivial exceptions, the roots of (s) all lie between 0 and 1. In the paper he stated that the zeta function had infinitely many nontrivial roots and that it seemed probable that they all have real part 1/2. This is the famous Riemann hypothesis which remains today one of the most important of the unsolved problems of mathematics. Riemann studied the convergence of the series representation of the zeta function and found a functional equation for the zeta function. The main purpose of the paper was to give estimates for the number of primes less than a given number. Many of the results which Riemann obtained were later proved by Hadamard and de la Vallée Poussin. In June 1862 Riemann married Elise Koch who was a friend of his sister. They had one daughter. In the autumn of the year of his marriage Riemann caught a heavy cold which turned to tuberculosis. He had never had good health all his life and in fact his serious heath problems probably go back much further than this cold he caught. In fact his mother had died when Riemann was 20 while his brother and three sisters all died young. Riemann tried to fight the illness by going to the warmer climate of Italy. The winter of 1862-63 was spent in Sicily and he then travelled through Italy, spending time with Betti and other Italian mathematicians who had visited Göttingen. He returned to Göttingen in June 1863 but his health soon deteriorated and once again he returned to Italy. Having spent from August 1864 to October 1865 in northern Italy, Riemann returned to Göttingen for the winter of 1865-66, then returned to Selasca on the shores of Lake Maggiore on 16 June 1866. Dedekind writes in [3]:His strength declined rapidly, and he himself felt that his end was near. But still, the day before his death, resting under a fig tree, his soul filled with joy at the glorious landscape, he worked on his final work which unfortunately, was left unfinished. Finally let us return to Weierstrass's criticism of Riemann's use of the Dirichlet's Principle. Weierstrass had shown that a minimising function was not guaranteed by the Dirichlet Principle. This had the effect of making people doubt Riemann's methods. Freudenthal writes in [1]:All used Riemann's material but his method was entirely neglected. ... During the rest of the century Riemann's results exerted a tremendous influence: his way of thinking but little. Weierstrass firmly believed Riemann's results, despite his own discovery of the problem with the Dirichlet Principle. He asked his student Hermann Schwarz to try to find other proofs of Riemann's existence theorems which did not use the Dirichlet Principle. He managed to do this during 1869-70. Klein, however, was fascinated by Riemann's geometric approach and he wrote a book in 1892 giving his version of Riemann's work yet written very much in the spirit of Riemann. Freudenthal writes in [1]:It is a beautiful book, and it would be interesting to know how it was received. Probably many took offence at its lack of rigour: Klein was too much in Riemann's image to be convincing to people who would not believe the latter.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.html (5 of 7) [2/16/2002 11:29:10 PM]

Riemann

In 1901 Hilbert mended Riemann's approach by giving the correct form of Dirichlet's Principle needed to make Riemann's proofs rigorous. The search for a rigorous proof had not been a waste of time, however, since many important algebraic ideas were discovered by Clebsch, Gordan, Brill and Max Noether while they tried to prove Riemann's results. Monastyrsky writes in [6]:It is difficult to recall another example in the history of nineteenth-century mathematics when a struggle for a rigorous proof led to such productive results. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (18 books/articles)

Some Quotations (3)

A Poster of Bernhard Riemann

Mathematicians born in the same country

Some pages from publications

First page of the paper Ueber die hypothesen, welche der Geometrie zu Grunde liegen (written in 1854, but not published until 1868) and another page from the paper.

Cross-references to History Topics

1. Non-Euclidean geometry 2. Topology enters mathematics 3. General relativity 4. An overview of the history of mathematics 5. Prime numbers

Other references in MacTutor

Chronology: 1850 to 1860

Honours awarded to Bernhard Riemann (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1866

Lunar features

Crater Riemann

Other Web sites

1. David R Wilkins 2. European Mathematical Society (Riemann's papers) 3. The Prime Pages (The Riemann Hypothesis) 4. Luneburg, Germany (in German) 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.html (6 of 7) [2/16/2002 11:29:10 PM]

Riemann

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Riemann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riemann.html (7 of 7) [2/16/2002 11:29:10 PM]

Ries

Adam Ries Born: 1492 in Staffelstein (near Bamberg), Saxony (now Germany) Died: 30 March 1559 in Annaberg, Saxony (now Annaberg-Buchholz, Germany)

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

German children today know the expression Das macht nach Adam Ries... (that gives according to Adam Ries) when doing arithmetic. Adam attended school in Zwickau then moved to Annaberg. In 1518 he moved to Erfurt and, although he did not attend university there, he had many contacts with academics from the university and he profited greatly from these contacts. Ries, the greatest son of Staffelstein was Rechenmeister (master of calculations) and Hofarithmeticus at many towns such as Zwickau and Erfurt in Middle Germany. In 1523 he became Bergbeamter (engineer and inspector of mines) in Annaberg in the Kingdom of Sachsen. In 1525 he became Rezessschreiber (recorder of yields), then in 1532 Gegenschreiber (recorder of ownership of mining shares) and, from 1533 to 1539, Zehnter auf dem Geyer (calculator of the ducal tithes). In 1539 Ries became court mathematician and was given the title Churfürstlich Sächsicher Hofarithmeticus. His income came mainly from his arithmetic textbooks. Rechenung nach der lenge, auff den Linihen vnd Feder is perhaps the most famous. It was published in 1550 and was a textbook written for everyone, not just for scientists and engineers. The book contains addition, subtraction, multiplication and, very surprisingly for that period, also division. At that time division could only be learnt at the University of Altdorf (near Nürnberg) and even most scientists did not know how to divide; so it is astonishing that Ries explained it in a textbook designed for everyone to use. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ries.html (1 of 2) [2/16/2002 11:29:12 PM]

Ries

Adam Ries taught both the old method, derived from the abacus, and the new, derived from the Indians, which at that time was forbidden in most countries! The importance of his book is that it was printed (book printing was invented just a short time before by Gutenberg) and many copies were therefore available. Among his other books were the algebra book Coss (1525), and Ein Gerechnet Büchlein auff den Schöffel, Eimer vnd Pfundgewicht... (1533) which contained tables to allow prices of several items to be found when the price of one was known. Brotordnung was a work which allowed one to calculate the weight of a loaf on the assumption that the price of grain varied and the price of a loaf remained constant. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (23 books/articles) A Poster of Adam Ries

Mathematicians born in the same country

Some pages from publications

The cover of Rechenung nach der lenge, auff den Linihen vnd Feder

Other references in MacTutor

Chronology: 1500 to 1600

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ries.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ries.html (2 of 2) [2/16/2002 11:29:12 PM]

Riesz

Frigyes Riesz Born: 22 Jan 1880 in Györ, Austria-Hungary (now Hungary) Died: 28 Feb 1956 in Budapest, Hungary

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Frigyes Riesz's father Ignácz Riesz was a medical man and Frigyes's his younger brother, Marcel Riesz, was himself a famous mathematician. Frigyes (or Frederic in German) Riesz studied at Budapest. He went to Göttingen and Zurich to further his studies and obtained his doctorate from Budapest in 1902. His doctoral dissertation was on geometry. He spent two years teaching in schools before being appointed to a university post. Riesz was a founder of functional analysis and his work has many important applications in physics. He built on ideas introduced by Fréchet in his dissertation, using Fréchet's ideas of distance to provide a link between Lebesgue's work on real functions and the area of integral equations developed by Hilbert and his student Schmidt. In 1907 and 1909 Riesz produced representation theorems for functional on quadratic Lebesgue integrable functions and, in the second paper, in terms of a Stieltjes integral. The following year he introduced the space of q-fold Lebesgue integrable functions and so he began the study of normed function spaces, since, for q 3 such spaces are not Hilbert spaces. Riesz introduced the idea of the 'weak convergence' of a sequence of functions ( fn(x) ). A satisfactory theory of series of orthonormal functions only became possible after the invention of the Lebesgue integral and this theory was largely the work of Riesz. Riesz's work of 1910 marks the start of operator theory. In 1918 his work came close to an axiomatic

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riesz.html (1 of 3) [2/16/2002 11:29:14 PM]

Riesz

theory for Banach spaces, which were set up axiomatically two years later by Banach in his dissertation. Riesz was appointed to a chair in Kolozsvár in Hungary in 1911. However, the Hungarian government was forced to sign the Treaty of Trianon on 4 June, 1920. Hungary was left with less than one third of the land that had previously been Hungary. Romania, Czechoslovakia and Yugoslavia all took over large areas but Austria, Poland and Italy also gained land from Hungary. Kolozsvár was no longer in Hungary after the Treaty of Trianon but rather it was in Romania and was renamed Cluj, so the Hungarian University there had to move within the new Hungarian borders and it moved to Szeged in 1920, where there had previously been no university. In Szeged in 1922 Riesz set up the János Bolyai Mathematical Institute in a joint venture with Haar. Of course the Institute was named after the famous Hungarian mathematician whose birthplace was Kolozsvár, the town from which the university had just been forced to move. Riesz became editor of the newly founded journal of the Institute Acta Scientiarum Mathematicarum which quickly became a major source of mathematics. Riesz was to publish many papers in this journal, the first in 1922 being on Egorov's theorem on linear functionals. It was published in the first part of the first volume. In 1945 Riesz was appointed to the chair of mathematics in the University of Budapest. Many of Riesz's fundamental findings in functional analysis were incorporated with those of Banach. His theorem, now called the Riesz-Fischer theorem, which he proved in 1907, is fundamental in the Fourier analysis of Hilbert space. It was the mathematical basis for proving that matrix mechanics and wave mechanics were equivalent. This is of fundamental importance in early quantum theory. Riesz made many contributions to other areas including ergodic theory where he gave an elementary proof of the mean ergodic theorem in 1938. He also studied orthonormal series and topology. Rogosinski, in [7], writes of Riesz's style:The work of F Riesz is not only distinguished by the genuine importance of his results, but also by his aesthetic discernment in mathematical taste and diction. ... The more leisurely mastership of F Riesz's style, whether he writes in his native Hungarian, or in French or German, conveys such pleasure and is to the older mathematician a nostalgic remainder of what we are in danger to lose. For him there was no mere abstraction for the sake of a structure theory, and he was always turning back to the applications in some concrete and substantial situation. His book Leçon's d'analyse fonctionnelle is one of the most readable accounts of functional analysis ever written. Rogosinski, in [7], describes this book which Riesz wrote jointly with his student B Szökefalvi-Nagy as follows:Here, in the first half written by himself, we find the old master picturing to us Real Analysis as he saw it, lovingly, leisurely, and with the discerning eye of an artist. This book, I have no doubt, will remain a classic in the treasure house of mathematical literature. With it, and with all his other work, will live the memory of Frederic Riesz as a great and fertile mathematician for long in the history of our art. Riesz received many honours for his work. He was elected to the Hungarian Academy of Science and, in 1949, he was awarded its Kossuth Prize. He was elected to the Paris Academy of Sciences and to the Royal Physiographic Society of Lund in Sweden. He received honorary doctorates from the universities http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riesz.html (2 of 3) [2/16/2002 11:29:14 PM]

Riesz

of Szeged, Budapest and Paris. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) A Poster of Frigyes Riesz

Mathematicians born in the same country

Cross-references to History Topics

1. Topology enters mathematics 2. Quantum theory

Other references in MacTutor

1. Chronology: 1900 to 1910 2. Chronology: 1920 to 1930

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Riesz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riesz.html (3 of 3) [2/16/2002 11:29:14 PM]

Riesz_Marcel

Marcel Riesz Born: 16 Nov 1886 in Györ, Hungary Died: 4 Sept 1969 in Lund, Sweden

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Marcel Riesz was the younger brother of Frigyes Riesz and won the Loránd Eötvös competition in 1904. He studied at Budapest and was influenced by Féjér. In a joint work with Hardy he introduced Riesz means. Invited by Mittag-Leffler to Sweden in 1908, he spent the rest of his life there. Appointed to Stockholm in 1911 he went to a chair at Lund in 1926. He has only one joint paper with his brother, written during World War I, on the boundary behaviour of an analytic function. His interests ranged from functional analysis to partial differential equations, mathematical physics, number theory and algebra. He also worked on Clifford algebras and spinors. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riesz_Marcel.html (1 of 2) [2/16/2002 11:29:15 PM]

Riesz_Marcel

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Riesz_Marcel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Riesz_Marcel.html (2 of 2) [2/16/2002 11:29:15 PM]

Ringrose

John Robert Ringrose Born: 21 Dec 1932

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Ringrose was educated at Buckhurst Hill County High School in Chigwell in the Epping Forest district of Essex, England. The school is on the northeastern perimeter of the metropolitan area of London. After leaving the school, Ringrose entered St John's College, Cambridge and he received an M.A., then later a Ph.D. from Cambridge in 1957. After completing his Ph.D., Ringrose was appointed a lecturer in mathematics at King's College of the University of Newcastle-upon-Tyne in 1957. He remained in Newcastle until 1961 when he returned to Cambridge as a lecturer in mathematics. He was also elected a fellow of his old College, St John's College, Cambridge. He remained at Cambridge for two years then, in 1963, Ringrose returned to Newcastle where he was appointed a senior lecturer in mathematics. In 1964 he was appointed to the chair of Pure Mathematics at the University of Newcastle-upon-Tyne and he held this post until he retired in 1993. On his retiral he became professor emeritus at Newcastle. He also served as Pro-Vice-Chancellor at the University of Newcastle-upon-Tyne from 1983 until 1988. Ringrose is a leading world expert on non-self-adjoint operators and operator algebras. He has written on operators of Volterra type, compact linear operators, the Neumann series of integral operators, algebras of operators, automorphisms and derivations of operator algebras, and the cohomology of operator algebras. He has written a number of influential texts including Compact non-self-adjoint operators (1971) and, with R V Kadison, Fundamentals of the theory of operator algebras in four volumes published in 1983, 1986, 1991 and 1992. The first of these volumes is an elementary approach to the theory of C*-algebras and von Neumann algebras. The authors state in the preface that their:... primary goal is to teach the subject and lead the reader to the point where the vast recent

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ringrose.html (1 of 3) [2/16/2002 11:29:17 PM]

Ringrose

literature, both in the subject proper and in its many applications, becomes accessible. The second volume treats advanced topics which Ringrose and Kadison consider to be fundamental for an understanding of current research in operator algebras. R S Duran, a reviewer of the text, writes that:The book [is] written by two eminent mathematicians each of whom has made major contributions to the theory of operator algebras. He concludes the review saying that:...the authors have presented [the material] in a fresh and attractive way which conveys the spirit and beauty of the subject. They are to be commended for writing a beautiful book which, in the reviewer's opinion, fulfills all of the promises made in the preface. These two volumes contain an outstanding collection of exercises which in many cases lead the reader to prove some further major results by skillfully breaking them down into manageable parts. The third and fourth volumes contain the solutions to the exercises. R S Duran says the authors' solutions:... which were developed from scratch specifically for this volume, are models of clarity and efficiency, reflecting their vast experience and insight into the subject. The Royal Society of London elected Ringrose as a fellow in 1977 and he has also been elected a fellow of the Royal Society of Edinburgh. He has served the London Mathematical Society in many different ways including holding the position of president of the Society from 1992 to 1994. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country Honours awarded to John Ringrose (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1977

Fellow of the Royal Society of Edinburgh London Maths Society President

1992 - 1994

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ringrose.html (2 of 3) [2/16/2002 11:29:17 PM]

Ringrose

JOC/EFR June 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ringrose.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ringrose.html (3 of 3) [2/16/2002 11:29:17 PM]

Roberts

Samuel Roberts Born: 15 Dec 1827 in Horncastle, Lincolnshire, England Died: 1913 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Samuel Roberts was educated at Queen Elizabeth's Grammar School in the town of his birth. In 1844 he entered Manchester New College and, the following year, he began his studies at the University of London. He graduated in mathematics with a BA in 1847, then two years later he was placed first in the examinations for his MA in mathematics and physics. After this he studied law, becoming a solicitor in 1853 in Lincoln. However his love of mathematics made him give up law after a few years, and he returned to London to take up his mathematical studies for a second time. In fact he had published mathematical papers from 1848 but, after the London Mathematical Society was founded in early 1865, Roberts could join in discussions with other mathematicians for he had no mathematical post. As Glaisher write in [1]:He was simply a private gentleman who pursued researches from pure love of investigation and desire to extend the boundaries of subjects that were attractive to him. And he never sought any recognition of his work; for him it was entirely its own reward. Robert's interests in mathematics were wide ranging. Glaisher, writing in [1], describes Roberts' contributions to mathematics as:... numerous and valuable, and they covered a somewhat wide range. Among the subjects to which his principal papers related were plane and solid geometry, theory of numbers, and link motion. He also wrote on the calculus of operations, interpolation etc. His writings on geometry included several important papers on parallel curves and surfaces. In theory of numbers he was interested in the Pellian equation and similar problems. Glaisher also records in [1] an interesting comment which Cayley made to him concerning Roberts. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roberts.html (1 of 2) [2/16/2002 11:29:19 PM]

Roberts

Cayley said that he regretted that joint papers were so rare in mathematics since he would have liked to have collaborated with Roberts, particularly on ideas which arose from Roberts' paper On the motion of a plane under given conditions. Roberts served on the Council of the London Mathematical Society from 1866 to 1892. He was Treasurer of the Society for 8 years from 1872 to 1880 and President from 1880 to 1882. He was awarded the De Morgan Medal by the Society in 1896. The greatest honour given to Roberts was his election as a Fellow of the Royal Society in 1878. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Honours awarded to Samuel Roberts (Click a link below for the full list of mathematicians honoured in this way) London Maths Society President

1880 - 1882

LMS De Morgan Medal

Awarded 1896

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Roberts.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roberts.html (2 of 2) [2/16/2002 11:29:19 PM]

Roberval

Gilles Personne de Roberval Born: 10 Aug 1602 in Senlis, France Died: 27 Oct 1675 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gilles Roberval began to study mathematics at the age of 14 years. He travelled widely visiting many places in France. At this time he earned his living teaching mathematics while he discussed advanced topics with university teachers in the towns he visited. On his travels he went to Bordeaux and met Fermat. Roberval was one of the group to meet with Mersenne. He arrived in Paris in 1628 and made contact with the group, particularly with Claude Hardy, Mydorge, Etienne Pascal and Blaise Pascal. In fact, Roberval became the only professional mathematician in the group. In 1632 Roberval was appointed professor of philosophy in the Collège Gervais in Paris and then, in 1634, was appointed to the Ramus chair of mathematics in the Collège Royale. This was a competitive appointment and Roberval had to compete for reappointment regularly. In 1655 he was appointed to Gassendi's chair of mathematics, in addition to the Ramus chair, and he held both chairs for the rest of his life. Roberval developed powerful methods in the early study of integration, writing Traité des indivisibles. He computed the definite integral of sin x, worked on the cycloid and computed the arc length of a spiral. Roberval is important for his discoveries on plane curves and for his method for drawing the tangent to a curve, already suggested by Torricelli. This method of drawing tangents makes Roberval the founder of kinematic geometry. He was elected to the Académie Royal des Sciences in 1666. In fact he was a founding member of the Académie. In 1669 he invented the Roberval balance which is now almost universally used for weighing scales of the balance type. He presented details to the Académie in that year. Roberval also worked with Jean Picard in cartography and wrote on mapping France. He studied the vacuum and designed apparatus which was used by Pascal in his experiments. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roberval.html (1 of 2) [2/16/2002 11:29:20 PM]

Roberval

List of References (14 books/articles) Mathematicians born in the same country Some pages from publications

Extract from Observations sur la composition des mouvemens et ... traitant des mouvemens composez (1693)

Cross-references to History Topics

The rise of Calculus

Cross-references to Famous Curves

1. Cissoid of Diocles 2. Cycloid 3. Folium of Descartes 4. Limaçon of Pascal 5. Right Strophoid

Other references in MacTutor

Chronology: 1625 to 1650

Honours awarded to Gilles Roberval (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Roberval (17th Arrondissement)

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Roberval.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roberval.html (2 of 2) [2/16/2002 11:29:20 PM]

Robins

Benjamin Robins Born: 1707 in Bath, England Died: 20 July 1751 in Madras, India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Benjamin Robins attended school in Bath, then went to London after leaving school. There he studied languages and mathematics without the help of tutors. He progressed quickly, publishing in the Philosophical Transactions of the Royal Society in 1727, the year he was elected a Fellow of the Royal Society. He wrote on Johann Bernoulli's laws of motion and impacts of rigid bodies, achieving considerable fame so that he was able to attract many pupils for mathematics tuition. He gave this as individual tuition, never teaching a class. Gradually Robins gave up teaching to become an engineer. He went on to construct bridges, mills and harbours. He also worked at making rivers navigable and draining fen land. In addition to this he began to study gunnery and fortifications. To gain experience he travelled through Flanders studying fortification work there, there was certainly much to see. On his return to England he published A discourse concerning the nature and certainty of Sir Isaac Newton's method of fluxions and prime and ultimate ratios. This was to support the differential calculus against attacks by Berkeley. He also published Remarks on M Euler's Treatise of Motion in 1739. His publications were not restricted to science however and around this time he published three famous political pamphlets. Robins published in 1742 New Principles of Gunnery which formed the basis for all subsequent work on the theory of artillery and projectiles. It was translated into German by Euler. In 1747 he received the Copley medal of the Royal Society. Robins also invented the ballistic pendulum which allowed precise measurements of the velocity of projectiles fired from guns. As described by Robins, a large wooden block is suspended in front of a gun. When a bullet is fired its momentum is transferred to the bob and can be determined from the amplitude of the pendulum. He experimented with rockets, publishing Rockets and the heights to which they ascend in 1750. He also improved the instruments at the Royal Observatory at Greenwich. Robins was sent to India in 1750 and there he prepared the defence of Madras. He contracted a fever in India and died, according to [2],

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Robins.html (1 of 2) [2/16/2002 11:29:22 PM]

Robins

with his pen in his hand while drawing up a report for the board of directors. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Benjamin Robins (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1727

Royal Society Copley Medal

Awarded 1746

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Robins.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Robins.html (2 of 2) [2/16/2002 11:29:22 PM]

Robinson

Abraham Robinson Born: 6 Oct 1918 in Waldenburg, Germany (now Walbrzych, Poland) Died: 11 April 1974 in New Haven, Connecticut, USA

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

We should note at the outset that in fact Abraham Robinson's family name was Robinsohn rather than Robinson, but we shall refer throughout this article to Robinson, the version which he used after 1940. Abraham Robinson's father was also named Abraham Robinson and his mother was Hedwig Lotte. He was the second child of the family, his older brother Saul Robinson also went on to have an outstanding career; he became an expert on comparative education. Abraham Robinson senior was also a highly talented man. After studying chemistry he became an important writer and philosopher but Abraham junior never knew his father for he died shortly before Abraham junior was born. It was a Jewish family and although Abraham Robinson senior was a Zionist he had never been to Palestine but he had accepted the position of head of the Hebrew National Library in Jerusalem just before he died. Hedwig Robinson was a teacher and she brought up her two sons in Germany until 1933 when Abraham was fourteen years old. Although little is known of Abraham during these years, some notebooks which he owned have survived [18]:... containing poems and plays, suggesting a sensitive observant child with an ambition to write. Clearly the family had always been attracted to Jerusalem but the anti-Jewish legislation introduced into Germany in 1933 indicated very clearly that it was time to leave. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the schools and universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Hedwig, Abraham and Saul Robinson avoided the problems that

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Robinson.html (1 of 4) [2/16/2002 11:29:24 PM]

Robinson

Jews would have in Germany from 1933 by starting a new life in Palestine. There Robinson completed his schooling and, in 1935, began studying mathematics under Fraenkel and Levitzki at the Hebrew University of Jerusalem. Robinson was a brilliant student and, after graduating in 1939, he was awarded a scholarship to allow him to study at the Sorbonne in Paris. After only a few months of study he was forced to flee when the Germans invaded France. After reaching England on one of the last small boats from Bordeaux to evacuate refugees, he changed his name from Robinsohn. As an undergraduate at the Hebrew University Robinson has been interested in both algebra and mathematical logic. However, once in England he enlisted in the Free French Air Force and, because he was a mathematician, he was sent in 1941 to the Royal Aircraft Establishment at Farnborough where he became a Scientific Officer. Rapidly he became an expert in aerodynamics and for the rest of World War II he worked on delta wings and supersonic flow. He was sent to Germany in 1945, still in his role as Scientific Officer, and then in 1946 he was appointed as a senior lecturer at the College of Aeronautics at Cranfield. Young writes in [20]:At Cranfield his interests in the aerodynamic theory of wings, both in subsonic and supersonic flow, broadened and became increasingly comprehensive. By now Robinson was a world leading authority in aerodynamics yet he continued with his interest in mathematical logic. In 1946 he was awarded a Master's Degree from the Hebrew University in Jerusalem and, following this, he began research at London University receiving a Ph.D. from London in 1949 for pioneering work in model theory and the metamathematics of algebraic systems. He went to the University of Toronto in 1951 to take up a chair of applied mathematics but left for Jerusalem in 1957 to fill Fraenkel's chair at the Hebrew University. He was Chairman of the Mathematics Department there until 1962 when he accepted the professorship of Mathematics and Philosophy at the University of California, Los Angeles. In 1967 he moved again, but remaining in the United States he went to Yale University as Professor of Mathematics. Other than changing his chair to the Sterling Professor of Mathematics at Yale in 1971 he remained there until his death. He was diagnosed as having cancer of the pancreas in 1973, underwent an operation in November of that year, but died a few months later. A collection of papers Model theory and algebra was published in 1975 as a memorial tribute to Robinson. The editors' foreword states:The sudden fatal illness of Abraham Robinson came as a great shock to many people around the world. For Robinson was more than an excellent mathematician. He was also a person whom one came very quickly to like very much. Those swift sad months of November 1973-April 1974 were for those at Yale tinged with a sense of unreality. He was gone before anyone could come to grips with what was happening. We sought a way of expressing our respect and our sense of personal loss. This volume was the best way we knew. Robinson was a leading expert in remarkably different areas of mathematics. The article [20] lists 130 papers and nine books which he wrote. Let us examine first his contributions to applied mathematics. Only one of his books deals with applied mathematics but it may surprise mathematicians who think of Robinson only as a mathematical logician to realise that almost half his papers are on applied mathematics, particularly on aerodynamics. The one applied mathematics book is Wing theory written jointly with J A Laurmann and published in 1956. Lighthill, reviewing the work, wrote:This is an admirable compendium of the mathematical theories of the aerodynamics of aerofoils and wings. Almost all the important results are referred to, even though there can http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Robinson.html (2 of 4) [2/16/2002 11:29:24 PM]

Robinson

be only a brief reference to literature in connection with the more difficult topics. ... It should be an invaluable introduction to wing aerodynamics for mathematically-minded students, as well as a solid stand-by for purposes of reference for all workers in this and allied fields. Robinson is best known, however, for his work on mathematical logic. His doctorate from the Hebrew University in 1949 was The metamathematics of algebraic systems and this became his first book published in 1951. He published Complete theories in 1956 which was written to study the properties of model-completeness and bounding transform. He applied these two concepts to the elementary theories of certain mathematical structures. Robinson's contributions to model theory were developed during his time at the University of Toronto. He weaved his many contributions and papers into a treatise Introduction to model theory and to the metamathematics of algebra published in 1963. Engeler wrote:This is ... the first attempt to write a connected exposition of the new subject of model theory. The main body of the work consists of rewritten versions of the author's main contributions to the subject, which are brought into a smooth and eminently readable sequence. ... there results as complete a survey as can be expected at this time from any single author. Robinson's most famous invention was non-standard analysis which he introduced in 1961. Kochen writes in [20]:I want to emphasise that non-standard analysis was not a sudden tangential direction in which Robinson moved. Rather, it was the systematic application of the same viewpoint which he earlier applied to algebra to the study of analysis. Fenyo has explained the ideas behind the theory:[Robinson's] theory is based on the metamathematical fact that the system of real numbers is incomplete. Thus, there exist extensions of the field of real numbers that possess all the properties of the system of real numbers that are formulated in the lower predicate calculus in terms of some given set of relations. Proper extensions of noncomplete theories are often referred to as non-standard models. A non-standard model for the system of real numbers has the feature of being a non-Archimedean totally ordered field which contains a copy of the real number system. In 1966 Robinson published his famous text Non-standard analysis. Kreisel wrote:This book, which appeared just 250 years after Leibniz's death, presents a rigorous and efficient theory of infinitesimals obeying, as Leibniz wanted, the same laws as the ordinary numbers. We end this biography by giving some comments on Robinson's personality. In [20] this appreciation is given:He had the humility and the kindness of the truly great, he was interested in people and he found it easy to like them and he patronised no-one. He was deeply concerned with most forms of human culture and creativity, and on all he could converse with the fascinating combination of logic, insight and knowledge that characterised his mathematics. Macintyre, in [18], writes:Robinson was a gentleman, unfailingly courteous, with inexhaustible enthusiasm. He took modest pleasure in his many honours. He was much respected for his willingness to listen,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Robinson.html (3 of 4) [2/16/2002 11:29:24 PM]

Robinson

and for the sincerity of his advice. Finally let us quote from Korner's tribute to Robinson during the memorial service at Yale University on 15 September 1974:When one considers the wealth, profundity and diversity of his interests and the continuous interplay in his thinking of pure mathematics, applied mathematics, logic and philosophy one is constantly reminded of Leibniz to whom he felt a natural affinity and for whom he had the deepest admiration. The one Leibnizian idea in which he could see little merit was Leibniz's 'principe de meilleur' according to which the world is the best of all possible worlds. I remember his asking me more than once in his gently ironic way whether I could make any sense of this principle. Today I should like to offer a partial answer: It cannot be a wholly bad worlds in which an Abraham Robinson could live and think; in which his wife and friends are able to cherish his memory; and in which his life's work will be remembered as long as logic, mathematics and philosophy matter to mankind. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (20 books/articles) Mathematicians born in the same country Other Web sites

G Don Allen

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Robinson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Robinson.html (4 of 4) [2/16/2002 11:29:24 PM]

Robinson_Julia

Julia Bowman Robinson Born: 8 Dec 1919 in St Louis, Missouri, USA Died: 30 July 1985 in USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Julia Bowman's mother died when she was two years old and her father, retiring a year later, moved to Arizona and then later to San Diego. Her schooling was disrupted by a year off school with scarlet fever at age nine. After graduating from San Diego High School she entered San Diego State College. Later she transferred to the University of California at Berkeley. There she became Neyman's assistant but after marrying an assistant professor of mathematics there, Raphael Robinson, she was no longer allowed to teach in the mathematics department. She left mathematics at this time. In 1946 she visited Princeton and took up mathematics again, working for a doctorate under Tarski's supervision. In her thesis she proved that the arithmetic of rational numbers is undecidable by giving an arithmetical definition of the integers in the rationals. Robinson was awarded a doctorate in 1948 and that same year started work on Hilbert's Tenth Problem: find an effective way to determine whether a Diophantine equation is soluble. Along with Martin Davis and Hilary Putman she gave a fundamental result which contributed to the solution to Hilbert's Tenth Problem. She also did important work on that problem with Matijasevic after he gave the solution in 1970. In addition to this work on Hilbert's Tenth Problem, Robinson also wrote on general recursive functions and on primitive recursive functions. In 1980 she gave the American Mathematical Society Colloquium

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Robinson_Julia.html (1 of 3) [2/16/2002 11:29:27 PM]

Robinson_Julia

Lectures on computability, Hilbert's Tenth Problem, decision problems for rings and fields, and non-standard models of arithmetic. She was the second woman to give the Colloquium Lectures, the first being Wheeler in 1927. Julia Robinson received many honours. She was the first woman to be elected to the National Academy of Sciences in 1975, the first woman officer of the American Mathematical Society in 1978 and the first woman president of the Society in 1982. She was elected to the American Academy of Arts and Sciences on 1984. She was awarded a MacArthur Fellowship in 1983 in recognition of her contributions to mathematics. Leon Henkin, writing in [2], describes her as follows:The style of quiet decorum she generally adopted was in contrast to the flashes of lively spirit that could be discerned in a wide range of bright or strong feelings when she spoke. Especially strong was her stubborn insistence that opportunity ought to be freely accessible to all - whether economic opportunity or opportunity for access to a mathematical career. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Julia B Robinson

Mathematicians born in the same country

Honours awarded to Julia B Robinson (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1983 - 1984

AMS Colloquium Lecturer

1980

Other Web sites

1. Agnes Scott College 2. AMS (A lecture by Julia Robinson's sister)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Robinson_Julia.html (2 of 3) [2/16/2002 11:29:27 PM]

Robinson_Julia

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Robinson_Julia.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Robinson_Julia.html (3 of 3) [2/16/2002 11:29:27 PM]

Rocard

Yves-André Rocard Born: 22 May 1903 in Vannes, France Died: 16 March 1992 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yves Rocard was awarded a doctorate in mathematics in 1927 from the Ecole Normale Supérieure, Paris. The following year he was awarded a doctorate in physical sciences. He was appointed to the chair of physics at Ecole Normale Supérieure. During World War II, Rocard was a member of a French Resistance group. In a highly dangerous mission, he was flown from France to England in a small airplane and, once in England, he became Head of the Research Department of the Free French Naval Forces. In fact this was to prove a significant time for Rocard in the development of his scientific ideas, for at this time he learnt that radars in England had been shown to have detected strong radio emission from the Sun. Of course this had not been detected during scientific work, rather the solar emission was detected as interfering with the 'proper' war time use of radar. After the war, Rocard returned to France and proposed that France set up a site to conduct radio astronomy. Rocard was even able to get his hands on equipment to start off such a project, providing two German radar mirrors of 'Wurzburg' type each having a 7.5 meter diameter. Using his wartime contacts, Rocard was able to give his scientists access to the Research Centre of the French Navy at Marcoussis. By 1952, despite the pioneering work in radio astronomy in France, it became clear that others were using more powerful instruments and the French could not compete. Rocard gave strong support to the project and the French Ministry of National Education gave 25 million Francs to the Ecole Normale Supérieure. A site was found for the radio astronomy observatory at Nancay in the Cher region, 200 km due south of Paris. In addition to his work on radio astronomy, Rocard contributed to the development of the French atomic bomb. He also undertook research into semiconductors. In the last part of his life he studied biomagnetism and dowsing which reduced his standing in the eyes of many of his colleagues. Article by: J J O'Connor and E F Robertson A Reference (One book/article)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rocard.html (1 of 2) [2/16/2002 11:29:29 PM]

Rocard

Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rocard.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rocard.html (2 of 2) [2/16/2002 11:29:29 PM]

Rogers

Claude Ambrose Rogers Born: 1 Nov 1920

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ambrose Rogers was educated at Berkhamsted School in Hertfordshire, England about 45 km northwest of London. From there he went to University College, London and Birkbeck College, London and was awarded his B.Sc. from the University of London in 1941. However, these were the years of World War II and Rogers did war work as an Experimental Officer with the Ministry of Supply from 1940 until the end of the war in 1945. Other mathematicians such as Kendall also held such posts with the Ministry of Supply. After the war ended Rogers returned to his mathematical studies. He was appointed a lecturer in mathematics at University College, London in 1946 and worked for his doctorate while lecturing at the College. He was awarded a Ph.D. in 1949 and a D.Sc. in 1952 having worked at this time with Davenport and publishing two joint papers with him in 1950. At University College, Rogers was promoted to reader before he moved to Birmingham in 1954 as the Professor of Pure Mathematics at Birmingham University. After four years in Birmingham, Rogers returned to London, this time as the Astor Professor of Mathematics at University College. He succeeded Harold Davenport as the Astor Professor of Mathematics who had moved to Cambridge in 1958. Rogers held this post until he retired in 1986, when he became professor emeritus and remained at University College. Roger's continues to produce a remarkable mathematical output having published to date over 170 papers. His early work was on number theory and he wrote on Diophantine inequalities and the geometry of numbers. Jointly with Erdös, he wrote The covering of n-dimensional space by spheres (1953) and Covering space with convex bodies (1961), writing many other articles on coverings and packings including Covering space with equal spheres with Coxeter.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rogers.html (1 of 2) [2/16/2002 11:29:30 PM]

Rogers

His later work covered a wide range of different topics in geomery and analysis including Borel functions, Hausdorff measure and local measure, topological properties of Banach spaces and upper semicontinuous functions. Rogers has written two important books, Packing and Covering in 1964 and Hausdorff Measures in 1970. Rogers wrote an obituary of Harold Davenport for the London Mathematical Society which was published in 1972, the year after Rogers wrote a survey of Davenport's work. He was one of the editors of the The collected works of Harold Davenport published in four volumes in 1977. Rogers was elected a fellow of the Royal Society in 1959 and served on the Council of the Royal Society for two spells, first from 1966 to 1968 and then again in 1983-84. He also served as the 55th President of the London Mathematical Society from 1970 to 1972 and was honoured with the award of the London Mathematical Society's De Morgan Medal in 1977. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

A Poster of Ambrose Rogers

Mathematicians born in the same country

Honours awarded to Ambrose Rogers (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1959

London Maths Society President

1970 - 1972

LMS De Morgan Medal

Awarded 1977

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rogers.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rogers.html (2 of 2) [2/16/2002 11:29:30 PM]

Rohn

Karl Rohn Born: 28 Jan 1855 in Schwanheim (near Bensheim), Hesse, Germany Died: 4 Aug 1920 in Leipzig, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karl Rohn attended the Polytechnikum at Darmstadt. He studied mathematics and engineering at the University of Leipzig, then at the University of Munich where he was awarded a doctorate in 1878. Rohn's dissertation was influenced by Klein and involved the relationship of Kummer's surface to hyperelliptic functions. In 1884 Rohn was appointed assistant professor at Leipzig, then in 1887 he became a full professor at Technische Hochschule in Dresden where he held the chair of descriptive geometry. From 1904 until his death Rohn held the chair of mathematics at the University of Leipzig. He made major contributions to algebraic geometry. He became an expert in this field with an outstanding ability to see geometric facts in algebraic equations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rohn.html (1 of 2) [2/16/2002 11:29:33 PM]

Rohn

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rohn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rohn.html (2 of 2) [2/16/2002 11:29:33 PM]

Rolle

Michel Rolle Born: 21 April 1652 in Ambert, Basse-Auvergne, France Died: 8 Nov 1719 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Michel Rolle had little formal education being largely self-educated. He worked as assistant to several attorneys around Ambert, then in 1675 he went to Paris. In Paris he worked as a scribe, and arithmetical expert. He was elected to the Académie Royal des Sciences in 1685, and became Pensionnaire Géometre of the Académie in 1699. Rolle worked on Diophantine analysis, algebra (using methods of Bachet involving use of the Euclidean algorithm) and geometry. He published a work Traité d'algèbre on the theory of equations. In 1682 he achieved a certain fame by solving a problem which had been publicly posed by Ozanam. Jean-Baptiste Colbert, controller general of finance and secretary of state for the navy under King Louis XIV of France, rewarded Rolle for this achievement. Colbert arranged a pension for Rolle and he also received a pension form the Académie in 1699, as mentioned above. Rolle is best remembered, however, for 'Rolle's Theorem' which was published in an obscure book in 1691, using a method of Hudde in the proof. If f(a) = f(b) = 0 then f '(x) = 0 for some x with a x b. Rolle described the calculus as a collection of ingenious fallacies. He invented the notation n x for the nth root of x. He also adopted the notion that if a > b then -b > -a in opposition to Descartes and others. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1675 to 1700

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rolle.html (1 of 2) [2/16/2002 11:29:34 PM]

Rolle

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rolle.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rolle.html (2 of 2) [2/16/2002 11:29:34 PM]

Roomen

Adriaan van Roomen Born: 29 Sept 1561 in Louvain, Belgium Died: 4 May 1615 in Mainz, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adriaan van Roomen is often known by his Latin name Adrianus Romanus. After studying at the Jesuit College in Cologne, Roomen studied medicine at Louvain. He then spent some time in Italy, particularly with Clavius in Rome in 1585. Roomen was professor of mathematics and medicine at Louvain from 1586 to 1592, he then went to Würzburg where again he was professor of medicine. He was also "Mathematician to the Chapter" in Würzburg. From 1603 to 1610 he lived frequently in both Louvain and Würzburg. He was ordained a priest in 1604. After 1610 he tutored mathematics in Poland. One of Roomen's most impressive results was finding to 16 decimal places. He did this in 1593 using 230 sided polygons. Roomen's interest in was almost certainly as a result of his friendship with Ludolph van Ceulen. Roomen proposed a problem which involved solving an equation of degree 45. The problem was solved by Viète who realised that there was an underlying trigonometric relation. After this a friendship grew up between the two men. Viète proposed the problem of drawing a circle to touch 3 given circles to Roomen (the Apollonian Problem) and Roomen solved it using hyperbolas, publishing the result in 1596. Roomen worked on trigonometry and the calculation of chords in a circle. In 1596 Rheticus's trigonometric tables Opus palatinum de triangulis were published, many years after Rheticus died. Roomen was critical of the accuracy of the tables and wrote to Clavius at the Collegio Romano in Rome pointing out that, to calculate tangent and secant tables correctly to ten decimal places, it was necessary to work to 20 decimal places for small values of sine, see [2]. In 1600 Roomen visited Prague where he met Kepler and told him of his worries about the methods employed in Rheticus's trigonometric tables. Among other contributions made by Roomen was one to figures of equal perimeter. Pappus had proved a number of results concerning the maximum area of polygons of equal perimeter. For example regular n-sided polygons have the maximum area amomg all n-sided polygons of fixed perimeter. Roomen generalised the results of Pappus and, again showing his presise thinking, realised that 'regular' had not been properly defined. His work in this area is discussed in detail in [4].

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roomen.html (1 of 2) [2/16/2002 11:29:35 PM]

Roomen

Roomen also wrote a commentary on al-Khwarizmi's Algebra but the only two known copies were destroyed in 1914 and 1944 (as a result of World War I and World War II). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Pi through the ages 2. A chronology of pi

Other references in MacTutor

Chronology: 1500 to 1600

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Roomen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roomen.html (2 of 2) [2/16/2002 11:29:35 PM]

Rosanes

Jakob Rosanes Born: 16 Aug 1842 in Brody, Austria-Hungary (now Ukraine) Died: 6 Jan 1922 in Breslau, Germany (now Wroclaw, Poland)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jakob Rosanes studied at the University of Berlin and the University of Breslau between 1860 and 1865. He obtained his doctorate from Breslau in 1865 and then taught there for the rest of his life. He became professor in 1876 and rector of the university during the years 1903-4. Rosanes worked on algebraic geometry and invariant theory, some of his work being joint with Pasch. Some of his work was on Cremona transformations and he proved a result which was proved independently by Max Noether. Both Rosanes' and Max Noether's proofs were incomplete and Castelnuovo, 30 years later, gave a final satisfactory form. Rosanes wrote on many aspects of algebraic geometry and invariant theory which were in fashion at that time. He also wrote an excellent chess book Theorie und Praxis des Schachspiels. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rosanes.html (1 of 2) [2/16/2002 11:29:37 PM]

Rosanes

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rosanes.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rosanes.html (2 of 2) [2/16/2002 11:29:37 PM]

Rosenhain

Johann Georg Rosenhain Born: 10 June 1816 in Königsberg, Prussia (now Kaliningrad, Russia) Died: 14 May 1887 in Königsberg, Prussia (now Kaliningrad, Russia) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Rosenhain studied at Königsberg where he was awarded a doctorate. In 1844 he was appointed as a lecturer at Breslau but, in 1848, because of his revolutionary activities he was forced to leave. Rosenhain went to the University of Vienna in 1851. Then, in 1857, he returned to Königsberg where he taught until he retired in 1886. At Königsberg as a student Rosenhain became a close friend of Jacobi, editing some of Jacobi's lectures while he was still a student. Rosenhain won the 1846 Paris Academy Prize (in 1851!) for work on elliptic functions with a beautiful piece of work. Göpel solved the problem too but he did not submit it for the prize. Rosenhain's methods are more closely linked to those of Jacobi than are the methods of Göpel. Rosenhain never published anything other than his Academy Prize paper despite having shown outstanding mathematical abilities. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rosenhain.html (1 of 2) [2/16/2002 11:29:38 PM]

Rosenhain

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rosenhain.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rosenhain.html (2 of 2) [2/16/2002 11:29:38 PM]

Rota

Gian-Carlo Rota Born: 27 April 1932 in Vigevano, Italy Died: 18 April 1999 in Cambridge, Massachusetts, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gian-Carlo Rota's father, Giovanni Rota, was a civil engineer and architect who specialised in anti-earthquake structures. Giovanni Rota was a prominent anti-fascist and his name appears on a death list constructed by Mussolini. Gian-Carlo was born into a talented family in Vigevano, many members of his family had achieved fame in their areas of expertise, for example one of Gian-Carlo's uncles, Flaiano, wrote scripts for Federico Fellini's films, including La Dolce Vita. Gian-Carlo was educated in Italy up to the age of thirteen in 1945. This was near the end of World War II and, due to Giovanni Rota's anti-fascist views, the family was forced to leave Vigevano to escape Mussolini's death squads. Giovanni Rota took his family to northern Italy where they hide for a time before crossing the border into Switzerland. The family eventually escaped to Ecuador where Gian-Carlo completed his secondary school education. The positive side to this remarkable escape story was that Rota was fluent in English, Italian, Spanish and French. Rota entered the United States in 1950 at the age of eighteen to undertake his university studies. He entered Princeton University in 1950 and received a BA summa cum laude in 1953. After graduating, Rota entered Yale University where he studied for his Master's Degree in Mathematics which was awarded in 1954. He then undertook doctoral studies, supervised by Jacob T Schwartz, and he was awarded a PhD from Yale in 1956 for his thesis Extension theory of differential operators. His thesis supervisor wrote:[Rota] was my first graduate student at Yale. He began as a functional analyst and after a few years moved on to combinatorics, where he became a leading national and international figure. He really loved mathematics all his life very passionately. He was also http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rota.html (1 of 4) [2/16/2002 11:29:40 PM]

Rota

a person of great cultural and literary attainment. He loved to write, loved to edit. He was a gourmet of mathematics. He was ebullient, a bit of a raconteur. In this same year that he was awarded his doctorate, Rota married Teresa Rondón and he received a Postdoctoral Research Fellowship to undertake research at the Courant Institute at New York University. After spending the year 1956-57 in New York, Rota was appointed as Benjamin Peirce Instructor at Harvard University. He held this post until 1959 when he joined the faculty at Massachusetts Institute of Technology. With the exception of two years, 1965 to 1967, when he was at the Rockefeller University, Rota remained at MIT for the rest of his career. Rota was given the title Professor of Applied Mathematics at MIT but in 1972 his title was changed to Professor of Applied Mathematics and Philosophy. He is the only professor at MIT ever to have such a title. However, he had many other roles outside MIT. Rota had a long association with the Los Alamos Scientific Laboratory where he enjoyed being with his friend Ulam and collaborating with him. He served as a consultant to the Laboratory from in 1966 and, in 1971, he was made a Senior Fellow of the Laboratory. M Waterman writes in [3]:Rota soon became part of Los Alamos. He gave lectures that were deeply informative, polished works of art that made him known throughout the Lab. The topics were wide-ranging: differential equations, ergodic theory, nonstandard analysis, probability, and of course, combinatorics. Rota was also a consultant with the Rand Corporation from 1966 to 1971 and with the Brookhaven National Laboratory from 1969 to 1973. As we have indicated above, Rota worked on functional analysis for his doctorate and, up to about 1960, he wrote a series of papers on operator theory. Two papers in 1959-60, although still in the area of operator theory, looked at ergodic theory which is an area which requires considerable combinatorial skills. These papers seem to have led Rota away from operator theory and into the area of combinatorics. His first major work on combinatorics, which was to change the direction of the whole subject, was On the Foundations of Combinatorial Theory which Rota published in 1964. Rota received the Steele Prize from the American Mathematical Society in 1988. The Prize citation singles out the 1964 paper On the Foundations of Combinatorial Theory as:... the single paper most responsible for the revolution that incorporated combinatorics into the mainstream of modern mathematics. This paper was the first of a series of ten papers with this main title, all ten have subtitles (for example this first one was subtitled Theory of Möbius functions ) and all the remaining nine have between one and three additional co-authors. Papers two to nine were all published between 1970 and 1974 with the tenth being published in 1992. Richard Guy, reviewing [1] in 1980, writes:Combinatorists owe much to Gain-Carlo Rota, already a "respectable" mathematician when he interested himself in combinatorics and embarked on his gallant crusade to unify the subject which almost everyone regarded as being at best a bag of clever isolated tricks; to reverse the tide of abstraction in mathematics; to return to the concreteness of a century ago. ... Rota observes that combinatorics is providing the essential continuing link between mathematics and the sciences: biology (structure of large molecules), linguistics http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rota.html (2 of 4) [2/16/2002 11:29:40 PM]

Rota

(context-free languages, automata theory), physics (statistical mechanics, phase transition problems, elementary particles). Rota received many awards for his outstanding contributions to numerous areas. In addition to the Steele Prize mentioned above, he was awarded the Medal for Distinguished Service from the National Security Agency in 1992. He was elected to the National Academy of Sciences in 1982, was vice-president of the American Mathematical Society in 1995-97 and the Society's Colloquium Lecturer in 1998. He was also a fellow of the American Academy of Arts and Sciences, a member of the Academia Argentina de Ciencias, a fellow of the Institute of Mathematical Statistics, the Heidegger Circle, the American Association for the Advancement of Science and the Husserl Circle. He received the 1996-97 James R Killian Jr. Faculty Achievement Award from MIT for his work as a:... leading innovator and theorist in the transformation of combinatorics from a disparate collection of facts and techniques unworthy of serious mathematical consideration into an active, systematic and profound branch of modern pure and applied mathematics. He held four honorary degrees from the University of Strasbourg (1984), the University L'Aquila (1990), the University of Bologna (1996), and Brooklyn Polytechnical University (1997). E F Beschler writes in [3]:Gian-Carlo Rota was a mathematician and a philosopher, and the richness of his writing in these fields was known to both communities. I also like to think of him as a poet - not in the formal sense, since to the best of my knowledge he never wrote a poem - but in the larger sense of a person who expresses himself with imaginative power and beauty of thought, even when many of these thoughts were sardonic reflections on people, ideas, institutions, and the general condition of humanity. His sense of humour was biting and deep - and full of truth. And his modes of expression poetic in a fundamental sense of the word. Rota died in his sleep and was found in bed on the afternoon of 19 April 1999. He had been due to give a series of three lectures at Temple University, the Groswald Memorial Lectures, on the previous day and, when he failed to arrive in Philadelphia, a check was made at his home. The cause of his death was atherosclotic cardiovascular disease. There were many tributes to Rota after his death, for example David Sharp wrote:The first thing you noticed about Gian-Carlo was his love for the life of the mind. He lived and breathed mathematics and philosophy. He had a passion for ideas, an uncompromising dedication to the truth, and a boundless curiosity. Brian D Taylor, one of the last of Rota's forty-two doctoral students, wrote:I remember Rota's characteristic generosity: The research problems he shared freely, not just with his graduate students, but with anyone he talked to or taught; the dinners out at good restaurants ... I remember travelling to his apartment in Harvard square to talk about my work even when he wasn't going to be at MIT. I remember telling him about results in my thesis, drawing tableaux on the slightly rickety easel he had up in his living room. And I remember a weekend spent at his apartment, mostly sitting at the Macintosh in his bedroom redrafting my first paper; when I shown him my first draft of our work, he asked "are you trying to hide your techniques from your readers?'' By the time that weekend was over, the techniques, and the paper were clear. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rota.html (3 of 4) [2/16/2002 11:29:40 PM]

Rota

Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles)

A Quotation

Mathematicians born in the same country Honours awarded to Gian-Carlo Rota (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1998

Other Web sites

1. MIT (Obituary) 2. MIT (An interview with Rota) 3. MIT ("Mathematical Snapshots" lecture) 4. Ohio State ("Ten lessons I wish I'd been taught" lecture) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rota.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rota.html (4 of 4) [2/16/2002 11:29:40 PM]

Roth

Leonard Roth Born: 29 Aug 1904 in Edmonton, London, England Died: 28 Nov 1968 in Pittsburgh, Pennsylvania, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Leonard Roth graduated from Cambridge in 1926 then undertook research under H F Baker. He spent a year in Rome (1930-31) learning from Castelnuovo, Enriques, Levi-Cività and Severi. He was greatly influenced by the work of these mathematicians and his future research directions were very much laid down at this time. Roth was appointed to Imperial College London in 1931 where he remained until 1965 when he went to Pittsburg, first as a visiting professor, then as Andrew Mellon Professor of Mathematics. Almost all his work was on geometry where he extended work begun in the Italian school. Article by: J J O'Connor and E F Robertson List of References (2 books/articles)

A Quotation

Mathematicians born in the same country Honours awarded to Leonard Roth (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh Other Web sites

Liouville-Roth Constants

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roth.html (1 of 2) [2/16/2002 11:29:42 PM]

Roth

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Roth.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roth.html (2 of 2) [2/16/2002 11:29:42 PM]

Roth_Klaus

Klaus Friedrich Roth Born: 29 Oct 1925 in Breslau, Germany (now Wroclaw, Poland)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Klaus Roth came to England when he was young and attended St Paul's School in London from 1939 to 1943. He then went to Peterhouse, Cambridge where he was awarded his BA in 1945. After graduating, Roth was appointed as an assistant master at the internationally famous Gordonstoun School, which lies 10 km north of Elgin in Scotland. The school had been founded in 1934 by the German educator Kurt Hahn as a boys' school that would emphasise the development of character in addition to academic excellence. The boys were expected to live in quite hard conditions without any of the luxuries of life. Roth returned to London in 1946 to undertake research at University College. He was awarded his master's degree in 1948 and appointed an assistant lecturer there in that year. He was awarded his doctorate two years later, becoming a lecturer, then a reader in 1956, and then a professor in 1961. In fact Roth made a remarkable mathematical breakthrough while still a lecturer at University College. He solved the major open problem of approximating algebraic numbers by rationals in 1955. It was for this work that Roth was awarded a Fields Medal in 1958. For any irrational number r it is easy to see that there are infinitely many rational numbers a/b with | a/b - r | < 1/b2 (the convergents for the continued fraction of r all satisfy this). For a given r let the exponents e such that there are infinitely many rational numbers a/b with | a/b - r | < 1/be.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roth_Klaus.html (1 of 3) [2/16/2002 11:29:44 PM]

(r) be the upper bound

Roth_Klaus

The above shows that

(r) 2 for all r.

Liouville showed in 1844 that if r is an algebraic number of degree n then The open question was then to find where in the range 2

(r) n the value of

Thue showed that

(r) n.

(r) was for an algebraic number of degree n.

(r) n/2 + 1 in 1908 and Siegel improved this in 1921 to

(r) 2 n.

Roth solved the problem completely in 1955 by showing that for any algebraic number r,

(r) = 2.

Davenport presented Roth with the Fields Medal at the International Congress in Edinburgh in 1958. Speaking of Roth's solution to this problem of approximating algebraic numbers Davenport said [2]:The achievement is one that speaks for itself: it closes a chapter, and a new chapter is now opened. Roth's theorem settles a question which is both of a fundamental nature and of extreme difficulty. It will stand as a landmark in mathematics for as long as mathematics is cultivated. Davenport, in his Fields Medal presentation, mentions another problem solved by Roth. This was Roth's proof in 1952 of a conjecture made in 1935 by Erdös and Turán. The conjecture concerned a sequence n1, n2, n3, ... of natural numbers satisfying np + nq 2nr unless p = q = r. If N(x) denotes the number of terms of the sequence less than x, Roth proved the conjecture that N(x)/x

0 as x

.

Davenport ends his address [2] by saying:The Duchess, in Alice in Wonderland, said that there is a moral in everything if only you can find it. It is not difficult to find the moral in Dr Roth's work. It is that the great unsolved problems of mathematics may still yield to direct attack however difficult and forbidding they appear to be, and however much effort has already been spent on them. Roth moved to the chair of Pure Mathematics in Imperial College, London in 1966 and held this chair until 1988. In that year he returned to Imperial College as a visiting professor a position he held until 1996 when he returned to the north of Scotland, not far from where he taught at Gordonstoun School before he began his research career. The Fields Medal was not the only honour to be bestowed on Roth. He received many other honours including fellowship of the Royal Society in 1960 and of the Royal Society of Edinburgh in 1993. He received the De Morgan Medal of the London Mathematical Society in 1983 and the Sylvester Medal of the Royal Society in 1991. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roth_Klaus.html (2 of 3) [2/16/2002 11:29:44 PM]

Roth_Klaus

Mathematicians born in the same country Honours awarded to Klaus Roth (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1958

Fellow of the Royal Society

Elected 1960

LMS De Morgan Medal

Awarded 1983

Royal Society Sylvester Medal winner

1991

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Roth_Klaus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Roth_Klaus.html (3 of 3) [2/16/2002 11:29:44 PM]

Routh

Edward John Routh Born: 20 Jan 1831 in Quebec, Canada Died: 7 June 1907 in Cambridge, Cambridgeshire, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edward Routh came to England in 1842 and studied under De Morgan in London. He entered Peterhouse at the same time as Maxwell but Maxwell transferred to Trinity (perhaps because he felt Routh was too strong competition). In 1854 Routh was Senior Wrangler, Maxwell was second, while the Smith Prize was divided between them (the first time the prize had been awarded jointly). Routh became the most famous of the Cambridge coaches and he published famous advanced treatises which became standard applied mathematics texts such as A Treatise on Dynamics of Rigid Bodies (1860), A Treatise on Analytic Statistics (1891) and A Treatise on Dynamics of a Particle (1898). Although he was most famed as a teacher, his work on mechanics was important and in 1877 Routh was awarded the Adams Prize for work on dynamic stability. He was also elected a Fellow of the Royal Society on the strength of this work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Edward J Routh (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1872

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Routh.html (1 of 2) [2/16/2002 11:29:45 PM]

Routh

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Routh.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Routh.html (2 of 2) [2/16/2002 11:29:45 PM]

Rudio

Ferdinand Rudio Born: 2 Aug 1856 in Wiesbaden, Germany Died: 21 June 1929 in Zurich, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ferdinand Rudio attended secondary school in his home town of Wiesbaden. He entered the Eidgenössische Polytechnikum Zürich in 1874 to study mathematics and physics. from 1877 until 1880 Rudio studied at the University of Berlin. He obtained a doctorate from the University of Berlin with a thesis on Kummer's problem of determining all surfaces of which the centres of curvature form second order cofocal surfaces. He reduced this problem to the problem of solving a differential equation. In 1881 Rudio returned to the Eidgenössische Polytechnikum Zürich as a lecturer and, in 1889, he was appointed professor of mathematics there. Rudio was to hold this professorship until he retired in 1928. Rudio worked on group theory, algebra and geometry. He is best remembered for his work in the history of mathematics, in particular he wrote a major article on squaring the circle and he also wrote biographies of mathematicians. One of his most important contributions to mathematics was editing the collected works of Euler. Rudio proposed the project in 1883 since this was the centenary of Euler's death. He continued to advocate the importance of this project and at the International Congress of Mathematicians at Zurich in 1897 he suggested it would be a suitable memorial for the year 1907 which was the bicentennial of Euler's birth. The project was not approved until 1909, twenty six years after Rudio first proposed it. Rudio was appointed general editor for the project. He edited two volumes himself and collaborated in the editing of three more. In fact he supervised the production of over 30 volumes in his role as general editor.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rudio.html (1 of 2) [2/16/2002 11:29:47 PM]

Rudio

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

A Quotation

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rudio.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rudio.html (2 of 2) [2/16/2002 11:29:47 PM]

Rudolff

Christoff Rudolff Born: 1499 in Jauer, Silesia (now Jawor, Poland) Died: 1545 in Vienna, Austria Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Christoff Rudolff studied algebra at the University of Vienna, between 1517 and 1521. He remained in Vienna after studying at the university and earned his living giving private lessons in mathematics. He did use the facilities offered by the university, being able to use books in the university library and talking with academics at the university. Rudolff's book Coss, written in 1525, is the first German algebra book. The reason for the title is that cosa is a thing which was used for the unknown. Algebraists were called cossists, and algebra the cossic art, for many years. Rudolff calculated with polynomials with rational and irrational coefficients and was aware that ax2 + b = cx has 2 roots. He used for square roots (the first to use this notation) and for cube roots and for 4 th roots. He has the idea that x0 = 1 which is important. The year after Coss appeared Rudolff produced Künstliche Rechnung mit der Ziffer und mit den Zahlpfennigen. This studied applications of mathematics to commerce. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1500 to 1600

Honours awarded to Christoff Rudolff (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Rima Rudolf

Other Web sites

The Galileo Project

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rudolff.html (1 of 2) [2/16/2002 11:29:49 PM]

Rudolff

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rudolff.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rudolff.html (2 of 2) [2/16/2002 11:29:49 PM]

Ruffini

Paolo Ruffini Born: 22 Sept 1765 in Valentano, Papal States (now Italy) Died: 10 May 1822 in Modena, Duchy of Modena (now Italy)

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paolo Ruffini's father, Basilio Ruffini, was a medical doctor in Valentano. As a young child Paolo was [4]:... of a mystical temperament and appeared to be destined for the priesthood... The family moved to Reggio, near Modena in the Emilia-Romagna region of northern Italy, when Paolo was a teenager. He entered the University of Modena in 1783 where he studied mathematics, medicine, philosophy and literature. Among his teachers of mathematics at Modena were Luigi Fantini, who taught Ruffini geometry, and Paolo Cassiani, who taught him calculus. The Este family ruled Modena and, in 1787, Cassiani was appointed as a councillor for the Este estates. Cassiani's course at Modena on the foundations of analysis was taken over by Ruffini in 1787-88 although he was still a student at this time. On 9 June 1788 Ruffini graduated with a degree in philosophy, medicine and surgery. Soon after this he graduated with a mathematics degree. Ruffini must have made a good job of the foundations of analysis course he took over from Cassiani for, on 15 October 1788, he was appointed professor of the foundations of analysis. Fantini, who had taught Ruffini geometry when he was an undergraduate, found his eyesight deteriorating and in 1791 he had to resign his post at Modena. Ruffini was appointed to fill the position of Professor of the Elements of Mathematics in 1791. However, Ruffini was not only a mathematician. He had trained in medicine and, also in 1791, he was granted a licence to practice medicine by the Collegiate Medical Court of Modena. This was a time of wars following the French Revolution. By early 1795 France had won victories on every front. In northern Italy the French army threatened Austrian-Sardinian positions, but its

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ruffini.html (1 of 5) [2/16/2002 11:29:51 PM]

Ruffini

commander failed to take the initiative. In March 1796 he was replaced by Napoleon Bonaparte who executed a brilliant campaign of manoeuvres. Taking the offensive on 12 April and successively defeated and separated the Austrian and the Sardinian armies and then marched on Turin. The King of Sardinia asked for an armistice and Nice and Savoy were annexed to France. Bonaparte continued the war against the Austrians and occupied Milan but was held up at Mantua. Before Mantua fell to his armies he signed armistices with the duke of Parma and the duke of Modena. Napoleon's troops occupied Modena and, much against his wishes, Ruffini found himself in the middle of the political upheaval. Napoleon set up the Cisalpine Republic consisting of Lombardy, Emilia, Modena and Bologna. Although not wishing to get involved, Ruffini found himself appointed as a representative to the Junior Council of the Cisalpine Republic. However, he soon left this position and, in early 1798, he returned to his scientific work at the University of Modena. He was required to swear an oath of allegiance to the republic and this Ruffini found he could not bring himself to do on religious grounds. By failing to swear the oath he lost his professorship and was barred from teaching. Ruffini did not seem greatly disturbed by the loss of his chair, in fact he was a very calm man who took all the dramatic events around him in his stride. The fact that he could not teach mathematics meant that he had more time to practise medicine and therefore help his patients to whom he was extremely devoted. On the other hand it gave him the chance to work on what was one of the most original of projects, namely to prove that the quintic equation cannot be solved by radicals. To solve a polynomial equation by radicals meant finding a formula for its roots in terms of the coefficients so that the formula only involves the operations of addition, subtraction, multiplication, division and taking roots. Quadratic equations (of degree 2) had been known to be soluble by radicals from the time of the Babylonians. The cubic equation had been solved by radicals by del Ferro, Tartaglia and Cardan. Ferrari had solved the quartic by radicals in 1540 and so 250 years had passed without anyone being able to solve the quintic by radicals despite the attempts of many mathematicians. Among those who had made serious attempts to understand the problem were Tschirnhaus, Euler, Bézout, Vandermonde, Waring and Lagrange. It appears that nobody before Ruffini really believed that the quintic could not be solved by radicals. Certainly no mathematician has published such a claim and even Lagrange in his famous paper Reflections on the resolution of algebraic equations says he will return to the question of the solution of the quintic and, clearly, he still hoped to solve it by radicals. In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals as the title shows: General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible. The introduction to the book begins:The algebraic solution of general equations of degree greater than four is always impossible. Behold a very important theorem which I believe I am able to assert (if I do not err): to present the proof of it is the main reason for publishing this volume. The immortal Lagrange, with his sublime reflections, has provided the basis of my proof. Ruffini used group theory in his work but he had to invent the subject for himself. Lagrange had used permutations and one can argue that groups appear in Lagrange's work but since Lagrange never composed permutations it is rather with hindsight that we now see the beginnings of group theory in his paper. Ruffini is the first to introduce the notion of the order of an element, conjugacy, the cycle

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ruffini.html (2 of 5) [2/16/2002 11:29:51 PM]

Ruffini

decomposition of elements of permutation groups and the notions of primitive and imprimitive. He proved some remarkable theorems (not of course with the modern terminology quoted below):The order of a permutation is the least common multiple of the lengths in the decomposition into disjoint cycles. An element of S5 of order 5 is a 5-cycle. If G < S5 has order divisible by 5 then G has an element of order 5. S5 has no subgroups of index 3, 4 or 8. It is remarkable work and, except for one gap, proves the theorem as Ruffini claimed. The proof is given in modern notation in [4]. However there was a strange lack of response to Ruffini's work from mathematicians. In 1801 Ruffini sent a copy of his book to Lagrange. He received no response and so he sent a second copy with a covering letter [4]:Because of the uncertainty that you may have received my book, I send you another copy. If I have erred in any proof, or if I have said something which I believed new, and which is in reality not new, finally if I have written a useless book, I pray you point it out to me sincerely. Again Ruffini received no reply and he wrote yet again in 1802:No one has more right ... to receive the book which I take the liberty of sending you. ... In writing this book, I had principally in mind to give a proof of the impossibility of solving equations of degree higher than four. Some mathematicians accepted Ruffini's proof although one would have to say that Pietro Paoli, the professor at Pisa, was influenced by patriotic motives when he wrote in 1799 [4]:I read with much pleasure your book ... and recommend greatly the most important theorem which excludes the possibility of solving equations of degree greater than four. I rejoice exceedingly with you and with our Italy, which has seen a theory born and perfected and to which other nations have contributed little... To understand this quotation one has to realise that Lagrange was born in Turin which was part of Italy at the time. This patriotic reaction apart, the world of mathematics seemed to almost ignore Ruffini's great result. So how did Ruffini react? He published a second proof in 1803 which he hoped might be more easily understood, writing in the introduction:In the present memoir, I shall try to prove the same proposition [insolubility of the quintic] with, I hope, less abstruse reasoning and with complete rigour. At least Ruffini received comments from Malfatti concerning this paper, but unfortunately Malfatti had not understood Ruffini's arguments and raised a fallacious objection. Ruffini published further proofs in 1808 and 1813. Of this last proof Ayoub writes in [4]:Can anything be more elegant? This proof is essentially what was later called the Wantzel modification of Abel's proof and was published in 1845. It is no surprise that it should resemble Ruffini's proof, since Wantzel says in his paper ..."using works of Abel and Ruffini...".

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ruffini.html (3 of 5) [2/16/2002 11:29:51 PM]

Ruffini

Ruffini did not stop trying to have his work recognised by the mathematical community. When Delambre wrote in a report on the state of mathematics since 1789:Ruffini proposes to prove that it is impossible ..., Ruffini replied:... I not only proposed to prove but in reality did prove ... . One has to feel desperately sorry for Ruffini. If some mathematician had written to him showing him there was an error or even a gap in the proof, then at least Ruffini would have had the chance to correct it. However, it seemed that nobody really wanted to know that quintics could not be solved by radicals. Ruffini asked the Institute in Paris to pronounce on the correctness of his proof and Lagrange, Legendre and Lacroix were appointed to examine it. Again they produced a report which was highly unsatisfactory as far as Ruffini was concerned:... if a thing is not of importance, no notice is taken of it and Lagrange himself, "with his coolness" found little in it worthy of attention. The Royal Society were also asked to pronounce on the correctness and Ruffini received a somewhat kinder reply which said that although they did not give approval of particular pieces of work they were quite sure that it proved what was claimed. The one person who did acknowledge the importance and correctness was Cauchy. This is all the more surprising since Cauchy was one of the worst of all mathematicians at giving credit to others. He wrote to Ruffini in 1821, less than a year before Ruffini's death [1]:... your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgement, proves completely the impossibility of solving algebraically equations of higher than the fourth degree. In fact Cauchy had written a major work on permutation groups between 1813 and 1815 and in it he generalised some of Ruffini's results. He had certainly been greatly influenced by Ruffini's ideas. This influence through Cauchy is perhaps the only way in which Ruffini's work was to make an impact on the development of mathematics. We left the story of Ruffini's career around 1799 when he began his publications on the quintic. He left the University of Modena to spend 7 years teaching applied mathematics in the military school in Modena. He continued to practice medicine and tend to patients from the poorest to the richest in society. After the fall of Napoleon, Ruffini became rector of the University of Modena in 1814. The political situation was still extremely complex and despite his personal skills, the great respect in which he was held, and his reputation for honesty, his time as rector must have been a very difficult one. As well as the rectorship, Ruffini held a chair of applied mathematics, a chair of practical medicine and a chair of clinical medicine in the University of Modena. In 1817 there was a typhus epidemic and Ruffini continued to treat his patients until he caught the disease himself. Although he made a partial recovery, he never fully regained his health and in 1819 he gave up his chair of clinical medicine. He did not give up his scientific work, however, and in 1820 he published a scientific article on typhus based on his own experience with the disease. There are further aspects of Ruffini's work which should be mentioned. He wrote several works on philosophy, one of which argues against some of Laplace's philosophical ideas. He also wrote on http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ruffini.html (4 of 5) [2/16/2002 11:29:51 PM]

Ruffini

probability and the application of probability to evidence in court cases. Given the information in this article about the insolubility of the quintic, it is reasonable to ask why Abel has been credited with proving the theorem while Ruffini has not. Ayoub suggests that [4]:... the mathematical community was not ready to accept so revolutionary an idea: that a polynomial could not be solved in radicals. Then, too, the method of permutations was too exotic and, it must be conceeded, Ruffini's early account is not easy to follow. ... between 1800 and 1820 say, the mood of the mathematical community ... changed from one attempting to solve the quintic to one proving its impossibility... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (18 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. The development of group theory 2. Arabic mathematics : forgotten brilliance?

Other references in MacTutor

Chronology: 1780 to 1800

Other Web sites

1. The Catholic Encyclopedia 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ruffini.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ruffini.html (5 of 5) [2/16/2002 11:29:51 PM]

Runge

Carle David Tolmé Runge Born: 30 Aug 1856 in Bremen, Germany Died: 3 Jan 1927 in Göttingen, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

At the age of 19, after leaving school, Carle Runge spent 6 months with his mother visiting the cultural centres of Italy. On his return to Germany he enrolled at the University of Munich to study literature. However after 6 weeks of the course he changed to mathematics and physics. Runge attended courses with Max Planck and they became close friends. In 1877 both went to Berlin but Runge turned to pure mathematics after attending Weierstrass's lectures. His doctoral dissertation (1880) dealt with differential geometry. After taking his secondary school teachers examinations he returned to Berlin where he was influenced by Kronecker. Runge then worked on a procedure for the numerical solution of algebraic equations in which the roots were expressed as infinite series of rational functions of the coefficients. Runge published little at that stage but after visiting Mittag-Leffler in Stockholm in September 1884 he produced a large number of papers in Mittag-Leffler's journal Acta mathematica (1885). Runge obtained a chair at Hanover in 1886 and remained there for 18 years. Within a year Runge had moved away from pure mathematics to study the wavelengths of the spectral lines of elements other than hydrogen (J J Balmer had constructed a formula for the spectral lines of helium.) Runge did a great deal of experimental work and published a great quantity of results. He succeeded in arranging the spectral lines of helium in two spectral series and, until 1897, this was thought to be evidence that hydrogen was a mixture of two elements.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Runge.html (1 of 2) [2/16/2002 11:29:53 PM]

Runge

In 1904 Klein persuaded Göttingen to offer Runge a chair of Applied Mathematics, a post which Runge held until he retired in 1925. Runge was always a fit and active man and on his 70 th birthday he entertained his grandchildren by doing handstands. However a few months later he had a heart attack and died. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) A Poster of Carle Runge

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1900 to 1910

Honours awarded to Carle Runge (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Runge

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Runge.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Runge.html (2 of 2) [2/16/2002 11:29:53 PM]

Russell

Bertrand Arthur William Russell Born: 18 May 1872 in Ravenscroft, Trelleck, Monmouthshire, Wales Died: 2 Feb 1970 in Penrhyndeudraeth, Merioneth, Wales

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bertrand Russell published a large number of books on logic, the theory of knowledge, and many other topics. He is one of the most important logicians of the 20th Century. Russell's Mathematical Contributions Over a long and varied career, Bertrand Russell made ground-breaking contributions to the foundations of mathematics and to the development of contemporary formal logic, as well as to analytic philosophy. His contributions relating to mathematics include his discovery of Russell's paradox, his defence of logicism (the view that mathematics is, in some significant sense, reducible to formal logic), his introduction of the theory of types, and his refining and popularizing of the first-order predicate calculus. Along with Kurt Gödel, he is usually credited with being one of the two most important logicians of the twentieth century. Russell discovered the paradox which bears his name in May 1901, while working on his Principles of Mathematics (1903). The paradox arose in connection with the set of all sets which are not members of themselves. Such a set, if it exists, will be a member of itself if and only if it is not a member of itself. The significance of the paradox follows since, in classical logic, all sentences are entailed by a contradiction. In the eyes of many mathematicians (including David Hilbert and Luitzen Brouwer) it therefore appeared that no proof could be trusted once it was discovered that the logic apparently underlying all of mathematics was contradictory. A large amount of work throughout the early part of this century in logic, set theory, and the philosophy and foundations of mathematics was thus prompted. Russell's paradox arises as a result of naive set theory's so-called unrestricted comprehension (or http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Russell.html (1 of 5) [2/16/2002 11:29:55 PM]

Russell

abstraction) axiom. Originally introduced by Georg Cantor, the axiom states that any predicate expression, P(x), which contains x as a free variable, will determine a set whose members are exactly those objects which satisfy P(x). The axiom gives form to the intuition that any coherent condition may be used to determine a set (or class). Most attempts at resolving Russell's paradox have therefore concentrated on various ways of restricting or abandoning this axiom. Russell's own response to the paradox came with the introduction of his theory of types. His basic idea was that reference to troublesome sets (such as the set of all sets which are not members of themselves) could be avoided by arranging all sentences into a hierarchy (beginning with sentences about individuals at the lowest level, sentences about sets of individuals at the next lowest level, sentences about sets of sets of individuals at the next lowest level, etc.). Using the vicious circle principle also adopted by Henri Poincaré, together with his so-called "no class" theory of classes, Russell was then able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function "x is a set", should not be applied to themselves since self-application would involve a vicious circle. On this view, it follows that it is possible to refer to a collection of objects for which a given condition (or predicate) holds only if they are all at the same level or of the same "type". Although first introduced by Russell in 1903 in the Principles, his theory of types finds its mature expression in his 1908 article Mathematical Logic as Based on the Theory of Types and in the monumental work he co-authored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913). Thus, in its details, the theory admits of two versions, the "simple theory" and the "ramified theory". Both versions of the theory later came under attack. For some, they were too weak since they failed to resolve all of the known paradoxes. For others, they were too strong since they disallowed many mathematical definitions which, although consistent, violated the vicious circle principle. Russell's response to the second of these objections was to introduce, within the ramified theory, the axiom of reducibility. Although the axiom successfully lessened the vicious circle principle's scope of application, many claimed that it was simply too ad hoc to be justified philosophically. Of equal significance during this same period was Russell's defence of logicism, the theory that mathematics was in some important sense reducible to logic. First defended in his Principles, and later in more detail in Principia Mathematica, Russell's logicism consisted of two main theses. The first is that all mathematical truths can be translated into logical truths or, in other words, that the vocabulary of mathematics constitutes a proper subset of that of logic. The second is that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of those of logic. Like Gottlob Frege, Russell's basic idea for defending logicism was that numbers may be identified with classes of classes and that number-theoretic statements may be explained in terms of quantifiers and identity. Thus the number 1 would be identified with the class of all unit classes, the number 2 with the class of all two-membered classes, and so on. Statements such as "there are two books" would be recast as "there is a book, x, and there is a book, y, and x is not identical to y". It followed that number-theoretic operations could be explained in terms of set-theoretic operations such as intersection, union, and the like. In Principia Mathematica, Whitehead and Russell were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory. A fourth

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Russell.html (2 of 5) [2/16/2002 11:29:55 PM]

Russell

volume on geometry was planned but never completed. In much the same way that Russell wanted to use logic to clarify issues in the foundations of mathematics, he also wanted to use logic to clarify issues in philosophy. As one of the founders of "analytic philosophy", Russell is remembered for his work using first-order logic to show how a broad range of denoting phrases could be recast in terms of predicates and quantified variables. Thus, he is also remembered for his emphasis upon the importance of logical form for the resolution of many related philosophical problems. Here, as in mathematics, it was Russell's hope that by applying logical machinery and insights one would be able to resolve otherwise intractable difficulties. Russell's Life and Public Influence Russell was born the grandson of Lord John Russell, who had twice served as Prime Minister under Queen Victoria. Following the death of his mother (in 1874) and of his father (in 1876), Russell and his brother went to live with their grandparents. (Although Russell's father had granted custody of Russell and his brother to two atheists, Russell's grandparents had little difficulty in getting his will overturned.) Following the death of his grandfather (in 1878), Russell was raised by his grandmother, Lady Russell. Educated at first privately, and later at Trinity College, Cambridge, Russell obtained first class degrees both in mathematics and in the moral sciences. Although elected to the Royal Society in 1908, Russell's career at Trinity appeared to come to an end in 1916 when he was convicted and fined for anti-war activities. He was dismissed from the College as a result of the conviction. (The details of the dismissal are recounted in Bertrand Russell and Trinity (1942) by G H Hardy.) Two years later Russell was convicted a second time. This time he spent six months in prison. It was while in prison that he wrote his well-received Introduction to Mathematical Philosophy (1919). He did not return to Trinity until 1944. Married four times and notorious for his many affairs, Russell also ran unsuccessfully for Parliament, in 1907, 1922, and 1923. Together with his second wife, he opened and ran an experimental school during the late 1920s and early 1930s. He became the third Earl Russell upon the death of his brother in 1931. While teaching in the United States in the late 1930s, Russell was offered a teaching appointment at City College, New York. The appointment was revoked following a large number of public protests and a judicial decision, in 1940, which stated that he was morally unfit to teach at the College. Nine years later he was awarded the Order of Merit. He received the Nobel Prize for Literature in 1950. During the 1950s and 1960s, Russell became something of an inspiration to large numbers of idealistic youth as a result of his continued anti-war and anti-nuclear protests. Together with Albert Einstein, he released the Russell-Einstein Manifesto in 1955, calling for the curtailment of nuclear weapons. In 1957, he was a prime organizer of the first Pugwash Conference, which brought together scientists concerned about the proliferation of nuclear weapons. He became the founding president of the Campaign for Nuclear Disarmament in 1958 and was once again imprisoned, this time in connection with anti-nuclear protests, in 1961. Upon appeal, his two-month prison sentence was reduced to one week in the prison hospital. He remained a prominent public figure until his death nine years later at the age of 97. Article by: A.D. Irvine, [email protected] or [email protected] Click on this link to see a list of the Glossary entries for this page http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Russell.html (3 of 5) [2/16/2002 11:29:55 PM]

Russell

List of References (44 books/articles)

Some Quotations (26)

A Poster of Bertrand Russell

Mathematicians born in the same country

Cross-references to History Topics

The beginnings of set theory

Other references in MacTutor

1. Chronology: 1900 to 1910 2. Chronology: 1910 to 1920

Honours awarded to Bertrand Russell (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1950

Fellow of the Royal Society

Elected 1908

Royal Society Sylvester Medal

Awarded 1934

LMS De Morgan Medal

Awarded 1932

Other Web sites

1. Bertrand Russell Archives 2. Bertrand Russell Editorial Project 3. The Bertrand Russell Society 4. Madison (Links to some of Russell's writings) 5. Nobel prizes site (A Biography of Russell, his Nobel lecture and his Nobel prize presentation speech) 6. Stanford Encyclopedia of Philosophy 7. Stanford Encyclopedia of Philosophy (Principia Mathematica) 8. Cut the Knot (The Russell Paradox) 9. Internet Encyclopedia of Philosophy (The Russell Paradox) 10. Eric Schechter (The Axiom of Choice) 11. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Russell.html (4 of 5) [2/16/2002 11:29:55 PM]

Russell

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Russell.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Russell.html (5 of 5) [2/16/2002 11:29:55 PM]

Russell_Scott

John Scott Russell Born: 9 May 1808 in Parkhead, near Glasgow, Scotland Died: 8 June 1882 in Ventnor, Isle of Wight, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Scott Russell was primarily an engineer and naval architect, rather than a mathematician; but his name is well-known to applied mathematicians today through his experimental discovery of the 'solitary wave'. This is now recognised as a fundamental ingredient in the theory of 'solitons', applicable to a wide class of nonlinear partial differential equations. Russell's father, David, was a parish school teacher and a graduate of Glasgow University. His mother, Agnes Clark Scott, died soon after John's birth. He was her only child. When John was three years old, his father became minister of Colinsburgh Relief Church, near Kirkcaldy, a congregation which had seceded from the Church of Scotland; but he soon moved on to a larger congregation in Hawick, where he remarried and had more children. Later, he moved again, to Errol, near Perth. John spent just one year at St Andrews University, before transferring to Glasgow University; there, he first added his mother's name, Scott, to his own. He graduated in 1825, aged 17. In 1825, he moved to Edinburgh, where he taught mathematics at a 'South Academy', perhaps founded by himself and a friend. Later, he also taught at the Leith Mechanics Institute, and gave courses on mathematics and natural philosophy to medical students, under the auspices of the Royal College of Surgeons. During 1832-33, following the death of John Leslie, he substituted for the professor of natural philosophy at Edinburgh University; but he did not apply for the vacant post, which he was sure would go to David Brewster. In fact, the successful candidate was the 23 year-old James D Forbes, who had strong political and academic supporters. In 1838, Russell was an unsuccessful applicant for the chair of Mathematics, which Philip Kelland secured. In the 1830's Russell developed a prototype passenger-carrying steam carriage, but it met with opposition http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Russell_Scott.html (1 of 3) [2/16/2002 11:29:57 PM]

Russell_Scott

from the road trustees, and the venture failed. He had more success with the Union Canal Company, investigating the feasability of steam-powered canal transport and studying the connection between resistance to motion and wave-generation. He spoke of his work at early meetings of the British Association for the Advancement of Science, founded in 1831; and the Association appointed him and Sir John Robison of Edinburgh to a 'Committee on Waves' to conduct observations and experiments. A substantial report by Robison and Russell appeared in 1837 and Russell alone wrote his major Report on Waves (1844) following Robison's death in 1843. These reports, in fact all Russell's own work, contain a remarkable series of observations, at sea, in rivers and canals, and in Russell's own wave tank constructed for the purpose. There, Russell's "Great Wave of Translation" is described in detail. Russell's experimental work helped to stimulate a revival in theoretical hydrodynamics in Britain. George Green, George Airy, Philip Kelland and Samuel Earnshaw all attempted theoretical descriptions of the solitary wave, but these were not successful. Airy objected to the emphasis placed by Russell on his "Great Primary Wave", wrongly arguing that it was neither great nor primary, but just one consequence of the linear shallow water theory which he (Airy) had given. G G Stokes was more cautious, but he also doubted that the solitary wave could propagate without change in form. A correct approximate theory was at last given by Boussinesq (1871) and Rayleigh (1876); but the issue was not really settled until the appearance of the important paper by Korteweg and de Vries (1895). The full significance of the solitary wave and its generalisation was finally uncovered in 1965: see [4] and [5]. After working for a shipbuilder in Greenock, Russell moved to London in 1844, with his wife and two young children. He first worked on a railway magazine, and he became secretary of the Society of Arts, a post which led to major involvement in planning the Great Exhibition of 1851. Thereafter, he became increasingly involved in the design of yachts, boats, barges and ships, and he became a director of a ship-building company. He collaborated with Isambard Kingdom Brunel on the construction of the pioneering iron vessels Great Britain and Great Eastern; and was alone responsible for the successful iron warship Warrior. In the 1860's, Russell's reputation suffered a number of setbacks from which it never recovered. He became embroiled in a lengthy financial dispute about an armaments contract; the Great Eastern suffered a serious breakdown; and he was controversially expelled from the Institute of Civil Engineers. His major published work, The Modern System of Naval Architecture, appeared in 1865. Much later, he revisited his work on solitary waves, published posthumously as The Wave of Translation in the Oceans of Water, Air and Ether (1885). Emmerson's biography of Russell [1] deals authoritatively with his contributions of naval architecture, concluding that Russell's great love was applied science, and it was perverse fate which channelled him into complicating it with the economic cares of a large and uncertain business... He was a great patriot and he sought to serve great ends in the national interest... He was not embittered by the shabby treatment he received from fate, from his fellows or from his country. Article by: Alex D D Craik, University of St Andrews. List of References (7 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Russell_Scott.html (2 of 3) [2/16/2002 11:29:57 PM]

Russell_Scott

Mathematicians born in the same country Other Web sites

1. C Eilbeck 2. BBC, UK

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Russell_Scott.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Russell_Scott.html (3 of 3) [2/16/2002 11:29:57 PM]

Rutherford

Daniel Edwin Rutherford Born: 4 July 1906 in Stirling, Scotland Died: 9 Nov 1966 in St Andrews, Fife, Scotland

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dan Rutherford was an undergraduate at St Andrews receiving his M.A. in 1928. Turnbull advised him to undertake research in Amsterdam and he obtained a doctorate from there with a thesis on modular invariants. His thesis appears as the Cambridge Tract Modular Invariants (1932) and was reprinted in New York in 1964. After one year in Edinburgh he joined the staff in St Andrews. He was appointed to the Gregory Chair of Applied Mathematics in 1964. Despite his mathematical interests, since Copson held the Regius Chair of Mathematics, Rutherford was appointed to the new chair of applied mathematics. Rutherford's most important work was Substitutional Analysis (1948) in which explicit representations of the symmetric group are given. Rutherford was a talented amateur artist. You can see a picture he drew of the Church of the Holy Rude in Stirling. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rutherford.html (1 of 2) [2/16/2002 11:29:59 PM]

Rutherford

Mathematicians born in the same country Honours awarded to Dan Rutherford (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rutherford.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rutherford.html (2 of 2) [2/16/2002 11:29:59 PM]

Rydberg

Johannes Robert Rydberg Born: 8 Nov 1854 in Halmstad, Sweden Died: 28 Dec 1919 in Lund, Sweden

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johannes Rydberg's father was Sven Rydberg while his mother was Maria Anderson. Johannes attended school in Halmstad which is in southwestern Sweden, on the eastern shore of the Kattegat, at the mouth of Nissan River. He completed his secondary school education at the Gymnasium in Halmstad in 1873 and in the same year he entered the University of Lund. The University of Lund, in the town of Lund in southern Sweden northeast of Malmo, is the second oldest university in Sweden being founded in 1666. Rydberg received his bachelor's degree in 1875 from the University of Lund. He continued his study of mathemtics and wrote a dissertation on conic sections for his doctorate in mathematics which was awarded in 1879. The following year he was appointed to the post of lecturer in mathematics at Lund but his interests were now turning towards mathematical physics rather than to pure mathematics. During his two years as a lecturer in mathematics he worked on problems relating to electricity. In 1882 Rydberg moved from a lectureship in mathematics to become a lecturer in physics at Lund. Ten years later he was promoted to assistant at the Physics Institute. In 1879 he was promoted to a professorship in physics but the was only a temporary position until it was confirmed as a permanent appointment in March 1901. From this time until his retirement in 1919 he held the chair of physics at Lund. However, his health deteriotated during the time that he held the chair and he became seriously ill in 1914. Although he continued to hold the chair he took sick leave in 1914 and was absent from the university from that time on. His final retirement came five years after he had ceased to be able to work and came only a few weeks before his death. Manne Siegbahn, who had been a student of Rydberg from 1906 to 1911, then Rydberg's assistant from 1911 to 1914, took over his teaching duties in 1914. He carried these out until Rydberg formally retired http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rydberg.html (1 of 2) [2/16/2002 11:30:01 PM]

Rydberg

in 1919, then early in 1920 he was appointed to Rydberg's chair of physics. Siegbahn wrote the biography of Rydberg [5]. Rydberg's most important work is on spectroscopy where he found a relatively simple expression relating the various lines in the spectra of the elements (1890). He hoped to determine the structure of the atom but, although his work did provide the basis for the structure theory, he himself did not reach his goal. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Honours awarded to Johannes Rydberg (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1919

Lunar features

Crater Rydberg

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Rydberg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Rydberg.html (2 of 2) [2/16/2002 11:30:01 PM]

SGravesande

Willem Jacob 'sGravesande Born: 26 Sept 1688 in 'sHertogenbosch, Netherlands Died: 28 Feb 1742 in Leiden, Netherlands

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Willem 'sGravesande was a practising lawyer and he is important as an exponent of Newton's philosophy in Europe. His early education was at home with a private tutor, the he studied law at Leiden writing a doctoral thesis on 'suicide'. He practiced law at The Hague. Appointed as secretary to the Dutch Embassy, he was sent to England in 1715 to congratulate George I on has accession to the throne. While in London he was elected a Fellow of the Royal Society. He got to know Newton, Desaguliers and John Keill at this time and, after returning to The Hague in 1716, he continued to correspond with Keill. In 1717 'sGravesande became professor of mathematics and astronomy at Leiden. He became professor of philosophy at Leiden in 1734. He taught and wrote many texts on Newtonian science and Keill's contributions. Like Keill he conducted physics experiments in his lectures. 'sGravesande wrote textbooks on mathematics and philosophy. He also published and edited works of others, for example work by Huygens, Keill and Newton. 'sGravesande's book Mathematical Elements of Physics was very influential. Article by: J J O'Connor and E F Robertson List of References (2 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/SGravesande.html (1 of 2) [2/16/2002 11:30:03 PM]

SGravesande

Mathematicians born in the same country Honours awarded to Willem 'sGravesande (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1715

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/SGravesande.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/SGravesande.html (2 of 2) [2/16/2002 11:30:03 PM]

Saccheri

Giovanni Girolamo Saccheri Born: 5 Sept 1667 in San Remo, Genoa (now Italy) Died: 25 Oct 1733 in Milan, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giovanni Saccheri entered the Jesuit Order at Genoa in 1685. Five years later he went to Milan where he studied philosophy and theology at the Jesuit College. While in this College he was encouraged to take up mathematics by Tommaso Ceva. Saccheri was ordained a priest in 1694 at Como and then taught at various Jesuit Colleges through Italy. He taught philosophy at Turin from 1694 to 1697, philosophy and theology at Pavia from 1697 until his death. He also held the chair of mathematics at Pavia from 1699 until his death. In Euclides ab Omni Naevo Vindicatus (1733) Saccheri did important early work on non-euclidean geometry, although he did not see it as such, rather an attempt to prove the parallel postulate of Euclid. He also worked on mathematical logic. His Logica Demonstrativa (1697) treats logic with definitions, postulates and demonstrations in the style of Euclid. Among his other publications was a work on statics Neo-statica published in 1708 and his first work Quaesita geometrica (1693) written with considerable advice and help from Tommaso Ceva. In fact through Tommaso Ceva he also made the acquaintance of Giovanni Ceva and Viviani. He corresponded with all three of these mathematicians. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) Mathematicians born in the same country Some pages from publications

The title page of Euclides ab omni naevo vindicatus (1733) and another page.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saccheri.html (1 of 2) [2/16/2002 11:30:04 PM]

Saccheri

Cross-references to History Topics

Non-Euclidean geometry

Other references in MacTutor

Chronology: 1720 to 1740

Other Web sites

1. Jesuit Scientists 2. The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Saccheri.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saccheri.html (2 of 2) [2/16/2002 11:30:04 PM]

Sacrobosco

Johannes de Sacrobosco Born: about 1195 in Holywood, Yorkshire, England Died: 1256 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John of Holywood or Johannes de Sacrobosco was educated at Oxford. He became a canon of the Order of St Augustine at the monastery of Holywood in Nithsdale. In 1220 Sacrobosco went to study in Paris. Although almost all dates for Sacrobosco are guesses we do know one date precisely for, on 5 June 1221, he was appointed a teacher at the University of Paris. Soon after this he became professor of mathematics at Paris. Sacrobosco promoted Arabic methods of arithmetic and algebra in his teachings. In De Algorismo he discusses calculating with positive integers. The work contains 11 chapters, one each on topics such as addition, subtraction, multiplication, division, square roots and cube roots. In 1220 Sacrobosco wrote Tractatus de Sphaera a book on astronomy in four chapters. The first chapter deals with the shape and place of the Earth within a spherical universe. The second chapter deals with various circles on the shy. The third chapter describes rising and setting of heavenly bodies from different geographical locations while the fourth chapter gives a brief introduction to Ptolemy's theory of the planets and of eclipses. The book, which predates Grosseteste's astronomy book, is well written and was widely used throughout Europe from the middle of the 13th Century. Clavius used in the 16th Century and it was still the basic astronomy text until the 17th Century. It was essentially the first astronomy text to be printed in 1472. Despite its long life as a teaching book Barocius had pointed out 84 errors in the book as early as 1570. Sacrobosco wrote De Anni Ratione in 1232. This book deals with time, studying the day, week, month, year as well as the Moon and the ecclesiastical calendar. He maintains that the Julian calendar is 10 days in error and should be corrected. He suggest a reform of the calendar achieved by omitting one day every 288 years. He wrote a number of other book including Tractatus de Quadrante on the quadrant. Article by: J J O'Connor and E F Robertson List of References (6 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sacrobosco.html (1 of 2) [2/16/2002 11:30:06 PM]

Sacrobosco

Mathematicians born in the same country Other references in MacTutor

Chronology: 1100 to 1300

Honours awarded to John Sacrobosco (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Sacrobosco

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sacrobosco.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sacrobosco.html (2 of 2) [2/16/2002 11:30:06 PM]

Saint-Venant

Adhémar Jean Claude Barré de Saint-Venant Born: 23 Aug 1797 in Villiers-en-Bière, Seine-et-Marne, France Died: 6 Jan 1886 in St Ouen, Loir-et-Cher, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean Claude Saint-Venant was a student at the Ecole Polytechnique, entering the school in 1813 when he was sixteen years old. He graduated in 1816 and spent the next 27 years as a civil engineer. For the first seven of these 27 years Saint-Venant worked for the Service des Poudres et Salpêtres, then he spent the next twenty years working for the Service des Ponts et Chaussées. Saint-Venant attended lectures at the Collège de France and the lecture notes he took in Liouville's 1839-40 class still survive. He taught mathematics at the Ecole des Ponts et Chaussées where he succeeded Coriolis. Saint-Venant worked mainly on mechanics, elasticity, hydrostatics and hydrodynamics. Perhaps his most remarkable work was that which he published in 1843 in which he gave the correct derivation of the Navier-Stokes equations. Anderson writes in [2]:Seven years after Navier's death, Saint-Venant re-derived Navier's equations for a viscous flow, considering the internal viscous stresses, and eschewing completely Navier's molecular approach. That 1843 paper was the first to properly identify the coefficient of viscosity and its role as a multiplying factor for the velocity gradients in the flow. He further identified those products as viscous stresses acting within the fluid because of friction. Saint-Venant got it right and recorded it. Why his name never became associated with those equations is a mystery. certainly it is a miscarriage of technical attribution.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saint-Venant.html (1 of 3) [2/16/2002 11:30:08 PM]

Saint-Venant

We should remark that Stokes, like Saint-Venant, correctly derived the Navier-Stokes equations but he published the results two years after Saint-Venant. Saint-Venant developed a vector calculus similar to that of Grassmann which he published in 1845. He then entered into a dispute with Grassmann about which of the two had thought of the ideas first. Grassmann had published his results in 1844, but Saint-Venant claimed (and there is little reason to doubt him) that he had first developed these ideas in 1832. Again it would appear that Saint-Venant was unlucky. Itard writes in [1]:Saint-Venant used this vector calculus in his lectures at the Institut Agronomique, which were published in 1851 as "Principes de méchanique fondés sur la cinématique". In this book Saint-Venant, a convinced atomist, presented forces as divorced from the metaphysical concept of cause and from the physiological concept of muscular effort, both of which, in his opinion, obscured force as a kinematic concept accessible to the calculus. Although his atomistic conceptions did not prevail, his use of the vector calculus was adopted in the French school system. In the 1850s Saint-Venant derived solutions for the torsion of non-circular cylinders. He extended Navier's work on the bending of beams, publishing a full account in 1864. In 1871 he derived the equations for non-steady flow in open channels. In 1868 Saint-Venant was elected to succeed Poncelet in the mechanics section of the Académie des Sciences. By this time he was 71 years old, but he continued his research and lived for a further 18 years after this time. At age 86 he translated (with A Flamant) Clebsch's work on elasticity into French and published it as Theorie de l'élasticité des corps solides and Saint-Venant added notes to the text which he wrote himself. Note that Saint-Venant's co-translator A Flamant was a co-author of the obituary notice [3] for Saint-Venant. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

Abstract linear spaces

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saint-Venant.html (2 of 3) [2/16/2002 11:30:08 PM]

Saint-Venant

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Saint-Venant.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saint-Venant.html (3 of 3) [2/16/2002 11:30:08 PM]

Saint-Vincent

Gregorius Saint-Vincent Born: 8 Sept 1584 in Bruges, Belgium Died: 27 Jan 1667 in Ghent, Belgium

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gregory of Saint-Vincent began his studies at the Jesuit College of Bruges in 1595. He went to Douai in northern France in 1601, studying mathematics and philosophy there. Saint-Vincent entered the Jesuit Order in 1607. He was a student of Clavius in Rome but he went to Louvain in 1612 to complete his theology degree. From 1613 he began to teach, first at Brussels where he taught Greek. Then he continued to teach Greek at a number of places, 'sHertogenbosch in the Netherlands in 1614, and Coutrai in 1615. The next year he was appointed chaplain to the Spanish troops stationed in Belgium which must have been a difficult job since this was the period of the Dutch revolt against Spain. He next taught at the Jesuit school in Antwerp, then from 1621 he spent four years teaching mathematics in Louvain. After six years in Prague as chaplain to the Holy Roman Emperor Ferdinand II from 1626 until 1632. However when the Swedish army attacked Prague, Saint-Vincent fled to Vienna leaving behind many of his important mathematical papers. He moved to the Jesuit College in Ghent where he taught from 1632 for the rest of his life. Eventually the papers he had been force to leave in Prague were returned to him and they were published. Saint-Vincent's main work is a book over 1250 pages long. There are many topics covered in the book including a study of circles, triangles, geometric series, ellipses, parabolas and hyperbolas. His book also contains his quadrature method which is related to that of Cavalieri but which he discovered independently. He gives a method of squaring the circle which we can now see is essentially integration. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saint-Vincent.html (1 of 2) [2/16/2002 11:30:10 PM]

Saint-Vincent

Saint-Vincent integrated x-1 in a geometric form that is easily recognised as the logarithmic function. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) Mathematicians born in the same country Cross-references to History Topics

Squaring the circle

Other Web sites

1. Jesuit Scientists 2. The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Saint-Vincent.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saint-Vincent.html (2 of 2) [2/16/2002 11:30:10 PM]

Saks

Stanislaw Saks Born: 30 Dec 1897 in Kalisz, Poland Died: 23 Nov 1942 in Warsaw, Poland

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stanislaw Saks was born into a Jewish family, his parents being Philip Saks and Ann Labedz Saks. Stanislaw was born in Kalisz in west-central Poland, not in Warsaw as stated in [1], but he did attend secondary school in Warsaw. He entered the University of Warsaw in November 1915. In fact this was a very special occasion for education in Warsaw and we should look at a little Polish history to explain just why this was so special. Poland had been partitioned in 1772 with the south called Galicia and under Austrian control while Russia controlled much of the rest of the country, in particular the Warsaw region. In a policy implemented between 1869 and 1874, all secondary schooling had to be conducted in the Russian language. The University of Warsaw was closed by the Russian administration in 1869 and from then until World War I there were no Polish language universities. In August 1915 the Russian forces, which had held Poland for many years, withdrew from Warsaw. Germany and Austria-Hungary took control of most of the country and a German governor general was installed in Warsaw. One of the first moves after the Russian withdrawal was the refounding of the University of Warsaw and it began operating as a Polish university in November 1915. Saks entered the newly refounded university on this occasion of great rejoicing for all patriotic Poles. Kuratowski tells us in [3] that:... in a short time [Saks] shone as one of the most talented students of mathematics. Mathematical analysis, and especially those of its branches which used modern methods of set theory and topology, became his main field of interest. At the Paris Peace Conference in 1919 Poland demanded the return of the former Prussian sector of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saks.html (1 of 4) [2/16/2002 11:30:12 PM]

Saks

Upper Silesia from Germany. There had already been an uprising against the Germans in December 1918 in Poznan. The Treaty of Versailles was signed in the summer of 1919 and it gave Poland part, but not all, of the Prussian sector. There were three insurrections in Upper Silesia as the Polish population rebelled against the German administration. Saks participated in the insurrections and was awarded the Cross of Valour for his patriotic actions. Saks continued to study mathematics for his doctorate, which was supervised by Mazurkiewicz, at the University of Warsaw. As well as his supervisor he was greatly influences by Sierpinski who was appointed to the university in 1919 and began to work closely with Mazurkiewicz, Saks' supervisor. In 1920 Mazurkiewicz and Sierpinski became editors of Fundamenta Mathematicae and Polish mathematics was flourishing. Certainly, therefore it was an exciting period during which Saks embarked on a research career and he was awarded his doctorate in 1922 for the thesis A contribution to the theory of surfaces and plane domains. Even before the doctorate was awarded, Saks began teaching at Warsaw Technical University and from 1926 he also lectured at the University of Warsaw. The following year, from 7 to 10 September, he attended the First Polish Mathematical Congress held at Lvov as part of the Warsaw contingent and lectured to the conference. Steinhaus recalls a contribution Saks made around the same time (see for example [2]):In 1927, a collaboration [between Steinhaus and] Banach resulted in a paper "Sur le principe de la condensation des singularités" published in Fundamenta Mathematicae 9. ... Stanislaw Saks helped edit the paper and later deepened the result by introducing in its proof the notion of category. This helped make the paper an important contribution to the Polish success between the two wars in the area of functional operations .... Banach and Saks collaborated on a joint paper Sur la convergence forte dans le champ Lp. Published in 1930 [2]:... the paper addressed the problem of summability in abstract spaces. This gave birth to a class of spaces that are still actively studied and that are now called spaces with the Banach-Saks property. Saks continued to teach at both Warsaw institutions until 1939. However, he did spend a year, namely academic year 1931-32, in the United States on a visit financed through a Rockefeller scholarship. He spent most of his time in the United States at Brown University. Zygmund had become a colleague and friend of Saks early in his career. He was appointed to Warsaw Technical University shortly after Saks began to lecture there, and the two began to collaborate on mathematical projects. One of the works for which Saks is most famous is their joint book Analytic functions which appeared in 1938 as volume eight in the Mathematical Monographs series. This book received a prize from the Polish Academy of Sciences in the year it was published. This was not Saks' first monograph, however, for he had already published an important volume in the Mathematical Monographs series. This earlier volume, the volume two in the series published in 1933, was his famous work Theory of the integral. This monograph was based on lecture courses Saks had given at the University of Warsaw. Hawkins, in [1], writes:In this highly original work Saks systematically developed the theory of integration and differentiation from the standpoint of countably additive set functions. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saks.html (2 of 4) [2/16/2002 11:30:12 PM]

Saks

In 1937 an English translation of Theory of the integral was published with Banach's article The Lebesgue integral in abstract spaces as an appendix. We have already mentioned Mazurkiewicz, Sierpinski, and Zygmund as major influences on Saks. We should mention, in addition, that he was also influenced by Luzin's work. Saks' contributions, including the important texts mentioned above [1]:... involved the theory of real functions, such as problems on the differentiability of functions and the properties of Denjoy-Perron integrals. Kuratowski in [3] describes of Saks as a teacher:Stanislaw Saks was a brilliant lecturer, universally respected and very popular with his colleagues and students. When war broke out in 1939 Saks joined the Polish army and retreated with them to Lvov which was by that time under Russian control. There he worked with Banach in the Soviet held town for two years, being appointed a professor at Lvov University which had been renamed the Ivan Franko University by the Russians. At this time he taught in the Department with Banach at its head. In Lvov, Saks joined the community of mathematicians working and drinking in the Scottish Café. He contributed problems to the Scottish Book, the famous book in which the mathematicians working in the Café entered unsolved problems. One problem on subharmonic functions was entered into the Book by Saks on 8 February 1940 with the promise of a kilo of bacon to the first person to solve it! You can see a picture of the Scottish Café. In June 1941 the German army entered Lvov and a systematic extermination of Jews began. Saks returned to Warsaw where he was arrested, put in prison and killed by the Gestapo (allegedly while attempting to escape from prison). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saks.html (3 of 4) [2/16/2002 11:30:12 PM]

Saks

JOC/EFR February 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Saks.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saks.html (4 of 4) [2/16/2002 11:30:12 PM]

Salem

Raphaël Salem Born: 7 Nov 1898 in Saloniki (now Thessaloniki), Greece Died: 20 June 1963 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Raphaël Salem was born in Saloniki, (now in Greece) to Jewish parents of Spanish origin. His father, Emmanuel Salem, was a well known lawyer who dealt with international problems while his mother was Fortunée Salem. Raphaël was brought up in a truly international atmosphere for he lived in the Ottoman Empire, in a Jewish family who followed the Spanish traditions of their ancestors but spoke Italian and French at home. From this mixture of influences it is perhaps surprising that he attended an Italian school in Saloniki. When Salem was 15 years old, his family moved to France and set up home in Paris. Salem attended the Lycée Condorcet for two years and then entered the Law Faculty of the University of Paris. It is not entirely clear how much the choice of law was that of Salem or that of his father. Certainly Salem's father wished his son to follow in his footsteps, and another factor here is that Salem always had a great respect for his father. It may well have been, therefore, that he did not take this course of study unwillingly, but at the same time it was clear that his interests were not in law but rather they were in mathematics and engineering. Of course there were few better places in the world to study mathematics than Paris, and Salem was soon taking mathematics courses with Hadamard. The tension between his interests and his official course of study became greater as the course progressed. He received his law degree, graduating as Licencié et lauréat in 1919. He even began working for a doctorate in law but quickly decided that he had to change direction to science, which he had been studying for years in parallel to his work in law. He received his Licencié ès sciences from the Sorbonne, also in 1919, then worked for a degree in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Salem.html (1 of 4) [2/16/2002 11:30:13 PM]

Salem

engineering. He received the degree of Ingénieur des Arts et Manufactures from the Ecole Centrale in 1921. Now, having made the change away from law to science and engineering one might have expected Salem to seek a profession using these qualifications, but in fact he went into banking. This again may well have been as a result as pressure from his family or he may just have been happy to make money in banking while he regarded mathematics as a hobby to enjoy. Whenever he had free time in the evenings he worked on Fourier series, a topic which interested him throughout his life. He worked for the Banque de Paris et des Pays-Bas from 1921. In 1923 he married Adriana and they had a daughter and two sons. Zygmund writes in [3]:In spite of his absorbing career Salem's interest in mathematics, which he acquired in his early studies, grew stronger and with time developed into an interest in mathematical research. He was spending on it most of his spare time, working mainly in the evenings. He became attracted to Fourier series, and the interest in the subject remained undiminished throughout his life. ... Although he read some of the current literature on Fourier series, he apparently worked all alone ... He did have some connections with Paris mathematicians, however, particularly with Denjoy who may have influenced him towards Fourier series. His career in the bank progressed well and in 1938 he became one of the managers of the Banque de Paris et des Pays-Bas. It was around this time, with a deteriorating political situation, that Salem finally made the decision to change career and become a mathematician. Denjoy was certainly a factor in this decision, for he well realised Salem's potential as a mathematician and tried to persuade him to take a doctorate in mathematics. Another factor was the arrival of Marcinkiewicz in Paris in the spring of 1939. Salem collaborated with this brilliant young Polish mathematician and of the mathematics papers he wrote while working for the bank, the one he wrote with Marcinkiewicz was his only joint work. World War II broke out in September 1939. Salem was called up for military work and attached to the Deuxième Bureau of the General Staff of the French Army. He had already taken Denjoy's advice and submitted his published papers, with some minor improvements, for a doctorate, and this was awarded in 1940. As part of his military duties, Salem was sent to England to assist the Head of the Franco-British Coordination Committee but he was demobilised in June 1940. By this time his family had managed to escape from France and they were in Canada. Salem left England in the autumn of 1940 and emigrated to the United States where he settled in Cambridge, Massachusetts. He went to Cambridge, Massachusetts via Canada where he spent a short time and met up with his family. In 1941, he was appointed as a lecturer in mathematics at the Massachusetts Institute of Technology. It was very fortunate for Salem that he had obtained his doctorate in mathematics in the previous year from the University of Paris. It is hard to see how he would have been appointed without this mathematics qualification for he had no experience as a lecturer. This did not seem a problem, however, for [3]:.. he was a born teacher and he enjoyed teaching though initially it was an effort for him to teach in English. He had a style of his own which combined verve and naturalness, precision and elegance. The way he could explain essential things without going into calculation were always admired by people attending his lectures and appreciating the difficulties inherent in mathematical presentations.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Salem.html (2 of 4) [2/16/2002 11:30:13 PM]

Salem

He was in the right place to carry on with his interest in Fourier series, and he collaborated on this topic with Norbert Wiener and Zygmund (with whom he wrote joint papers). Zygmund, reviewing [2], puts Salem's contributions to Fourier series into perspective:For the last few decades two problems were central in the field: convergence almost everywhere of Fourier series and the nature of the sets of uniqueness for trigonometric series. With recent results ... the first problem is essentially solved. But the second problem still challenges, in spite of progress here. Much of this progress is due to [Salem] and people influenced by his ideas, and acquaintance with his work seems to be a prerequisite for those who would like to contribute to the solution of the problem. Another direction in which [Salem] did a lot was applications of the calculus of probability to Fourier series and, curiously enough, this has connection with problems of uniqueness. Moreover, it seem that, far from being incidental, as it might have appeared some 30 or so years ago, the calculus of probability is becoming a standard method of attacking problems of trigonometric series. Going through the papers of [Salem] one sees this growing role of the calculus of probability. Incidentally, the reader will greatly appreciate the elegance and lucidity of [Salem's] style. We should also note that Salem introduced the idea of a random measure into harmonic analysis. This began an area of research which is still very active today. Returning to details of Salem's life we should emphasise how difficult the war years were for him. His father died in Paris in 1940 while his mother, his sister, his sister's husband, and his sister's son, were all arrested and deported by the Nazis to a concentration camp where they died. Salem's older son however, survived the war. He enlisted in the free French Forces and took part in Allied landings in the South of France in 1944. Salem was rapidly promoted at MIT where he became an assistant professor in 1945, and an associate professor in 1946. However, happy as he was in the United States, once the war had ended and his country was again free, Salem longed to return to France. It is worth noting how, despite his multi-national upbringing, he was a Frenchman with a true love of France. For some years he split his time between Paris and Cambridge, Massachusetts, spending one semester in each. In 1950 he was appointed Professor at MIT and also Professor at the University of Caen in France. He continued to split his time between the two countries until 1958 when he was appointed as Professor at the Sorbonne. He lived in Paris from that time on until his death. After Salem died his wife established an international prize for outstanding contributions to Fourier series [1]:The list of those who have received the prize, first awarded in 1968, is impressive and testifies to the explosive activity in this area of mathematics. As to Salem's personality we quote from [3]:[Salem's] intellectual brilliance ... attracted people, the more so that it was accompanied by personal charm and natural friendliness. He had a great sense of humour, a vivid way of speaking and was always interested in conversation. But he also had a warm personality and was sensitive to human hardships ... he extended [hospitality] easily and it was gladly accepted by his many friends ... Finally we should mention his interests outside mathematics. He loved music and played the violin, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Salem.html (3 of 4) [2/16/2002 11:30:13 PM]

Salem

preferring to play in quartets. He was also interested in the arts, particularly French and Italian literature, while his sporting interests were skiing and horse riding. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2001 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Salem.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Salem.html (4 of 4) [2/16/2002 11:30:13 PM]

Salmon

George Salmon Born: 25 Sept 1819 in Cork, Ireland Died: 22 Jan 1904 in Dublin, Ireland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Salmon's father, Michael Salmon, was a linen merchant while his mother, Helen Weekes, was the daughter of the Reverend Edward Weekes. George attended school in his home town of Cork, in the south of Ireland, and then entered Trinity College, Dublin in 1833. He studied mathematics and classics at Dublin, and graduated with a degree in mathematics in 1838. At this time Trinity College, Dublin restricted its degrees, fellowships, and scholarships to Anglicans. It was some time after this, in 1873, that all religious requirements were removed from the university. This was not a problem for Salmon, who was an Anglican, but in order to take up the Fellowship that Trinity offered him in 1841 he was required to take holy orders in the Church of Ireland, which indeed he did. In 1844 Salmon married Frances Anne Salvador, the daughter of the Reverend J L Salvador. They would have six children, four boys and two girls. The mathematics department in Trinity College had some outstanding mathematicians on the staff when Salmon joined them in 1841. Hamilton, MacCullagh were on the staff there as was Charles Graves and Humphrey Lloyd. Although the main topic of interest was in synthetic geometry, Salmon only worked in this area for a short time before moving into the area of algebraic geometry. Salmon became interested in the algebraic approach to geometry taken by Cayley, Sylvester, Hermite and later by Clebsch. He became a close friend of Cayley and Sylvester. Salmon discovered, together with Cayley, the 27 lines on the cubic surface. To be more accurate, Cayley discovered these but they were enumerated by Salmon. He also made many discoveries about ruled surfaces and other surfaces,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Salmon.html (1 of 3) [2/16/2002 11:30:15 PM]

Salmon

including the idea of the normal singularities of an algebraic surface. As was typical of the time Salmon's work showed a lack of concern with complete rigour. In [5] Gow gives many interesting quotations by Hirst and Poncelet and several other of Salmon's contemporaries and successors. Many of these comments are surprisingly critical of Salmon's style. Salmon is best remembered for his four textbooks written between 1848 and 1862 which did much to breath life into British mathematics. McConnell writes in [1]:These four treatises on conic sections, higher plane curves, modern higher algebra, and the geometry of three dimensions not only gave a comprehensive treatment of their respective fields but also were written with a clarity of expression and an elegance of style that made them models of what a textbook should be. They were translated into every western European language and ran into many editions (each incorporating the latest developments); they remained for many years the standard advanced textbooks in their respective subjects. The famous four textbooks referred to in this quote are A treatise on conic sections (1848), A treatise on higher plane curves: Intended as a sequel to a treatise on conic sections (1852), Lessons introductory to the modern higher algebra (1859), and A treatise on the analytic geometry of three dimensions (1862). A characteristic of Salmon's work was his love of carrying out lengthy calculations. He calculated an invariant of a curve of degree six and published the resulting calculation, which ran to thirteen pages, in the second edition of his treatise on higher algebra which appeared in 1866. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of George Salmon

Mathematicians born in the same country

Honours awarded to George Salmon (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1863

Royal Society Copley Medal

Awarded 1889

Royal Society Royal Medal

Awarded 1868

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Salmon.html (2 of 3) [2/16/2002 11:30:15 PM]

Salmon

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Salmon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Salmon.html (3 of 3) [2/16/2002 11:30:15 PM]

Samoilemko

Anatoly Samoilemko Born: 2 Jan 1938 in Potiivka, Radomyshl, Zhytomyr oblast, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Anatoly Samoilemko was born in western Ukraine. He graduated from Kiev University in 1960 and was appointed to the Institute of Mathematics of the Academy of Sciences of the Ukraine in Kiev. He also taught at the University of Kiev from 1967. In 1988 Samoilemko was appointed Director of the Institute of Mathematics in Kiev. Samoilemko worked on both linear and nonlinear ordinary differential equations. In the 1960s he studied nonlinear ordinary differential equations with impulsive action. With contributions from M Perestiuk, Samoilemko put the application of asymptotic methods to solve discontinuous and impulsive systems on a rigorous foundation. This work continued over a long period and was written up in an important joint monograph in 1987. Petryshyn writes in [1]:His most original contribution was the numeric-analytic method for the study of periodic solutions of differential equations with periodic right hand side. A monograph on the method of accelerated convergence, written jointly by Samoilemko, N Bogoliubov, and Yu Mytropolshy in 1969, gives an exhaustive analysis of the speed of convergence, error estimates, stability, and applications. Samoilemko was to undertake several joint projects with Mytropolshy who was the Director of the Institute of Mathematics in Kiev where he worked. In addition to the work mentioned above they worked jointly on the theory of multifrequency oscillation, then later on a system of evolutionary equations with periodic and conditional periodic coefficients. This last work was done in collaboration with D Martyniuk and the three of them published a monograph on their results in 1984. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Samoilemko.html (1 of 2) [2/16/2002 11:30:17 PM]

Samoilemko

Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace Maps

Search Form

Glossary index

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Samoilemko.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Samoilemko.html (2 of 2) [2/16/2002 11:30:17 PM]

Samoilenko

Anatoly Samoilenko Born: 2 Jan 1938 in Potiivka, Radomyshl, Zhytomyr oblast, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Anatoly Samoilenko was born in western Ukraine. He graduated from Kiev University in 1960 and was appointed to the Institute of Mathematics of the Academy of Sciences of the Ukraine in Kiev. He also taught at the University of Kiev from 1967. In 1988 Samoilenko was appointed Director of the Institute of Mathematics in Kiev. Samoilenko worked on both linear and nonlinear ordinary differential equations. In the 1960s he studied nonlinear ordinary differential equations with impulsive action. With contributions from M Perestiuk, Samoilenko put the application of asymptotic methods to solve discontinuous and impulsive systems on a rigorous foundation. This work continued over a long period and was written up in an important joint monograph in 1987. Petryshyn writes in [1]:His most original contribution was the numeric-analytic method for the study of periodic solutions of differential equations with periodic right hand side. A monograph on the method of accelerated convergence, written jointly by Samoilenko, N Bogoliubov, and Yu Mytropolshy in 1969, gives an exhaustive analysis of the speed of convergence, error estimates, stability, and applications. Samoilenko was to undertake several joint projects with Mytropolshy who was the Director of the Institute of Mathematics in Kiev where he worked. In addition to the work mentioned above they worked jointly on the theory of multifrequency oscillation, then later on a system of evolutionary equations with periodic and conditional periodic coefficients. This last work was done in collaboration with D Martyniuk and the three of them published a monograph on their results in 1984.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Samoilenko.html (1 of 2) [2/16/2002 11:30:19 PM]

Samoilenko

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Samoilenko.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Samoilenko.html (2 of 2) [2/16/2002 11:30:19 PM]

Sang

Edward Sang Born: 30 Jan 1805 in Kirkcaldy, Fife, Scotland Died: 23 Dec 1890

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edward Sang, mathematician, engineer and actuary, was the 6th of 11 children of Edward Sang (1771-1862), nurseryman and sometime Provost of Kirkcaldy, and his wife Jean Nicol (b.1773) a sister of William Nicol (b.1768) who invented the Nicol prism. He attended a subscription school, founded by his father and others, under a gifted headmaster Edward Irving. At Edinburgh University during 1818 1824, he impressed professors William Wallace and John Leslie in mathematics and natural philosophy, despite periods of illness. Small for his age, he was first mocked by fellow-students, then admired for his precocious talent. Sang first worked in Edinburgh as surveyor, civil engineer and mathematics teacher and lectured on natural philosophy. During 1841-43 he was Professor of Mechanical Sciences at the nonconformist Manchester New College, then went to Constantinople to establish engineering schools, plan railways and an ironworks. He lectured (in Turkish) at the Imperial School, Muhendis-hana Berii and gained fame by predicting the solar eclipse of 1847, thereby dispelling superstition. He resigned against the Sultan's wishes, returning to Edinburgh in 1854 to teach mathematics. An active Fellow of the Royal Scottish Society of Arts and the Royal Society of Edinburgh, he received awards from both and from the Institution of Civil Engineers, London (1879). He was a founder member and first official lecturer of the Faculty of Actuaries in Scotland, a corresponding member of the Royal Tunis Academy, an Ll.D. of Edinburgh University and an honorary member of the Franklin Institute, Philadelphia. He married Isabella Elmslie in 1832 and had at least three daughters and at least one son. Mainly in Edinburgh-based journals, Sang wrote extensively on mathematical, mechanical, optical and actuarial topics including vibration of wires, a theory of toothed wheels, an improved lighthouse light, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sang.html (1 of 2) [2/16/2002 11:30:21 PM]

Sang

railways, bridges, manufacturing and life insurance. He published actuarial, annuity and astronomical tables, books on Elementary and Higher Arithmetic and much-used tables of 7-place logarithms (1871). But his most remarkable achievement is his massive unpublished compilation of 26- and 15-place logarithmic, trigonometric and astronomical tables, filling 47 manuscript volumes. Compiled over forty years, latterly with assistance from two daughters Flora and Jane, these perhaps surpass in accuracy the (also unpublished) French 'Cadastre' tables of 1801. They were gifted to the nation in 1907 by Anna and Flora Sang. Article by: A D D Craik, St Andrews List of References (10 books/articles) A Poster of Edward Sang

Mathematicians born in the same country

Honours awarded to Edward Sang (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sang.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sang.html (2 of 2) [2/16/2002 11:30:21 PM]

Sankara

Sankara Narayana Born: about 840 in India Died: about 900 in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sankara Narayana (or Shankaranarayana) was an Indian astronomer and mathematician. He was a disciple of the astronomer and mathematician Govindasvami. His most famous work was the Laghubhaskariyavivarana which was a commentary on the Laghubhaskariya of Bhaskara I which in turn is based on the work of Aryabhata I. The Laghubhaskariyavivarana was written by Sankara Narayana in 869 AD for the author writes in the text that it is written in the Shaka year 791 which translates to a date AD by adding 78. It is a text which covers the standard mathematical methods of Aryabhata I such as the solution of the indeterminate equation by = ax c (a, b, c integers) in integers which is then applied to astronomical problems. The standard Indian method involves using the Euclidean algorithm. It is called kuttakara ("pulveriser") but the term eventually came to have a more general meaning like "algebra". The paper [2] examines this method. The reader who is wondering what the determination of "mati" means in the title of the paper [2] then it refers to the optional number in a guessed solution and it is a feature which differs from the original method as presented by Bhaskara I. Perhaps the most unusual feature of the Laghubhaskariyavivarana is the use of katapayadi numeration as well as the place-value Sanskrit numerals which Sankara Narayana frequently uses. Sankara Narayana is the first author known to use katapayadi numeration with this name but he did not invent it for it appears to be identical to a system invented earlier which was called varnasamjna. The numeration system varnasamjna was almost certainly invented by the astronomer Haridatta, and it was explained by him in a text which many historians believe was written in 684 but this would contradict what Sankara Narayana himself writes. This point is discussed below. First we should explain ideas behind Sankara Narayana's katapayadi numeration. The system is based on writing numbers using the letters of the Indian alphabet. Let us quote from [1]:... the numerical attribution of syllables corresponds to the following rule, according to the regular order of succession of the letters of the Indian alphabet: the first nine letters represent the numbers 1 to 9 while the tenth corresponds to zero; the following nine letters also receive the values 1 to 9 whilst the following letter has the value zero; the next five represent the first five units; and the last eight represent the numbers 1 to 8. Under this system 1 to 5 are represented by four different letters. For example 1 is represented by the letters ka, ta, pa, ya which give the system its name (ka, ta, pa, ya becomes katapaya). Then 6, 7, 8 are

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sankara.html (1 of 3) [2/16/2002 11:30:22 PM]

Sankara

represented by three letters and finally nine and zero are represented by two letters. The system was a spoken one in the sense that consonants and vowels which are not vocalised have no numerical value. The system is a place-value system with zero but one may reasonably ask why such an apparently complicated numeral system might ever come to be invented. Well the answer must be that it lead to easily remembered mnemonics. In fact many different "words" could represent the same number and this was highly useful for works written in verse as the Indian texts tended to be. Let us return to the interesting point about the date of Haridatta. Very unusually for an Indian text, Sankara Narayana expresses his thanks to those who have gone before him and developed the ideas about which he is writing. This in itself is not so unusual but the surprise here is that Sankara Narayana claims to give the list in chronological order. His list is Aryabhata I Varahamihira Bhaskara I Govindasvami Haridatta [Note that we have written Bhaskara I where Sankara Narayana simply wrote Bhaskara. The more famous Bhaskara II lived nearly 300 years after Sankara Narayana.] Now the chronological order in the list agrees with the dates we have for the first four of these mathematicians. However, putting Haridatta after Govindasvami would seem an unlikely mistake for Sankara Narayana to make if Haridatta really did write his text in 684 since Sankara Narayana was himself a disciple of Govindasvami. If the dating given by Sankara Narayana is correct then katapayadi numeration had been invented only a few years before he wrote his text. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sankara.html (2 of 3) [2/16/2002 11:30:22 PM]

Sankara

JOC/EFR November 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Sankara.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sankara.html (3 of 3) [2/16/2002 11:30:22 PM]

Sasaki

Shigeo Sasaki Born: 18 Nov 1912 in Yamagata Prefecture, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Shigeo Sasaki's father was a farmer who lived in a small village in the Yamagata Prefecture of Japan. Shigeo was the second of his parents sons but he never knew his mother who died when he was only two years old. His uncle, who was a superior of a Buddhist temple, had no children of his own and offered to help by bringing up one of the two boys. So Shigeo was brought up by his uncle. He attended the Sakata Middle School from 1925 where he was first introduced to mathematics. Shigeo lived in a dormitory, rather than at home, and the mathematics teacher at the school looked after the boys in the dormitory. He loved to explain mathematics to Shigeo and there were many opportunities. In 1929 Shigeo moved from middle school to high school, entering the Second High School at Sendai. There were academies in Japan for the brightest pupils who went to the one corresponding to the area in which they lived in order to prepare for a university education. Sasaki therefore, after showing great talents at middle school, made the natural progression to Sendai where he studied for three years. Although in earlier years there were no mathematics texts in Japanese, by the time Sasaki attended High School there were Japanese texts on algebra, analytic geometry, trigonometry and calculus, all of which he studied. The book he read at this stage of his education which he found most attractive was a Japanese translation of Salmon's A treatise on conic sections. Sasaki graduated form the Second High School and entered Tohoku Imperial University at Sendai in April 1932. He was particularly interested in the courses taught by T Kubota, one of the professors. These included several different geometry courses, including projective geometry, conformal geometry, non-Euclidean geometry, differential geometry, and synthetic geometry. Sasaki writes [3]:Although his lectures were not so systematic, he presented important theorems and interesting ones and proved them with elegant ideas and attracted students. In addition Sasaki, who was by now becoming fascinated by differential geometry, read some classic http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sasaki.html (1 of 3) [2/16/2002 11:30:24 PM]

Sasaki

differential geometry texts including ones by Blaschke, Eisenhart, Schouten, and Cartan. He graduated in March 1935 and remained at Tohoku University to undertake research on differential geometry under Kubota's supervision. In January 1937, Sasaki began his career as a lecturer at Tohoku University while he continued to undertake research for his doctorate. He writes [3]:During these years, I also read papers from mathematical journals and wrote several papers, although they were not far from being exercises. I wrote somewhat better papers five years after graduating. One of them is a series of three papers on the relations between the structure of spaces with normal conformal connections and their holonomy groups. It was this last series of three papers which formed the basis of Sasaki's doctoral thesis which he presented in 1943, receiving his doctorate in July of that year. A year later he was promoted to assistant professor. There were major difficulties in carrying out research in these war years since, quite apart from military reasons and problems caused by bombing, international mathematical journals and texts were not reaching Japan. Sasaki studied various classic papers which had reached Japan before the war including ones by G D Birkhoff, Morse, Seifert and Threlfall, and Rado. During the early 1940s Sasaki wrote a major text Geometry of Conformal Connection in Japanese, completing the manuscript of the book in 1943. However, it was impossible to publish the book immediately after it was written due to problems caused by the war. It was eventually published in 1948. K Yano, who undertook research on the same topic, explains the context of the book:Weyl opened the way to the conformal differential geometry of Riemannian spaces in which one studies the properties of the spaces invariant under the so-called conformal transformation of the Riemannian metric. He discovered a tensor, now called Weyl's conformal curvature tensor, whose vanishing is a necessary condition that the space be conformally flat, that is to say, that the space can be mapped conformally on the Euclidean space. That this is also sufficient was proved by Schouten. ... writers... studied exclusively the conformal properties of a Riemannian space itself and paid only slight attention to the conformal properties of curves and surfaces immersed in a Riemannian space. S Sasaki, Y Muto, and K Yano have developed, since 1938, the conformal theory of curves and surfaces in a conformally connected space as well as in a Riemannian space. Sasaki has obtained also a result on the structure of a conformally connected space whose group of holonomy fixes a point or a hypersphere. ... This book contains almost all the results mentioned above in the geometry of conformal connection. Not long after the end of the war, Kubota retired and in December 1946 Sasaki was appointed to fill the vacant chair. He spent a period at the Institute for Advanced Study at Princeton from September 1952 to May 1954. He collaborated with Veblen and Morse during this time. He also visited Chern at Chicago where he spent June and July of 1954. In 1974 Chern visited Sasaki at Tohoku University. He writes [2]:In 1974 I was a visiting professor at Tohoku University when my wife and I stayed at the University Guest House ... Professor Sasaki's hospitality was the main factor in making my visit a profitable and enjoyable one. Sasaki remained in the chair at Tohoku University until he retired in March 1976, at which time he took up an appointment as professor at the Science University of Tokyo. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sasaki.html (2 of 3) [2/16/2002 11:30:24 PM]

Sasaki

Among the topics Sasaki contributed to over a long research career were Lie geometry of circles, conformal connections, projective connections, holonomy groups, Hermitian manifolds, geometry of tangent bundles and almost contact manifolds (now called Sasaki manifolds), global problems on curves and surfaces in various spaces. He wrote a major text Differential geometry : Theory of surfaces which, S Funabashi, writes:... is a guide to differential geometry, illustrating the topics with the theory of surfaces. The author's aim is to describe the method of study of global differential geometry, especially of the theory of two-dimensional surfaces immersed isometrically in a three-dimensional Euclidean space R3. Most of the features for surfaces appearing in this book are closely related to topological geometry. The book is written in a clear style and avoids unnecessary generalizations. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sasaki.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sasaki.html (3 of 3) [2/16/2002 11:30:24 PM]

Saurin

Joseph Saurin Born: 1 Sept 1659 in Courthézon, Vaucluse, France Died: 29 Dec 1737 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Saurin was educated at home, being brought up a Calvinist by his father who was a Calvinist minister. Joseph entered the Calvinist ministry in 1684 but was soon in trouble for his outspoken views. He tried to escape his troubles by going to Switzerland where he became a minister at Bercher, Yverdon. However he continued to be too outspoken for his own good and, in 1685, refused despite strong pressure being exerted on him, to sign the Consensus of Geneva. In 1690, unhappy with the Calvinist ways, he became a Roman Catholic. He went to Paris in 1690 and began to learn and to teach mathematics. He became friends with de L'Hôpital, Malebranche and Varignon but, by 1702, he was in dispute with Rolle over the calculus. He appealed to the Académie Royal des Sciences but they had no wish to come out against Rolle who was a member. Perhaps to be diplomatic, Saurin was elected to the Académie Royal des Sciences in 1707. He spent several months in jail for writing libellous poems about Rousseau. Then he retired to spend the rest of his life working on mathematics. Saurin made contributions to the calculus, wrote on Jacob Bernoulli's problem of quickest descent and Huygens' theory of the pendulum. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saurin.html (1 of 2) [2/16/2002 11:30:26 PM]

Saurin

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Saurin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Saurin.html (2 of 2) [2/16/2002 11:30:26 PM]

Savage

Leonard Jimmie Savage Born: 20 Nov 1917 in Detroit, Michigan, USA Died: 1 Nov 1971 in New Haven, Connecticut, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Savage was educated at Central High School, Detroit and the entered the University of Michigan, Ann Arbor to study mathematics. He received his BS in 1938, then three years later received his PhD with a doctoral thesis was on metric and differential geometry. His doctoral thesis The Application of Vectorial Methods to the Study of Distance Spaces was supervised by Sumner Myers. He spent session 1941-42 at the Institute for Advanced Study at Princeton where he continued to work on pure mathematics. In 1944 he joined the Statistical Research Group at Columbia University - this move into statistics was suggested by von Neumann who had recognised his talents when Savage was at Princeton. Savage wrote on the foundations of statistics which led him into deep philosophical questions both about statistics and knowledge in general. The other main direction of his work was to study gambling as a source to stimulate problems in probability and decision theory. Savage's book The Foundations of Statistics (1954) is perhaps his greatest achievement. It shows von Neumann's influence and also that of Ramsey. The book considers subjective probability and utility. It starts with six axioms, which are both motivated and discussed, and from these are deduced the existence of a subjective probability and a utility function. A special case of a utility function had been introduced by von Neumann and Morgenstern in their theory of games. Another important work by Savage is How to gamble if you must : Inequalities for stochastic processes in 1965, written jointly with L Dubins.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Savage.html (1 of 2) [2/16/2002 11:30:28 PM]

Savage

Other articles written by Savage relate to statistical inference, in particular the Bayesian approach. He introduced Bayesian hypothesis tests and Bayesian estimation. His Bayesian approach, however, opposed the views of Fisher and Neyman. In his later years he wrote on the philosophy of statistics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Savage.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Savage.html (2 of 2) [2/16/2002 11:30:28 PM]

Savart

Felix Savart Born: 30 June 1791 in Mézières, France Died: 16 March 1841 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Felix Savart taught at the Collège de France from 1828, becoming a professor there in 1836. He collaborated with Biot on a theory of magnetism. Magnetic fields produced by electric currents can be calculated using the law discovered in 1820 by Savart in his joint work with Biot. They took magnetism as the fundamental property rather than the Ampère approach which treated it as derived from electric circuits. Savart also carried out experiments on sound which became important for later students of acoustics. He developed the Savart disk, a device which produced a sound wave of known frequency, using a rotating cog wheel as a measuring device. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Felix Savart (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1839

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Savart.html (1 of 2) [2/16/2002 11:30:29 PM]

Savart

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Savart.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Savart.html (2 of 2) [2/16/2002 11:30:29 PM]

Savary

Felix Savary Born: 4 Oct 1797 in Paris, France Died: 15 July 1841 in Estagel (near Perpignan), France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Félix Savary was a student at the Ecole Polytechnique, then taught there, becoming a professor of astronomy and geodesy in 1831. There he became a founder of studies into surveying and machines. Savary also served as librarian at the Bureau des Longitudes from 1823 to 1829. Then in 1832 he was elected the Académie des Sciences. He worked on electromagnetism and electrodynamics, some work being done jointly with Ampère. In particular, on this topic, he wrote Mémoire sur l'application du calcul aux phenomènes élecro-dynamique (1823). Savary also developed a theorem (named after him) on the curvature of a roulette, the curve traced out by a point on a fixed curve which rolls on a second curve. He wrote on the rotation of magnets, applied the laws of gravity to determine the orbits of double stars in close orbit round each other (1827), and studied the intensity of magnetism through an electrical discharge (1827). Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Savary.html (1 of 2) [2/16/2002 11:30:30 PM]

Savary

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Savary.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Savary.html (2 of 2) [2/16/2002 11:30:30 PM]

Savile

Sir Henry Savile Born: 30 Nov 1549 in Bradley (near Halifax), Yorkshire, England Died: 19 Feb 1622 in Eton, Berkshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henry Savile entered Brasenose College Oxford in 1561 and he was elected a Fellow of Merton College Oxford in 1565. He graduated with an B.A. in 1566 and an M.A. in 1570. On 10 October 1570 he began to lecture at Oxford on Ptolemy's Almagest and we are fortunate in that his lecture notes for this course have survived. We shall now describe the content of these lectures more fully. The lectures are far more than Ptolemy's text with added explanation. Savile introduced his students to the new ideas of Regiomontanus and Copernicus. He mentions both classical authors of mathematics, giving their biographies, and the leading mathematicians of the day whose works he had clearly studied. In the introduction to the lectures Savile gives his views on why students should study mathematics. The study of mathematics, argues Savile, turns a student into an educated, civilised human being. As an example he quotes the classical story of Aristippus who, on being shipwrecked on Rhodes, realised that the inhabitants were civilised when he saw a mathematical figure drawn in the sand. It is worth noting, however, that twenty years later, when Savile was trying to make sure his subject received proper funding, he argued for mathematics because of its practical uses. It is interesting to read Savile's comments in these lectures on why he felt that mathematics at that time was not flourishing. Students did not understand the importance of the subject, Savile wrote, there were no teachers to explain the difficult points, the texts written by the leading mathematicians of the day were not studied, and no overall approach to the teaching of mathematics had been formulated. Of course, as we shall see below, fifty years later Savile tried to rectify these shortcomings by setting up two chairs at

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Savile.html (1 of 3) [2/16/2002 11:30:32 PM]

Savile

the University of Oxford. In 1578 Savile set out on a major European tour. First he visited France, and he also visited Wroclaw and Rome and, in addition, many other places on his travels between these cities. He met and exchanged ideas with the leading European mathematicians of the time. On his return to Oxford in 1582 he became a Greek tutor to Queen Elizabeth. In 1585 he became Warden of Merton and, in addition, he became Provost of Eton in 1596 [4]:He continued to hold the Wardenship of Merton while residing at Eton, and governed both with uncompromising strictness, not to say arrogance; this, and his constant absence from Merton, made him rather unpopular amongst the Fellows. In 1592 Queen Elizabeth I visited the University of Oxford and Savile was much occupied in making sure that the University impressed her so that it would continue to receive financial support. In fact, as well as giving a speech of welcome, he summed up a debate which Queen Elizabeth attended. In his summing up he defended the usefulness of mathematics saying that it has important applications to setting the calendar (quoting Plato as his authority), and was vital in military affairs (this time using the example of Archimedes to make his point). However Savile is most famous for founding two chairs at Oxford in 1619. Savile said that he established the Chairs to remedy the fact that:... geometry is almost totally unknown and abandoned in England. He gave the first geometry lecture himself to a large number of students with the first holders of the chairs in his audience, and again we shall spend a moment looking at its contents. This lecture was designed to be an introduction to the first book of Euclid's Elements. It digressed into other areas, however, such as how mathematics should be taught. It also contained what Savile saw as his own contributions to the subject, the most important of which he considered was his demonstration that Euclid, the author of the Elements, was distinct from Euclid of Megara. This indeed is an important contribution to the history of mathematics The Savilian chair of Geometry was first occupied by Briggs and Savile ended his lecture with the words (see for example [4]):I hand on the lamp to my successor, a most learned man, who will lead you to the innermost mysteries of geometry. Many famous mathematicians have held this chair, see the list of those who have occupied the Savilian Chair of Geometry. The second chair was the Savilian Chair of Astronomy, first occupied by John Bainbridge. One of the most famous people to be appointed to this chair was Christopher Wren in 1661. Savile did not found these chairs so that those appointed could follow their own ideas. Far from it. He laid down very precise conditions on how the subjects were to be taught and required the professors to undertake research in their disciplines. The professor of geometry was required to teach the whole of Euclid's Elements, Apollonius's Conics and the complete works of Archimedes having first provided all the necessary mathematical background for an understanding of the texts. His course notes had to be deposited in the University Library. He was also required to show the practical applications of mathematics, teach arithmetic, mechanics and the theory of music. Perhaps more unusual, especially to those thinking in terms of mathematics taught in universities today, was the requirement that field work

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Savile.html (2 of 3) [2/16/2002 11:30:32 PM]

Savile

was to be undertaken in the country when the weather allowed such activities, and the students would there study practical geometry. The professor of astronomy had to meet similar requirements, but in this case the text was to be Ptolemy's Almagest but full details of the newer theories had also to be presented such as those of Jabir ibn Aflah in his Correction of the Almagest and Copernicus's heliocentric point of view. Other requirements for the astronomy professor was to teach spherics, calculation with sexagesimal numbers, optics, geography and navigation. Although this sounds a very classical course, this was not the attitude that Savile took. In fact he required the professor of astronomy to carry out research and, although this may not sound unusual by today's standards, at that time many professors did no more than teach. The professor had to make his own instruments and carry out his own observations with them which, like the lecture notes, had to be deposited in the library. These conditions were given to ensure that astronomy was be a subject that would develop and not be simply that fixed by the classical writers. One final, but very strongly put, condition was that the teaching of astrology in any shape or form was banned. Aubrey [1] described Savile's appearance with these words:He was an extraordinarily handsome man; no lady had a finer complexion. Outside the area of mathematics Savile is best known for his contributions to the preparation of the King James Version of the Bible. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Savile.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Savile.html (3 of 3) [2/16/2002 11:30:32 PM]

Schatten

Robert Schatten Born: 28 Jan 1911 in Lvov, Poland (now Ukraine) Died: 26 Aug 1977 in New York, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Schatten received the Magister degree from John Casimir University in Lvov in 1933. After emigrating to the United States, he enrolled in the graduate school of Columbia University, receiving an M.A. degree in 1939. He continued his research under the direction of Francis J. Murray, and was awarded the Ph.D. degree in 1942. He held a brief appointment as a lecturer in the College of Pharmacy at Columbia University in 1942 before joining the U.S. Army, in which he served from 1942 to 1943. He suffered a broken back during training at Fort Benning, Georgia, and this injury gave him much pain for the rest of his life. In the academic year 1943-1944, he had an appointment as assistant professor at the University of Vermont. He then won a two-year appointment as a Fellow of the National Research Council, and divided his time during this period between the Institute for Advanced Study and Yale University. He collaborated during these years with John von Neumann and with Nelson Dunford. In 1946, Schatten began a long association with the University of Kansas, first as associate professor (1946-1952) and then as professor (1952-1961). This tenure was interrupted by leaves in 1950 and in 1952-1953, both of which he spent at the Institute for Advanced Study at Princeton. The year 1960-1961 was spent as a visiting professor at the University of Southern California, and in 1961-1962 he served as professor at the State University of New York at Stony Brook. In 1962 he became professor at Hunter College, in New York where he remained until his death. During the years 1964-1972 he was also a member of the doctoral faculty of the Graduate School of the City University of New York. At the time of his death he had no immediate survivors, all his known relatives in Poland having been killed during the war. To his former students, Schatten will be remembered as a dedicated teacher who was genuinely concerned with the intellectual development of his students. They will certainly not forget his unique style of lecturing. He always spoke without a book or notes, and rarely used the blackboard. His lectures were extremely clear and well-organized; he never lost his way in complicated arguments. The pace was such that the students could (and were expected to) take notes verbatim; if they did so, their notes would read like a polished book, except for some linguistic idiosyncracies such as, "Given is a set...". He left nothing to chance in his dictation; for example, he invariably ended an argument with "This concludes http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schatten.html (1 of 2) [2/16/2002 11:30:33 PM]

Schatten

the proof." Schatten had his own way of making abstract concepts memorable to his elementary classes. Who could forget what a sequence was after hearing Schatten describe a long corridor, stretching as far as the eye could see, with hooks regularly spaced on the wall and numbered 1, 2, 3, ...? "Then," Schatten would say, "I come along with a big bag of numbers over my shoulder, and hang one number on each hook." This of course was accompanied by suitable gestures for emphasis. Schatten had some eccentricities which endeared him to his friends. He hated noise, especially when it interrupted his sleep. In Lawrence, Kansas, he was seen early one morning in his garden, clad in pyjamas, trying to shoo away the grackles from a tree near his bedroom. Cars were also his bêtes noires: although he owned a car at one time, he never fully mastered the art of driving. He once got a nasty bruise from attempting to put his head out of the car window before lowering the glass. Bachelor life also presented various pitfalls such as having to contend with laundries that insisted on ironing his socks. He kept his unpublished mathematical researches in a bank's safe-deposit box. Schatten's principal mathematical achievement was that of initiating the study of tensor products of Banach spaces. The concepts of crossnorm, associate norm, greatest crossnorm, least crossnorm, and uniform crossnorm, all either originated with him or at least first received careful study in his papers. He was mainly interested in the applications of this subject to linear transformations on Hilbert space. In this subject, the Schatten Classes perpetuate his name. Article by: E W Cheney, Austin, Texas, USA Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schatten.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schatten.html (2 of 2) [2/16/2002 11:30:33 PM]

Schauder

Juliusz Pawel Schauder Born: 21 Sept 1899 in Lvov, Galicia, Austria-Hungary (now Ukraine) Died: Sept 1943 in Lvov, Galicia, Austria-Hungary (now Ukraine)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Julius Schauder was born into a Jewish family. His father, Samuel Schauder, was a lawyer and Julius attended secondary school in Lvov. At this time Lvov was in Galicia, the part of partitioned Poland which was under Austrian control. Schauder was still at school when World War I started but when he graduated from school in 1917 he was drafted into the Austro-Hungarian army. He was with the army in the north of Italy when he was taken prisioner by the Italians. After the collapse of Austria, Schauder joined a new Polish army which was being organised in France. He returned to Poland with this army in 1919, by this time the country of Poland had been reestablished and it was to a Polish Lvov that Schauder went. He entered the Jan Kazimierz University in Lvov and worked for his doctorate under Steinhaus. After obtaining his doctorate in 1923 with a thesis The theory of surface measure, he taught both in a secondary school and worked for an insurance firm. In 1927 he wrote the paper Contributions to the theory of continuous mappings on function spaces and was allowed to teach at the university on the strength of this work. His first courses were given at the University in Lvov during 1928-29 but he continued also with his position as a secondary school teacher. As Ulam writes in [3]:Although Lvov was a remarkable centre for mathematics, the number of professors both at the Institute and at the University was extremely limited and their salaries were very small. ... Schauder had to teach in high school in order to supplement a meager income as lecturer ... Schauder married Emilia Löwenthal in 1929. She also came from a Jewish family although her grandfather had been expelled from the Jewish community on the grounds that he was an atheist. They http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schauder.html (1 of 3) [2/16/2002 11:30:35 PM]

Schauder

would have one daughter Eva Schauder. Brouwer published his fixed point theorems in 1911 for finite dimensional spaces. Schauder published fixed point theorems for Banach spaces in 1930. In 1932 he was awarded a Rockefeller scholarship which enabled him to spend part of 1932-33 Leipzig. Still financed by the scholarship in May 1933 he moved to Paris to work with Hadamard. While Schauder was in Paris he collaborated with J Leray and their joint work led to a paper Topologie et équations fonctionelles published in the Annales scientifiques de l'Ecole normale Supérieure. This 1934 paper on topology and partial differential equations is of major importance [1]:In this paper what is now known as Leray-Schauder degree (a homotopy invariant) is defined. This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations. His last work was to generalise results of Courant, Friedrichs and Lewy on hyperbolic partial differential equations. In 1938 he received the Grand Prix Internationaux de Mathématique Malaxa (jointly with Leray) for the work of the 1934 paper. In fact by the time Schauder received this prize his final publication (1937) had appeared in print. His short career was about to come to an end with the start of World War II but, despite his publications spanning only 10 years, he had written 33 works. Kuratowski, in [2], sums up Schauder's main mathematical contributions:Schauder's main achievement consists in transferring some topological notions and theorems to Banach spaces (the fixed point theorem, invariance of domain, the concept of index). In particular, Schauder's formulation of a fixed point theorem originated a new, extremely fruitful method in the theory of differential equations, known as Schauder's method ... Forster, in [1], writes:Schauder's fixed point theorem and his skillful use of function space techniques to analyse elliptic and hyperbolic partial differential equations are contributions of lasting quality. Existence proofs for complicated nonlinear problems using his fixed point theorem have become standard. The topological method developed in the 1934 Leray-Schauder paper ... is now utilised not only to obtain qualitative results but also to solve problems numerically on computers. In 1939, at the beginning of the World War II, Soviet troops occupied Lvov. Schauder was treated well by the new Soviet administration. He was appointed to professor at the university, now renamed the Ivan Franko University. In June 1941 the German army entered Lvov and a systematic extermination of Jews began. Schauder sent pleas for help to Hopf and Heisenberg saying he had many important results but no paper to write them on. There are two versions of how he died and it is impossible to tell which is correct. One version states that he was betrayed to the Gestapo who then arrested him and, like many Jews, he was never seen again. The second version of his death (thought by Forster the author of [1] to be more likely) is the he was shot by the Gestapo in September 1943 in one of their regular searches for Jews. Schauder's wife Emilia was hidden in Lvov by the Polish resistance for some time after her husbands death. She was hidden in the sewers of Lvov with her daughter Eva. However, eventually she http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schauder.html (2 of 3) [2/16/2002 11:30:35 PM]

Schauder

surrendered to the Germans and was sent to Lublin concentration camp where she died. Eva, Schauder's daughter, survived until the end of the war when she went to Italy to live with Schauder's brother who lived there. We should comment on the picture of Schauder. Ulam [3] explains that Leray, the French mathematician with whom Schauder collaborated, wrote to him several years after the end of World War II:Leray wanted to have a photograph of [Schauder] for himself and for Schauder's daughter who survived the war and lives in Italy. But he could not find any in Poland or anywhere and he wrote to me asking whether I maght have a snapshot. Some months after Johnny von Neumann's death I was looking at some of the books in his library and a group photo of the participants in the [1935] Moscow conference fell out. Schauder was there, as were Aleksandrov, Lefschetz, Borsuk, and some dozen other topologists. I sent this photograph to Leray. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Other Web sites

1. H M Schaerf 2. R S Ingarden

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schauder.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schauder.html (3 of 3) [2/16/2002 11:30:35 PM]

Scheffe

Henry Scheffé Born: 11 April 1907 in New York, USA Died: 5 July 1977 in Berkeley, California, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henry Scheffé's parents were German but had emigrated to the United States and lived in New York. Henry's father was a baker by trade but lst his job during the depression and he was forced to sell apples on the street corner to save the family from starvation. It was a time of great suffering and [1]:... the memory of this injustice and of his father's suffering remained with Scheffé throughout his life. Mathematics was not the first area to interest Scheffé and he began working as an engineer. However, he soon decided to go to university to study mathematics and took a course in pure mathematics at the University of Wisconsin, receiving his BA in 1931. He wrote a doctoral thesis on differential equations and was awarded his PhD in 1935. Before completing his doctoral studies he married Miriam Knott in 1934. They would have one daughter and one son. Scheffé's doctoral dissertation The Asymptotic Solutions of Certain Linear Differential Equations in Which the Coefficient of the Parameter May Have a Zero was supervised by Rudolph E Langer. Immediately after completing his doctorate, Scheffé began a career as a university teacher and, having trained as a pure mathematician, it was naturally the subject which he taught. He was on the Faculty at the University of Wisconsin from 1935 to 1937, then spent three of the next four years at Oregon State University with the year 1939-40 spent at Reed College. In 1941 Scheffé's interests moved from pure mathematics to statistics, and he join Wilks at Princeton where a statistics team had grown up. Having retrained as a statistician, he began a second career as a university teacher, but this time he taught statistics rather than mathematics. He was on the Faculty at Syracuse in session 1944-45, and at University of California at Los Angeles from 1946 until 1948. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Scheffe.html (1 of 3) [2/16/2002 11:30:37 PM]

Scheffe

Immediately after leaving Los Angeles he was appointed to Columbia University where he became chair of the statistics department. After five years at Columbia, Scheffé went to Berkeley as professor of statistics in 1953. He was to remain at Berkeley for the rest of his life, retiring from his chair in 1974. He was chairman of department at Berkeley as he had been at Columbia University. Scheffé, perhaps not surprisingly given his route into statistics, was interested in the more mathematical areas of statistics. He was particularly interested in optimal properties and he extended the Neyman Pearson theory of best similar test. His research was influenced by certain consultancy positions he held, such as one at the Office of Scientific Research and Development from 1943 to 1946. Later consultancy appointments were with the Consumer's Union and with Standard Oil. After 1950 Scheffé's research was concerned with aspects of linear models, particularly the analysis of variance. One of his most important papers appeared in 1953 on the S-method of simultaneous confidence intervals for estimable functions in a subspace of the parameter space. Although he did not show the optimality of the S-method, this was proved by R A Wijsman in the late 1970s. He also studied other aspects of analysis of variance such as paired comparisons which he studied in 1952, then mixed models studied two years later. In 1958 and again in 1963 he published on experiments on mixtures and in 1973 he wrote on calibration methods. His mixture designs were of fundamental importance and led to a major theory of mixtures being built later on Scheffé work. A complete list of Scheffé's publications is given in [4]. His most important work was a comprehensive review of nonparametric statistics in 1943 and his book The Analysis of Variance (1959). Lehmann, who worked jointly with Scheffé on a general theory of similar tests, describes the book The Analysis of Variance in [1]:Its careful exposition of the different principal models, their analyses, and the performance of the procedures when the model assumptions do not hold is exemplary, and the book continues to be a standard text and reference. Scheffé was in the middle of revising this book for a new edition when he died. He had retired from Berkeley and, following retirement, spent three years at the University of Indiana continuing his research work. Then, only a few weeks after his post at the University of Indiana had ended and he had returned to Berkeley, he was involved in a bicycle accident which resulted in his death. Scheffé was elected to many statistical societies. He became a fellow of the Institute of Mathematical Statistics in 1944, the American Statistical Association in 1952 and the International Statistical Institute in 1964. He achieved high office in these organisations, being elected as president of the International Statistical Institute and vice president of the American Statistical Association. In [1] Scheffé's interests outside mathematics and statistics are described:Throughout his life Scheffé enjoyed reading, music (as an adult he learned to play the recorder), and travelling. He was also physically active. At Wisconsin he was an intercollegiate wrestler, and he liked to cycle, swim, and backpack with his family. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Scheffe.html (2 of 3) [2/16/2002 11:30:37 PM]

Scheffe

Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Henry Scheffé

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Scheffe.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Scheffe.html (3 of 3) [2/16/2002 11:30:37 PM]

Scheffers

Georg Scheffers Born: 21 Nov 1866 in Altendorf (near Holzminden), Germany Died: 12 Aug 1945 in Berlin, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georg Scheffers studied at Leipzig from 1884 to 1888 obtaining his doctorate in 1890. From 1896 he lectured at Darmstadt becoming a full professor there in 1900. In 1907 he was appointed to Charlottenburg where he held the chair of mathematics until he retired in 1935. Lie was at Leipzig from 1886 until 1896 and he greatly influenced Scheffers' work. Lie suggested the topic for his doctoral thesis on plane contact transformations and also the topic for his Habilitationsschrift on complex number systems. Scheffers' most important work, also inspired by Lie, was a paper in 1903 on Abel's theorem. Later in life Scheffers wrote many popular mathematics textbooks and edited Lie's works. His favourite topic was differential geometry and here he discovered many properties of particular curves and surfaces. He was an outstanding writer on his subject giving many excellent accounts of his work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Scheffers.html (1 of 2) [2/16/2002 11:30:38 PM]

Scheffers

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Scheffers.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Scheffers.html (2 of 2) [2/16/2002 11:30:38 PM]

Schickard

Wilhelm Schickard Born: 22 April 1592 in Herrenberg (near Tübingen), Württemberg (now Germany) Died: 24 Oct 1635 in Tübingen, Württemberg (now Germany)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Schickard was educated at the University of Tübingen. After receiving his first degree, B.A. in 1609 and M.A. in 1611, he continued to study theology and oriental languages at Tübingen until 1613. In 1613 he became a Lutheran minister at towns around Tübingen. He continued this work with the church until 1619 when he was appointed professor of Hebrew at the University of Tübingen. Schickard was a universal scientist and taught biblical languages such as Aramaic as well as Hebrew at Tübingen. In 1631 he had rather a change of subject being appointed professor of astronomy at the University of Tübingen. His research was broad and included astronomy, mathematics and surveying. He invented many machines like one to calculate astronomical dates and one for Hebrew grammar. He also made significant advances in mapmaking, showing how to produce maps which were far more accurate than those which were currently available. Long before Pascal and Leibniz, Schickard invented a calculating machine in 1623 which was used by Kepler. He wrote to Kepler suggesting a mechanical means to calculate ephemerides. Schickard corresponded with many scientists including Boulliau, Gassendi and Kepler. Among his other skills, Schickard was renowned as an engraver both in wood and in copperplate. Schickard died of the plague either on the day given above or, possibly, one day earlier. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schickard.html (1 of 2) [2/16/2002 11:30:40 PM]

Schickard

Article by: J J O'Connor and E F Robertson List of References (5 books/articles) A Poster of Wilhelm Schickard

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1600 to 1625

Other Web sites

1. The Galileo Project 2. Schickard's calculator 3. A stamp of Schickard's calculator

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schickard.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schickard.html (2 of 2) [2/16/2002 11:30:40 PM]

Schlafli

Ludwig Schläfli Born: 15 Jan 1814 in Grasswil, Bern, Switzerland Died: 20 March 1895 in Bern, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ludwig Schläfli first studied theology, then turned to science. He worked for ten years as a school teacher in Thun. During this period he studied advanced mathematics in his spare time. Schläfli was an expert linguist speaking many languages including Sanskritt and Rigveda. In 1843 Steiner, Jacobi and Dirichlet travelled to Rome and took Schläfli as an interpreter. He gained greatly from discussions with these mathematicians. Schläfli's work was in geometry, arithmetic and function theory. He gave the integral representation of the Bessel function and of the gamma function. He also worked on elliptic modular functions. Schläfli made an important contribution to non- Euclidean (elliptic) geometry when he proposed that spherical three-dimensional space could be regarded as the surface of a hypersphere in Euclidean four-dimensional space. In 1853 Schläfli became professor of mathematics at Bern. His major work Theory of continuous manifolds was published in 1901 after his death and only then did his importance become fully appreciated. He received the Steiner Prize from the Berlin Academy for his discovery of the 27 lines and the 36 double six on the general cubic surface. Schläfli also made significant contributions to celestial mechanics. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schlafli.html (1 of 2) [2/16/2002 11:30:42 PM]

Schlafli

Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) A Poster of Ludwig Schläfli

Mathematicians born in the same country

Other Web sites

SuperAm

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schlafli.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schlafli.html (2 of 2) [2/16/2002 11:30:42 PM]

Schlomilch

Oscar Xavier Schlömilch Born: 13 April 1823 in France Died: 7 Feb 1901

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oscar Schlömilch taught at Jena and Dresden. Cauchy's techniques in analysis became well known in Germany through Schlömilch's textbook. In 1847 he gave a general remainder formula for the remainder in Taylor series. He discovered an important series expansion of an arbitrary function in terms of Bessel functions in 1857. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schlomilch.html (1 of 2) [2/16/2002 11:30:44 PM]

Schlomilch

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schlomilch.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schlomilch.html (2 of 2) [2/16/2002 11:30:44 PM]

Schmidt

Erhard Schmidt Born: 13 Jan 1876 in Dorpat, Germany (now Tartu, Estonia) Died: 6 Dec 1959 in Berlin, Germany

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Erhard Schmidt's father was a medical biologist called Alexander Schmidt. Erhard's university career followed a pattern which was common in Germany at this time, namely that students studied at several different universities as their course progressed. He attended his local university in Dorpat before going to Berlin where he studied with Schwarz. His doctorate was obtained from the University of Göttingen in 1905 under Hilbert's supervision. His doctoral dissertation was entitled Entwickelung willkürlicher Funktionen nach Systemen vorgeschriebener and was a work on integral equations. The main ideas of this thesis appeared in Schmidt's 1907 paper which we describe below. After obtaining his doctorate he went to Bonn where he was awarded his habilitation in 1906. After leaving Bonn, Schmidt held positions in Zurich, Erlangen and Breslau before he was appointed to a professorship at the University of Berlin in 1917. The appointment was to fill the chair left vacant by Schwarz's retirement. Schmidt arrived at the University of Berlin shortly after the death of Frobenius, who had jointly led the department with Schwarz. The other full professor was Schottky. Carathéodory was appointed in 1918 to fill Frobenius's chair and to jointly head mathematics in Berlin with Schmidt. However Carathéodory was to spend only one year in Berlin before leaving. Schmidt now had the main responsibility for filling the vacant chair. This proved a difficult task. Schmidt drew up an impressive list of candidates: Brouwer, Weyl, and Herglotz in that order. The professorship was offered to each of these in turn, with each turning it down. The next person to be offered the chair was Hecke who also turned it down. The position was not filled until 1921 when Bieberbach was offered the post and accepted it. In this same year http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schmidt.html (1 of 4) [2/16/2002 11:30:46 PM]

Schmidt

Schottky retired and Schur, who was already an extraordinary professor in Berlin, was promoted to full professor. The appointments we have discussed were on the pure mathematics side. When Schmidt arrived in Berlin there was no applied mathematics there, the subject being considered more suitable for technical colleges. However Schmidt was the main person who pushed for the founding of an Institute of Applied Mathematics in Berlin. After the Institute was set up Schmidt had to fill the new chair of applied mathematics and the post of Director of the Institute of Applied Mathematics. He was able to engineer a superb appointment in 1920 when von Mises accepted the two positions. Ostrowski wrote in 1965:Only with the appointment of Richard von Mises to the University of Berlin did the first mathematically serious German school of applied mathematics with a broad sphere of influence come into existence. Credit for bringing Berlin to this leading role in applied mathematics must chiefly go to Schmidt. Clearly his abilities were recognised outside mathematics for he was appointed Dean for the academic year 1921-22 and the vice-chancellor of the University of Berlin during the years 1929-30. Despite the university wide nature of this post his wish to continue to promote mathematics is seen from the inaugural address he gave when taking up the post of vice-chancellor: it was entitled On certainty in mathematics. The 1930s were difficult years for Schmidt. With the Nazi rise to power in 1933 life became increasingly difficult for Schmidt's Jewish colleagues and Schur, von Mises and several others were forced out of their posts. In 1951 a meeting was held in Berlin to celebrate Schmidt's 75th birthday. Hans Freudenthal, himself a Jew who had survived the Nazi years, spoke of Schmidt's difficulties through the 1930s (see for example [2]):It is so easy to practise the honesty that mathematics demands in mathematics itself. If you don't, you will be punished quickly and bitterly. It is so much more difficult to stick to this virtue, proven with numbers and figures, against humans and friends. That we outside, excluded for years from a hostile Germany, know this, and never doubted on you, this is evident from the large number of contributions from abroad that have reached the editors of the Festschrift. In his reply to Freudenthal's address Schmidt spoke of his love of the University of Berlin (see for example [2]):I simply loved my students. And exactly the same is true of the university as a whole. I love the University of Berlin, whether it happens to be in happy conditions or not - this does not change anything. I have loved it from the time I have been in Berlin and I will remain faithful to it. In 1936, when the problems were very difficult, Schmidt was made head of the German delegation to the International Congress of Mathematicians at Oslo. Schmidt held positions of authority at the University of Berlin through these difficult years of Nazi rule. He had to carry through the resolutions against Jews but one of Bieberbach's assistants reported in 1938:I think that Schmidt does not at all understand the Jewish question. After the end of World War II Schmidt was appointed as Director of the Mathematics Research Institute of the German Academy of Science. He remained in that role until 1958. By that time he had retired from

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schmidt.html (2 of 4) [2/16/2002 11:30:46 PM]

Schmidt

his chair, which he did in 1950, and he has ceased as joint head of the mathematics department, which happened in 1952. Another role which he took on after the end of the war was as the first editor of Mathematische Nachrichten. He had co-founded the journal in 1948. Schmidt's main interest was in integral equations and Hilbert space. He took various ideas of Hilbert on integral equations and combined these into the concept of a Hilbert space around 1905. Hilbert had studied integral equations with symmetric kernel in 1904. He showed that in this case the integral equation had real eigenvalues, Hilbert's word, and the solutions corresponding to these eigenvalues he called eigenfunctions. He also expanded functions related to the integral of the kernel function as an infinite series in a set of orthonormal eigenfunctions. Schmidt published a two part paper on integral equations in 1907 in which he reproved Hilbert's results in a simpler fashion, and also with less restrictions. In this paper he gave what is now called the Gram-Schmidt orthonormalisation process for constructing an orthonormal set of functions from a linearly independent set. He then went on to consider the case where the kernel is not symmetric and showed that in that case the eigenfunctions associated with a given eigenvalue occurred in adjoint pairs. We should note, however, that Laplace presented the Gram-Schmidt process before either Gram or Schmidt. In 1908 Schmidt published an important paper on infinitely many equations in infinitely many unknowns, introducing various geometric notations and terms which are still in use for describing spaces of functions and also in inner product spaces. Schmidt's ideas were to lead to the geometry of Hilbert spaces and he must certainly be considered as a founder of modern abstract functional analysis. Schmidt defined a space H whose elements are square summable sequences of complex numbers. If w = {wn} and z = {zn} are two elements of H, Schmidt defined an inner product by (w,z) =

wnzn.

He defined the norm ||z|| of the element z to be the square root of the inner product of z with its complex conjugate. He defined orthogonal elements showing that a set consisting of pair-wise orthogonal elements was linearly independent. Again he gave the Gram-Schmidt orthonormalisation process in this setting. He also studied projections and spectral resolutions. What are today called Hilbert-Schmidt operators also appear in this 1908 paper. Bernkopf writes in [1]:Schmidt's work on Hilbert spaces represents a long step toward modern mathematics. He was one of the earliest mathematicians to demonstrate that the ordinary experience of Euclidean concepts can be extended meaningfully beyond geometry into the idealised constructions of more complex abstract mathematics. After Schmidt moved to Berlin his interests turned towards topology. He found a new proof of the Jordan curve theorem which quickly became a classic. Schmidt's interest in topology influenced Hopf and, in 1929, he was an examiner of Hopf's doctoral thesis. Later still Schmidt became interested in isoperimetric inequalities, publishing an important paper on this topic in 1949.

Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schmidt.html (3 of 4) [2/16/2002 11:30:46 PM]

Schmidt

Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) A Poster of Erhard Schmidt

Mathematicians born in the same country

Cross-references to History Topics

1. Topology enters mathematics 2. Abstract linear spaces

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schmidt.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schmidt.html (4 of 4) [2/16/2002 11:30:46 PM]

Schoenberg

Isaac Jacob Schoenberg Born: 21 April 1903 in Galatz, Romania Died: 21 Feb 1990

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Isaac Schoenberg was known as Iso to his friends. His father was a medical doctor but he was fascinated by mathematical puzzles, an interest he soon gave to Isaac. Schoenberg studied at the University of Jassy (Iasi, Moldavia) and received his M.A. in 1922. During 1922-25 he studied at Berlin, where he came under Schur's influence, and at Göttingen. During this time he was engaged in research on a topic in analytic number theory suggested by Schur. He presented his thesis to the University of Jassy and was awarded his Ph.D. in 1926. At Göttingen Schoenberg met Edmund Landau and it was Landau who arranged a visit for Schoenberg to the Hebrew University of Jerusalem which he made in 1928. It was during this visit that [1]:... Schoenberg became interested in estimating the number of real zeros of a polynomial and so began his very influential work on Total Positivity and Variation diminishing linear transformations... Landau, however, proved important in other ways in Schoenberg's life. In 1930, after his return from Jerusalem, Schoenberg married Landau's daughter Charlotte in Berlin. This was not his only mathematical connection by marriage since his sister married Hans Rademacher. In 1930 Schoenberg was awarded a Rockefeller fellowship which enabled him to go to the United States. There he was a postdoctoral worker at Chicago, where he collaborated with Bliss, and then at Harvard. In 1933 he became a member of the Institute for Advanced Studies at Princeton and he remained there until 1935. At Princeton he began working on distance geometry, namely [1]:...the isometric imbedding of metric spaces into Hilbert space and positive definite functions. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schoenberg.html (1 of 3) [2/16/2002 11:30:48 PM]

Schoenberg

After Princeton, Schoenberg held posts at Swarthmore College and Colby College. Then, in 1941, he was appointed to the faculty at the University of Pennsylvania. During the years 1943-45 he was released from the University of Pennsylvania for war work as a mathematician at the Army's Ballistic Research Laboratory in Aberdeen, Maryland (the Aberdeen Proving Ground). It was during this war work that he initiated the work for which he is most famous, the theory of splines. Karlin writes in [4]:... Schoenberg is noted worldwide for his realisation of the importance of spline functions for general mathematical analysis and in approximation theory, their key relevance in numerical procedures for solving differential equations with initial and/or boundary conditions, and their role in the solution of a whole host of variational problems. The fundamental papers by Schoenberg [two papers in 1946] form a monument in the history of the subject as well as its inauguration. The authors of [1] state:For the next 15 years, Schoenberg had splines all to himself. This changed around 1960, when computers became more widespread and splines first assumed their role as the premier tool for data fitting and computer-aided geometric design. Schoenberg's more than 40 papers on splines after 1960 gave much impetus to the rapid development of the field. Schoenberg made further outstanding contributions in a series of papers between 1950 and 1959 on the theory of Pólya frequency functions. His work here extended that begun by Pólya, Laguerre and Schur on approximating functions by polynomials with only real zeros. This work led Schoenberg to discover remarkable properties of polynomials all of whose zeros are negative and real. In 1966 Schoenberg moved from the University of Pennsylvania to the University of Wisconsin where he became a member of the Mathematics Research Center. He remained at Wisconsin-Madison until he retired in 1973. However, he continued to produce important works after he retired and of his 174 papers and books, over 50 appeared after his retirement. During his time at Wisconsin, Schoenberg introduced another concept of major importance, namely cardinal splines. He investigated their wide applications in approximation theory in a series of three papers between 1969 and 1973. Schoenberg published joint papers with a number of mathematicians including his brother-in-law Rademacher. He also collaborated with Besicovitch, Erdos, Curry, von Neumann and Szego. Although he never produced a joint publication with his father-in-law Landau, he did spend a great deal of his time working on problems that Landau had considered. In [4], written at the time he retired in 1973, his interests were described:Schoenberg is a man of broad culture, fluent in several languages, addicted to art, music and world literature, sensitive, gracious and giving in all ways. [He] frequently builds physical models related to his mathematical enquiries. ... The working desk at his home where he engages in research is actually a draftsman's bench complete with T-square, etc. and a tall stool. Mobiles, artistic works, models of ruled surfaces, icosahedrons and other objects are strewn throughout the room. English, French and German novels, numerous paintings and artefacts are scattered on all the nearby easy chairs. He buys and collects books of all vintages with passion. Historical mathematical discourses especially fascinate him and his articles frequently reflect this interest. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schoenberg.html (2 of 3) [2/16/2002 11:30:48 PM]

Schoenberg

Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schoenberg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schoenberg.html (3 of 3) [2/16/2002 11:30:48 PM]

Schonflies

Arthur Moritz Schönflies Born: 17 April 1853 in Landsberg an der Warthe, Germany (now Gorzów, Poland) Died: 27 May 1928 in Frankfurt am Main, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Arthur Schönflies was a student at the University of Berlin from 1870 until 1875 working under Kummer and Weierstrass. He obtained a doctorate from Berlin in 1877 and the following year he obtained a post as a teacher at a school in Berlin. In 1880 he went to Colmar in Alsace to teach. He then wrote his Habilitation thesis which he presented to Göttingen, the qualification being awarded in 1884. Klein worked to set up a chair of applied mathematics at Göttingen and in 1892 Schönflies was appointed to this chair. He left Göttingen in 1899 to take up a chair at Königsberg, then in 1911 he became professor at the Academy for Social and Commercial Sciences in Frankfurt. This Academy became a University in 1914. Schönflies ended his career at the University of Frankfurt where he served as professor from 1914 until 1922 being rector of the University in the session 1920-21. Schönflies worked first on geometry and kinematics but became best known for his work on set theory and crystallography. Klein suggested the problem of finding the crystallographic space groups in the late 1880's. By 1891 he had found the complete list of 230 such groups. His presentation of crystallographic space groups published in 1892 used the latest aspects of group theory and became a classic on the subject. In fact the classification of the crystallographic space groups was made independently by E S Fedorov. Schönflies corresponded with Fedorov and corrected some minor errors in his classification. He

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schonflies.html (1 of 2) [2/16/2002 11:30:49 PM]

Schonflies

republished his classification in 1923 and in the same year he published a book on crystallography. In around 1895 Schönflies turned his attention towards set theory and topology. He wrote many works which were important at the time they were published but they were rather superseded by Hausdorff's Grundzüge der Mengenlehre in 1914. Three important papers on plane topology proved the topological invariance of the dimension of the square. His work contains gaps and errors which were investigated by Brouwer who made some deep discoveries from studying these errors. Schönflies also wrote on kinematics and projective geometry. He wrote textbooks on descriptive geometry and analytic geometry and a calculus textbook jointly with Nernst. In 1895 Schönflies edited Plücker's complete works. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Arthur Schönflies

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1890 to 1900

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schonflies.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schonflies.html (2 of 2) [2/16/2002 11:30:49 PM]

Schooten

Frans van Schooten Born: 1615 in Leiden, Netherlands Died: 29 May 1660 in Leiden, Netherlands

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Frans van Schooten should not be confused with his father, Frans van Schooten (the elder), who was professor at the engineering school in Leiden. He enrolled at the University of Leiden in 1631 and he studied mathematics there. In 1637 Descartes visited Leiden and met van Schooten. This proved important for van Schooten since Descartes provided contacts for van Schooten to become acquainted with Mersenne's circle in Paris. Some time after this he went abroad, travelling first to Paris and then to London where he stayed from 1641 to 1643. He discussed mathematics in these two centres and he continued to correspond with the mathematicians he met in these towns after his return to Leiden, but unfortunately this correspondence is now lost. While in Paris he obtained manuscripts Viète's work and he later published them in Leiden. In 1643 van Schooten became assistant to his father and when his father died two years later he was appointed to his father's chair. Van Schooten was one of the main people to promote the spread of Cartesian geometry. He studied Stifel's edition of Rudolff's Coss and printed the first Latin version of Descartes La géométrie in 1649. His own book Geometria a Renato Des Cartes appeared in two volumes 1659-1661. It contained appendices by three of van Schooten disciples, Jan de Witt, Johan Hudde, and Hendrick van Heuraet. He also published Exercitationes mathematicae in 1657. Van Schooten also taught Huygens who became a friend. It was on Descartes's recommendation that van

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schooten.html (1 of 2) [2/16/2002 11:30:51 PM]

Schooten

Schooten replaced Stampioen as tutor to Huygens and his brother. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) A Poster of Frans van Schooten

Mathematicians born in the same country

Some pages from publications

Frontispiece of Geometria a Renato Des Cartes (1649)

Other references in MacTutor

1. Chronology: 1625 to 1650 2. Chronology: 1650 to 1675

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schooten.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schooten.html (2 of 2) [2/16/2002 11:30:51 PM]

Schottky

Friedrich Hermann Schottky Born: 24 July 1851 in Breslau, Germany (now Wroclaw, Poland) Died: 12 Aug 1935 in Berlin, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Friedrich Schottky attended the Humanistisches Gymnasium St Magdalenen in Breslau. After graduating from the Gymnasium he entered the University of Breslau in 1870, graduating in 1874. After leaving Breslau he studied at the University of Berlin under Weierstrass and Helmholtz and obtained his doctorate in 1875. After obtaining his doctorate, Schottky taught at the University of Berlin from 1875 until 1882 when he was appointed professor of mathematics in Zurich. He held this appointment in Zurich for ten years before moving to another chair at the University of Marburg in 1892. Keeping up his move every ten years he went to a chair at the University of Berlin in 1902 but remained there for twenty years until he retired in 1922. Most of Schottky's work concerns elliptic, abelian and theta functions. His doctoral thesis is an important contribution to conformal mappings of multiply connected plane domains. This was the origin of the mapping of a domain bounded by three disjoint circles which provides an example of an automorphic function with a Cantor set boundary. Schottky's thesis also discusses conformal mappings of domains bounded by circular and conic arcs. This work was published in 1877. Schottky's Theorem (1904) is related to Picard's Theorem. Schottky published 55 papers and, in 1880, a book Abriss einer Theorie der Abel'schen Functionen von drei Variablen. In [1] Hans Freudenthal writes:His work is difficult to read. Although he was a student of Weierstrass, his approach to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schottky.html (1 of 2) [2/16/2002 11:30:53 PM]

Schottky

function theory was Riemannian in spirit, combined with Weierstrassian rigor. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schottky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schottky.html (2 of 2) [2/16/2002 11:30:53 PM]

Schoute

Pieter Hendrik Schoute Born: 21 Jan 1846 in Wormerveer, Netherlands Died: 18 April 1923 in Groningen, Netherlands Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pieter Hendrik Schoute came from a family of industrialists who had a business near Amsterdam. He studied at Polytechnic in Delft, graduating as a civil engineer in 1867. After graduating, Schoute went to Leiden to undertake research in mathematics. His doctorate was awarded by Leiden in 1870 for a dissertation Homography applied to the theory of quadric surfaces. After being awarded his doctorate, Schoute taught mathematics in a secondary school in Nijmegen from 1871 until 1874 when he moved to The Hague and again taught mathematics in a secondary school there until 1881. From 1881 until his death he was professor of mathematics at the University of Groningen. Johann Bernoulli was professor of mathematics at Groningen from 1695 until 1705 when he left the University disillusioned. Sierksma in [3] explains that after Bernoulli left Groningen there were a number of professors of mathematics who were to some extent involved in mathematics but none of them was really interested in it. The situation changed markedly for the better with the appointment of Schoute in 1881 and from that time the fortunes of mathematics at Groningen improved greatly. In particular there was the second excellent appointment of Floris de Boer as professor of mathematics to Groningen to join Schoute three years after he was appointed. Schoute studied various topics in geometry such as quadrics and algebraic curves. Struik writes in [1]:Schoute was a typical geometer. In his early work he investigated quadrics, algebraic curves, complexes, and congruences in the spirit of nineteenth-century projective, metrical, and enumerative geometry. From 1891 Schoute studied Euclidean geometry of more than 3 dimensions, writing 28 papers, some jointly with Alica Boole Stott the daughter of George Boole. It was research on regular polytopes, which generalise the concept of regular polyhedra, that led to his collaboration with Stott. The article [3] is published in the journal Nieuw Archief Wiskunde. Schoute was an editor of this journal from 1898 until his death in 1923. He was also a founding editor of Revue semestrielle des publications mathématique from 1893, when the journal was founded, again until his death in 1923. Schoute was elected to the Royal Netherlands Academy of Sciences in 1886. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schoute.html (1 of 2) [2/16/2002 11:30:54 PM]

Schoute

Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other Web sites

1. Nijmegen, Netherlands (in Dutch, but with some pictures) 2. Nijmegen, Netherlands

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schoute.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schoute.html (2 of 2) [2/16/2002 11:30:54 PM]

Schouten

Jan Arnoldus Schouten Born: 28 Aug 1883 in Nieuweramstel (now part of Amsterdam), Netherlands Died: 20 Jan 1971 in Epe, Netherlands

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jan A Schouten studied electrical engineering at the Technische Hogeschool in Delft and then for several years he was an electrical engineer. However he came from a family with substantial amounts of money and when he inherited money he gave up his electrical engineering job and went to Leiden University to study mathematics. Schouten's doctoral thesis, presented in 1914, was on tensor analysis, a topic he worked on all his life. That same year he became professor of mathematics at Delft and held the post for nearly 30 years. However, A Nijenhuis one of his co-workers, wrote:In 1943 Schouten resigned the post, divorced his wife and remarried. From then on, he lived in semiseclusion at Epe. Nijenhuis summarises Schouten's life in the following words:A descendant of a prominent family of shipbuilders, Schouten grew up in comfortable surroundings. He became not only on of the founders of the "Ricci calculus" but also an efficient organiser (he was a founder of the Mathematical Center at Amsterdam in 1946) and an astute investor. A meticulous lecturer and painfully accurate author, he instilled the same standards in his pupils. From 1948 until 1953 Schouten was professor of mathematics at the University of Amsterdam but he did not teach. He was director of the Mathematical Research Centre at Amsterdam for five years. Schouten produced 180 papers and 6 books on tensor analysis. He applied it to Lie groups, relativity, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schouten.html (1 of 2) [2/16/2002 11:30:56 PM]

Schouten

unified field theory and systems of differential equations. In 1919 he made the independent discovery of connections in Riemannian manifolds discovered by Levi-Civita earlier. Influenced by Weyl and Eddington, Schouten investigated affine, projective and conformal mappings. Klein's Erlanger Programm of 1872 looked at geometry as properties invariant under the action of a group. This approach had a large influence on Schouten's approach to his topic. An important figure in the development of the tensor calculus, Schouten was president of the 1954 International Congress of Mathematicians at Amsterdam. All the obituaries listed below were all written by co-workers of Schouten who were strongly influenced by him. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Jan A Schouten

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schouten.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schouten.html (2 of 2) [2/16/2002 11:30:56 PM]

Schreier

Otto Schreier Born: 3 March 1901 in Vienna, Austria Died: 2 June 1929 in Hamburg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Otto Schreier graduated from high school in Vienna in July 1919. He entered the University of Vienna in 1920 to study mathematics. At Vienna he attended lecture courses by Wirtinger, Furtwängler, Hahn, Reidemeister, Rella, Lense and Vietoris. His doctorate, supervised by Furtwängler, was awarded for a thesis Über die Erweiterung von Gruppen (On the extension of groups) on 8 November 1923. After receiving his doctorate, Schreier went to Hamburg and worked until his death at the Mathematische Seminar. He was appointed to the post of assistant in the summer 1925 and worked for his habilitation. In fact Schreier gave lecture courses, at the request of the mathematical faculty, before completing his habilitation. This was formally awarded on 1 December 1926 for a thesis entitled Die Untergruppen der freien Gruppe (The subgroups of free groups). Schreier was offered a professorship at the University of Rostock in 1928 and decided to accept the position but he preferred to wait until the summer of 1929 before taking up the post. During the beginning of the 1928/29 session Schreier lectured on function theory giving parallel courses in Hamburg and Rostock. However, around Christmas of 1928, an illness which had been steadily worsening prevented him from continuing with his lectures. He died five months later at the age of 28 of a 'general sepsis'. The sulpha drugs discovered a few years later probably would have saved his life and therefore would have greatly changed the development of combinatorial group theory. Schreier was much influenced by Furtwängler and Reidemeister. His first paper in 1924 gave a simple algebraic proof of a theorem on knot groups, which generalised a theorem given by Dehn 10 years earlier. He may have been directed towards the main theorem, which proves that certain torus knots were http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schreier.html (1 of 3) [2/16/2002 11:30:58 PM]

Schreier

not isomorphic to their mirror images, by Reidemeister. These knots gave rise to groups which were free products with an amalgamated subgroup and Schreier studied this property in detail in a 1927 paper. Schreier will be best remember for his work on subgroups of free groups which he studied in his habilitation thesis. He published the results in 1927 in the paper Die Untergruppen der freien Gruppe which is described in [1] as ... one of the most important papers ever published on combinatorial group theory. It took a long time for all its aspects to become effective, and it contains much more than the title indicates. In January 1926 Schreier attended a lecture given by Reidemeister in Hamburg on finding presentations for normal subgroups of finitely presented groups. Reidemeister published his method later in 1926. Schreier, who took a more algebraic approach compared to Reidemeister's geometrical approach, was able to extend Reidemeister's method to arbitrary subgroups and, by cleverly choosing generators for the subgroup, was able to greatly simplify the presentation obtained. Schreier published his method in his 1927 paper Die Untergruppen der freien Gruppe. Other work of Schreier is described in [1] as follows. ... Schreier made important contributions to other parts of group theory. The classical Lie groups ... can be considered as topological spaces. Schreier (1927) showed that the fundamental group of such a space is always abelian. Schreier (1928) found an important refinement of the fundamental Jordan-Hölder theorem, 39 years after the publication of Hölder's paper. It is rare that such a widely used and basic theorem can be deepened after such a long time. (In this case, something even more unusual happened. Zassenhaus (1934) discovered a second improvement of the theorem.) Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Otto Schreier

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1920 to 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schreier.html (2 of 3) [2/16/2002 11:30:58 PM]

Schreier

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schreier.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schreier.html (3 of 3) [2/16/2002 11:30:58 PM]

Schroder

Friedrich Wilhelm Karl Ernst Schröder Born: 25 Nov 1841 in Mannheim, Germany Died: 16 June 1902 in Karlsruhe, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernst Schröder's important work is in the area of algebra, set theory and logic. He studied under Hesse and Kirchhoff then under Franz Neumann. His work on ordered sets and ordinal numbers is fundamental to the subject. In 1877 in Der Operations-kreis des Logikkalkuls Schröder, influenced by Boole and Grassmann, emphasised the duality of conjunction (intersection) and disjunction (union) showing how dual theorems could be found. He seems to be the first to use the term mathematical logic and he compares algebra and Boole's logic saying: There is certainly a contrast of the objects of the two operations. They are totally different. In arithmetic, letters are numbers, but here, they are arbitrary concepts. In Vorlesungen über die Algebra der Logik, a large work published between 1890 and 1905 (it was completed by E. Müller after his death), Schröder gave a detailed account of algebraic logic, provided a source for Tarski to develop the modern algebraic theory and gave an extensive bibliography of the history of logic. Lattice theory also grew out of this work. In addition to his work on logic he wrote an important article Über iterirte Functionen (1871) often cited as a basis of modern dynamical systems theory. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schroder.html (1 of 2) [2/16/2002 11:31:00 PM]

Schroder

List of References (13 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schroder.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schroder.html (2 of 2) [2/16/2002 11:31:00 PM]

Schrodinger

Erwin Rudolf Josef Alexander Schrödinger Born: 12 Aug 1887 in Erdberg, Vienna, Austria Died: 4 Jan 1961 in Vienna, Austria

Click the picture above to see eight larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Erwin Schrödinger's father Rudolf Schrödinger ran a small linoleum factory which he had inherited from his own father. Erwin's grandmother, Emily Bauer, was half English, this side of the family coning from Leamington Spa, and half Austrian with her father coming from Vienna. Schrödinger learnt English and German almost at the same time due to the fact that both were spoken in the household. He was not sent to elementary school, but received lessons at home from a private tutor up to the age of ten. He entered the Akademisches Gymnasium in the autumn of 1898, rather later than was usual since he spent a long holiday in England around the time he might have entered the school. He wrote later about his time at the Gymnasium:I was a good student in all subjects, loved mathematics and physics, but also the strict logic of the ancient grammars, hated only memorising incidental dates and facts. Of the German poets, I loved especially the dramatists, but hated the pedantic dissection of this works. In [15] there is the following quotation from a student in Schrödinger's class at school:Especially in physics and mathematics, Schrödinger had a gift for understanding that allowed him, without any homework, immediately and directly to comprehend all the material during the class hours and to apply it. After the lecture ... it was possible for [our professor] to call Schrödinger immediately to the blackboard and to set him problems, which he solved with playful facility. Schrödinger graduated in 1906 and, in that year, entered the University of Vienna. In theoretical physics http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schrodinger.html (1 of 6) [2/16/2002 11:31:02 PM]

Schrodinger

he studied analytical mechanics, applications of partial differential equations to dynamics, eigenvalue problems, Maxwell's equations and electromagnetic theory, optics, thermodynamics and statistical mechanics. It was Fritz Hasenöhrl's lectures on theoretical physics which had the greatest influence on Schrödinger. In mathematics he was taught calculus and algebra by Franz Mertens, function theory, differential equations and mathematical statistics by Wilhelm Wirtinger (who he found uninspiring as a lecturer). He also studied projective geometry, algebraic curves and continuous groups in lectures by Gustav Kohn. On 20 May 1910, Schrödinger was awarded his doctorate with a doctoral dissertation On the conduction of electricity on the surface of insulators in moist air. After this he undertook voluntary military service in the fortress artillery. Then he was appointed to an assistantship at Vienna but, rather surprisingly, in experimental physics rather than theoretical physics. He later said that his experiences conducting experiments proved an invaluable asset to his theoretical work since it gave him a practical philosophical framework in which to set his theoretical ideas. Having completed the work for his habilitation, he was awarded the degree on 1 September 1914. That it was not an outstanding piece of work is shown by the fact that the committee was not unanimous in recommending him for the degree. As Moore writes in [7]:Schrödinger's early scientific work was inhibited by the absence of a group of first-class theoreticians in Vienna, against whom he could sharpen his skills by daily argument and mutual criticism. In 1914 Schrödinger's first important paper was published developing ideas of Boltzmann. However, with the outbreak of World War I, Schrödinger received orders to take up duty on the Italian border. His time of active service was not useless as far as research was concerned and he continued his theoretical work, submitting another paper from his position on the Italian front. In 1915 he was transferred to duty in Hungary and from there he submitted further work for publication. After being sent back to the Italian front, Schrödinger received a citation for outstanding service commanding a battery during a battle. In the spring of 1917 Schrödinger was sent back to Vienna, assigned to teach a course in meteorology. He was able to continue research and he published his first results on quantum theory. After the end of the war he continued working at Vienna. From 1918 to 1920 he made substantial contributions to colour theory. Schrödinger had worked at Vienna on radioactivity, proving the statistical nature of radioactive decay. He had also made important contributions to the kinetic theory of solids, studying the dynamics of crystal lattices. On the strength of his work he was offered an associate professorship at Vienna in January 1920 but by this time he wished to marry Anny Bertel. They had become engaged in 1919 and Anny had come to work as a secretary in Vienna on a monthly salary which was more than Schrödinger's annual income. He was offered an associate professorship, still not at a salary large enough to support a non-working wife so he declined. Schrödinger accepted instead an assistantship in Jena and married Anny on 24 March 1920. After only a short time there, he moved to a chair in Stuttgart where he became friendly with Hans Reichenbach. He then moved to a chair Breslau, his third move in eighteen months. Soon however he was to move yet again, accepting the chair of theoretical physics at Zurich in late 1921. During these years of changing from one place to another, Schrödinger studied physiological optics, in particular he continued his work http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schrodinger.html (2 of 6) [2/16/2002 11:31:02 PM]

Schrodinger

on the theory of colour vision. Weyl was Schrödinger's closest colleague in his first years in Zurich and he was to provide the deep mathematical knowledge which would prove so helpful to Schrödinger in his work. The intellectual atmosphere in Zurich suited Schrödinger and Zurich was to be the place where he made his most important contributions. From 1921 he studied atomic structure. Then in 1924 he began to study quantum statistics, soon after this he read de Broglie's thesis which was to have a major influence on his thinking. On 3 November 1925 Schrödinger wrote to Einstein:A few days ago I read with great interest the ingenious thesis of Louis de Broglie, which I finally got hold of... On 16 November, in another letter, Schrödinger wrote:I have been intensely concerned these days with Louis de Broglie's ingenious theory. It is extraordinarily exciting, but still has some very grave difficulties. One week later Schrödinger gave a seminar on de Broglie's work and a member of the audience, a student of Sommerfeld's, suggested that the there should be a wave equation. Within a few weeks Schrödinger had found his wave equation. Schrödinger published his revolutionary work relating to wave mechanics and the general theory of relativity in a series of six papers in 1926. Wave mechanics, as proposed by Schrödinger in these papers, was the second formulation of quantum theory, the first being matrix mechanics due to Heisenberg. The relation between the two formulations of wave mechanics and matrix mechanics was understood by Schrödinger immediately as this quotation from one of his 1926 papers shows:To each function of the position- and momentum- coordinates in wave mechanics there may be related a matrix in such a way that these matrices, in every case satisfy the formal calculation rules of Born and Heisenberg. ... The solution of the natural boundary value problem of this differential equation in wave mechanics is completely equivalent to the solution of Heisenberg's algebraic problem. The work was indeed received with great acclaim. Planck described it as epoch-making work. Einstein wrote:... the idea of your work springs from true genius... Then, ten days later Einstein wrote again:I am convinced that you have made a decisive advance with your formulation of the quantum condition... Ehrenfest wrote:I am simply fascinated by your [wave equation] theory and the wonderful new viewpoint it brings. Every day for the past two weeks our little group has been standing for hours at a time in front of the blackboard in order to train itself in all the splendid ramifications. Schrödinger accepted an invitation to lecture at the University of Wisconsin, Madison leaving in December 1926 to give his lectures in January and February 1927. Before he left he was told he was the leading candidate for Planck's chair in Berlin. After giving a brilliant series of lectures in Madison he was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schrodinger.html (3 of 6) [2/16/2002 11:31:02 PM]

Schrodinger

offered a permanent professorship there but [7]:... he was not at all tempted by an American position, and he declined on the basis of a possible commitment to Berlin. The list of candidates to succeed Planck in the chair of theoretical physics at Berlin was impressive. Sommerfeld was ranked in first place, followed by Schrödinger, with Born as the third choice. When Sommerfeld decided not to leave Munich, the offer was made to Schrödinger. He went to Berlin, taking up the post on 1 October 1927 and there he became a colleague of Einstein's. Although he was a Catholic, Schrödinger decided in 1933 that he could not live in a country in which persecution of Jews had become a national policy. Alexander Lindemann, the head of physics at Oxford University, visited Germany in the spring of 1933 to try to arrange positions in England for some young Jewish scientists from Germany. He spoke to Schrödinger about posts for one of his assistants and was surprised to discover that Schrödinger himself was interested in leaving Germany. Schrödinger asked for a colleague, Arthur March, to be offered a post as his assistant. To understand Schrödinger's request for March we must digress a little and comment on Schrödinger's liking for women. His relations with his wife had never been good and he had had many lovers with his wife's knowledge. Anny had her own lover for many years, Schrödinger's friend Weyl. Schrödinger's request for March to be his assistant was because, at that time, he was in love with Arthur March's wife Hilde. Many of the scientists who had left Germany spent the summer of 1933 in the South Tyrol. Here Hilde became pregnant with Schrödinger's child. On 4 November 1933 Schrödinger, his wife and Hilde March arrived in Oxford. Schrödinger had been elected a fellow of Magdalen College. Soon after they arrived in Oxford, Schrödinger heard that, for his work on wave mechanics, he had been awarded the Nobel prize. In the spring of 1934 Schrödinger was invited to lecture at Princeton and while there he was made an offer of a permanent position. On his return to Oxford he negotiated about salary and pension conditions at Princeton but in the end he did not accept. It is thought that the fact that he wished to live at Princeton with Anny and Hilde both sharing the upbringing of his child was not found acceptable. The fact that Schrödinger openly had two wives, even if one of them was married to another man, did not go down too well in Oxford either but his daughter Ruth Georgie Erica was born there on 30 May 1934. In 1935 Schrödinger published a three-part essay on The present situation in quantum mechanics in which his famous Schrödinger's cat paradox appears. This was a thought experiment where a cat in a closed box either lived or died according to whether a quantum event occurred. The paradox was that both universes, one with a dead cat and one with a live one, seemed to exist in parallel until an observer oped the box. In 1936 Schrödinger was offered the chair of physics at the University of Edinburgh in Scotland. He may have accepted that post but for a long delay in obtaining a work permit from the Home Office. While he was waiting he received an offer from the University of Graz and he went to Austria and spent the years 1936-1938 in Graz. Born was then offered the Edinburgh post which he quickly accepted. However the advancing Nazi threat caught up with Schrödinger again in Austria. After the Anschluss the Germans occupied Graz and renamed the university Adolf Hitler University. Schrödinger wrote a letter to the University Senate, on the advice on the new Nazi rector, saying that he had:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schrodinger.html (4 of 6) [2/16/2002 11:31:02 PM]

Schrodinger

... misjudged up to the last the true will and the true destiny of my country. I make this confession willingly and joyfully... It was a letter he was to regret for the rest of his life. He explained the reason to Einstein in a letter written about a year later:I wanted to remain free - and could not do so without great duplicity. The Nazis could not forget the insult he had caused them when he fled from Berlin in 1933 and on 26 August 1938 he was dismissed from his post for 'political unreliability'. He went to consult an official in Vienna who told him that he must get a job in industry and that he would not be allowed to go to a foreign country. He fled quickly with Anny, this time to Rome from where he wrote to de Valera as President of the League of Nations. De Valera offered to arrange a job for him in Dublin in the new Institute for Advanced Studies he was trying to set up. From Rome Schrödinger went back to Oxford, and there he received an offer of a one year visiting professorship at the University of Gent. After his time in Gent, Schrödinger went to Dublin in the autumn of 1939. There he studied electromagnetic theory and relativity and began to publish on a unified field theory. His first paper on this topic was written in 1943. In 1946 he renewed his correspondence with Einstein on this topic. In January 1947 he believed he had made a major breakthrough [7]:Schrödinger was so entranced by his new theory that he threw caution to the winds, abandoned any pretence of critical analysis, and even though his new theory was scarcely hatched, he presented it to the Academy and to the Irish press as an epoch-making advance. The Irish Times carried an interview with Schrödinger the next day in which he said:This is the generalisation. Now the Einstein Theory becomes simply a special case... I believe I am right, I shall look an awful fool if I am wrong. Einstein, however, realised immediately that there was nothing of merit in Schrödinger's 'new theory' [7]:[Schrödinger] was even thinking of the possibility of receiving a second Nobel prize. In any case, the entire episode reveals a lapse in judgment, and when he actually read Einstein's comment, he was devastated. Einstein wrote immediately breaking off the correspondence on unified field theory. Unified field theory was, however, not the only topic to interest him during his time at the Institute for Advanced Study in Dublin. His study of Greek science and philosophy is summarised in Nature and the Greeks (1954) which he wrote while in Dublin. Another important book written during this period was What is life (1944) which led to progress in biology. On the personal side Schrödinger had two further daughters while in Dublin, to two different Irish women. He remained in Dublin until he retired in 1956 when he returned to Vienna and wrote his last book Meine Weltansicht (1961) expressing his own metaphysical outlook. During his last few years Schrödinger remained interested in mathematical physics and continued to work on general relativity, unified field theory and meson physics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schrodinger.html (5 of 6) [2/16/2002 11:31:02 PM]

Schrodinger

List of References (19 books/articles)

Some Quotations (2)

A Poster of Erwin Schrödinger

Mathematicians born in the same country

Cross-references to History Topics

The quantum age begins

Honours awarded to Erwin Schrödinger (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1933

Fellow of the Royal Society

Elected 1949

Lunar features

Crater Schrodinger

Other Web sites

1. Nobel prizes site (A biography of Schrödinger and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schrodinger.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schrodinger.html (6 of 6) [2/16/2002 11:31:02 PM]

Schroeter

Heinrich Eduard Schroeter Born: 8 Jan 1829 in Königsberg, Germany (now Kaliningrad, Russia) Died: 3 Jan 1892 in Breslau, Germany (now Wroclaw, Poland)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Heinrich Schroeter attended secondary school in Königsberg, then in 1845 he entered the University of Königsberg to study mathematics and physics. There he was taught by Friedrich Richelot. After doing military service he studied at the University of Berlin for two years. At Berlin he was taught by Dirichlet and Steiner. Steiner was a major influence on Schroeter who spent most of his life working on geometry. However his doctorate was obtained from Königsberg in 1854 where his dissertation was supervised by Richelot on elliptic functions. Schroeter then lectured at the University of Breslau, becoming professor there in 1858. He remained there for the rest of his life. His last few years were badly affected by ill health and he suffered from paralysis. Schroeter edited Steiner's lectures on synthetic geometry, then published a major book Die Theorie der Oberflächen in 1880. This work on the theory of second order surfaces and third order space curves continues Steiner's work. In 1888 Schroeter published on third-order plane curves and in 1890 he published his study on fourth-order space curves. Among his students were Rudolf Sturm. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schroeter.html (1 of 2) [2/16/2002 11:31:05 PM]

Schroeter

List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Heinrich Schroeter (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Schroter

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schroeter.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schroeter.html (2 of 2) [2/16/2002 11:31:05 PM]

Schubert

Hermann Cäsar Hannibal Schubert Born: 22 May 1848 in Potsdam, Germany Died: 20 July 1911 in Hamburg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hermann Schubert worked on enumerative geometry, winning a prize in 1874 for solving a question posed by Zeuthen. Enumerative geometry considers those parts of algebraic geometry that involve a finite number of solutions. Using methods of Chasles, with Schröder's logical calculus as a model, he set up a system to solve such problems, he called it the principal of conservation of the number. Hilbert, in 1900, asked for a proof, which was given by Severi in 1912. Some remarkable counting results of Schubert were neglected for many years for their lack of rigour but recently many of them have been confirmed. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of Hermann Schubert

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schubert.html (1 of 2) [2/16/2002 11:31:06 PM]

Schubert

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schubert.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schubert.html (2 of 2) [2/16/2002 11:31:06 PM]

Schur

Issai Schur Born: 10 Jan 1875 in Mogilyov, Mogilyov province, Belarus Died: 10 Jan 1941 in Tel Aviv, Palestine

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Although Issai Schur was born in Mogilyov on the Dnieper, he spoke German without a trace of an accent, and nobody even guessed that it was not his first language. He went to Latvia at the age of 13 and there he attended the Gymnasium in Libau, now called Liepaja. In 1894 Schur entered the University of Berlin to read mathematics and physics. Frobenius was one of his teachers and he was to greatly influence Schur and later to direct his doctoral studies. Frobenius and Burnside had been the two main founders of the theory of representations of groups as groups of matrices. This theory proved a very powerful tool in the study of groups and Schur was to learn the foundations of this subject from Frobenius. Schur then made major steps forward, both in work of his own and work done in collaboration with Frobenius. In 1901 Schur obtained his doctorate with a thesis which examined rational representations of the general linear group over the complex field. Functions which Schur introduced in his thesis are today called S-functions, where the S stands for Schur. Interest in the results of Schur's thesis continues today, for example J A Green published an account of these results in a modern setting in 1980. In 1903 Schur became a lecturer at Berlin University and then, from 1911 until 1916, he held a professorship in mathematics at the University of Bonn. He returned to Berlin in 1916 and there he built his famous school and spent most of the rest of his life there. He was promoted to full professor in Berlin in 1919, three years after he returned there, and he held this chair until he was dismissed by the Nazis in 1935.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schur.html (1 of 5) [2/16/2002 11:31:08 PM]

Schur

Schur is mainly known for his fundamental work on the representation theory of groups but he also worked in number theory, analysis and other topics described below. Between 1904 and 1907 he worked on projective representations of groups and group characters. One of the most fundamental results which he discovered at this time is today called Schur's Lemma. In a series of papers he introduced the concept now known as the 'Schur multiplier'. This is an extremely important abstract concept which arose from the concrete problems that Schur was studying. Much later, in 1949, Eilenberg and MacLane defined cohomology groups. There were unaware at that time that the second cohomology group with coefficients in the nonzero complex numbers is the Schur multiplier, and therefore that Schur had made some of the first steps forty years earlier. Around 1914 Schur's interest in representations of groups was put to one side while he worked on other topics but, around 1925, developments in theoretical physics showed that groups representations were of fundamental importance in that subject. Schur returned to work on representation theory with renewed vigour and he was able to complete the programme of research begun in his doctoral dissertation and give a complete description of the rational representations of the general linear group. Schur was also interested in reducibility, location of roots and the construction of the Galois group of classes of polynomials such as Laguerre and Hermite polynomials. In [1] an indication of the other topics that Schur worked on is given:First there was pure group theory, in which Schur adopted the surprising approach of proving without the aid of characters, theorems that had previously been demonstrated only by that means. Second, he worked in the field of matrices. Third, he handled algebraic equations, sometimes proceeding to the evaluation of roots, and sometimes treating the so-called equation without affect, that is, with symmetric Galois groups. He was also the first to give examples of equations with alternating Galois groups. Fourth, he worked in number theory; Fifth in divergent series; Sixth in integral equations; and lastly in function theory. The school which Schur built at Berlin was of major importance not only for the representation theory of groups but, as indicated above, for other areas of mathematics. The school partly worked through the Schur's lecturing [7]:...there are [many] mathematicians who went to Schur's lectures and seminars in Berlin and were strongly influenced by him... The school also worked with collaborations [1]:A lively interchange with many colleagues led Schur to contribute important memoirs .... Some of these were published as collaborations with other authors, although publications with dual authorship were almost unheard of at that time. This school was certainly the most coherent and influential group of mathematicians in Berlin, and among the most important in all of Germany. Schur's charismatic leadership inspired those around him to push forward with research on group representations. Schur's own impressive contributions were extended by his students in a number of different directions. They worked on topics such as soluble http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schur.html (2 of 5) [2/16/2002 11:31:08 PM]

Schur

groups, combinatorics, and matrix theory. Among the students who completed their doctorates under Schur were Richard Brauer, Alfred Brauer (Richard Brauer's brother), Robert Frucht, Bernhard Neumann, Richard Rado, and Helmut Wielandt. There were others who worked under Schur such as Kurt Hirsch, Walter Ledermann, Hanna Neumann and Menahem Max Schiffer. Ledermann in [7] describes Schur as a teacher:Schur was a superb lecturer. His lectures were meticulously prepared... [and] were exceedingly popular. I remember attending his algebra course which was held in a lecture theatre filled with about 400 students. Sometimes, when I had to be content with a seat at the back of the lecture theatre, I used a pair of opera glasses to get at least a glimpse of the speaker. In 1922 Schur was elected to the Prussian Academy, proposed by Planck, the secretary of the Academy. Planck's address which listed Schur's outstanding achievements had been written by Frobenius, at least five years earlier, as Frobenius died in 1917. From 1933 events in Germany made Schur's life increasingly difficult. Hirsch spoke of the events of 1 April 1933 when posters carried the message 'Germans defend yourselves against jewish atrocity propaganda : buy only at German shops':That was the so-called 'Boycott Day', the day on which Jewish shops were boycotted and Jewish professors and lecturers were not allowed to enter the university. Everybody who was there had to make a little speech about the rejuvenation of Germany etc. And Bieberbach did this quite nicely and then he said 'A drop of remorse falls into my joy because my dear friend and colleague Schur is not allowed to be among us today'. On 7 April 1933 the Nazis passed a law which, under clause three, ordered the retirement of civil servants who were not of Aryan descent, with exemptions for participants in World War I and pre-war officials. Schur had held an appointment before World War I which should have qualified him as a civil servant, but the facts were not allowed to get in the way, and he was 'retired'. Schiffer wrote [8]:When Schur's lectures were cancelled there was an outcry among the students and professors, for Schur was respected and very well liked. The next day Erhard Schmidt started his lecture with a protest against this dismissal and even Bieberbach, who later made himself a shameful reputation as a Nazi, came out in Schur's defence. Schur went on quietly with his work on algebra at home. Schur saw himself as a German, not a Jew, and could not comprehend the persecution and humiliation he suffered under the Nazis. In fact Schur's dismissal was revoked and he was able to carry out some of his duties for a while. By November 1933 when Walter Ledermann took his Staatsexamen he was examined by Schur together with Bieberbach who was wearing Nazi uniform. There were invitations to Schur to go to the United States and to Britain but he declined them all, unable to understand how a German was not welcome in Germany. For example Ledermann obtained a scholarship to go to St Andrews in Scotland in the spring of 1934 and he tried unsuccessfully to persuade Schur to join him in St Andrews.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schur.html (3 of 5) [2/16/2002 11:31:08 PM]

Schur

Schur continued to suffer the humiliation that was heaped on him. Schiffer recalls an event in [8] relating to Schur's 60th birthday on 10 January 1935:Schur told me that the only person at the Mathematical Institute in Berlin who was kind to him was Grunsky, then a young lecturer. Long after the war, I talked to Grunsky about that remark and he literally started to cry: "You know what I did? I sent him a postcard to congratulate him on his sixtieth birthday. I admired him so much and was very respectful in that card. How lonely he must have been to remember such a small thing." Later in 1935 Schur was dismissed from his chair in Berlin but he continued to work there suffering great hardship and difficulties. A Brauer, the brother of Richard Brauer, writes in [6]:When Landau died in February 1938, Schur was supposed to give an address at his funeral. For that reason he was in need of some mathematical details from the literature. He asked me to help him in this matter. Of course I was not allowed to use the library of the mathematical institute which I had built up over many years. Finally I got an exemption for a week and could use the library of the Prussian Staatsbibliothek for a fee. ... So I could answer at least some of Schur's questions. Pressure was put on Schur to resign from the Prussian Academy to which he had been honoured to be elected in 1922. On 29 March 1938 Bieberbach wrote below Schur's signature on a document of the Prussian Academy:I find it surprising that Jews are still members of academic commissions. Just over a week later, on 7 April 1938, Schur resigned from Commissions of the Academy. However, the pressure on him continued and later that year he resigned completely from the Academy. Schur left Germany for Palestine in 1939, broken in mind and body, having the final humiliation of being forced to find a sponsor to pay the 'Reichs flight tax' to allow him to leave Germany. Without sufficient funds to live in Palestine he was forced to sell his beloved academic books to the Institute for Advanced Study in Princeton. He died two years later on his 66th birthday. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) A Poster of Issai Schur

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schur.html (4 of 5) [2/16/2002 11:31:08 PM]

Schur

JOC/EFR October 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schur.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schur.html (5 of 5) [2/16/2002 11:31:08 PM]

Schwartz

Laurent Schwartz Born: 5 March 1915 in Paris, France

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Laurent Schwartz entered the Ecole Normale Supérieure in Paris in 1934. He graduated with the Agrégation de Mathématique in 1937 and studied for his doctorate in the Faculty of Science at Strasbourg which he was awarded in 1943. Schwartz spent the year 1944-45 lecturing at the Faculty of Science at Grenoble before moving to Nancy where he became a professor at the Faculty of Science. It was during this period of his career that he produced his famous work on the theory of distributions described below. In 1953 Schwartz returned to Paris where he became professor, holding this position until 1959. He taught at the Ecole Polytechnique in Paris from 1959 to 1980. He then spent three years at the University of Paris VII before he retired in 1983. The outstanding contribution to mathematics which Schwartz made in the late 1940s was his work in the theory of distributions. The first publication in which he presented these ideas was Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathématique et physiques which appeared in 1948. The theory of distribution is a considerable broadening of the differential and integral calculus. Heaviside and Dirac had generalised the calculus with specific applications in mind. These, and other similar methods of formal calculation, were not, however, built on an abstract and rigorous mathematical foundation. Schwartz's development of the theory of distributions put methods of this type onto a sound basis, and greatly extended their range of application, providing powerful tools for applications in numerous areas.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwartz.html (1 of 3) [2/16/2002 11:31:10 PM]

Schwartz

In the article in Analysis in [6] François Treves describes Schwartz's work as follows:... Schwartz's idea (in 1947) was to give a unified interpretation of all the generalized functions that had infiltrated analysis as (continuous) linear functionals on the space Cc of infinitely differentiable functions vanishing outside compact sets. He provided a systematic and rigorous description, entirely based on abstract functional analysis and on duality. It is noteworthy that such an approach had a precedent, in the presentation by André Weil of the integration of locally compact groups ... Because of the demands of differentiability in distribution theory, the spaces of test-functions and their duals are somewhat more complicated. This has led to extensive studies of topological vector spaces beyond the familiar categories of Hilbert and Banach spaces, studies that, in turn, have provided useful new insights in some areas of analysis proper, such as partial differential equations or functions of several complex variables. Schwartz's ideas can be applied to many other spaces of test-functions besides Cc, as he himself and others have shown ... Harald Bohr presented a Fields Medal to Schwartz at the International Congress in Harvard on 30 August 1950 for his work on the theory of distributions. Harald Bohr [2] described Schwartz's 1948 paper as one:... which certainly will stand as one of the classical mathematical papers of our times. ... I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus to which the theory leads and on understanding how essential an advance its application may mean to many parts of higher analysis, such as spectral theory, potential theory, and indeed the whole theory of linear partial differential equations ... Schwartz has received a long list of prizes, medals and honours in addition to the Fields Medal. He received prizes from the Paris Academy of Sciences in 1955, 1964 and 1972. In 1972 he was elected a member of the Academy. He has been awarded honorary doctorates from many universities including Humboldt (1960), Brussels (1962), Lund (1981), Tel-Aviv (1981), Montreal (1985) and Athens (1993). Later work by Schwartz on stochastic differential calculus is described by him in the survey article [5], see also [4]. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1940 to 1950

Honours awarded to Laurent Schwartz (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1950

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwartz.html (2 of 3) [2/16/2002 11:31:10 PM]

Schwartz

Other Web sites

1. AMS 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schwartz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwartz.html (3 of 3) [2/16/2002 11:31:10 PM]

Schwarz

Hermann Amandus Schwarz Born: 25 Jan 1843 in Hermsdorf, Silesia (now Poland) Died: 30 Nov 1921 in Berlin, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hermann Schwarz's father was an architect. He studied at the Gymnasium in Dortmund where his favourite subject was chemistry. When he left school he intended to take a degree in chemistry and he entered the Gewerbeinstitut, later called the Technical University of Berlin, with this aim. Schwarz began his study of chemistry at Berlin but it was not long before Kummer and Weierstrass had influenced him to change to mathematics. The first of his teachers to influence the direction that his research would eventually take was Karl Pohlke. Through him Schwarz became interested in geometry. Schwarz attended Weierstrass's lectures on The integral calculus in 1861 and the notes that Schwarz took at these lectures still exist. His interest in geometry was soon combined with Weierstrass's ideas of analysis. As Bölling writes in [3]:... ideas coming from geometrical considerations were translated [by Schwarz] into the language of analysis. He continued to study in Berlin, being supervised by Weierstrass, until 1864 when he was awarded his doctorate. His doctoral dissertation was examined by Kummer. While in Berlin, Schwarz worked on minimal surfaces (surfaces of least area), a characteristic problem of the calculus of variations. Plateau published a famous memoir on the topic in 1866 and in the same year Weierstrass established a bridge between the theory of minimal surfaces and the theory of analytic functions. Schwarz had made an important contribution in 1865 when he discovered what is now known as the Schwarz minimal surface. This minimal surface has a boundary consisting of four edges of a regular tetrahedron. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwarz.html (1 of 4) [2/16/2002 11:31:12 PM]

Schwarz

Schwarz continued studying in Berlin for his teacher's training qualification which he completed by 1867. In that year he was appointed as a Privatdozent to the University of Halle. In 1869 he was appointed as professor of mathematics at the Eidgenössische Technische Hochschule in Zurich then, in 1875, he accepted appointment to the chair of mathematics at Göttingen University. Perhaps surprisingly after Schwarz succeeded Weierstrass accepting a professorship in Berlin in 1892, the balance in favour of the most eminent university in Germany for mathematics, which had undoubtedly been Berlin, began to shift towards Göttingen. There were several reasons for this. Firstly Schwarz failed to keep up his output of mathematical research after his move. Bieberbach in [2] put it rather well when he wrote that Schwarz retired to Berlin in 1892. That this was the case should not have come as a complete surprise to those making the appointment for Schwarz had published his Complete Works in 1890, two years earlier. Boerner writes in [1] that:... teaching duties and concern for [Schwarz's] many students took so much of his time that he published very little more. A contributing element may have been his propensity for handling both the important and the trivial with the same thoroughness, a trait also evident in his mathematical papers. We should not give the impression that the only reason for Berlin moving down from being the leading German university for mathematics to become its second university was due to Schwarz. The other effect was Klein whose dynamic leadership in Göttingen made it prosper at the expense of Berlin where Frobenius and Schwarz could not provide the same inspired approach. Perhaps the final sign that Göttingen had overtaken Berlin came in 1902 when Frobenius and Schwarz chose Hilbert to succeed to the Berlin chair which had become vacant on the death of Fuchs. Hilbert turned down the offer, preferring to remain at Göttingen. The Berlin chair was then filled by Schottky but, like Schwarz before him, he had moved to Berlin after his best days for mathematical research were behind him. Schwarz continued teaching at Berlin until 1918. We shall describe some of his very fine mathematical achievements in a moment, but first we note that he had several interests outside mathematics, although his marriage was a mathematical one since he married Kummer's daughter. Outside mathematics he was the captain of the local Voluntary Fire Brigade and, more surprisingly, he assisted the stationmaster at the local railway station by closing the doors of the trains. One important area which Schwarz worked on was that of conformal mappings. In 1870 he produced work related to the Riemann mapping theorem. Although Riemann had given a proof of the theorem that any simply connected region of the plane can be mapped conformally onto a disc, his proof involved using the Dirichlet problem. Weierstrass had shown that Dirichlet's solution to this was not rigorous, see [10] for details. Schwarz's gave a method to conformally map polygonal regions to the circle. Then, by approximating an arbitrary simply connected region by polygons he was able to give a rigorous proof of the Riemann mapping theorem. Schwartz also gave the alternating method for solving the Dirichlet problem which soon became a standard technique. This aspect of Schwarz's work is examined in detail in [10]. His most important work is a Festschrift for Weierstrass's 70th birthday. Schwarz answered the question of whether a given minimal surface really yields a minimal area. An idea from this work, in which he constructed a function using successive approximations, led Emile Picard to his existence proof for

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwarz.html (2 of 4) [2/16/2002 11:31:12 PM]

Schwarz

solutions of differential equations. It also contains the inequality for integrals now known as the 'Schwarz inequality', see [9] for details. The fact that Schwrz should have come up with a special case of the general result now known as the Cauchy-Schwarz inequality (or the Cauchy- Bunyakovsky- Schwarz inequality) is not surprising for much of his work is characterised by looking at rather specific and narrow problems but solving them using methods of great generality which have since found widespread applications. That he found such general methods says much for his great intuition which was perhaps based on a deep feeling for geometry. For example the Cauchy-Schwarz inequality appears in arithmetic, geometric and function-theoretic formulations in works of mathematicians such as Bunyakovsky, Cauchy, Grassmann, von Neumann and Weyl. The form in which the inequality is usually presented today 0 with its standard modern proof seems to have been first given by Weyl in 1918. In answering the problem of when Gauss's hypergeometric series was an algebraic function Schwarz, as he had done so many times, developed a method which would lead to much more general results. It was in this work that he defined a conformal mapping of a triangle with arcs of circles as sides onto the unit disc which is now known as the 'Schwarz function'. This function is an early example of an automorphic function and in this work Schwarz was looking at ideas which led Klein and Poincaré to develop the theory of automorphic functions. Let us end with quoting Bölling [3] on Schwarz's character. He writes:Schwarz was deeply influenced by Weierstrass. From their correspondence one finds that Schwarz addressed his teacher often with an accuracy going down to the last detail, sometimes almost timidly. Schwarz's demeanour has been described as naive, dramatic, coarse. In spite of giving the impression of self-confidence, he was, in fact, rather insecure and besides, not efficient in business matters. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) A Poster of Hermann Schwarz

Mathematicians born in the same country

Cross-references to History Topics

The beginnings of set theory

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwarz.html (3 of 4) [2/16/2002 11:31:12 PM]

Schwarz

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Schwarz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwarz.html (4 of 4) [2/16/2002 11:31:12 PM]

Schwarz_Stefan

Stefan Schwarz Born: 18 May 1914 in Nové Mesto nad Váhom, Slovakia Died: 1996

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stefan Schwarz was born in Nové Mesto, meaning new town, of Váhom in central Slovakia. He attended secondary school in the town of his birth and already at this stage his talent for mathematics was clear to all his teachers who were impressed by the depth of his interest in the subject. In 1932 he entered the Charles University of Prague, the Universita Karlova which had been founded in 1348 by the Holy Roman emperor Charles IV. Schwarz continued his studies at the Charles University after his undergraduate studied to work for his doctorate under the supervision of Karel Petr. Schwarz submitted his doctoral thesis On the reducibility of polynomials over finite fields in 1937. However 1937 was the year in which the situation in Czechoslovakia became serious. In September 1937 Hitler began his program of eastward expansion. In November 1937 he informed his military chiefs of his intention to invade Austria and Czechoslovakia. After the annexation of Austria in March 1938 Czechoslovak knew that they were next in line. Hitler, Mussolini, Chamberlain, and Daladier met at Munich on September 29-30. They agreed the Munich treaty requiring the Prague government to cede to Germany all of Bohemia and Moravia with populations that were more than half German. On 15 March 1939 Bohemia and Moravia were occupied by Hitler's armies and proclaimed a protectorate of the Third Reich. Schwarz knew that his life would be in danger if he remained in Prague until the Nazis arrived so, immediately after Bohemia and Moravia were occupied, Schwarz left Prague and returned to Slovakia where he felt more safe. He was able to find a post in the new Slovak Technical University in Bratislava and he taught there until 1944 publishing a 64 page work Theory of Semigroups in 1943. However the Germans had taken over the whole country after May 1942 and mass executions followed. Many sent to concentration camps and,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwarz_Stefan.html (1 of 3) [2/16/2002 11:31:15 PM]

Schwarz_Stefan

in November 1944, Schwarz was betrayed to the SS by some local informers. He was arrested and sent him to the Oranienburg-Sachsenhausen concentration camp north-west of Berlin. Of the total of around 200000 prisoners who were sent to Sachsenhausen, about 100000 died there. Many of the remainder were transferred to other camps and indeed Schwarz was transferred to Buchenwald, a camp which complemented the Sachsenhausen concentration camp. There were no gas chambers at Buchenwald but hundreds still died there through disease, starvation, assaults and executions. In April 1945, when Schwarz was near death, Buchenwald camp was liberated and his life was saved. Schwarz's two sisters did not survive the war however, one dying in the concentration camp at Auschwitz and the other in Bergen-Belsen camp. Despite his horrific experiences during the war, Schwarz quickly began to devote his energies to rebuilding the education system in his country. In 1946 he was appointed to the Comenius University in Bratislava, then the following year he was appointed professor at the Slovak Technical University in Bratislava. He continued teaching there until he retired in 1982. He was Head of the Mathematical Institute of the Slovak Academy of Sciences from 1966 until 1987. In [2] the direction of Schwarz's research towards semigroup theory is described:... in his dissertation Schwarz considered reducibility of polynomials over finite fields. Continuing his research, he studied arithmetic in the ring of integers in algebraic number fields. This led him to a necessity of investigating systems closed under an associative operation and the abstract theory of ideals in such systems. In short, his interest in classical algebra and number theory brought him to abstract semigroups. In addition to his work on semigroups, number theory and finite fields, Schwarz contributed to the theory of non-negative and Boolean matrices. Schwarz organised the first International Conference on Semigroups in 1968. At this conference setting up the journal Semigroup Forum was discussed and Schwarz became an editor from Volume 1 which appeared in 1970, continuing as editor until 1982. This was not his first editorial role since he had been an editor of the Czechoslovak Mathematical Journal from 1945 and continued to edit this journal until he was nearly 80 years old. He also founded the Mathematico-Physical Journal of the Slovak Academy of Sciences in 1950 and continued as an editor of the mathematics part of the journal when it split from the physics part to become Mathematica Slovaca until 1990. Among the many honours which were awarded to Schwarz there were memberships of the Czechoslovak Academy of Sciences in 1952 and the Slovak Academy of Sciences in 1953. He served as President of the Slovak Academy of Sciences from 1965 to 1970 and vice-president of the Czechoslovak Academy of Sciences from 1965 to 1970. Schwarz was awarded the 1980 National Prize of the Slovak Socialist Republic. Schwarz had a fine reputation as a teacher and was very well liked by his students who appreciated the help and guidance he gave them. The authors of [2] write:Anyone who has listened to lectures by Schwarz could not fail to notice that he is an exceptionally clear expositor. That is why his lectures were so popular among students and graduate engineers. The following student saying is well known among the initiated "If you don't understand Schwarz, go and study something far from mathematics ..." ...

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwarz_Stefan.html (2 of 3) [2/16/2002 11:31:15 PM]

Schwarz_Stefan

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schwarz_Stefan.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwarz_Stefan.html (3 of 3) [2/16/2002 11:31:15 PM]

Schwarzschild

Karl Schwarzschild Born: 9 Oct 1873 in Frankfurt am Main, Germany Died: 11 May 1916 in Potsdam, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karl Schwarzschild published his first paper on the theory of orbits at the age of 16, then studied at Strasbourg, then at Munich where he obtained his doctorate with a dissertation on an application of Poincaré's theory of stable configurations of rotating bodies to tidal deformation of moons and to Laplace's origin of the solar system. At a meeting of the German Astronomical Society in Heidelberg in 1900 he discussed the possibility that space was non-Euclidean. In the same year he published a paper giving a lower limit for the radius of curvature of space as 2500 light years. From 1901 until 1909 he was professor at Göttingen where he collaborated with Klein, Hilbert and Minkowski. Schwarzschild published on electrodynamics and geometrical optics during his time at Göttingen. In 1906 he studied the transport of energy through a star by radiation. From Göttingen he went to Potsdam but in 1914 he volunteered for military service. He served in Belgium, France and Russia. While in Russia he wrote two papers on Einstein's relativity theory and one on Planck's quantum theory. The quantum theory paper explained that the Stark effect, namely the splitting of the spectral lines of hydrogen by an electric field (the amount being proportional to the field strength), could be proved from the postulates of quantum theory. This was proved independently by a P Epstein from Munich at almost the same time. Schwarzschild's relativity papers give the first exact solution of Einstein's general gravitational equations, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwarzschild.html (1 of 2) [2/16/2002 11:31:16 PM]

Schwarzschild

giving an understanding of the geometry of space near a point mass. He also made the first study of black holes showing that bodies of sufficiently large mass would have an escape velocity exceeding the speed of light and so could not be seen. However he contracted an illness while in Russia and died soon after returning home. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

General relativity

Honours awarded to Karl Schwarzschild (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Schwarzschild

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Schwarzschild.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwarzschild.html (2 of 2) [2/16/2002 11:31:16 PM]

Schwinger

Julian Seymour Schwinger Born: 12 Feb 1918 in New York, USA Died: 16 July 1994 in Los Angeles, California, USA

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Julian Schwinger progressed rapidly through the public school system of New York City. He was an undergraduate at the City College of New York where he published his first physics paper at the age of sixteen. Isidor I Rabi, the professor who led the molecular beam laboratory at Columbia University at this time, persuaded Schwinger to study for his doctorate at Columbia. He received his doctorate in 1939 at the age of 21 for a dissertation in physics On the Magnetic Scattering of Neutrons. However, his thesis had been written two or three years before he was awarded the degree, there being problems in completing the formalities. Uhlenbeck [4] explained about Schwinger's problems in obtaining his doctorate:I was in Columbia in 1938 and Schwinger was in trouble; he couldn't get his Ph.D. because he didn't go to lectures of the mathematicians and he didn't have enough credits. So Rabi had told Schwinger that he had to go to my lectures at Columbia; of course, he didn't because it was early in the morning, and I asked Rabi, 'What shall I do?'. I was of course perfectly willing to give him an 'A' on the course because he needed the credits. ... He clearly knew as much as I did - we talked as complete equals. ... Rabi said, 'No, you shouldn't do that, you should give him an exam and make it a tough one.' So I did. We made an appointment, and of course he knew everything. He somehow had got the notes. After the award of his doctorate Schwinger worked at the University of California, Berkeley from 1939 to 1941. During the first year he was a National Research Council Fellow and then he became J Robert Oppenheimer's assistant. In 1941 he was appointed as an instructor in physics at Purdue University and the following year he was promoted to Assistant Professor.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwinger.html (1 of 4) [2/16/2002 11:31:18 PM]

Schwinger

During World War II, beginning in 1943, Schwinger was given leave of absence from Purdue and was sent to the Radiation Laboratory in the Massachusetts Institute of Technology. Later he was sent to the Metallurgical Laboratory (atom-bomb project) of the University of Chicago where Wigner was working. Schwinger did not like the work on atomic bombs so he got in his car and drove to Boston where Uhlenbeck was working on radar at the Radiation Laboratory. Schwinger asked Uhlenbeck if he could work there and it was agreed. Uhlenbeck said [4]:And that he liked; he was in my group and he did all these mathematical problems on wave guides which was very good, of course. ... He was a real computer, really remarkable. He was mathematically and technically really remarkably good. Schwinger worked at night beginning [4]:... in the evening at about 4 o'clock. I finally got him to give a seminar at 4.30. Julian was always out of breath when he came in, but then I had a certain influence on him so he did it and very conscientiously. In 1961 Schwinger was awarded an honorary doctorate from Purdue. In nominating him for this degree Hubert M James wrote on 6 December 1960 about Schwinger's contributions. Speaking of his time in the Radiation Laboratory James wrote:While at the Radiation Laboratory Schwinger invented important methods in electromagnetic field theory, which were extensively employed in the development of the theory of wave guides. He developed variational techniques that produced major advances in several fields of mathematical physics. Still more important were his contributions to the development of the modern form of quantum electrodynamics, through introduction of the "renormalization" technique. For this work he received the Nature of Light Award of the National Academy of Science, and shared with Kurt Gödel the first award of the $15 000 Albert Einstein Prize for achievement in Natural Science. Despite being on leave of absence at Purdue he was promoted to Research Professor in Theoretical Physics there. However, when he had finished his war work he resigned his position at Purdue to take up a post at Harvard. He worked at Harvard University from 1945 to 1972, first as an Associate Professor but being promoted to full Professor in 1947. The year he became a full professor he married Clarice Carrol of Boston. Schwinger was one of the inventors in the 1940s of the theory of renormalization, mentioned above. This theory allows individual particles to be considered from a distant viewpoint. Virtual particle pairs are not considered individually but rather surrounding virtual particles influence the appearance of the original particle. In 1951 he proposed, what is today called the Schwinger effect in quantum electrodynamics, where electron-positron pairs are sucked out of a vacuum by an electric field. This has not yet been confirmed by experiment. In 1957 his theoretical work led him to conclude that there were two different neutrinos one associated with the electron and one with the muon. Later experimental work has verified these theoretical conclusions. He invented source theory, which deals uniformly with strongly interacting particles, photons, and gravitons. His development of these ideas provided a general framework for all physical phenomena. Schwinger was joint winner of the Nobel Prize for Physics (1965) for his work in formulating quantum

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwinger.html (2 of 4) [2/16/2002 11:31:18 PM]

Schwinger

electrodynamics and thus reconciling quantum mechanics with Einstein's special theory of relativity. This topic, originating with the work of Dirac, was independently studied by Feynman who was a joint winner of the prize. The Presentation Speech given by Ivar Waller on the occasion when Schwinger received the Nobel Prize put his work in context as follows:The electrons of an atom move according to the laws of quantum mechanics established in 1925 and the next following years. For the hydrogen atom, which has only one electron and consequently is the simplest atom to investigate theoretically, the calculation of the motion of the electron in the electric field of the nucleus led to results of such accuracy that 20 years elapsed until any error of the theory could be found experimentally. This occurred, however, in 1947 when Lamb and his collaborator Retherford discovered that some energy levels of hydrogen which should coincide theoretically were in fact somewhat shifted relative to each other. One important result of the work of this year's Nobel Prize winners ... was the explanation of the Lamb-shift. ... Almost simultaneously with the discovery of the Lamb-shift another peculiarity was found by Kusch and his collaborator Foley, which made it clear that the magnetic moment of the electron is somewhat larger than had been assumed before. Using the method of renormalization which he also developed Schwinger was able to prove that a small anomalous contribution should be added to the value of the magnetic moment accepted until then. His calculation agreed with the experiments. Schwinger's calculation was indeed earlier than and very important for the proper interpretation of these measurements. Schwinger had developed the formalism of the new quantum electrodynamics in several fundamental papers .... He has also made this formalism more useful for practical calculations. From 1972 until his death in 1994 Schwinger worked at the University of California, Los Angeles. He was enormously respected, was a highly gifted lecturer, and supervised a string of impressive graduate students. Over his career he supervised over 70 doctoral students, 3 of whom have received Nobel prizes. Schwinger gave his students much more than guidance on their research. He gave them a depth of understanding and a mastery of the field which permitted each to become not a Schwinger disciple, but an independent scientist. Despite this remarkable record of achievements, he tended to become more and more solitary in his work as he grew older. This meant that he did not have as much impact on the later developments as one would have expected. The cover notes of [2] give this summary of his contributions:Schwinger was one of the most important and influential scientists of the twentieth century. The list of his contributions is staggering, from his early work leading to the Schwinger action principle, Euclidean quantum field theory, and the genesis of the standard model, to later valuable work on magnetic charge and the Casimir effect. In [5] he is described as follows:Julian Schwinger's legacy goes far beyond his published work. His lectures were elegant, lucid and original (he never did anything the same way twice), works of art and physics both. The Nobel Prize for Physics was certainly not the only honour Schwinger received. On the contrary he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwinger.html (3 of 4) [2/16/2002 11:31:18 PM]

Schwinger

received many honours, some of which we have already mentioned above, including the first Einstein Prize (1951), the National Medal of Science (1964), honorary doctorates from Purdue University (1961) and Harvard University (1962), and the Nature of Light Award of the National Academy of Sciences of the United States (1949). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) A Poster of Julian Schwinger

Mathematicians born in the same country

Honours awarded to Julian Schwinger (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1965

AMS Gibbs Lecturer

1960

Other Web sites

1. Nobel prizes site (A biography of Schwinger and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Schwinger.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Schwinger.html (4 of 4) [2/16/2002 11:31:18 PM]

Scott

Charlotte Angas Scott Born: 8 June 1858 in Lincoln, England Died: 10 Nov 1931 in Cambridge, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charlotte Angas Scott's father was a Congregational Church minister and provided tutors for her from the age of seven. From these tutors Charlotte Scott was first introduced to mathematics. She won a scholarship in 1876 to Hitchin College, soon to be renamed Girton College, of the University of Cambridge. Four years later she was placed eighth Wrangler but, as a woman, she was not allowed to graduate. Kenschaft, writing in [2], quotes a report of the graduation ceremony:The man read out the names and when he came to 'eighth', before he could say the name, all the undergraduates called out 'Scott of Girton', and cheered tremendously, shouting her name over and over again with tremendous cheers and raising of hats. Scott continued research at Girton on algebraic geometry under Cayley's supervision receiving her doctorate in 1885. In this year Bryn Mawr College in Pennsylvania, United States opened. On Cayley's recommendation Scott was appointed there and became the first head of the mathematics department there. In 1894 Scott published an important textbook An Introductory Account of Certain Modern Ideas and Methods in Plane Analytical Geometry. In 1899 Scott became an editor of the American Journal of Mathematics and continued an impressive publication record. She also served on the Council of the American Mathematical Society and served as its vice-president in 1905. Scott retired from teaching in 1924 and, after spending one further year at Bryn Mawr to complete the supervision of her last doctoral student, she returned to England.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Scott.html (1 of 2) [2/16/2002 11:31:20 PM]

Scott

Macaulay wrote in [4]:Miss Scott was a geometer who whenever possible brought to analytical geometry the full resources of pure geometrical reasoning. Alfred North Whitehead, speaking in 1922 at a meeting of the American Mathematical Society held at Bryn Mawr in Scott's honour, said:A friendship of peoples is the outcome of personal relations. A life's work such as that of Professor Charlotte Angas Scott is worth more to the world than many anxious efforts of diplomatists. She is a great example of the universal brotherhood of civilisations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Other Web sites

Agnes Scott College

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Scott.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Scott.html (2 of 2) [2/16/2002 11:31:20 PM]

Segner

Johann Andrea von Segner Born: 9 Oct 1704 in Pressburg, Hungary (now Bratislava, Slovakia) Died: 5 Oct 1777 in Halle, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

The Hungarian version of Segner's name is Jan Andrej Segner while in German it is often given as Johann Andrea von Segner. Segner attended school at Bratislava's Lyceum where he showed special talents for medicine and mathematics. In 1725 he entered the University of Jena, studying medicine there. He did not find being a doctor of medicine to his liking and he returned to the academic world accepting a chair at the University of Jena. He had the great distinction of becoming the first professor of mathematics at Göttingen taking up the chair in 1735. Segner's was therefore the first to fill what was to become one of the foremost chairs of mathematics in the world. In 1743 Segner was put in charge of the construction of the university observatory which was finished in 1751. While at Göttingen Segner discovered that every solid body has 3 axes of symmetry. He used Daniel Bernoulli's theoretical work on the 'reaction effect' to produce a horizontal waterwheel using the same principle which drives a modern lawn sprinkler. Segner's work, which influenced Euler to work on turbines, is described in [6]:Segner's wheel established the basic principles on which the jet turbine was developed decades later. It works on the principle of a stream of water coming out of a cylinder which at its lowest part has several horizontal paddles bent in one direction. The water streaming through the paddles produces a counter-pressure able to turn the cylinder in the opposite direction. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Segner.html (1 of 2) [2/16/2002 11:31:22 PM]

Segner

In 1751 Segner introduced the concept of the surface tension of liquids and made an unsuccessful attempt to give a mathematical description of capillary action. He left Göttingen in 1755 and, with Euler's assistance, became professor at Halle. Other work which he undertook included the theory of spinning tops. His publications include Elements of Arithmetic and Geometry and Nature of Liquid Surfaces. Segner received many honours for his work. He was made a member of the Academy of Science in St Petersburg, the Academy in Berlin and the Royal Society in London. Recently he has been honoured with a crater on the Moon being named after him. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) A Poster of Jan A Segner

Mathematicians born in the same country

Honours awarded to Jan A Segner (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1738

Lunar features

Crater Segner

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Segner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Segner.html (2 of 2) [2/16/2002 11:31:22 PM]

Segre_Beniamino

Beniamino Segre Born: 16 Feb 1903 in Turin, Italy Died: 22 Oct 1977 in Frascati, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Beniamino Segre's teachers at Turin University included Peano, Fano, Fubini and Corrado Segre (a not too close relative). Beniamino graduated from Turin in 1923 having written a dissertation on geometry. He was appointed to a post in Turin where he remained until 1926. After studying in Paris with Cartan for a year, Beniamino became Severi's assistant in Rome. By 1931 when he was appointed to the chair at Bologna he already had 40 publications in algebraic geometry, differential geometry, topology and differential equations. However he was of Jewish background and the Fascist Italian Government forced him out and he went to England. After being interned as an alien in 1940 he was appointed to a teaching post in Manchester with Mordell in 1942. In 1946 he returned to Bologna succeeding Severi in Rome in 1950. His output of research papers on geometry and related topics reached nearly 300 not counting a long list of other publications. Segre's contributions to geometry are many but, particularly in the latter part of his life, he is remembered for his study of geometries over fields other than the complex numbers. He gave a series of lectures in London in 1950 which were published as Arithmetical questions on algebraic varieties in 1951. Many questions were asked in these lectures about how the results would change if the ground field were different. By 1955 Segre was concentrating on geometries over a finite field and was producing results which we would now class as combinatorics rather than geometry. He collected many major results into a 100 page paper Le geometrie di Galois (1959) and a further 200 page paper in 1965 was devoted to the case where

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Segre_Beniamino.html (1 of 2) [2/16/2002 11:31:24 PM]

Segre_Beniamino

the order of the ground field is a perfect square. In [10] it is recounted how many of Segre's publications came from answering questions arising from lectures he attended. Some anecdotes are recounted in [10] about Segre's participation in lectures of others:... a lecture by Hodge in Oxford ... ended with Segre and another member of the audience occupying opposite ends of the blackboard and holding forth quite independently. ... a lecture by Severi in Harvard ... was constantly interrupted by Lefschetz in strong disagreement: the situation developed with Segre at the blackboard, firmly explaining what he thought was the resolution of the difference, while Severi and Lefschetz continued to shout each other down in French. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Segre_Beniamino.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Segre_Beniamino.html (2 of 2) [2/16/2002 11:31:24 PM]

Segre_Corrado

Corrado Segre Born: 20 Aug 1863 in Saluzzo, Italy Died: 18 May 1924 in Turin, Italy

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Corrado Segre studied at Turin under Enrico D'Ovidio. Gino Loria, who was to write famous texts on the history of mathematics, was a fellow student of Segre's and they remained friends throughout their lives. In 1883 Segre was awarded his doctorate for a thesis on quadrics in higher dimensional spaces and was appointed as an assistant to the professor of algebra and to the professor of geometry at Turin. In 1885 he was appointed as assistant in descriptive geometry. In 1888 Segre succeeded D'Ovidio to the chair of higher geometry in Turin and he continued to hold this post for the next 36 years until his death. Plücker's ideas on the geometry of ruled surfaces had been extended by Klein, and D'Ovidio lectured on this topic in session 1881-82. D'Ovidio also included in these lectures results of Veronese on projective geometry and of Weierstrass on bilinear and quadratic forms. This must have been a truly inspiring lecture course by D'Ovidio since it set the scene for all of Segre's research. He spent the rest of his working life on problems which arose directly or indirectly from this lecture course. Before Segre had written his thesis he submitted a joint paper with Loria to Mathematische Annalen. It was published in 1883 but perhaps the most important outcome was that the paper greatly interested Klein who then began to correspond with Segre, a correspondence which was to continue over many years. Segre worked on geometric properties invariant under linear transformations, algebraic curves and ruled surfaces studying transformations already considered by Brill, Clebsch, Gordan and Max Noether. In [1] http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Segre_Corrado.html (1 of 3) [2/16/2002 11:31:27 PM]

Segre_Corrado

P Speziale says that through this work of Segre's:... it thus became possible to reduce the classification of surfaces to that of curves. The insufficiencies of the earlier theories proposed by A Möbius, Grassmann, Cayley and Cremona were thus soon revealed. Using the methods which he had introduced, Segre was able to study Kummer's surface in a much simpler way. This surface, which had been discovered by Kummer in 1864, is a fourth order surface with sixteen double points. In a paper published in 1896, Segre found an invariant of surfaces under birational transformations which had appeared in a different form in a 1871 article by Zeuthen: this invariant is now called the Zeuthen-Segre invariant. In 1890 Segre looked at properties of the Riemann sphere and was led to a new area of representing complex points in geometry. He introduced bicomplex points into geometry. Motivated by the works of von Staudt, Segre considered a different type of complex geometry in 1912. Among other important work which Segre produced was an extension of ideas of Darboux on surfaces defined by certain differential equations. In [1] his clarity of writing is mentioned and illustrated with these comments:Segre wrote a long article on hyperspaces for the Encyklopädie der mathematischen Wissenschaften, containing all that was then known about such spaces. A model article, it is notable for its clarity and elegance. Finally we quote the summary of Segre's importance as described in the [1] article:Through his teaching and publications, Segre played an important role in reviving an interest in geometry in Italy. His reputation and the new ideas he presented in his courses attracted many Italian and foreign students to Turin. Segre's contribution to the knowledge of space assures him a place after Cremona in the ranks of the most illustrious members of the new Italian school of geometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Segre_Corrado.html (2 of 3) [2/16/2002 11:31:27 PM]

Segre_Corrado

Glossary index

Quotations index

Mathematicians of the day JOC/EFR April 1997

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Segre_Corrado.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Segre_Corrado.html (3 of 3) [2/16/2002 11:31:27 PM]

Seidel

Philipp Ludwig von Seidel Born: 24 Oct 1821 in Zweibrücken, Germany Died: 13 Aug 1896 in Munich, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Philipp von Seidel's mother was Julie Reinhold. His father, Justus Christian Felix Seidel, worked for the German post office and his job entailed him moving frequently. This meant that Philipp attended several different schools during his upbringing. The first of these schools was in Nördlingen, the next was in Nuremberg, then finally he completed his school education in Hof. Although Seidel completed his school studies in the autumn of 1839 he did not enter university immediately but received private coaching in mathematics before beginning his university career. He was coached by L C Schnürlein who was a mathematics teacher at the Gymnasium in Hof. This was valuable coaching for Seidel, particularly since Schnürlein was a good mathematician who had studied under Gauss. Seidel entered the University of Berlin in 1840 and studied under Dirichlet and Encke. The custom among German students at this time was to spend time at different universities and, following the usual custom, Seidel moved to Königsberg in 1842 where he studied under Bessel, Jacobi and Franz Neumann. In the autumn of 1843 Jacobi left Königsberg on the grounds of ill health and set off for Italy with Borchardt, Dirichlet, Schläfli and Steiner. Bessel certainly seems to have expected Jacobi to be away from Königsberg for a long while since he advised Seidel to go to Munich to study for his doctorate. He obtained his doctorate from Munich in 1846 for a thesis Uber die beste Form der Spiegel in Teleskopen. Six months later he submitted his habilitation dissertation Untersuchungen über die Konvergenz und Divergenz der Kettenbrüche and qualified to become a lecturer at Munich. It is worth noting that these two theses, submitted only six months apart, were on two completely different topics - the first was on astronomy while the second was on mathematical analysis. Like these two theses, Seidel worked on dioptics and mathematical analysis throughout his career. His work on lenses, and identified mathematically five coefficients describing the aberration of a lens, now called 'Seidel sums'. These Seidel sums correspond to spherical aberration, coma, astigmatism, Petzval curvature and distortion. He also introduced the concept of nonuniform convergence and applied probability to astronomy. Seidel progressed rapidly at Munich. He was appointed as an extraordinary professor in Munich in 1847 and then an ordinary professor in 1855. He received many honours such as appointment as a Royal Privy Councillor. He received many medals for his work and, in 1851, was elected to the Bavarian Academy of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Seidel.html (1 of 2) [2/16/2002 11:31:28 PM]

Seidel

Sciences. Other academies also honoured him, for example he was elected to the academies of Göttingen and of Berlin. An interesting aspect of Seidel's astronomical work involved, as we mentioned above, the use of probability theory. However, he did not restrict his use of this mathematical discipline to astronomy, for he also applied his skills in this area to study the frequency of certain diseases and also looked at certain questions relating to the climate. He lectured on probability theory, and also on the method of least squares. Problems with his eyesight forced Seidel into early retirement. Since he had never married he had no immediate family to help him when he became ill, but he had an unmarried sister Lucie Seidel who looked after him until 1889. By this time he certainly could not care for himself, yet had no family who could help out. In his last seven years he was looked after by the widow of a clergyman. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Philipp von Seidel (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Seidel

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Seidel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Seidel.html (2 of 2) [2/16/2002 11:31:28 PM]

Seifert

Karl Johannes Herbert Seifert Born: 27 May 1907 in Bernstadt, Saxony Died: 1 Oct 1996 in Princeton, New Jersey, USA

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Herbert Seifert's father was a middle ranking official at court. He moved with his family from Bernstadt, where Herbert was born, to Bautzen which is another town in Saxony. Herbert attended the Knabenbürgerschule, the local primary school, in Bautzen. He then attended secondary school, the Oberrealschule, in Bautzen but there was little evidence of the brilliant mathematical career that he would achieve. that is not to say that he performed badly at school, just that he appeared simply to be a good pupil close to the top of the class. Seifert took his Abitur in early 1926 at the age of eighteen and, leaving school, he entered the Technische Hochschule in Dresden to study mathematics and physics.. it was in 1927 that his whole life took a new turn when he attended a topology course by William Threlfall who was a Privatdozent at the technical university. Not only did Threlfall turn Seifert into an enthusiastic student of topology, but far more than that, they became firm friends and mathematical collaborators. We should note in passing the considerable age difference between the two with Seifert being twenty years younger than his friend and teacher. It was common practice for German students to spend time at a number of different universities and Seifert spent part of the session 1928-29 at Göttingen University. At this time Göttingen was the leading mathematics centre of the whole world so it was a good choice for Seifert. More than this, however, as well as world leaders in mathematics such as Hilbert, it had some of the leading topologists in the world. Hopf was a Privatdozent at Göttingen and he had just returned from an exciting year at Princeton with Aleksandrov. During Seifert's time at Göttingen, Aleksandrov was again a visitor and Seifert's visit only heightened his knowledge of, and passion for, topology. He returned to Dresden for the summer term of 1929 and his friendship with Threlfall was now so close that he lived in Threlfall's very fine house in Dresden. Seifert took his examinations to become a school

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Seifert.html (1 of 4) [2/16/2002 11:31:30 PM]

Seifert

teacher which he passed on 17 July 1930. He had already written a thesis on 3-dimensional closed manifolds and having submitted this he was awarded a doctorate a month later. At this stage Seifert was awarded a scholarship by the Technische Hochschule in Dresden to allow him to continue to study for a doctorate of philosophy. He chose to use the scholarship to allow him to go to Leipzig University where he was supervised by van der Waerden. Now although this was the official position, in fact Seifert returned to Dresden every weekend and he worked with Threlfall, so certainly Threlfall was an unofficial supervisor. Seifert and Threlfall also spent vacations together working on mathematics but of course it was to Leipzig that Seifert submitted his dissertation Topology of 3-dimensional fibred spaces on 1 February 1932 and he was awarded his doctorate of philosophy after his oral examination on 3 March. In this paper Seifert introduced the term "fibre space" for the first time, although its definition was not quite the same as the one used today. Much of Seifert and Threlfall's collaboration at this time was working on making a textbook out of Threlfall's lecture notes on topology. The book Lectures on topology was published in 1934. Threlfall wrote a preface which reads (see for example [1]):This textbook arose from a course which one of us gave to the other at the Technische Hochschule in Dresden. But soon the student contributed new ideas to such an extent and changed the presentation so fundamentally that it would be more justifiable to omit on the title page the name of the original author than his. There is little doubt that Threlfall had honestly described Seifert's enormous contribution but Seifert was far too modest to allow such a preface. After some discussion, the two friends settled on the following compromise which appears in the preface of the published work:The first step towards writing this textbook was a course which one of us (Threlfall) taught at the Technische Hochschule in Dresden. But only part of the course was included in the book. The main part of its contents originated later from daily discussions between the two authors. Puppe describes the merits of the text in [1]:The book gives an excellent account of what was known in topology at that time. It was superior in contents and in ways of presentation to other books in the field not only when it appeared but for a long time to come. it was translated into several languages, and generation of topologists in all countries of the world studied it. Even now, more than 60 years later, it is worth reading because of its lucid style and because, for some special problems, it is still the best source of information ... For his habilitation Seifert submitted his paper Continuous vector fields and by the beginning of 1934 he was ready to become a university teacher. Turning down an offer of a post from the University of Greifswald, he became an extraordinary professor at the Technische Hochschule in Dresden before the end of 1934. This was of course a difficult time in German universities. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Now although he did not realise what was happening at first, this worked in a certain way to Seifert's advantage. The two professors of mathematics, Heinrich Liebmann and Artur Rosenthal, at Heidelberg University were both Jewish and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Seifert.html (2 of 4) [2/16/2002 11:31:30 PM]

Seifert

were dismissed under the Nazi laws. This, although effectively correct, is not strictly true since Liebmann chose to take ill-health retirement, knowing what was coming. Seifert was offered Liebmann's in November 1935 (again according to Threlfall this is not strictly true as he claims that Seifert was ordered to take the chair). Certainly Seifert was no supporter of the Nazis and it was a situation with which he was very unhappy. In fact the authorities, knowing that he was not supporting the cause in the way they wanted, delayed confirming his appointment even though he was in Heidelberg carrying out the job. In August 1936 Seifert attended the International Mathematical Congress in Oslo. While there he contracted poliomyelitis and was taken to hospital in Oslo. While in hospital he received the formal offer of his Heidelberg chair. Strangely though, by the time he returned to Heidelberg the chair had been transferred from mathematics to another subject and Seifert spent the war years as an extraordinary professor although he was given the rights of an ordinary professor. From the time that Seifert took up his duties in Heidelberg until the start of World War II in 1939, he continued his collaboration was Threlfall. they exchanged letters and, as in previous years, spent holidays together working on mathematics. they published their second joint book in 1938 which was the monograph Variational calculus in the large which was a text on Morse theory. The book was accepted by Blaschke for the Hamburg monograph series but the two authors ran into problems with a Latin epigraph which they wished to put at the beginning. It was a quote from Kepler which reads in translation:Today it is very hard to write mathematical books. Blaschke, who went along with the Nazi ideas, objected on the grounds that it looked like a political statement, and of course so it was meant to be. It is remarkable that Threlfall and Seifert risked their positions by insisting that the epigram remain. they won their case, the epigram appeared in the book when published, and Blaschke wrote a letter to Seifert expressing fury that the quote had not been deleted. When war broke out Seifert volunteered for war work with the Institut für Gasdynamik which was a research centre attached to the German Air Force. This was a clever move which meant that he avoided what would almost certainly been far worse and he was able to continue with research into mathematics throughout the war. Seifert, on leave from Heidelberg University, became Head of a department in the Institut für Gasdynamik. He was successful in getting Threlfall appointed in his department. Seifert, still able to do mathematical research, worked on differential equations and wrote a series of papers on the topic through the war years. At the end of the war the University of Heidelberg was closed down while the Allies ensured that the Nazis were removed from the staff. Seifert was one of only a very few professors accepted by the Allies and he returned to the university when it reopened in 1946. He now tried to continue his collaboration with Threlfall by pressing for him to be appointed to Heidelberg. Indeed he achieved his aim and after accepting an invitation from Marston Morse to spend the winter term of 1948-49 at Princeton he intended to return to Heidelberg and continue his collaboration with Threlfall. Sadly Threlfall died at age 60 before they could restart their joint work. Not long after he returned to Heidelberg in 1949, Seifert married Katharina Korn. For three years he was the only ordinary professor of mathematics at Heidelberg but from 1952 the department rapidly expanded. Seifert retired in 1975 and enjoyed gardening and entertaining his former colleagues and students. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Seifert.html (3 of 4) [2/16/2002 11:31:30 PM]

Seifert

We have already mentioned some of Seifert's work. other important work related to knot invariants. In particular, in 1934 he published results, using surfaces today called Seifert surfaces, which he used to calculate homological knot invariants. Another topic which Seifert worked on was the homeomorphism problem for 3-dimensional closed manifolds. In a paper on this problem which he published in 1932, the results of which also appear in his 1934 book with Threlfall, he writes:... instead of investigating a complete system of topological invariants of 3-dimensional manifolds, we search for a system of invariants for fibre preserving maps of fibred 3-manifolds. This problem is completely solved in this paper. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Seifert.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Seifert.html (4 of 4) [2/16/2002 11:31:30 PM]

Seki

Takakazu Seki Kowa Born: March 1642 in Fujioka, Kozuke, Japan Died: 24 Oct 1708 in Edo (now Tokyo), Japan

Click the picture above to see three larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Takakazu Seki was born into a samurai warrior family. However at an early age he was adopted by a noble family named Seki Gorozayemon. The name by which he is now known, Seki, derives from the family who adopted him rather than from his natural parents. Seki was an infant prodigy in mathematics. He was self-educated in mathematics having been introduced to the topic by a servant in the household who, when Seki was nine years old, realised the talent of the young boy. Seki soon built up a library of Japanese and Chinese books on mathematics and became acknowledged as an expert. He was known as 'The Arithmetical Sage', a term which is carved on his tombstone, and soon had many pupils. His position in life is described in [18] as follows:In due time he, as a descendant of the samurai class, served in public capacity, his office being that of examiner of accounts to the Lord of Koshu, just as Newton became master of the mint under Queen Anne. When his lord became heir to the Shogun, Seki became Shogunate samurai and in 1704 was given a position of honor as master of ceremonies in the Shogun's household. In 1674 Seki published Hatsubi Sampo in which he solved fifteen problems which had been posed four years earlier. The work is remarkable for the careful analysis of the problems which Seki made and this certainly was one of the reasons for his great success as a teacher. Seki anticipated many of the discoveries of Western mathematics. Seki was the first person to study determinants in 1683. Ten years later Leibniz, independently, used determinants to solve simultaneous equations although Seki's version was the more general.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Seki.html (1 of 2) [2/16/2002 11:31:32 PM]

Seki

Seki also discovered Bernoulli numbers before Jacob Bernoulli. He studied equations treating both positive and negative roots but had no concept of complex numbers. He wrote on magic squares, again in his work of 1683, having studied a Chinese work by Yank Hui on the topic in 1661. This was the first treatment of the topic in Japan. In 1685, he solved the cubic equation 30 + 14x - 5x2 - x3 = 0 using the same method as Horner a hundred years later. He discovered the Newton or Newton-Raphson method for solving equations and also had a version of the Newton interpolation formula. Among other problems studied by Seki were Diophantine equations. For example, in 1683, he considered integer solutions of ax - by = 1 where a, b are integers. Secrecy surrounded the schools in Japan so it is hard to determine the contributions made by Seki, but he is also credited with major discoveries in the calculus which he passed on to his pupils. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (20 books/articles)

A Quotation

A Poster of Takakazu Seki

Mathematicians born in the same country

Cross-references to History Topics

1. Matrices and determinants 2. A chronology of pi

Other references in MacTutor

Chronology: 1675 to 1700

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Seki.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Seki.html (2 of 2) [2/16/2002 11:31:32 PM]

Selberg

Atle Selberg Born: 14 June 1917 in Langesund, Norway

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Atle Selberg's interest in mathematics began when he was a schoolboy. He read Ramanujan's collected papers and was not only greatly impressed by the mathematics he read but also he was intrigued by Ramanujan's personality which he described as having "the air of mystery". Inspired by reading about Ramanujan and reading his work, Selberg began to make his own mathematical explorations. Another major influence on Selberg's mathematical development was a lecture by Hecke at the International Mathematical Conference in Oslo in 1936. Selberg undertook his doctoral research at the University of Oslo. He was appointed a research fellow in 1942, the year before the award of his doctorate. He remained in this post until 1947 when he married and went to the United States. Selberg spent the academic year 1947-48 at the Institute for Advanced Study at Princeton. The following year he spent as associate professor of mathematics at Syracuse University, returning to the Institute for Advanced Study at Princeton in 1949 as a permanent member. In 1951 Selberg was promoted to professor at Princeton. In 1950 Selberg was awarded a Fields Medal at the International Congress of Mathematicians at Harvard. The Fields Medal was awarded for his work on generalisations of the sieve methods of Viggo Brun, and for his major work on the zeros of the Riemann zeta function where he proved that a positive proportion of its zeros satisfy the Riemann hypothesis. Selberg is also well known for his elementary proof of the prime number theorem, with a generalisation to prime numbers in an arbitrary arithmetic progression. The history of the prime number theorem is very interesting. The theorem stating:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Selberg.html (1 of 3) [2/16/2002 11:31:34 PM]

Selberg

The number of primes n tends to

as n/loge n,

was conjectured in the 18th century. Riemann came close to proving the result, but the theory of functions of a complex variable was not sufficiently developed to enable him to complete the proof. The necessary analytic tools were known by 1896 when Hadamard and de la Vallée Poussin independently proved the theorem using complex analysis. The successful proof of this result was seen as one of the greatest achievements of analytic number theory. In 1949 Selberg and Erdös found an elementary proof that makes no use of complex function theory. Subsequent events are not entirely clear but Selberg published two papers An elementary proof of the prime number theorem and An elementary proof of Dirichlet's theorem about primes in an arithmetic progression in volume 50 of the Annals of Mathematics. The following year he published An elementary proof of the prime number theorem for arithmetic progressions. In [2] Bombieri explains the source of Selberg's number theory sieve and shows that the idea of Selberg's l method and of his l2 sieve has its origin in Selberg's work on the analytic theory of the Riemann zeta function. In this work Selberg also introduced so-called mollifiers by the l2 method. Probably Selberg's best and most important work is his trace formula for SL2(R), which was done several years after the work for which he was awarded the Fields Medal. Selberg used his trace formula to prove that the "Selberg zeta function" of a Riemann surface satisfies an analogue of the Riemann hypothesis. Among the many outstanding mathematical contributions Selberg has made, there is his work on:... the Rankin-Selberg method, the "mollifier" device in the theory of Riemann's zeta function with its deep applications to zeros on or near the critical line and with Selberg's sieve as a by-product, ... Selberg's trace formula, Selberg's zeta function, ... automorphic functions, Dirichlet series. Selberg's collected papers were published in two volumes (1989, 1991). Matti Jutila, reviewing these, writes:The publication of the collected papers of Atle Selberg is most warmly welcomed by the mathematical community for several reasons. First of all, the author is a living classic who has profoundly influenced mathematics, especially analytic number theory in a broad sense, for about fifty years. Secondly, his papers up to 1947, which appeared mostly in Norwegian series or journals of limited distribution and partly even during World War II, are now at last easily accessible. And thirdly, a lot of highly interesting mathematics comes into daylight via the two volumes of Selberg's collected papers... Selberg was one of the four editors of Axel Thue's Selected mathematical papers published in Oslo in 1977. In 1989 Selberg published Reflections around the Ramanujan centenary which is the text of a talk which he gave at the conclusion of the Ramanujan Centenary Conference in January 1988 at the Tata Institute in Bombay. This tribute to Ramanujan, on the 100th aniversary of his birth, shows the important influence that Ramanujan had in Selberg's mathematical development. Selberg has received many distinctions for his work in addition to the Fields Medal. He has been elected to the Norwegian Academy of Science, the Royal Danish Academy of Sciences and Letters and the American Academy of Arts and Sciences.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Selberg.html (2 of 3) [2/16/2002 11:31:34 PM]

Selberg

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other references in MacTutor

1. Prime Number Theorem 2. Chronology: 1940 to 1950

Honours awarded to Atle Selberg (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1950

Other Web sites

1. 2. AMS (Selberg's eigenvalue conjecture) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Selberg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Selberg.html (3 of 3) [2/16/2002 11:31:34 PM]

Selten

Reinhard Selten Born: 5 Oct 1930 in Breslau, Germany (now Wroclaw, Poland)

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Reinhard Selten was awarded a Master's degree in mathematics from the Johann- Wolfgang- GoetheUniversity in Frankfurt in 1957. For 10 years he worked as an assistant in Frankfurt University, being awarded his doctorate from Frankfurt in 1961. After spending a year visiting the University of California, Berkeley he submitted his Habilitationsschrift in economics to Frankfurt University, the award being made in 1968. In 1969 Selten was appointed to a chair of economics at the Free University in Berlin. Then, in 1972 he moved to the Institute for Mathematical Economics of the University of Bielefeld. After twelve years in Bielefeld he moved to a chair at the University of Bonn. In the late 1940's he became interested in game theory. In 1965 he published important work on distinguishing between reasonable and unreasonable decisions in predicting the outcome of games. For his work in game theory Selten was, jointly with Nash, awarded the 1994 Nobel Prize in Economic Science for their pioneering analysis of equilibria in the theory of non-cooperative games. The respective contributions of Nash and Selten are as follows. Nash divided game theory into two parts, cooperative games, in which binding agreements can be made, and non-cooperative games, where binding agreements are not possible. Nash made a significant contribution with his equilibrium concept for non-cooperative games. It is now called Nash equilibrium. Selten worked on this concept and he refined the Nash equilibrium concept for analysing dynamic strategic interaction. Selten has also applied his refined version of these concepts to other problems such as analysing http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Selten.html (1 of 2) [2/16/2002 11:31:36 PM]

Selten

competition when there are only a small number of sellers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1960 to 1970

Honours awarded to Reinhard Selten (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1994

Other Web sites

1. Bonn, Germany 2. Nobel prizes site (An autobiography of Selten and his Nobel prize presentation speech) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Selten.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Selten.html (2 of 2) [2/16/2002 11:31:36 PM]

Semple

John Greenlees Semple Born: 10 June 1904 in Belfast, Ireland Died: 23 Oct 1985 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Semple (known as Jack to his friends and colleagues) was born in Northern Ireland and took his first degree in Belfast graduating in 1925, then studied for a doctorate from St John's College Cambridge under Baker. After lecturing at Edinburgh for a year (1929-30) he was appointed to the chair at Belfast where he remained for 6 years before taking up the chair of Pure Mathematics at King's College London where he was to remain for the rest of his life. In London he quickly became close friends with the Head of the Mathematics Department, George Temple. The two worked closely on the running of mathematics in London. Semple began a collaboration with Roth and together they wrote the first of three famous texts which Semple was to co-author. Introduction to algebraic geometry was published in 1949. Roth and Semple also worked together setting up and running the London Geometry Seminar which operated for 40 years and provided one of the major focal points for geometry research throughout the world. Semple worked with Du Val who joined the London Geometry Seminar but they only wrote one joint paper. Semple's work is on various aspects of geometry, in particular work on Cremona transformations and work extending results of Severi. He wrote two famous texts Algebraic projective geometry (1952) and Algebraic curves (1959) both written jointly with G T Kneebone.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Semple.html (1 of 2) [2/16/2002 11:31:38 PM]

Semple

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Jack Semple

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Semple.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Semple.html (2 of 2) [2/16/2002 11:31:38 PM]

Serenus

Serenus Born: about 300 in Antinoupolis, Egypt Died: about 360 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Very little is known of Serenus's life. In fact the article [4] claims that Serenus was born in Antissa but this has been shown by more recent historians of mathematics to be based on an error. That he was born in Antinoupolis is confirmed from two sources. The information has been added to one of the manuscripts of his works by at a later stage but we have no reason to doubt the authority of the addition. It can also be deduced from a copy of the second treatise of Serenus which has survived. Serenus was a commentator on the texts of others but, unlike some commentators, he was a fine mathematician in his own right. He wrote two original mathematical works which show that he was indeed a mathematician of considerable ability. The two treatises by Serenus are On the Section of a Cylinder and On the Section of a Cone both of which have survived. In the preface to the first of these Serenus gives his reasons for writing the work [2]:... many persons who were students of geometry were under the erroneous impression that the oblique section of a cylinder was different from the oblique section of a cone known as an ellipse, whereas it is of course the same curve. The work consists of 33 propositions. Typical of these are the following two problems. (i) Suppose we are given a cone and an ellipse E on the cone. Serenus shows how to find the cylinder which is cut in the ellipse E. (ii) Given a cone, find a cylinder so that when both are cut by the same plane the sections of the cuts form similar ellipses. The final five propositions, involving rays of light, are designed to support his friend Peithon who wrote a tract giving what he considered a better definition of parallels to that given by Euclid. It appears that Peithon's work treated as a bit of a joke and Serenus tries to in these propositions to show that Peithon's ideas are mathematically sound. Peithon [1]:... had defined parallels to be such lines as are cast on a wall or a roof by a pillar with a light behind it. As Heiberg comments in [3], even though Greek geometry was in decline by this time mathematicians were sufficiently knowledgeable to find this definition funny.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Serenus.html (1 of 2) [2/16/2002 11:31:39 PM]

Serenus

In the first 57 propositions in On the Section of a Cone Serenus examined triangular sections of right and scalene cones made by planes passing through the vertex. He also considered some problems relating to maximising areas. The remaining 12 propositions deal with the volumes of right cones given its height, its base, and the area of a triangular section through its axis. Serenus wrote a commentary on Apollonius's Conics unfortunately is lost, except for a fragment preserved by Theon of Smyrna. That he wrote such a work is confirmed by Serenus in his own writings. The result described by Theon of Smyrna is introduced with the words (see for example [1]):From Serenus the philosopher out of the lemmas. The result is that if a number of angles are subtended at points on a diameter of a circle so that the arcs of the circle subtended by the angles are all equal, then the closer to the centre of the circle is the point on the diameter, the greater is the angle. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Serenus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Serenus.html (2 of 2) [2/16/2002 11:31:39 PM]

Serre

Jean-Pierre Serre Born: 15 Sept 1926 in Bages, France

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean-Pierre Serre was educated at the Lycée de Nimes and then the Ecole Normale Supérieure in Paris from 1945 to 1948. Serre was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1954 Serre went to the University of Nancy where he worked until 1956. From 1956 he held the chair of Algebra and Geometry in the Collège de France until he retired in 1994 when he became an honorary professor. His permanent position in the Collège de France allowed Serre to spend quite a lot of time making research visits. In particular he spent time at the Institute for Advanced Study at Princeton and at Harvard University. Serre's early work was on spectral sequences. A spectral sequence is an algebraic construction like an exact sequence, but more difficult to describe. Serre did not invent spectral sequences, these were invented by the French mathematician Jean Leray. However, in 1951, Serre applied spectral sequences to the study of the relations between the homology groups of fibre, total space and base space in a fibration. This enabled him to discover fundamental connections between the homology groups and homotopy groups of a space and to prove important results on the homotopy groups of spheres. Serre's work led to topologists realising the importance of spectral sequences. The Serre spectral sequence provided a tool to work effectively with the homology of fibrings. For this work on spectral sequences and his work developing complex variable theory in terms of sheaves, Serre was awarded a Fields Medal at the International Congress of Mathematicians in 1954. Serre's theorem led to rapid progress not only in homotopy theory but in algebraic topology and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Serre.html (1 of 3) [2/16/2002 11:31:41 PM]

Serre

homological algebra in general. Over many years Serre has published many highly influential texts covering a wide range of mathematics. Among these texts, which show the topics Serre has worked on, are Homologie singulière des espaces fibrés (1951), Faisceaux algébriques cohérents (1955), Groupes d'algébriques et corps de classes (1959), Corps locaux (1962), Cohomologie galoisienne (1964), Abelian l-adic representations (1968), Cours d'arithmétique (1970), Représentations linéaires des groupes finis (1971), Arbres, amalgames, SL2 (1977), Lectures on the Mordell-Weil theorem (1989) and Topics in Galois theory (1992). These books are outstanding and led to Serre being honoured. In 1995 he was awarded the Steele Prize for mathematical exposition and the citation for the award reads [2]:It is difficult to decide on a single work by a mathematician of Jean-Pierre Serre's stature which is most deserving of the Steele Prize. Any one of Serre's numerous other books might have served as the basis of this award. Each of his books is beautifully written, with a great deal of original material by the author, and everything smoothly polished. It would be hard to make any significant improvement on his expositions; many are the everyday standard references in their areas, both for working mathematicians and graduate students. Serre brings his whole mathematical personality to bear on the material of these books; they are alive with the breadth of real mathematics and are an example to all of how to write for effect, clarity, and impact. The references [4] and [5] provide a fascinating view of Serre's views on some aspects of his career up to 1985:Presently, the topic which amuses me most is counting points on algebraic curves over finite fields. It is a kind of applied mathematics: you try to use any tool in algebraic geometry and number theory that you know of, ... and you don't quite succeed! The interview in [4] and [5] also provides a chance to examine Serre's views on mathematics. Serre has received numerous awards. In addition to the Fields Medal in 1954 he was elected a Fellow of the Royal Society in 1974. He has also been made an Officer Légion d'Honneur and Commander Ordre National du Mérite. He has been elected to many national academies in addition to the Royal Society, in particular the academies of France, Sweden, United States and the Netherlands. He was awarded the Prix Gaston Julia in 1970, the Balzan Prize in 1985, and the Steele Prize, described above, from the American Mathematical Society in 1995. He has been awarded honorary degrees from the University of Cambridge in 1978, the University of Stockholm in 1980 and the University of Glasgow in 1983. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1950 to 1960

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Serre.html (2 of 3) [2/16/2002 11:31:41 PM]

Serre

Honours awarded to Jean-Pierre Serre (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1954

Fellow of the Royal Society

Elected 1974

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Serre.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Serre.html (3 of 3) [2/16/2002 11:31:41 PM]

Serret

Joseph Alfred Serret Born: 30 Aug 1819 in Paris, France Died: 2 March 1885 in Versailles, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Serret graduated from the Ecole Polytechnique in Paris in 1840. He became an entrance examiner for the Ecole Polytechnique in 1848. In 1861 he became professor of celestial mechanics at Collège de France, then two years later he was appointed to the chair of differential and integral calculus at the Sorbonne. Serret joined the Bureau des Longitudes in 1873. Serret did important work in differential geometry. Together with Bonnet and Bertrand he made major advances in this topic. The fundamental formulas in the theory of space curves are the Frenet-Serret formulas. In 1860 Serret succeeded Poinsot in the Académie des Sciences. In 1871 he retired to Versailles as his health began to deteriorate. Serret also worked in number theory, calculus and mechanics. He edited the works of Lagrange which were published in 14 volumes between 1867 and 1892. He also edited the 5th edition of Monge in 1850. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Serret.html (1 of 2) [2/16/2002 11:31:44 PM]

Serret

Mathematicians born in the same country Cross-references to History Topics

The development of group theory

Honours awarded to Joseph A Serret (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Serret (15th Arrondissement)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Serret.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Serret.html (2 of 2) [2/16/2002 11:31:44 PM]

Servois

François Joseph Servois Born: 19 July 1768 in Mont-de-Laval (N of Morteau), Doubs, France Died: 17 April 1847 in Mont-de-Laval, Doubs, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

François-Joseph Servois's father, Jacques-Ignance Servois, was a merchant and his mother was Jeanne-Marie Jolliet. Servois's first intention was to join the priesthood and he began by following this aim and was ordained at Besançon. However, this was in the early days of the French Revolution and a time of great political and military activity in France. Servois soon changed his mind about following a career in the Church, and left in 1793 to join the army. He was at the artillery school in Châlons-sur-Marne in 1794 and immediately after this he was promoted to lieutenant. There were numerous military campaigns by the French army shortly after this and Servois was in the thick of the action serving as a staff officer. However, he had a great love of mathematics and while on the military campaigns Servois spent all his free time studying. Legendre realised that Servois had considerable mathematical talents and he supported a move to have him appointed to the artillery school of Besançon as professor of mathematics. Appointed to this post in July 1801, Servois went on to hold similar positions over the next few years. His first move was only a few months after his first appointment at Besançon when he moved to the artillery school in Châlons-sur-Marne where he had begun his military career. Then in 1802 he made his second move, this time to the artillery school in Metz. In comparison with his earlier appointments, Servois spent quite a while in Metz at the artillery school, remaining there until 1808. His next move was to the artillery school La Fère where he remained until 1816 when he moved to the artillery and engineering school at Metz. Hardly had he arrived in Metz when a position as curator of the artillery museum in Paris fell vacant. Servois was appointed as curator in 1816 and he held this post until he retired in 1827. After he retired Servois returned to his home town of Mont-de-Laval where he lived for nearly twenty further years. Servois worked in projective geometry, functional equations and complex numbers. He introduced the word pole in projective geometry. He also came close to discovering the quaternions before Hamilton. Petrova, in [5], describes a paper by Servois on differential operators written in the Annales de mathématique in November 1814. Servois introduced the terms "commutative" and "distributive" in this paper describing properties of operators, and he also gave some examples of noncommutativity. Although he does not use the concept of a ring explicitly, he does verify that linear commutative

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Servois.html (1 of 2) [2/16/2002 11:31:45 PM]

Servois

operators satisfy the ring axioms. In doing so he showed why operators could be manipulated like algebraic magnitudes. This work initiates the algebraic theory of operators. Servois was critical of Argand's geometric interpretation of the complex numbers. He wrote to Gergonne telling him so in November 1813 and Gergonne published the letter in the Annales de mathématique in January 1814. Servois wrote:I confess that I do not yet see in this notation anything but a geometric mask applied to analytic forms the direct use of which seems to me simple and more expeditious. Considered as a leading expert by many mathematicians of his day, he was consulted on many occasions by Poncelet while he was writing his book on projective geometry Traité des propriétés projective. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles)

A Quotation

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Servois.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Servois.html (2 of 2) [2/16/2002 11:31:45 PM]

Severi

Francesco Severi Born: 13 April 1879 in Arezzo, Italy Died: 8 Dec 1961 in Rome, Italy

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Francesco Severi studied at Turin University where he had very little money and had to tutor privately in order to make enough to live. The hardship came from the sad loss of his father when he was nine years old but one feels that he should not have had to suffer in this way as his wealthy relatives could easily have funded his studies but chose not to help him at all. His ideas of becoming an engineer soon changed after studying under Corrado Segre at Turin. Severi became fascinated by geometry and, under Corrado Segre's supervision he went on to obtain his doctorate in 1900. His doctoral thesis, together with a series of other papers which Severi published at this time, deal with enumerative geometry, a subject which had been started by Schubert. After being awarded his doctorate, Severi accepted a post in Turin as assistant to D'Ovidio. From there he moved to Bologna where he acted as assistant to Enriques. His final post as assistant was held in Pisa where this time he was assistant to Bertini. In 1904 Severi was appointed to the Chair of Projective and Descriptive Geometry at Parma. He only worked at Parma for one year, accepting the chair at Padua in 1905. World War I interrupted Severi's tenure of the chair at Padua and during the war he served with distinction in the artillery. From 1922 Severi worked at the University of Rome. His most important contributions are to algebraic geometry. He criticised the work of his contemporaries as lacking rigour and relying too heavily on intuition. Roth, in [8], summarises Severi's contributions in the following way:Severi's scientific work presents several features which, when taken together, must make his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Severi.html (1 of 3) [2/16/2002 11:31:47 PM]

Severi

career a rarity. To begin with, there is the uniformly high level of his very considerable scientific production: as a rule Severi attacks only important questions of general character and usually of great difficulty. ... In the second place, one cannot fail to observe an essential unity of outlook. Severi maintains a balance between geometry and analysis - he has actually made outstanding contributions to function theory. But within his geometrical work itself the same unity is manifest... After work on enumerative geometry, Severi turned to birational geometry of surfaces, a topic which Castelnuovo and Enriques has spent ten years developing before Severi began to work on it. Severi introduced many concepts into geometry, for example he notion of algebraic equivalence. He gave necessary and sufficient conditions for the linear equivalence of two curves on a surface F in 1905. Some rate Severi's discovery of a base of algebraically independent curves on any surface as his most important contribution. He published this in Mathematische Annalen in 1906 and Max Noether wrote to Severi concerning these results saying:You have shed a great light on geometry. In 1907 Enriques and Severi won the Prix Bordin from the French Academy for a work on hyperelliptic surfaces. It is impossible to give any real indication of the contribution which Severi made in a short article. In [9] Beniamino Segre lists over 400 publications by Severi. Roth describes Severi's teaching abilities in [8] writing:... it was as a teacher of geometry that Severi excelled. His lectures on his own work were unforgettable, the style was beautifully simple ... and the presentation masterly. He was greatly interested in teaching for its own sake, and his didactic skill found an outlet in a whole stream of books... Despite the incredible output of mathematics from Severi, he had an amazing number of outside interests. Again we quote [8]:As he approached middle age, mathematics came to occupy less and less of his time, it had to compete with a host of other occupations. For Severi by the was (among other things) President of an Arezzo bank, head of the engineering faculty at Padua, an expert agriculturist who managed his own estate. His most impressive work came before he went to Rome but, despite spending less time on mathematics, after this he still managed to produce work of the greatest importance like the solution of the Dirichlet problem and his development of the theory of rational equivalence. For his character we again quote Roth [8]:Personal relationships with Severi, however complicated in appearance, were always reducible to two basically simple situations: either he had just taken offence or else he was in the process of giving it - and quite often genuinely unaware that he was doing so. Paradoxically, endowed as he was with even more wit than most of his fellow Tuscans, he showed a childlike incapacity either for self-criticism or for cool judgement. Thus he meddled in politics, whereas it would have been far better had he left them alone.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Severi.html (2 of 3) [2/16/2002 11:31:47 PM]

Severi

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) A Poster of Francesco Severi

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Severi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Severi.html (3 of 3) [2/16/2002 11:31:47 PM]

Shanks

William Shanks Born: 25 Jan 1812 in Corsenside, Northumberland, England Died: 1882 in Houghton-le-Spring, Durham, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Shanks kept a boarding school at Houghton-le-Spring in a coal mining area near Durham, England. Shanks is famed for his calculation of to 707 places in 1873, which unfortunately was only correct for the first 527 places. He used the formula /4 = 4 tan-1(1/5) - tan-1(1/239). Shanks also calculated e and Euler's constant to a great many decimal places. He published a table of primes up to 60 000, found the natural logarithms of 2, 3, 5 and 10 to 137 places and the values of 212m+1 for m = 1, 2, 3, ..., 60. In 1944 Ferguson calculated

using the formula

/4 = 3 tan-1(1/4) + tan-1(1/20) + tan-1(1/1985). He found that his value disagreed with that of Shanks in the 528th place. Ferguson discovered that Shanks had omitted two terms which caused his error. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Cross-references to History Topics

Mathematicians born in the same country 1. Memory, mental arithmetic and mathematics 2. Pi through the ages 3. A chronology of pi

Other references in MacTutor

Previous

Chronology: 1850 to 1860

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shanks.html (1 of 2) [2/16/2002 11:31:48 PM]

Shanks

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Shanks.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shanks.html (2 of 2) [2/16/2002 11:31:48 PM]

Shannon

Claude Elwood Shannon Born: 30 April 1916 in Gaylord, Michigan, USA Died: 24 Feb 2001 in Medford, Massachusetts, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Claude E Shannon's father was also named Claude Elwood Shannon and his mother was Mabel Catherine Wolf. Shannon was a graduate of the University of Michigan, being awarded a degree in mathematics and electrical engineering in 1936. He then went to the Massachusetts Institute of Technology where he obtained a Master's Degree in electrical engineering and his Ph.D. in mathematics in 1940. Shannon wrote a Master's thesis A Symbolic Analysis of Relay and Switching Circuits on the use of Boole's algebra to analyse and optimise relay switching circuits. His doctoral thesis was on theoretical genetics. At the Massachusetts Institute of Technology he also worked on the differential analyser, an early type of mechanical computer developed by Vannevar Bush for obtaining numerical solutions to ordinary differential equations. Shannon published Mathematical theory of the differential analyzer in 1941. In the introduction to the paper he writes:The most important results [mostly given in the form of theorems with proofs] deal with conditions under which functions of one or more variables can be generated, and conditions under which ordinary differential equations can be solved. Some attention is given to approximation of functions (which cannot be generated exactly), approximation of gear ratios and automatic speed control. Shannon joined AT&T Bell Telephones in New Jersey in 1941 as a research mathematician and remained at the Bell Laboratories until 1972. Johnson writes in [4] that Shannon:... became known for keeping to himself by day and riding his unicycle down the halls at night. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shannon.html (1 of 4) [2/16/2002 11:31:50 PM]

Shannon

D Slepian, a colleague at the Bell Laboratories wrote:Many of us brought our lunches to work and played mathematical blackboard games but Claude rarely came. He worked with his door closed, mostly. But if you went in, he would be very patient and help you along. He could grasp a problem in zero time. He really was quite a genius. He's the only person I know whom I'd apply that word to. Working with John Riordan, Shannon published a paper in 1942 on the number of two-terminal series-parallel networks. This paper extended results obtained by MacMahon who had published his early contribution in the Electrician in 1892. Shannon published A Mathematical Theory of Communication in the Bell System Technical Journal (1948). This paper founded the subject of information theory and he proposed a linear schematic model of a communications system. This was a new idea. Communication was then thought of as requiring electromagnetic waves to be sent down a wire. The idea that one could transmit pictures, words, sounds etc. by sending a stream of 1's and 0's down a wire, something which today seems so obvious as we take the information from a server in St Andrews, Scotland and view it anywhere in the world, was fundamentally new. Shannon considers a source of information which generates words composed of a finite number of symbols. These are transmitted through a channel, with each symbol spending a finite time in the channel. The problem involved statistics with the assumption that if xn is the nth symbol produced by the source the xn process is a stationary stochastic process. He gave a method of analysing a sequence of error terms in a signal to find their inherent variety, matching them to the designed variety of the control system. In A Mathematical Theory of Communication , which introduced the word "bit" for the first time, Shannon showed that adding extra bits to a signal allowed transmission errors to be corrected. Slepian, in the introduction to [2], writes:Probably no single work in this century has more profoundly altered man's understanding of communication than C E Shannon's article, "A mathematical theory of communication", first published in 1948. The ideas in Shannon's paper were soon picked up by communication engineers and mathematicians around the world. They were elaborated upon, extended, and complemented with new related ideas. The subject thrived and grew to become a well-rounded and exciting chapter in the annals of science. On 27 March 1949 Shannon married Mary Elizabeth Moore. They had three sons and one daughter; Robert, James, Andrew Moore, and Margarita. He continued his work showing how Boolean algebra could be used to synthesise and simplify relay switching circuits. He also proved results on colouring the edges of a graph so that no two edges of the same colour meet at a vertex. Another important paper, published in 1949, was Communication theory of secrecy systems. In 1952 Shannon devised an experiment illustrating the capabilities of telephone relays. He had held a position as a visiting professor of communication sciences and mathematics at the Massachusetts Institute of Technology in 1956, then from 1957 he was appointed to the Faculty there, but remained a consultant with Bell Telephones. In 1958 he became Donner Professor of Science. R G Gallager, a colleague who worked at the Massachusetts Institute of Technology, wrote:Shannon was the person who saw that the binary digit was the fundamental element in all of communication. That was really his discovery, and from it the whole communications

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shannon.html (2 of 4) [2/16/2002 11:31:50 PM]

Shannon

revolution has sprung. His later work looked at ideas in artificial intelligence. He devised chess playing programs and an electronic mouse which could solve maze problems. The chess playing program appeared in the paper Programming a computer for playing chess published in 1950. This proposal led to the first game played by the Los Alamos MANIAC computer in 1956. This was the year that Shannon published a paper showing that a universal Turing machine may be constructed with only two states. Latterly he felt that the communications revolution, which he had played a major role in starting, was going too far. He wrote:Information theory has perhaps ballooned to an importance beyond its actual accomplishments. Marvin Minsky described Shannon as follows:Whatever came up, he engaged it with joy, and he attacked it with some surprising resource which might be some new kind of technical concept or a hammer and saw with some scraps of wood. For him, the harder a problem might seem, the better the chance to find something new. Shannon received many honours for his work. Among a long list of awards were the Alfred Nobel American Institute of American Engineers Award in 1940, the National Medal of Science in 1966, and the Audio Engineering Society Gold Medal in 1985. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Claude E Shannon

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1940 to 1950

Honours awarded to Claude E Shannon (Click a link below for the full list of mathematicians honoured in this way) AMS Gibbs Lecturer Other Web sites

1963 1. New York Times (Obituary) 2. Guardian (Obituary) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shannon.html (3 of 4) [2/16/2002 11:31:50 PM]

Shannon

Glossary index

Quotations index

Mathematicians of the day JOC/EFR March 2001

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Shannon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shannon.html (4 of 4) [2/16/2002 11:31:50 PM]

Sharkovsky

Oleksandr Mikolaiovich Sharkovsky Born: 7 Dec 1936 in Kiev, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oleksandr Sharkovsky attended his local university of Kiev, graduating in 1958. In 1961 he was appointed to the Institute of Mathematics of the Academy of Sciences of the Ukraine in Kiev. He also taught at the University of Kiev from 1967. Sharkovsky's main areas of interest are the theory of dynamical systems, the theory of stability and the theory of oscillations. He also works in the theory of functional and functional differential equations, and the study of difference equations and their application. He is perhaps best known for an important theorem on continuous functions which he proved in 1964. Although the result did not attract a great deal of interest at the time of its publication, during the 1970s other surprising results were proved which turned out to be special cases of Sharkovsky's theorem. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sharkovsky.html (1 of 2) [2/16/2002 11:31:52 PM]

Sharkovsky

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sharkovsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sharkovsky.html (2 of 2) [2/16/2002 11:31:52 PM]

Shatunovsky

Samuil Osipovich Shatunovsky Born: 25 March 1859 in Velyka Znamenka, Tavricheskaya gubernia, Ukraine Died: 27 March 1929 in Odessa, USSR Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Samuil Shatunovsky attended lectures by Chebyshev in St Petersburg. He was a student at several technological and engineering insitutes in St Petersburg and taught at some of the institutes. However he could not enrol formally at the university since he had no school certificate. Shatunovsky decided to try to obtain a university education in Switzerland and travelled to that country. However, his attempt to gain a higher education in Switzerland failed from lack of funds. Returning to Russia Shatunovsky sent some of his mathematical work to Odessa University. The quality of the work was seen immediately and Shatunovsky began studies at Odessa. After obtaining a degree from Odessa he was appointed to the staff there in 1905. In 1917 he was promoted to the rank of professor at Odessa University and continued to work there for the rest of his life. Shatunovsky's research was on several topics from analysis and algebra. In particular he produced good work in group theory, the theory of numbers and geometry. He is perhaps best known, however, as one of the founders of the constructive approach to contemporary mathematics. Working on the foundations of mathematics, he produced an axiomatic theory independently of, but similar to, that of Hilbert. He used the axiomatic method to lay the logical foundations of geometry, algebraic fields, Galois theory and analysis. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shatunovsky.html (1 of 2) [2/16/2002 11:31:54 PM]

Shatunovsky

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Shatunovsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shatunovsky.html (2 of 2) [2/16/2002 11:31:54 PM]

Shen

Shen Kua Born: 1031 in Ch'ien-t'ang, Chekiang province (now Zhejiang), China Died: 1095 in Ching-k'ou, China Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Shen Kua commanded troops which defended his province from attacks of the Khitan tribes. His troops lost a battle and sustained heavy losses so he was relieved from his command and banished. This seems to have been fortunate as far as science is concerned since after he was banished he wrote his scientific works. He is famous for the first reference to a magnetic compass which occurs in his work Meng ch'i pi t'an (Dream Pool Essays). This book also contains his work on mathematics, astronomy, cartography, optics and medicine. Shen is also said to have constructed a celestial sphere and a bronze gnomon, a pointer whose shadow gives the time. In 1074 Shen devised a new calendar. Around 1080 Shen Kua claimed that fossilised plants were evidence for changes in climate. He recognised fossils of certain sea creatures in rock far from the sea and understood what this meant. Observing seashells in strata of the T'ai-hang Shan mountains, he deduced that these mountains, though now far from the sea, must once have been a sea shore. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 900 to 1100

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shen.html (1 of 2) [2/16/2002 11:31:55 PM]

Shen

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Shen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shen.html (2 of 2) [2/16/2002 11:31:55 PM]

Shewhart

Walter Andrew Shewhart Born: 18 March 1891 in New Canton Illinois, USA Died: 11 March 1967 in Troy Hills, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Most of Walter Shewhart's career was spent in the Bell Telephone Laboratories. He worked on statistical tools to examine when a corrective action must be applied to a process. Shewhart wrote on statistical control of industrial processes and applications to measurement processes in science. His control chart techniques have been widely adopted.

Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Other Web sites

University of Minnesota

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shewhart.html (1 of 2) [2/16/2002 11:31:57 PM]

Shewhart

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Shewhart.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shewhart.html (2 of 2) [2/16/2002 11:31:57 PM]

Shields

Allen Lowell Shields Born: 1927 in USA Died: 16 Sept 1989 in Ann Arbor, Michigan, USA

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Allen Shields was an undergraduate at the City College New York where his lecturers included Emil Post and Lee Lorch. His doctoral studies were at M.I.T. under Witold Hurewicz's supervision and he received his doctorate in 1952. Despite the fact that Hurewicz was his official supervisor, Shields was unofficially supervised by Raphaël Salem who, at that time, was half the time in Paris and half in M.I.T. Shields' first post was at Tulane where he learnt a lot of functional analysis and wrote papers on topological semigroups. He was appointed to Ann Arbor in 1955 and remained there except for two years (1959-61) spent in New York. Shields worked on a wide range of mathematical topics including measure theory, complex functions, functional analysis and operator theory. F W Gehring described his work in these words Allen's standards were high, his taste impeccable, and his ideas deep. He was one of the world's most versatile practitioners in the art of applying functional analysis to gain insight on and solve problems of classical function theory. He had a profound effect on the development of this subject through his research and his personal contacts with colleagues and students. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Allen Shields

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shields.html (1 of 2) [2/16/2002 11:31:59 PM]

Shields

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Shields.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shields.html (2 of 2) [2/16/2002 11:31:59 PM]

Shnirelman

Lev Genrikhovich Shnirelman Born: 2 Jan 1905 in Gomel, Belarus Died: 24 Sept 1938 in Moscow, USSR Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lev Shnirelman entered the University of Moscow in 1921 at the age of 16. There he was taught by Khinchin, Luzin and Urysohn. He started research in algebra, geometry and topology as a student but did not consider his results sufficiently important to merit publication. He was appointed to the chair of mathematics in Novocherkassk in 1929. Then in 1930 he returned to Moscow. From 1934 he worked at the Mathematical Institute of the Academy. Shnirelman made significant contributions to topological methods in the calculus of variations. In 1930 he introduced important new ideas into number theory. Using these he was able to prove a weak form of the Goldbach's conjecture showing that every number is the sum of 20 primes. The Goldbach conjecture that every number is the sum of at most 3 primes still stands. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shnirelman.html (1 of 2) [2/16/2002 11:32:00 PM]

Shnirelman

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Shnirelman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shnirelman.html (2 of 2) [2/16/2002 11:32:00 PM]

Shoda

Kenjiro Shoda Born: 25 Feb 1902 in Tatebayashi, Gunma Prefecture, Japan Died: 20 March 1977 in Ashikaga, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kenjiro Shoda was born in Tatebayashi in the Gunma Prefecture of Japan but he underwent schooling in Tokyo until he completed middle school. There were academies in Japan for the brightest pupils with the task of preparing them for a university education. Shoda, after showing great talents at middle school, attended the Eighth National Senior High School in Nagoya. After graduating from the Eighth High School, Shoda entered Tokyo Imperial University (the title 'Imperial' would soon be dropped from the name of all Japanese universities) and there he was taught by Takagi. This was an exciting period to study at Tokyo University for Takagi had published his famous paper on class field theory in 1920. Takagi lectured on group theory, representation theory, Galois theory, and algebraic number theory. When Shoda was in his final undergraduate year, his studies were supervised by Takagi and he inspired Shoda to work on algebra. Shoda graduated from the Department of Mathematics at Tokyo University in 1925 and began his graduate studies under Takagi's supervision. During his first year of graduate studies he read works on the theory of group representations by Frobenius and Schur. Then in 1926, his second year of study, he was awarded a scholarship to allow him to study in Germany and he set off for Berlin to work with Schur. While in Berlin he attended Schur's lectures and had his first mathematical success in research discovering an interesting result on matrices. After a year in Berlin, Shoda went to Göttingen where he joined Emmy Noether's school, attending her lectures on hypercomplex systems and representation theory. Nagao writes [1]:This particular year seems to mark the most significant period in his mathematical growth. There, near Noether, he witnessed the remarkable process of creation of great mathematical ideas and theory, and youthful Shoda buried himself in enthusiastic pursuit of mathematics in a wonderful creative atmosphere generated by the many young, able mathematicians who http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shoda.html (1 of 4) [2/16/2002 11:32:03 PM]

Shoda

had come from all over the world to Göttingen, attracted by Emmy Noether. Shoda returned to Japan in 1929 and almost immediately began to write his algebra book. Abstract algebra, an advanced-level textbook of modern algebra, was first published in 1932 and it proved a very significant work for Japanese mathematics. The twelfth printing of the book was published in Tokyo in 1971 with the chapter headings: Basic concepts; Field theory; Galois theory; Elimination theory; General ideal theory; Valuation theory. This fine work, published in 1932, must have been a significant factor in Shoda being appointed as professor in the Faculty of Science at Osaka University in 1933. He had also published twelve papers on groups and rings before he was appointed to this post. The war years were particularly difficult ones in Japan and many Japanese mathematicians failed to keep their research going through these difficult times. Shoda, however, managed to continue to undertake research and in 1946 he was elected the first Chairman of the Mathematical Society of Japan. In this role he had the task of reconstructing Japanese mathematics and he did this in many ways, one of which was to lead by example with some fine and important publications. In 1947 he published his text General Algebra. This text attempted to unify many existing algebraic systems. Here are some details from a review of the work by T Nakayama:In this book a systematic and consistent treatment of general algebraic systems is given ... In the first chapter the fundamental notions are introduced and discussed. An algebraic system is defined as a set possessing a family of compositions (where a composition may not have meaning for all pairs of elements), a primitive algebraic system as one in which every composition is defined for every pair of elements and which admits certain identities with respect to compositions, while an elementary algebraic system is a weakening of the latter in which the identities are supposed to be valid as soon as both sides have meaning. ... Lattices, groups, groupoids, mixed groups (of Loewy) are considered. For instance, the notion of group is shown to be primitive by taking division as its composition. The second chapter is on the theory of free systems, including the fundamental theorem and the theorem of change of generators (of Tietze). A theory of independence is given, making use of a certain notion of valuation so as to take care of algebraic and linear dependence, the latter being distinguished in that an element is (linearly) dependent on a set of elements if and only if it is contained in the subsystem generated by the set. Free lattices, free groups, free Lie and associative rings are treated, emphasizing the relationship among them. The third chapter starts by proving the author's sufficient condition for a lattice of congruences to be modular ... Then the author ... develops generalized theories of normal chains, composition series, of direct and subdirect products, and generalizations of the Jordan-Holder and the Remak-Schmidt-Ore theorems. After completely reducible systems, the notions of solvable and nilpotent systems are discussed, where general identities are considered instead of the usual commutativity. Further, a set of endomorphisms of an algebraic system is considered as an algebraic system in terms of the usual multiplication (of mappings) and the compositions induced from those of the original system. Here the multiplication is of course associative, and distributive for the latter compositions, presenting the notion of ring-systems as a generalization of the ring notion. Structural

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shoda.html (2 of 4) [2/16/2002 11:32:03 PM]

Shoda

theory of abstract ring-systems is developed, under chain conditions, including (generalized) Peirce decompositions and Wedderburn's theorem; for the latter the notion of matrices is also generalized. The last chapter gives the theory of representing (primitive) algebraic systems as systems of endomorphisms of some other systems called representation systems. Reduction and direct decomposition are discussed in connection with representation systems. Particular observations are made to the cases of ring-systems, rings, and groups. In 1949 Shoda was awarded the Japan Academy Prize in recognition of his fine achievements. In the same year he became Dean of the Faculty of Science at Osaka [1]:It was at this time, after the war, that Japan was going through a difficult period of transition from the old to the new educational system. Under his headship a foundation for the new Faculty of Science and the Graduate School Division for research of Science was firmly established. In 1955 Shoda was appointed as President of Osaka University, a post he held for six years. One of his achievements as President was the setting up of a Faculty of Engineering Science at Osaka and, in 1961 after his term as President ended, he became Dean of the new Faculty. After he retired from Osaka University, Shoda continued to work for a better educational system in Japan taking on many roles where he was able to use his long experience to give advice to many education Committees. His death from a heart attack was very unexpected, occurring while he was driving his family to Ashikaga to see the plum blossom. Nagao pays this tribute to Shoda in [1]:He loved the scholarly life and he loved his fellow man. His discipline was strict, yet his heart was warm and big. His faith in any man whom he came to know never wavered or changed. I know that the memory of this man's warm and rich humanity will live in the heart of many for a long time to come. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shoda.html (3 of 4) [2/16/2002 11:32:03 PM]

Shoda

Mathematicians of the day

Anniversaries for the year

JOC/EFR September 2001 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Shoda.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shoda.html (4 of 4) [2/16/2002 11:32:03 PM]

Shtokalo

Josif Zakharovich Shtokalo Born: 16 Nov 1897 in Skomorokhy, Sokal, Galicia (now Ukraine) Died: 5 Jan 1987 in Kiev, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Josif Shtokalo attended Dnipropetrovsk Institute in south-central Ukraine. He graduated in 1931 and then was appointed to Kharkov University. In 1942 Shtokalo was appointed to the Institute of Mathematics of the Academy of Sciences of Ukraine in Kiev. In 1944 he also received an appointment at Kiev University. Then in 1949 he went to Lvov when he became the Chairman of branch of the Academy of Sciences of Ukraine. In 1956 Shtokalo returned to Kiev where he again took up his posts at the Institute of Mathematics of the Academy of Sciences of Ukraine and his chair at Kiev University. At the Institute of Mathematics Shtokalo set up a seminar on the history of mathematics. In 1963, while retaining his position at Kiev University he moved from the Institute of Mathematics to the Institute of History of the Academy of Sciences of Ukraine in Kiev. Shtokalo worked mainly in the areas of differential equations, operational calculus and the history of mathematics. Petryshyn, writing in [3] describes Shtokalo's work:After 1945 he became particularly interested in the qualitative and stability theory of solutions of systems of linear ordinary differential equations in the Lyapunov sense and in the 1940s and 1950 published a series of articles and three monographs in these areas. Shtokalo's work had a particular impact on linear ordinary differential equations with almost periodic and quasi-periodic solutions. He extended the applications of the operational method to linear ordinary differential equations with variable coefficients.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shtokalo.html (1 of 2) [2/16/2002 11:32:05 PM]

Shtokalo

In [3] Petryshyn describes Shtokalo's work on the history of mathematics:He is regarded as one of the founders of the history of Soviet mathematics and particularly of the history in Ukraine and articles about M Ostrogradski and H Voronoy, he edited the three volume collections of Voronoy's (1952-3) and Ostrogradski's works (1959-61), a Russian-Ukrainian mathematical dictionary (1960) and approximately eighteen other Russian-Ukrainian terminology dictionaries. Shtokalo received the prestigious Koyré award from the International Academy of the History of Sciences in 1970. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Shtokalo.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Shtokalo.html (2 of 2) [2/16/2002 11:32:05 PM]

Siacci

Francesco Siacci Born: 20 April 1839 in Rome, Italy Died: 31 May 1907 in Naples, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Francesco Siacci graduated from the University of Rome in 1860 having already shown himself to have remarkable mathematical abilities. However this was a period of instability in Italian affairs and Siacci was deeply involved with the political events of the time. The year 1860 was the year of Italian unification. Rome and its surroundings were to remain under the control of the Pope while the rest of the Italian peninsula was to become one kingdom with a moderate constitution. Victor Emmanuel II was declared "king of Italy" on 26 October 1860. The parliament in Turin declared that the Kingdom of Italy had come into being on 17 March 1861. The position of Rome presented problems and, for Siacci based in Rome, this was not the place to be. He moved to Turin in 1861 and enlisted in the army when he arrived there. Since Siacci was already an excellent mathematician it was felt that he could make the best contribution to the army at the Military Academy in Turin. He was made an officer in the army and remained at the Academy until the outbreak of trouble in 1866. In June 1866 a war broke out between Austria and Prussia and the Italian government saw it as a good opportunity to attack the Austrians in Venetia. Siacci took part in this campaign against the Austrians but was quickly sent back to teach ballistics at the Military Academy in Turin before the campaign ended. It was not a successful military campaign for the Italians and it was perhaps fortunate for Siacci that he returned to an academic post. Siacci taught mechanics at the University of Turin from 1871. In 1872 Siacci was promoted to Professor of Ballistics at the Military Academy and he continued to hold this post until his army career ended in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Siacci.html (1 of 2) [2/16/2002 11:32:07 PM]

Siacci

1892 by which time he had reached the rank of major general. From 1875 he held a Professorship at the University of Turin in Higher Mechanics. In 1893 Siacci was rewarded for his support for Italy by being appointed to the office of Senator in Rome. Turin was too long a journey from Rome to make it possible for Siacci to act as a Senator and hold his professorship in Turin. He therefore requested a transfer to the University of Naples. This was agreed: Siacci was made an Honorary Professor at Turin where Volterra was invited to fill his position, and Siacci taught at Naples for the rest of his life. In [1] his mathematical contributions are given:Siacci is said to have been an excellent teacher, both at the university and at the Military Academy. He left some hundred publications, the most important being those concerned with analytic mechanics. In the application of mechanics to artillery - ballistics - he was a master. His treatise on this subject, especially the second edition of 1888, which had a French translation in 1891, was considered masterly. He received many honours during his lifetime, including election to most of the important scientific academies of Italy. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Siacci.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Siacci.html (2 of 2) [2/16/2002 11:32:07 PM]

Siegel

Carl Ludwig Siegel Born: 31 Dec 1896 in Berlin, Germany Died: 4 April 1981 in Göttingen, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Carl Siegel's father worked for the post office. Siegel entered the University of Berlin in 1915, in the midst of World War I, and attended lectures by Frobenius and Planck. Siegel wrote [15]:By conducting [beginners' classes] personally the professors could see, after only a few lectures, which of the students were the more gifted by the work they handed in, and the professors could direct their work accordingly. This was the way I myself first came into contact with my teachers Frobenius and Planck ... Initially his intention had been to study astronomy, but Frobenius's influence took him towards number theory which would became the main research topic of his career. In 1917, however, he had to interrupt his studies when he was called for military service. Most certainly military life did not suit Siegel and he was eventually discharged from the army as one of their failures, for despite their best efforts they had failed to have him adapt to army life. One would have to believe that Siegel would have classed this as a success rather than a failure. After the war had ended, Siegel continued his studies at Göttingen, beginning in 1919. His doctoral dissertation at Göttingen was supervised by Landau and Siegel then continued to study for his habilitation. His dissertation, written in 1920, [1]:... was a landmark in the history of Diophantine approximations. It extended an idea first noted by Liouville, then pushed forward by Thue who proved that, given a rational number q and any e > 0 there are only finitely many rational numbers p/q (in their lowest terms) such that http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Siegel.html (1 of 6) [2/16/2002 11:32:09 PM]

Siegel

|q - p/q| 1/(q2+1+e). Siegel improved this by showing that there are only finitely many rational numbers p/q such that if q is an algebraic number of degree n |q - p/q| 1/qm, where m = 2 n. Schönflies had been appointed as professor at the Johann-Wolfgang-Goethe-University of Frankfurt in 1914, the year in which the new university opened. He was aged 61 when he was appointed and when he retired in 1922 Siegel was appointed as professor to succeed him at Frankfurt. Although Schönflies spent the six years of his retirement in Frankfurt, his days as an active mathematician were over by the time Siegel took up the professorship. There were, however, several young mathematicians on the staff at Frankfurt who would with Siegel create an excellent centre for mathematics. Hellinger, like Schönflies, had been appointed as a professor to the new university of Frankfurt when it opened in 1914, and Szasz had been appointed as a Privatdozent in the same year. Szasz was promoted to professor in 1921, Epstein was appointed in 1919, and Dehn in 1921. It was a strong and exciting department which Siegel joined in 1922. There were a number of activities on which the four mathematicians Siegel, Hellinger, Epstein, and Dehn collaborated. One was the history of mathematics seminar instigated by Dehn in 1922. Siegel wrote in [15]:As I look back now, those communal hours in the seminar are some of the happiest memories of my life. Even then I enjoyed the activity which brought us together each Thursday afternoon from four to six. And later, when we had been scattered over the globe, I learned through disillusioning experiences elsewhere what rare good fortune it is to have academic colleagues working unselfishly together without thought to personal ambition, instead of just issuing directives from their lofty positions. The history of mathematics seminar was to last for thirteen years. They made a rule that they would study all the mathematical works in their original languages and although this reduced the number of students who participated in the seminar, there was never less than six. They studied the works of mathematicians including Euclid, Archimedes, Fibonacci, Cardan, Stevin, Viète, Kepler, Desargues, Descartes, Fermat, Huygens, Barrow, and Gregory. The aim of the seminar was [5]:... to increase the understanding of the participating students for the results presented in lectures and to provide the teachers with aesthetic satisfaction of examining the outstanding works of past times in close detail. The history of mathematics seminar was not the only one which Siegel participated in at Frankfurt, for the professors organised also a proseminar and a seminar. Student numbers rapidly built up after Siegel was appointed. At first he taught only a few students and [15]:... I remember having only two in one of the advanced courses. One day they were both late for class, having been delayed at the university bursar. When they arrived, they were shocked to find I had begun without them and had already filled a whole section of the blackboard. By 1928 Siegel was teaching 143 students in the differential and integral calculus course, and had to put in many hours work correcting students exercises. It was at this time that the student numbers reached a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Siegel.html (2 of 6) [2/16/2002 11:32:09 PM]

Siegel

maximum, then they began to drop again. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities. This did not affect Siegel who was an Aryan (to use the terminology of the time which Siegel hated) and, at this stage it did not affect Epstein, Hellinger or Dehn who, although Jewish, fell under a clause which exempted non-Aryans who had fought for Germany in World War I. Szasz, however, was dismissed from his post. Although Siegel was not affected by the Civil Service Law, he hated the Nazi regime and this was the beginning of a very unhappy time for him. In 1935 Siegel spent a year at the Institute for Advanced Study at Princeton in the United States. He returned to Frankfurt to find that the problems of his Jewish colleagues had become much worse. After decisions at the Nuremberg party congress in the autumn of 1935, Epstein, Hellinger and Dehn were forced from their posts. They remained in Frankfurt, unable to teach. In late 1937 Siegel accepted a professorship at Göttingen and he moved there in early 1938. At Göttingen he [15]:... led a somewhat retiring life. Life in Göttingen was still influenced by the Nazi policies and mathematicians reacted in different ways to the political pressures. For example Hasse in Göttingen wanted to accept the habilitation thesis of his assistant, but Siegel and Herglotz felt that this was a political rather than mathematical decision by Hasse and stopped the habilitation being accepted. The Nazi regime had taken Germany to war in 1939 and Siegel felt that he could no longer remain in his native land. In early 1940 he left Germany, lecturing first in Denmark and then in Norway. In March 1940 he met up with Dehn in Norway. Dehn had fled from Germany in fear of his life and was teaching in Trondheim when Siegel visited him. Siegel saw German merchant ships in the harbour and only later, having left Norway for the United States, did he discover that the ships he had seen were the advanced party of the German invasion force. Siegel described his time in the United States as [15]:... self imposed exile in America. He worked at the Institute for Advanced Study at Princeton from 1940 until 1951, being appointed to a permanent professorship there in 1946. However, in 1951 he returned to Germany and again worked at Göttingen for the rest of his career. The paper [8] lists Siegel's impressive contributions to mathematics under seven headings. These are: 1. Approximation of algebraic numbers by rationals and applications thereof to Diophantine equations. 2. Transcendence questions, in particular values of certain functions at algebraic points. 3. Zeta functions including applications to class numbers. 4. Geometry of numbers and its applications to algebraic number theory. 5. Hardy-Littlewood method, including Waring-type problems for algebraic numbers. 6. Quadratic forms: analytic theory and modular forms. 7. Celestial mechanics. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Siegel.html (3 of 6) [2/16/2002 11:32:09 PM]

Siegel

Siegel is especially famed for his work on the theory of numbers where he held an eminent role. Schneider, who was a student of Siegel's, gave three lectures on Siegel's contributions to number theory to the German Mathematical Union in 1982. These are reproduced in [13] and describe Siegel's most important results in number theory. These include his improvement of Thue's theorem, described above, given in his 1920 dissertation, and its application to certain polynomial Diophantine equations in two unknowns, proving an affine curve of genus at least 1 over a number field has only a finite number of integral points in 1929. Perhaps his two part paper which appeared in 1929 is [1]:.. his deepest and most original. In the 1929 paper Siegel made a substantial contribution to transcendence theory, especially a new method for the algebraic independence of values of certain E-functions. He proved that if J0 is the Bessel function of index 0, then for any non-zero algebraic integer r he showed that J0(r) is transcendental. He had earlier than this in 1922, written papers on the functional equation of Dedekind's zeta functions of algebraic number fields and in 1921/23 made contributions to additive questions such as Waring type problems for algebraic number fields. He made further contributions to this latter topic in 1944. Siegel's research on the analytic theory of quadratic forms in 1935/37 was of fundamental importance and he broke new ground in considering quadratic forms in which the coefficients were from an algebraic number field. Klingen, in [9], discusses Siegel's contributions to complex analysis. In particular he studied automorphic functions in several complex variables, Siegel's modular functions, which have led to a much deeper understanding. In this general area Siegel considered the theory of discontinuous groups and their fundamental domains, algebraic relations between modular functions and between modular forms, and Fourier series of modular forms. Siegel's work in celestial mechanics, which came next to number theory in his list of favourite topics, is discussed by Rüssmann in [12]. The paper lists eight major contributions which Siegel made to the subject. He studied: i. the n-body problem and the theorem of Bruns on algebraic integrals. ii. the restricted problem of three bodies and their integrals, which used the results Siegel had proved in (i). iii. the orbit of the moon, again essentially a three-body problem. Siegel gave a much improved version of lunar theory as developed by Hill. iv. the Lagrangian solutions for the three-body problem. Siegel developed general methods to determine periodic orbits near the equilibrium points. v. the problem of small divisors, where Siegel first obtained convergence results. vi. Birkhoff normal forms. He examined Birkhoff's work on perturbation theory solutions for analytical Hamiltonian differential equations near an equilibrium point using formal power series. Siegel gave examples of systems which did not possess convergent transformations into a normal form. vii. contributions to stability theory. An interesting episode, which tells us a lot about Siegel's approach to mathematics, occurred in the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Siegel.html (4 of 6) [2/16/2002 11:32:09 PM]

Siegel

1960s. Serge Lang published Diophantine geometry in 1962 and Mordell wrote a critical review of it two years later. Siegel then wrote to Mordell [11]:When I first saw [Lang's Diophantine geometry], about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible. My feeling is very well expressed when you mention Rip van Winkle! The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory - Lagrange, Gauss, or on a smaller scale, Hardy, Landau. Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one. I see a pig broken into a beautiful garden and rooting up all flowers and trees. Unfortunately there are many "fellow-travellers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched. These people remind me of the impudent behaviour of the national socialists who sang: "Wir werden weiter marschieren, bis alles in Scherben zerfällt!'' I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction - as I call it: theory of the empty set - cannot be blocked up. ... Dieudonné, writes in [1]:Siegel, who never married, devoted his life to research. But Dieudonné explains why he believes that Siegel had few doctoral students:... the perfection and thoroughness of his papers did not leave much room for improvement with the same technique, [and this] discouraged many research students because to do better than he required new methods. Siegel enjoyed teaching, however, even elementary courses, and he published textbooks on the theory of numbers, celestial mechanics, and the theory of functions of several complex variables. He was awarded many honours, perhaps the most prestigious of which was the Wolf Prize in 1978. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1920 to 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Siegel.html (5 of 6) [2/16/2002 11:32:09 PM]

Siegel

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Siegel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Siegel.html (6 of 6) [2/16/2002 11:32:09 PM]

Sierpinski

Waclaw Sierpinski Born: 14 March 1882 in Warsaw, Poland Died: 21 Oct 1969 in Warsaw, Poland

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Waclaw Sierpinski's father was a doctor. He attended school in Warsaw where his talent for mathematics was quickly spotted by his first mathematics teacher. This was a period of Russian occupation of Poland and it was a difficult time for the gifted Sierpinski to be educated in Poland. The Russians had forced their language and culture on the Poles in sweeping changes to all secondary schools implemented between 1869 and 1874. The Russian aim was to keep illiteracy in Poland as high as possible, so they discouraged learning and the number of students fell. Despite the difficulties, Sierpinski entered the Department of Mathematics and Physics of the University of Warsaw in 1899. It would be more accurate to describe it as the Czar's University since this was the official name of the University which had become a Russian university in 1869. The lectures at the University were all in Russian and the staff were entirely Russian. It is not surprising therefore that it would be the work of a Russian mathematician, one of his teachers Voronoy, that first attracted Sierpinski. In 1903 the Department of Mathematics and Physics offered a prize for the best essay from a student on Voronoy's contribution to number theory. Sierpinski was awarded the gold medal in the competition for his dissertation. He described the events (see [12]):... I was awarded a gold medal by the university for work in a competition on the theory of numbers. It was my first scientific work. It was accepted for publication in the 'Izvestia' of Warsaw University. However, in the following year there was a strike to produce a boycott of Russian Schools in Poland and I did not want to have my first work printed in the Russian language and that is why I had it withdrawn from print in Warsaw's 'Izvestia'. That is why it http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sierpinski.html (1 of 5) [2/16/2002 11:32:11 PM]

Sierpinski

was not printed until 1907 in the mathematical magazine 'The works of Mathematics and Physics' published by Samuel Dickstein. Fifty years after he graduated from the University of Warsaw Sierpinski looked back at the problems that he had as a Pole taking his degree at the time of the Russian occupation:... we had to attend a yearly lecture on the Russian language. ... Each of the students made it a point of honour to have the worst results in that subject. ... I did not answer a single question ... and I got an unsatisfactory mark. ... I passed all my examinations, then the lector suggested I should take a repeat examination, otherwise I would not be able to obtain the degree of a candidate for mathematical science. ... I refused him saying that this would be the first case at our University that someone having excellent marks in all subjects, having the dissertation accepted and a gold medal, would not obtain the degree of a candidate for mathematical science, but a lower degree, the degree of a 'real student' (strangely that was what the lower degree was called) because of one lower mark in the Russian language. Sierpinski was lucky for the lector changed the mark on his Russian language course to 'good' so that he could take his degree. As he says:The policeman was human. The results in the prize essay that Sierpinski wrote in 1904 were a major contribution to a famous problem on lattice points. Suppose R(r) denotes the number of points (m, n), m, n Z contained in a circle centre O, radius r. There exists a constant C and a number k with | R(r) -

r2 | < Crk.

Let d be the minimal value of k. Gauss proved in 1837 that d 1. Sierpinski's major contribution was to show that it was possible to improve the inequality to d proof and described the result as profound.

2/ . 3

In 1913 Landau shortened Sierpinski's

Let us digress for a moment to discuss some further work which flowed from this result of Sierpinski on what is often called the 'Gauss circle problem'. In 1915 Hardy and Landau proved that d > 1/2, while in 1923 van der Corput proved that d < 2/3. The following year Littlewood and Walfisz proved that d 37/

56,

this being improved to d

163/

247

the following year. Slight further improvements were made by

Vinogradov in 1932 and Titchmarsh in 1934. The best result I [EFR] know is d

7/

11.

Sierpinski graduated in 1904 and worked for a while as a school teacher of mathematics and physics in a girls school in Warsaw. However when the school closed because of a strike, Sierpinski decided to go to Krakóv to study for his doctorate. At the Jagiellonian University in Krakóv he attended lectures by Zaremba on mathematics, studying in addition astronomy and philosophy. He received his doctorate and was appointed to the University of Lvov in 1908. In fact it was in 1907 that Sierpinski first became interested in set theory. It happened when he came across a theorem which stated that points in the plane could be specified with a single coordinate. He wrote to Banachiewicz, who was at Göttingen at the time, asking him how such a result was possible. He received a one word reply 'Cantor'. Sierpinski began to study set theory and in 1909 he gave the first ever lecture course devoted entirely to set theory. Throughout his life Sierpinski maintained an incredible output of research papers and books. During the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sierpinski.html (2 of 5) [2/16/2002 11:32:11 PM]

Sierpinski

years 1908 to 1914, when he taught at the University of Lvov, he published three books in addition to many research papers. These books were The theory of irrational numbers (1910), Outline of Set Theory (1912) and The theory of numbers (1912). When World War I began in 1914, Sierpinski and his family happened to be in Russia. At this time the governments of Austria and Russia tried to use the Polish question as a political weapon. Sierpinski was interned in Viatka. However Egorov and Luzin heard that he had been interned and arranged for him to be allowed to go to Moscow. Sierpinski spent the rest of the war years in Moscow working with Luzin. Together they began the study of analytic sets. In 1916, during his time in Moscow, Sierpinski gave the first example of an absolutely normal number, that is a number whose digits occur with equal frequency in whichever base it is written. Borel had proved such numbers exit but Sierpinski was the first to give an example. When World War I ended in 1918, Sierpinski returned to Lvov. However shortly after taking up his appointment again in Lvov he was offered a post at the University of Warsaw which he accepted. In 1919 he was promoted to professor at Warsaw and he spent the rest of his life there. In 1920 Sierpinski, together with his former student Mazurkiewicz, founded the important mathematics journal Fundamenta Mathematica. Sierpinski edited the journal which specialised in papers on set theory. From this period Sierpinski worked mostly is in the area of set theory but also on point set topology and functions of a real variable. In set theory he made important contributions to the axiom of choice and to the continuum hypothesis. He studied the Sierpinski curve which describes a closed path which contains every interior point of a square. The length of the curve is infinity, while the area enclosed by it is 5/12 that of the square. Sierpinski continued to collaborate with Luzin on investigations of analytic and projective sets. His work on functions of a real variable include results on functional series, differentiability of functions and Baire's classification. Sierpinski was also highly involved with the development of mathematics in Poland. He had been honoured with election to the Polish Academy in 1921 and he was made dean of the faculty at the University of Warsaw in the same year. In 1928 he became vice-chairman of the Warsaw Scientific Society and, in the same year was elected chairman of the Polish Mathematical Society. In 1939 life in Warsaw changed dramatically with the advent of World War II. Sierpinski continued working in the 'Underground Warsaw University' while his official job was a clerk in the council offices in Warsaw. His publications continued since he managed to send papers to Italy. Each of these papers ended with the words:The proofs of these theorems will appear in the publication of Fundamenta Mathematica which everyone understood meant 'Poland will survive'. After the uprising of 1944 the Nazis burned his house destroying his library and personal letters. Sierpinski spoke of the tragic events of the war during a lecture he gave at the Jagiellonian University in Krakóv in 1945 (see [13]). He spoke of his students who had died in the war:In July 1941 one of my oldest students Stanislaw Ruziewicz was murdered. He was a retired professor of Jan Kazimierz University in Lvov ... an outstanding mathematician and an

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sierpinski.html (3 of 5) [2/16/2002 11:32:11 PM]

Sierpinski

excellent teacher. In 1943 one of my most distinguished students Stanislaw Saks was murdered. He was an assistant professor at Warsaw University, one of the leading experts in the world in the theory of the integral... In 1942 another student of mine was Adolf Lindenbaum was murdered. He was an assistant professor at Warsaw University and a distinguished author of works on set theory. After listing colleagues who were murdered in the war such as Schauder and others who died as a result of the war such as Dickstein and Zaremba, Sierpinski continued:Thus more than half of the mathematicians who lectured in our academic schools were killed. It was a great loss for Polish mathematics which was developing favourably in some fields such as set theory and topology ... In addition to the lamented personal losses Polish mathematics suffered because of German barbarity during the war, it also suffered material losses. They burned down Warsaw University Library which contained several thousand volumes, magazines, mathematical books and thousands of reprints of mathematical works by different authors. Nearly all the editions of Fundamenta Mathematica (32 volumes) and ten volumes of Mathematical Monograph were completely burned. Private libraries of all the four professors of mathematics from Warsaw University and also quite a number of manuscripts of their works and handbooks written during the war were burnt too. Sierpinski was the author of the incredible number of 724 papers and 50 books. He retired in 1960 as professor at the University of Warsaw but he continued to give a seminar on the theory of numbers at the Polish Academy of Sciences up to 1967. He also continued his editorial work, as editor-in-chief of Acta Arithmetica which he began in 1958, and as an editorial board member of Rendiconti dei Circolo Matimatico di Palermo, Composito Matematica and Zentralblatt für Mathematik. He received so many honours that it would be impossible to mention them all here. We list a few. He was awarded honorary degrees from the universities Lvov (1929), St Marks of Lima (1930), Amsterdam (1931), Tarta (1931), Sofia (1939), Prague (1947), Wroclaw (1947), Lucknow (1949), and Lomonosov of Moscow (1967). He was elected to the Geographic Society of Lima (1931), the Royal Scientific Society of Liège (1934), the Bulgarian Academy of Sciences (1936), the national Academy of Lima (1939), the Royal Society of Sciences of Naples (1939), the Accademia dei Lincei of Rome (1947), the German Academy of Science (1950), the American Academy of Sciences 1959), the Paris Academy (1960), the Royal Dutch Academy (1961), the Academy of Science of Brussels (1961), the London Mathematical Society (1964), the Romanian Academy (1965) and the Papal Academy of Sciences (1967). Rotkiewicz, who was a student of Sierpinski's wrote in [12]:Sierpinski had exceptionally good health and a cheerful nature. ... He could work under any conditions. ... He did not like any corrections to his papers. When someone suggested a correction he added a line to it: 'Mr X remarked that ...' He was a creative mind and liked creative mathematics. He was the greatest and most productive of Polish mathematicians. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sierpinski.html (4 of 5) [2/16/2002 11:32:11 PM]

Sierpinski

List of References (18 books/articles) A Poster of Waclaw Sierpinski

Mathematicians born in the same country

Other references in MacTutor

1. Chronology: 1910 to 1920 2. Chronology: 1920 to 1930

Honours awarded to Waclaw Sierpinski (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Sierpinski

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sierpinski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sierpinski.html (5 of 5) [2/16/2002 11:32:11 PM]

Siguenza

Carlos de Siguenza y Gongora Born: 20 Aug 1645 in Mexico City, Mexico Died: 22 Aug 1700 in Mexico City, Mexico Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Siguenza y Gongora's father was a tutor to the royal family in Spain before sailing to Mexico (then known as New Spain) to make his life in the new world. Siguenza was educated at home, his father being well able to provide the education his son needed. In 1662 Siguenza took his first religious vows in the Jesuit College in Tepoztlan where he had begun his training for the Church. From there he moved to Puebla to continue his religious studies at the College of the Holy Spirit. The town of Puebla, 130 km southeast of Mexico City, was characteristically Spanish and had been founded over one hundred years before as Puebla de los Angeles in 1532. It was a great religious centre and a natural place for Siguenza to study. However, in 1667 Siguenza was expelled from the College for failing to observe the correct discipline. He returned to Mexico City to study at the university there. The University of Mexico was founded in Mexico City as the Royal Pontifical University by Antonio de Mendoza who was the first viceroy of New Spain. The university was controlled by the Roman Catholic church so, in entering this university, Siguenza was still maintaining his connections with the Church despite the expulsion from the College in Puebla. He also remained a priest, despite the expulsion, and was chaplain to the Amor de Dios Hospital in Mexico City. In 1672 Siguenza was appointed to the chair of mathematics at the University. He was to hold this appointment for 20 years and contribute not only to mathematics but also to astronomy and cosmography. Siguenza's writings did not always find favour with the Church. Father Eusebio Kino was a Jesuit missionary. He had been educated in Germany, studying mathematics and astronomy, and then entered the Society of Jesus before being sent as a missionary to Mexico City in 1681. Siguenza had published a work in that year which was intended to calm public fears over the appearance of a comet. In reply to criticism of this work, he also published El Belerofonte matematico in the same year. Father Kino strongly disagreed with Siguenza's ideas and published his criticisms. Not one to be intimidated by Jesuit arguments, Siguenza made a strong response to Father Kino's criticism, publishing Libra astronomica y philosophica in 1670. This was [1]:... a short book of great significance for its sound mathematical background, anti-Aristotelian outlook, and familiarity with modern authors: Copernicus, Galileo, Descartes, Kepler, and Tycho Brahe. In cosmography, Siguenza made maps of New Spain (the first by someone born in that country), a map of the lakes in the Valley of Mexico, and a map of Pensacola Bay, now northwestern Florida in the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Siguenza.html (1 of 2) [2/16/2002 11:32:13 PM]

Siguenza

United States. There had been Spanish settlement on the coast of the bay from 1559 but the settlement was abandoned in 1661. Siguenza's map of the bay was made in 1693 at a time when New Spain were still interested in the area. In fact Siguenza was sent there to investigate the bay with an admiral, and the diary he kept and charts he made are of great value. Siguenza intended to write a history of ancient Mexico and spent much time collecting material for this work. It was very unfortunate that his early death prevented him from writing the history, and the extremely valuable manuscripts which he collected were sadly lost. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Siguenza.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Siguenza.html (2 of 2) [2/16/2002 11:32:13 PM]

Simplicius

Simplicius Born: about 490 in Cilicia, Anatolia (now Turkey) Died: about 560 in probably Athens, Greece Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Simplicius was born in Cilicia in southern Anatolia which had been a Roman province from the first century BC. We know that he went to Alexandria where he studied philosophy at the school of Ammonius Hermiae. Ammonius himself was a pupil of Proclus and Eutocius dedicated his commentary on Book I of Archimedes' On the sphere and cylinder to him. Ammonius's main work throughout his life was writing critical works on Aristotle and this clearly influenced Simplicius who was himself to write extensive commentaries on Aristotle. After studying under Ammonius in Alexandria, Simplicius went to Athens where he studied with the Neoplatonist philosopher Damascius. Damascius had written Problems and Solutions about the First Principles which develops the Neoplatonist philosophy as expounded by Proclus. Again Simplicius was exposed to similar views to those he had learnt in Alexandria and his philosophy was built up in a consistent way. Damascius had been made head of Plato's Academy in about 520 and he was still head when the Christian emperor Justinian closed it in 529. At the same time Justinian closed all the other pagan schools. When Justinian had become emperor, his troops were fighting on the Euphrates River against the armies of the Persian king. It was therefore natural that Damascius, Simplicius and five other members of the Academy, when forced out of Athens, went to Persia to serve at the court of the Persian king Khosrow I. Khosrow was a great patron of culture and Simplicius was well received by the ruler. However, Khosrow and Justinian signed the Treaty of Eternal Peace which was ratified in 532 which led to Simplicius being able to return to Athens. It is not entirely clear what the terms of the treaty were in regard to Simplicius and the other philosophers who had gone to Persia. Agathias, the Byzantine poet and author of a history of his own times, wrote of these events after the death of Justinian in 565. He wrote that (see [1]):... the terms of the treaty would have guaranteed to the philosophers full security in their own environment: they were not to be compelled to accept anything against their personal conviction, and they were never to be prevented from living according to their own philosophical doctrine. The accuracy of this view by Agathias has been challenged by Cameron in [4], however, and things may

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Simplicius.html (1 of 3) [2/16/2002 11:32:14 PM]

Simplicius

not have been as easy for Simplicius after his return to Athens as Agathias suggested. There is some evidence that Simplicius did remain in Athens for the rest of his days, writing and undertaking research but certainly not being allowed to lecture. This is reflected by the nature of his writings which are not of the lecture course style, but instead are carefully constructed interpretations of the writings of Aristotle and in particular they attempt to harmonise the views of Plato and Aristotle. Of the writings of Simplicius that have survived, it is thought that the earliest is his commentary on Epictetus's Enchiridion which some historians believe was written by Simplicius while he was still in Alexandria. This, however, seems highly unlikely and the argument by Cameron in [4] that it was written during Simplicius's time in Persia seems much more convincing. Certainly this seems to have been written before the commentaries on Aristotle, the first of which is the commentary on De Caelo. This is followed by Simplicius's commentary on Aristotle's Physics and both these works are particularly important for the history of mathematics. In the commentary on De Caelo Simplicius gives a detailed account of the concentric spheres of Eudoxus and he also relates the modifications to the theory made later by Callippus. Simplicius is quoting from Eudemus's History of Astronomy in giving these details, but he does not quote directly from that work, rather quoting from Sosigenes (who wrote in the second century AD) who in turn quotes from Eudemus. In his commentary on Aristotle's Physics Simplicius quotes at length from Eudemus's History of Geometry which is now lost. In particular Simplicius quotes the writing on Eudemus on Antiphon's attempts to square the circle and also the attempts of Hippocrates when he squared certain lunes. Also in this commentary on Aristotle's Physics Simplicius gives important quotations from Geminus's summary of Posidonius's Meteorologica. Simplicius wrote a commentary on Euclid's Elements which survives in an Arabic translation. This is discussed in [6] where the author discusses the fact that the commentary does not contain an attempt at a proof of the parallel postulate by Simplicius himself, despite the evidence that indeed Simplicius did attempt such a proof. In [6] there is a discussion of how Simplicius's attempted proof of the parallel postulate entered Arabic mathematics and was first criticised, then incorporated into a new 'proof' designed to take the criticism into account. The importance of Simplicius as a commentator is described in [1]:[Simplicius] did not overestimate his own contributions but was quite aware of his debt to other philosophers, especially to Alexander, Iamblichus, and Porphyry. He did not hesitate to call his own commentaries a mere introduction to the writings of these famous masters, nor did he cling fanatically to his own interpretations; he was happy to exchange them for better explanations. On the other hand, the work of the commentator is far from being a neutral undertaking or a question of mere erudition; it is chiefly an opportunity to become more familiar with the text under consideration and to elucidate some intricate passages; hence Simplicius's constant concern to obtain reliable documents and to check the historical value of this information... Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Simplicius.html (2 of 3) [2/16/2002 11:32:14 PM]

Simplicius

Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Simplicius.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Simplicius.html (3 of 3) [2/16/2002 11:32:14 PM]

Simpson

Thomas Simpson Born: 20 Aug 1710 in Market Bosworth, Leicestershire, England Died: 14 May 1761 in Market Bosworth, Leicestershire, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Simpson is best remembered for his work on interpolation and numerical methods of integration. His first job was as a weaver. At this time he taught mathematics privately and from 1737 he began to write texts on mathematics. He also worked on probability theory and in 1740 published The Nature and Laws of Chance. Much of Simpson's work in this area was based on earlier work of De Moivre. Simpson was the most distinguished of a group of itinerant lecturers who taught in the London coffee-houses. He worked on the Theory of Errors and aimed to prove that the arithmetic mean was better than a single observation. Simpson published the two volume work The Doctrine and Application of Fluxions in 1750. It contains work of Cotes. In 1754 he became editor of the Ladies Diary. The following description of Simpson by Charles Hutton (made 35 years after Simpson's death) is interesting It has been said that Mr Simpson frequented low company, with whom he used to guzzle porter and gin: but it must be observed that the misconduct of his family put it out of his power to keep the company of gentlemen, as well as to procure better liquor. It would be fair to note that others described Simpson's conduct as irreproachable. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Simpson.html (1 of 2) [2/16/2002 11:32:16 PM]

Simpson

Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Thomas Simpson

Mathematicians born in the same country

Other references in MacTutor

1. Work on minimal paths 2. Chronology: 1720 to 1740 3. Chronology: 1740 to 1760

Honours awarded to Thomas Simpson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1745

Other Web sites

Rouse Ball

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Simpson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Simpson.html (2 of 2) [2/16/2002 11:32:16 PM]

Simson

Robert Simson Born: 14 Oct 1687 in West Kilbride, Ayrshire, Scotland Died: 1 Oct 1768 in Glasgow, Scotland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Simson was the eldest son of John Simson and Agnes Simson (née Simson). Simson's mother had 17 children, all boys, only six of whom reached manhood. Simson entered the University of Glasgow as a student on 3 March 1702, being 14 years old at the time, and studied under the regent John Tran. He distinguished himself in classics, oriental languages and botany. His father had intended that Robert should enter the church, and it was only by accident that his interests turned to mathematics. As a student of theology he was required to produce written work for his teachers. This he found unsatisfying, since he felt the arguments to be inconclusive and speculative. For recreation he turned first to a book on oriental philology, where he found statements that could be shown to be true or false, but this was not wholly satisfying and at that stage he had recourse to mathematics and Euclid's Elements. He then set to work to study mathematics seriously, but he had to do this on his own, since at that time, for some reason, there were no lectures given on the subject by the professor Robert Sinclair. During his first year as a student in Glasgow Robert Simson was involved in a rather interesting incident which shows that, even before the Union of the Parliaments, Scottish as well as English students celebrated the 5th of November with fireworks. On emerging from a close that night Robert was hurt in the face by shot from a pistol belonging to a fellow student Arthur Tran, who with a group of other students was firing a pistol and letting off squibs. They were hauled before the Faculty where Tran agreed that he had fired the pistol, and that he had said he would shoot it in some old wife's lug. Tran himself was fined half-a-crown, his friends lesser amounts and were publicly rebuked in the Common Hall. Tran may have been the son of the regent John Tran. It is generally supposed that Robert Simson attended the University as a student for a period of about eight years until 1710, this was not an unusual period of study at the time. In 1710 Professor Robert Sinclair resigned, and Simson was offered the chair. He was disinclined to accept it immediately, and asked permission to spend some time in London, where he might have the opportunity to become acquainted with some of the most eminent mathematicians in England. This was granted and he went immediately to London where he met several well-known mathematicians, such as Edmond Halley, John Caswell (d. 1712), Savilian Professor of Geometry at Oxford, William Jones, and finally Humphrey Ditton (1675-1715), Mathematical Master at Christ's Hospital, with whom Simson was particularly http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Simson.html (1 of 5) [2/16/2002 11:32:18 PM]

Simson

friendly. While in London he was informed that, by the Faculty minute of 8 March 1711, he had been nominated to the Glasgow chair providing always, that he gives a satisfactory proof to the faculty of his skill in the Mathematics before his admission. He returned to Glasgow, where on 10 November 1711 he was given two geometrical problems to resolve. These he dealt with to the satisfaction of the Faculty after which he gave a satisfactory specimen of his skill in mathematicks and dexterity in teaching geometry and algebra, he also produced sufficient testimonials from Mr Caswell the Professor of astronomy at Oxford and from others in London well skilled in the mathematicks, upon all which the faculty resolve he shall be admitted the nineteenth day of this instant November. Three days before his admission he graduated Master of Arts. It was while he was in England that Edmond Halley suggested to him that he might devote his considerable talents to the restoration of the work of the early Greek geometers, such as Euclid and Apollonius of Perga. These are works that only survive in abbreviated accounts given by later mathematicians such as Pappus of Alexandria. He first studied Euclid's so-called porisms. Playfair's 1792 definition of porism is a proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions. Simson's work on Euclid's porisms was published in 1723 in the Philosophical Transactions of the Royal Society, and his restoration of the Loci Plani of Apollonius appeared in 1749. Further work of his on porisms and other subjects including logarithms was published posthumously in 1776 by Lord Stanhope at his own expense. Simson also set himself the task of preparing an edition of Euclid's Elements in as perfect a form as possible, and his edition of Euclid's books 1-6, 11 and 12 was for many years the standard text and formed the basis of textbooks on geometry written by other authors. The work ran through more than 70 different editions, revisions or translations published first in Glasgow in 1756, with others appearing in Glasgow, Edinburgh, Dublin, London, Cambridge, Paris and a number of other European and American cities. Recent editions appeared in London and Toronto in 1933 under the editorship of Isaac Todhunter, and in Sao Paolo in 1944. Simson's lectures were delivered in Latin, at any rate at the beginning of his career. His most important writings were written in that language, however, his edition of Euclid, after its first publication in Latin, appeared in English, as did a treatise on conic sections that he wrote for the benefit of his students. His reputation as a geometer has always been very high, although, as a critic wrote the additions and alterations which Simson made by way of restoring the text to its 'original accuracy' are certainly not all of them improvements, and the notes he appended show with what reverence he regarded the great geometers of antiquity. There was a feeling in some quarters that, by limiting his efforts to the attainment of he perfect text, he lost an opportunity of applying his own considerable talents and insight to a more useful exposition of his subject. For Simson the best vehicle for presenting a mathematical argument was geometry and, although he was familiar with the recent developments in algebra and the infinitesimal calculus, he preferred to express himself in geometrical terms wherever possible. He was not, of course, alone in this, as Newton http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Simson.html (2 of 5) [2/16/2002 11:32:18 PM]

Simson

had adopted the same viewpoint when writing his Principia. That Simson's work was not restricted to Greek geometry is illustrated by Tweddle's paper [3]., in which he discusses an early manuscript of Simson dealing with inverse tangent series and their use in calculating . For fifty years Simson lectured five days a week during term time to his two main classes. By all accounts he was a good lecturer, although better in his younger days than towards the end of his life, when his absence of mind made him the victim of practical jokes. Several of his pupils achieved distinction in mathematics, notably Maclaurin, Stewart, John Robison who became Professor of Natural Philosophy in the University of Edinburgh and Trail. Simson was a good looking man, tall of stature and favouring light coloured cloths. He was unmarried and so had no use of the commodious house in the College to which he was entitled, but lived in rooms there. He ate all his meals, including breakfast, at a small tavern opposite the College gate kept by a Mrs Millar. He delighted in showing visitors round the College and was very knowledgeable about the large collection of Roman antiquities coming from the Antonine Wall and its neighbourhood. According to the Reverend Alexander Carlyle He was particularly averse to the company of ladies, and, except one day in the year, when he drank tea at Principal Campbell's and conversed with gaiety and ease with his daughter Mally, who was always his first toast, he was never in company with them. It is said to have been otherwise with him in his youth, and that he had been much attached to one lady, to whom he had made proposals, but on her refusing him he became disgusted with the sex. The lady was dead before I became acquainted with the family, but her husband I knew, and must confess that in her choice the lady preferred a satyr to Hyperion. Simson was a sociable man. On Friday evenings he would meet with friends at a club in a nearby tavern, where he would play whist, a game in which he excelled. This was followed by a period of conversation and singing. He was fond of singing Greek odes set to contemporary music. Every Saturday he would walk, alone or in company, to an inn in the nearby village of Anderston, where he played host to his particular friends and any visitors to Glasgow that he had invited to join him for dinner. He was a most methodical man and, on daily walks in the College garden and elsewhere, he counted the number of paces from one place to another. Robert Simson was the first person to be appointed to the office of Clerk (later known as Clerk of Senate), which he took up in 1728 and only demitted when he retired in 1761. In 1761 he retired from his chair having held it for fifty years. He kept his rooms in the tower until his death, but gave up his College house, which he had never lived in. Simson remained in good health until a few years before his death, during which period he had to employ an amanuensis to assist him in revising his geometrical writings. A year before he retired from the Chair of Mathematics, Simson had proposed that his colleague James Buchanan, the Professor of Oriental Languages, should relieve him of his teaching duties on condition of succeeding to the Chair when he retired, but Buchanan died before any action was taken. Before Simson retired in 1761 he stipulated that his Assistant James Williamson should succeed him, and this was agreed. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Simson.html (3 of 5) [2/16/2002 11:32:18 PM]

Simson

Robert Simson died in his eight-first year and was buried in the neighbouring Blackfriars burial ground, where a marble tombstone, bearing a long laudatory Latin inscription, was raised in his memory. Later, through the efforts of a Mr Fullerton of Overton in the parish of West Kilbride, an imposing monument fifty feet high was erected in West Kilbride cemetery, bearing the inscription in which he is described as the Restorer of Grecian Geometry, and by his Works the Great Promoter of its Study in the Schools. Simson bequeathed his very large library of books and papers to the University of Glasgow, where they comprise the valuable Simson Bequest. The collection includes the sixteen volumes of his daily notebooks, the Adversaria. These cover the years 1715-1765 and consist of numerous geometrical problems interspersed with exercises in algebra and astronomy, as well as occasional accounts of financial transactions. Additional comments Simson also made many discoveries of his own in geometry and the Simson line is named after him. However the Simson line does not appear in his work but Poncelet in Propriétés Projectives says that the theorem was attributed to Simson by Servois in the Gergonne's Journal. It appears that the theorem is due to William Wallace. The University of St Andrews awarded Simson an honorary Doctorate of Medicine in 1746. In 1753 Simson noted that, as the Fibonacci numbers increased in magnitude, the ratio between adjacent numbers approached the golden ratio, whose value is (1 + 5)/2 = 1.6180 . . . . Article by: R A Rankin, Glasgow Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other references in MacTutor

1. The Simson line configuration. 2. Chronology: 1740 to 1760

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Simson.html (4 of 5) [2/16/2002 11:32:18 PM]

Simson

Mathematicians of the day JOC/EFR August 1995

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Simson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Simson.html (5 of 5) [2/16/2002 11:32:18 PM]

Sinan

Abu Said Sinan ibn Thabit ibn Qurra Born: about 880 Died: 943 in Baghdad, (now in Iraq) Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sinan ibn Thabit ibn Qurra was the son of Thabit ibn Qurra and the father of Ibrahim ibn Sinan. Although Sinan was extremely eminent in medicine his contributions to mathematics were somewhat less major but he still deserves a place in this archive as a contributor to mathematics in this remarkable family of scholars. Thabit ibn Qurra, Sinan's father, was a member of the Sabian sect. The Sabian religious sect were star worshippers from Harran. Of course being worshipers of the stars meant that there was strong motivation for the study of astronomy and the sect produced many quality astronomers and mathematicians such as Thabit himself. Sinan was trained in medicine, a topic which his father had studied in Baghdad. His father's patron was the Caliph, al-Mu'tadid, one of the greatest of the 'Abbasid caliphs, and Sinan was brought up at the court where his father held the role of court astronomer. Sinan's father Thabit died in 901 and the caliph al-Mu'tadid died the following year. Al-Mu'tadid had shown great skill in playing the various factions off against each other during his period of power but after his troops were defeated by the Qarmatians, a schismatic sect and political movement. Historians argue whether al-Mu'tadid was poisoned in a palace intrigue, but even if he was not this is an indication of the atmosphere in the court where Sinan lived. By this time Sinan was a man of about 22 years of age but, despite having great medical skills, he seems to have held no positions at this time. On al-Mu'tadid death, his son al-Muktafi became caliph and succeeded in defeating the Qarmatian sect which had lead to his father's downfall. He ruled until 908 and Sinan certainly enjoyed a period of great cultural activity in Baghdad which was home to many intellectuals. However in 908 al-Muqtadir, who was only a boy at the time, became Caliph. He was a weak leader but his coming to power saw Sinan achieve his first major position in which he directed the hospitals and all medical activities in Baghdad. Although the government in Baghdad slowly lost control, Sinan achieved the respect of all the factions. He was, as we mentioned at the beginning of the article, a Sabian and not a Muslim. However, he was totally fair in his treatment of people regardless of which religious group they belonged to and for this he gained respect. By 931 he had gained such authority in Baghdad that all doctors had to be tested by him before being allowed to practise. Al-Muqtadir's reign ended in 932 and he was replaced by al-Qahir. Now Sinan faced a totally different type of regime, for al-Qahir persecuted the Sabians. Sinan tried to preserve his position by becoming Muslim but this was not sufficient to allow him to continue in Baghdad and he fled to Khurasan. The http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sinan.html (1 of 2) [2/16/2002 11:32:19 PM]

Sinan

Abbasid caliphs were rapidly losing control and al-Qahir only survived for two years before ar-Radi became caliph in 934. This allowed Sinan to return to Baghdad but, in 935, the final political crisis occurred and ar-Radi was forced to hand over most of his power to the ambitious general ibn Ra'iq. Ar-Radi died in 940 after a five year struggle to retain power and the problems only became worse as military leaders struggled for control. Sinan left Baghdad again to move this time to Wasit on the Tigris. Despite his high profile medical career, Sinan seems not to have written any works on medicine. He wrote mainly on three topics, political history, mathematics and astronomy. However Sinan's political work in which he set out his ideas for a government modelled on Plato's Republic were criticised by the historian and traveller al-Mas'udi who known as the "Herodotus of the Arabs". Al-Mas'udi stated that Sinan should have [1]:... occupied himself with topics within his competence, such as the science of Euclid, the Almagest, astronomy, the theories of meteorological phenomena, logic, metaphysics, and the philosophical systems of Socrates, Plato, and Aristotle. No works which can definitely be attributed to Sinan have survived although it is claimed that he wrote four mathematical works, although in [1] the author points out that only two of the four could have beed written by Sinan, one on Archimedes work On triangles and one On the elements of geometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sinan.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sinan.html (2 of 2) [2/16/2002 11:32:19 PM]

Sintsov

Dmitrii Sintsov Born: 21 Nov 1867 in Viatka (now Kirov), Russia Died: 28 Jan 1946 in Kharkov, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dmitrii Sintsov attended Kazan University, graduating in 1890. He was appointed to the staff at Kazan and taught there from 1894 to 1899. After leaving Kazan, Sintsov taught at the Odessa Higher Mining School, then, in 1903, he was appointed to Kharkov University where he taught until his death in 1946. He took a leading role in the development of mathematics at Kharkov University and, for many years, he was President of the Kharkov Mathematical Society. This Society is one of the early mathematics societies and was founded in 1879. Sintsov's main areas of interest were in the theory of conics and applications of this geometrical theory to the solution of differential equations and to the theory of nonholonomic differential geometry. At Kharkov University, Sintsov created a school of geometry which has continued to flourish through the years and is today a leading centre. There he studied the geometry of Monge equations and he introduced the important ideas of asymptotic line curvature of the first and second kind. Sintsov also took an interest in the history of mathematics and one of the major projects which he undertook in this area was the detailed study of the work of previous mathematicians at Kharkov University. This work provides a fascinating account of the development of mathematics there from the founding of the university in 1805. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sintsov.html (1 of 2) [2/16/2002 11:32:21 PM]

Sintsov

Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sintsov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sintsov.html (2 of 2) [2/16/2002 11:32:21 PM]

Sitter

Willem de Sitter Born: 6 May 1872 in Sneek, Netherlands Died: 20 Nov 1934 in Leiden, Netherlands

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Willem De Sitter studied mathematics at Groningen and then joined the Groningen astronomical laboratory. He worked at the Cape Observatory in South Africa (1897-99) then, in 1908, de Sitter was appointed to the chair of astronomy at Leiden. From 1919 he was director of the Leiden Observatory. In 1913 de Sitter produced an argument based on observations of double star systems which proved that the velocity of light was independent of the velocity of the source. It put to rest attempts which had been made up until this time to find emission theories of light which depended on the velocity of the source but were not in conflict with experimental evidence. De Sitter corresponded with Ehrenfest in 1916, and he proposed that a four- dimensional space- time would fit in with cosmological models based on general relativity. He published a series of papers (1916-17) on the astronomical consequences of Einstein's general theory of relativity. He found solutions to Einstein's field equations in the absence of matter. This was significant since Mach had stated a principle that local inertial frames of reference were determined by the large scale distribution of mass in the universe. De Sitter asked:If no matter exists other than the test body, does it have inertia. De Sitter's work led directly to Eddington's 1919 expedition to measure the gravitational deflection of light rays passing near the Sun, results which, at that time, could only be obtained during an eclipse. De Sitter, unlike Einstein, maintained that relativity actually implied that the universe was expanding, theoretical results which were later verified observationally and accepted by Einstein.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sitter.html (1 of 3) [2/16/2002 11:32:23 PM]

Sitter

In fact Einstein had introduced the cosmological constant in 1917 to solve the problem of the universe which had troubled Newton before him, namely why does the universe not collapse under gravitational attraction. This rather arbitrary constant of integration which Einstein introduced admitting it was not justified by our actual knowledge of gravitation was later said by him to be the greatest blunder of my life. However de Sitter wrote in 1919 that the term ... detracts from the symmetry and elegance of Einstein's original theory, one of whose chief attractions was that it explained so much without introducing any new hypothesis or empirical constant. In 1932 Einstein and de Sitter published a joint paper with Einstein in which they proposed the Einstein-de Sitter model of the universe. This is a particularly simple solution of the field equations of general relativity for an expanding universe. They argued in this paper that there might be large amounts of matter which does not emit light and has not been detected. This matter, now called 'dark matter', has since been shown to exist by observing is gravitational effects. However the dark matter postulated by Einstein and de Sitter in 1932 still remains a mystery in that its nature is still unknown but is the subject of major research efforts today. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Willem De Sitter (Click a link below for the full list of mathematicians honoured in this way) ASP Bruce Medallist

1931

Lunar features

Crater Sitter

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sitter.html (2 of 3) [2/16/2002 11:32:23 PM]

Sitter

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sitter.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sitter.html (3 of 3) [2/16/2002 11:32:23 PM]

Skolem

Albert Thoralf Skolem Born: 23 May 1887 in Sandsvaer, Norway Died: 23 March 1963 in Oslo, Norway

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thoralf Skolem worked on Diophantine equations, mathematical logic, group theory, lattice theory and set theory. In 1912 he produced a description of a free distributive lattice. He made refinements to Zermelo's axiomatic set theory, publishing work in 1922 and 1929. Skolem extended work by Löwenheim (1915) to give the Löwenheim- Skolem theorem, which states that if a theory has a model then it has a countable model. From 1933 he did pioneering work in metalogic and constructed a nonstandard model of arithmetic. He also developed the theory of recursive functions as a means of avoiding the so-called paradoxes of the infinite. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Skolem.html (1 of 2) [2/16/2002 11:32:25 PM]

Skolem

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Skolem.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Skolem.html (2 of 2) [2/16/2002 11:32:25 PM]

Slaught

Herbert Ellsworth Slaught Born: 21 July 1861 in Seneca Lake, Watkins, New York, USA Died: 21 May 1937 in Chicago, Illinois, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Herbert Slaught was brought up on a farm on Seneca Lake but, when he was 13 years old, his family lost their farm and were forced to move. Perhaps, as Bliss relates in [1], this was a blessing in disguise:... Slaught himself has said that he would probably have spent his life working on the farm if it had not happened, and the farm was apparently not a very good one. The family moved to Hamilton, New York, in 1875 and Slaught attended Colgate Academy until he graduated in 1879 and entered Colgate University. After an outstanding undergraduate career he graduated with an A.B. in 1883. He was offered the post of instructor in mathematics at the Peddie Institute in Hightstown, New Jersey and very quickly impressed everyone with his abilities to teach and his administrative abilities. Slaught was quickly promoted at the Peddie Institute, first to assistant principal in 1886, then to principal in 1889. However, despite his success or perhaps because of it, he decided to aim higher. Bliss writes [1]:He had married Miss Mary L Davis, the instructor in music at Peddie, in 1885, and she sympathized with and encouraged his desire to enter the field of university mathematical work, even though such a course meant a serious sacrifice for them for some time to come. So in 1892 Slaught accepted a two-year appointment to one of the first three fellowships awarded by the Department of Mathematics at the University of Chicago, which was just then opening its doors. After two years of research at Chicago, his fellowship ended and he was appointed onto the teaching staff. He did not complete his doctorate until 1898 because of the high teaching load that he had. His http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Slaught.html (1 of 2) [2/16/2002 11:32:27 PM]

Slaught

research had been supervised by Eliakim Moore and he was awarded his doctorate for a thesis entitled The Cross Ratio Group of 120 Quadratic Cremona Transformations of the Plane. Slaught soon received promotion. In 1900 he was made an assistant professor, them associate professor in 1908 and full professor in 1913. Slaught remained at Chicago for the rest of his career, retiring from his chair in 1931. During 1902-3 Slaught travelled in Europe attending lectures by the leading mathematicians. Perhaps he felt that he could never achieve the depth of research he was exposed to at this time for, after a worrying time of indecision, he decided that he was not cut out for a research career but could give most to the world of mathematics by concentrating on teaching. After seeking Dickson's advice on the best way to serve the mathematical community, he accepted Dickson's suggesting of becoming co-editor of the American Mathematical Monthly. He also became active in the organisation of the Mathematical Association of America, the National Council of Teachers of Mathematics, and the Chicago section of the American Mathematical Society. He served as secretary of the last named Society from 1906 to 1916. Bliss [1] describes Slaught as:... one of the men most widely known by teachers and students of mathematics... His lifelong devotion to... the promotion of the study of mathematics, his skill as a teacher, his effective leadership in the mathematical organizations which he sponsored, and his influence with teachers of mathematics the country over, were remarkable. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Slaught.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Slaught.html (2 of 2) [2/16/2002 11:32:27 PM]

Sleszynski

Ivan Sleszynski Born: 23 July 1854 in Lysianka, Cherkasy, Kiev gubernia, Ukraine Died: 9 March 1931 in Krakóv, Poland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ivan Sleszynski graduated from Odessa University in 1875. He then travelled to Germany where he studied under Weierstrass in Berlin, receiving his doctorate in 1882. Returning to Odessa, he became professor of mathematics there from 1883 to 1909. Sleszynski went to Poland where he was appointed to the University of Krakóv in 1911. He continued to work at Krakóv until he retired in 1924. Sleszynski's main work was on continued fractions, least squares and axiomatic proof theory based on mathematical logic. In 1898 A Pringsheim proved that the condition |bn| |an| + 1, an 0, n 1, ensures the convergence of the continued fraction K(an/bn), where an and bn are complex numbers; a result known as the Pringsheim criterion. W J Thron states in [2] that this result was established ten years earlier by Sleszynski. Thron demonstrates that Pringsheim was aware of Sleszynski's work, though Pringsheim himself claims that he only became aware of Sleszynski after his article was completed. Six papers by Sleszynski on continued fractions are discussed in [2] where a complete bibliography of Sleszynski's mathematical papers is given. His work on continued fractions is also discussed in [1]. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sleszynski.html (1 of 2) [2/16/2002 11:32:28 PM]

Sleszynski

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sleszynski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sleszynski.html (2 of 2) [2/16/2002 11:32:28 PM]

Slutsky

Evgeny Evgenievich Slutsky Born: 19 April 1880 in Novoe, Yaroslavskaya guberniya, Russia Died: 10 March 1948 in Moscow, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

After leaving school Evgeny Slutsky entered the University of Kiev in 1899 to study mathematics. He was involved in student politics and he participated in student unrest at the university. Student trouble makers were dealt with by giving them a spell in the army. That is precisely what happened to Slutsky in 1901, but he was not given a particularly long spell and he was soon back at Kiev University. The following year he was in trouble again and this time he was expelled and there was no chance to complete his studies at Kiev. Slutsky decided on getting an education abroad and he entered Munich Polytechnikum were he was able to complete a degree and return to Kiev in 1905. This time he went for course more in keeping with his political interests, taking a degree in political economics in the Faculty of Law. He graduated with the Gold Medal in 1911. From 1913 until 1926 he taught at the Kiev Institute of Commerce, then in 1926 he moved to the government statistics offices in Moscow. After eight years there during which time he published important statistical papers, he began teaching at the University of Moscow. From 1938 onwards he worked at the Institute of Mathematics of the U.S.S.R. Academy of Sciences. As a statistician, Slutsky was influenced by Pearson's work and he was interested in both the mathematical background of the statistical methods he studied as well as their application to economics and, later in his career, to natural sciences. While at the Kiev Institute of Commerce, Slutsky gave the fundamental equation of value theory to

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Slutsky.html (1 of 2) [2/16/2002 11:32:30 PM]

Slutsky

economics. Slutsky introduced stochastic concepts of limits, derivatives and integrals from 1925 to 1928 while he worked at the government statistics offices. In 1927 he showed that subjecting a sequence of independent random variables to a sequence of moving averages generated an almost periodic sequence. This work stimulated the creation of stationary stochastic processes. He also studied correlations of related series for a limited number of trials. He obtained conditions for measurability of random functions in 1937. Slutsky applied his theories widely, in addition to economics mentioned above he also studied solar activity using data from 500 BC onwards. Other applications were to diverse topics such as the pricing of grain and the study of chromosomes. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) A Poster of Evgeny Slutsky

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Slutsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Slutsky.html (2 of 2) [2/16/2002 11:32:30 PM]

Sluze

René François Walter de Sluze Born: 2 July 1622 in Visé, Principality of Liège (now Belgium) Died: 19 March 1685 in Liège, Principality of Liège (now Belgium) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

René de Sluze studied at the University of Louvain from 1638 to 1642. He then went to Rome where he received a law degree from the University of Sapienza in 1643. Sapienza was the name of the building which the University of Rome occupied at this time and it gave its name to the University. After this de Sluze studied a large number of subjects in Rome including many languages, mathematics and astronomy. De Sluze became a canon in the church in 1650, then went to Liège also as a cannon. His knowledge of law meant that he progressed rapidly within the Church and rose to positions of influence quickly. By 1659 he was a member of the privy council of the Bishop of Liège, then he became abbot of Amay in 1666. De Sluze wrote many books on mathematics. He worked on calculus, having studied the works of Cavalieri and Torricelli while in Rome, in particular he worked on the equation for the cycloid. In his works de Sluze discusses spirals, points of inflection and the finding of geometric means. He extended work by Descartes and Fermat on drawing tangents and finding turning points of functions. Because de Sluze's position in the church prevented him going to meet with other mathematicians he did most of his mathematical contacts by correspondence. He corresponded with many mathematicians in England, France and other European countries, for example Pascal, Huygens, Wallis and Ricci were among those with whom he was in regular contact. De Sluze was elected a Fellow of the Royal Society in 1674. The family of curves yn = k(a-x)pxm for positive integer exponents, are called the pearls of Sluze. De Sluze did not write exclusively on mathematics. He also wrote on astronomy, physics, natural history, history and theological matters connected with his work in the church. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sluze.html (1 of 2) [2/16/2002 11:32:31 PM]

Sluze

Mathematicians born in the same country Cross-references to Famous Curves

Pearls of Sluze

Other references in MacTutor

Chronology: 1650 to 1675

Honours awarded to René de Sluze (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1674

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sluze.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sluze.html (2 of 2) [2/16/2002 11:32:31 PM]

Smale

Stephen Smale Born: 15 July 1930 in Flint, Michigan, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stephen Smale was born in Flint in east Michigan, a town famed as the site of General Motors, the automobile company. He attended the University of Michigan, obtaining his BS in 1952 and his MS the following year. Smale continued to work for his doctorate at the University of Michigan, Ann Arbor under R Bott's supervision and he was awarded his Ph D in 1957 for the thesis Regular Curves on Riemannian Manifolds. In his thesis he generalised results proved by Whitney in 1937 for curves in the plane to curves on an n-dimensional manifold. Smale was an instructor at the University of Chicago in 1956-58. In 1958 Smale learnt about Pontryagin's work on structurally stable vector fields and he began to apply topological methods to study the these problems. He the spent the years 1958-60 at the Institute for Advanced Study at Princeton on a National Science Foundation Postdoctoral Fellowship. The last six months of this fellowship he spent at the Instituto de Mathematica Pura e Aplicada in Rio de Janeiro. Here he received a letter from Levinson which led him to study chaotic phenomena; we describe these ideas below. In 1960 Smale was appointed an associate professor of mathematics at the University of California at Berkeley, moving to a professorship at Columbia University the following year. In 1964 he returned to a professorship at the University of California at Berkeley where he has spent the main part of his career. He retired from Berkeley in 1995 and took up a post as professor at the City University of Hong Kong. Smale was awarded a Fields Medal at the International Congress at Moscow in 1966. The work which led to this award was described by René Thom, see [9]. One of Smale's impressive results was his work

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Smale.html (1 of 3) [2/16/2002 11:32:33 PM]

Smale

on the generalised Poincaré conjecture. The Poincaré conjecture, one of the famous problems of 20th-century mathematics, asserts that a simply connected closed 3-dimensional manifold is a 3-dimensional sphere. The higher dimensional Poincaré conjecture claims that any closed n-dimensional manifold which is homotopy equivalent to the n-sphere must be the n-sphere. When n = 3 this is equivalent to the Poincaré conjecture. Smale proved the higher dimensional Poincaré conjecture in 1961 for n at least 5. (Michael Freedman proved the conjecture for n = 4 in 1982 but the original conjecture remains open.) Another area in which Smale has made a very substantial contribution is in Morse theory which he has applied to multiple integral problems. In fact Smale attacked the generalised Poincaré conjecture using Morse theory. Another discovery of Smale's related to strange attractors. An attractor in classical mechanics is a geometrical way of describing the behaviour of a dynamical system. There are three classical attractors, a point which characterises a steady state system, a closed loop which characterises a periodic system, and a torus which combines several cycles. Smale discovered strange attractors which lead to chaotic dynamical systems. Strange attractors have detailed structure on all scales of magnification and were one of the early fractals to be studied. Smale's career changed direction in the late 1960s, see [2]:By the late sixties Smale had moved into applications. He modelled physical processes by dynamical systems, opening new lines of inquiry. The n-body problem and electric circuit theory were among the applications that Smale framed in the language of dynamical systems. For much of the seventies Steve focused on economics, injecting topology and dynamics into the study of general economic equilibria. Having established the nature of equilibria, Smale began to think algorithms for their computation. While traditional approaches to the convergence theory of algorithms were local, Smale introduced a global perspective to the problems. Was the algorithm reasonably reliable, and how many iterations were to be expected? ... Smale's recent work has been on theoetical computer science. With co-workers L Blum and M Shub, he has developed a model of computation which includes both the Turing machine approach and the numerical methods of numerical analysis. Smale has received many honours for his work. In addition to the Fields Medal described above, he was awarded the Veblen Prize for Geometry by the American Mathematical Society in 1966:... for his contributions to various aspects of differential topology. In 1996 Smale received the National Medal of Science ([2]) for:... four decades of pioneering work on basic research questions which have led to major advances in pure and applied mathematics. Smale's contribution is nicely summed up in [2]:Throughout his career Smale has approached mathematical problems with the scholarship to learn from others, the audacity to be unconstrained by conventional wisdom, and the power and vision to employ new methods and construct original frameworks. After the fact, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Smale.html (2 of 3) [2/16/2002 11:32:33 PM]

Smale

a Smale development seems so natural, yet no one else thought of it. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) A Poster of Stephen Smale

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1960 to 1970

Honours awarded to Stephen Smale (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1966

AMS Colloquium Lecturer

1972

Other Web sites

1. Geometry Center (Turning a sphere inside out) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Smale.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Smale.html (3 of 3) [2/16/2002 11:32:33 PM]

Smirnov

Vladimir Ivanovich Smirnov Born: 10 June 1887 in St Petersburg, Russia Died: 11 Feb 1974 in Leningrad, USSR

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vladimir Smirnov attended the 2nd Gymnasium, the oldest secondary school in St Petersburg, and there he won the gold medal for mathematics. From school he entered the Physics and Mathematics Faculty of St Petersburg University. Smirnov had become friends with a number of outstanding mathematicians while at the 2nd Gymnasium. He was particularly friendly with Friedmann and Tamarkin, who were in the class below him at school. Valentina Doinikova, who was a friend of Friedmann, describes how the three went around together while undergraduates at St Petersburg University:Friedmann, Tamarkin and Smirnov often came together, and they were called 'the boys from the second Gymnasium'. They were always smart and neatly dressed, and always called each other - in public - by their first name and patronymic. In 1910 Smirnov graduated from St Petersburg and remained at the University to study for the higher degrees which would allow him to become a university teacher. At the University a circle was formed in 1911 to study mathematical analysis and mechanics. Smirnov was a very active member of this circle, for example lecturing on the theory of algebraic equations, particularly the work of Goursat and Appell. In session 1911-12 he gave nine lectures on Goursat's books. Smirnov worked jointly with his friends from the 2nd Gymnasium. He published a joint paper with Friedmann in 1913 which was published in the Journal of the Russian Physico-Chemical Society (Physics Section). He wrote the first volume of his major five volume work A Course in Higher Mathematics jointly with Tamarkin. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Smirnov.html (1 of 3) [2/16/2002 11:32:35 PM]

Smirnov

From 1912 Smirnov taught at the St Petersburg Institute of Railway Engineering. He taught at Simferopol University in the southern Ukraine from 1919 to 1922, then he returned to St Petersburg (by now Leningrad). Smirnov was awarded his doctorate in 1936 and he became head of the Institute of Mathematics and Mechanics. He became the head of the Mathematics School at the University of Leningrad and was elected to the USSR Academy of Sciences. In 1953 Smirnov organised the Leningrad Mathematical Seminar. To some extent this Seminar also filled the gap left when the Leningrad Mathematical Society disbanded due to political pressure in the late 1920s. Smirnov had been an active member of the Leningrad Mathematical Society through the 1920s and he was a strong believer in relaunching the Society. In 1959, mainly due to the efforts of Smirnov, it became possible to restart the Leningrad Mathematical Society and Smirnov was elected the honorary president of the Society. Smirnov's mathematical activity was in both pure and applied mathematics. He wrote the five volume work A Course in Higher Mathematics referred to above, which was widely used in Russia. He worked on conjugate functions in multidimensional euclidean space and the theory of functions of a complex variable. With Sobolev he devised a method for obtaining solutions on the propagation of waves in elastic media with plane boundaries. Other applied mathematical work resulted in him developing methods for studying oscillations of elastic spheres. In [1] the authors write:... V I Smirnov was not only an outstanding mathematician and a famous historian of science, but also a person of exceptional nobility, benevolence and culture. All these qualities left a lasting impression even on those who seldom had occasion to meet this remarkable man in person, still more on his pupils and associates. Their love and respect for their teacher's memory were reflected in a three-day scientific conference which was held in Leningrad in June 1987 and was dedicated to the centenary of the scientist's birth. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (18 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Smirnov.html (2 of 3) [2/16/2002 11:32:35 PM]

Smirnov

JOC/EFR December 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Smirnov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Smirnov.html (3 of 3) [2/16/2002 11:32:35 PM]

Smith

Henry John Stephen Smith Born: 2 Nov 1826 in Dublin, Ireland Died: 9 Feb 1883 in Oxford, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henry Smith attended Rugby public school from the age of 15 as a boarder. He was outstanding over a range of subjects and his ambition was a scholarship to Balliol College, Oxford. This was made harder since his health was poor (a brother and sister had both died) and he was taken to Italy instead of completing his final year at Rugby. He undertook private reading while in Italy and was still able to win the scholarship. At 19 he became a student at Balliol, but while spending the vacation in Italy his health problems became acute. He could not return to Oxford but this had the advantage that he was able to study with some of the top mathematicians at the Sorbonne and the Collège de France. After his health recovered he returned to Oxford and in 1849 was awarded a double first in mathematics and classics. Smith became a fellow, then a tutor at Balliol. In 1860 he was appointed Savilian professor of geometry despite a strong field of applicants including George Boole. While on the continent he had been learnt French, German and Italian and read widely. He had been most influenced by the work of Gauss. Smith said: If we except the great name of Newton (and the exception is one that the great Gauss himself would have been delighted to make) it is probable that no mathematician of any age or country has ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute vigorousness in demonstration... Influenced by Gauss, Smith's most important contributions are in number theory where he worked on elementary divisors. He proved that any integer can be expressed as the sum of 5 squares and as the sum http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Smith.html (1 of 2) [2/16/2002 11:32:37 PM]

Smith

of 7 squares. Eisenstein had proved the result for 3 squares and Jacobi for 2, 4 and 6 squares. Smith also extended Gauss's theorem on real quadratic forms to complex quadratic forms. From 1859 to 1865 he prepared a report in five parts on the Theory of Numbers. In it Smith analyses the work of other mathematicians but adds much of his own. This work has been described as the the most complete and elegant monument ever erected to the theory of numbers. Smith also wrote on geometrical topics. His first two papers were on geometry and, in 1868, he wrote Certain cubic and biquadratic problems which won him the Steiner prize of the Royal Academy of Berlin. Smith is remembered for the Smith normal form for matrices. It appears to be less well known that, around 1875, he gave examples of discontinuous sets which are similar to the Sierpinski gasket, see [3]. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles)

Some Quotations (3)

Mathematicians born in the same country Honours awarded to Henry Smith (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1861

London Maths Society President

1874 - 1876

Savilian Professor of Geometry

1861

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Smith.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Smith.html (2 of 2) [2/16/2002 11:32:37 PM]

Sneddon

Ian Naismith Sneddon Born: 8 Dec 1919 in Glasgow, Scotland Died: 4 Nov 2000 in Glasgow, Scotland

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ian Sneddon attended Hyndland Secondary School in Glasgow and from there he entered the University of Glasgow. His interests at university were divided between mathematics and physics and he graduated in 1940 with First Class Honours in these two topics. After graduating from Glasgow, Sneddon was awarded a Bryce Fellowship which allowed him continue his studies at Trinity College, Cambridge. However, as was the custom at that time, he took the undergraduate courses at Cambridge leading to the Mathematical Tripos. Of course World War II meant that much of Europe was on a war footing even before Sneddon entered Cambridge and it was inevitable that he would soon end up undertaking war work. Indeed after taking Part II of the Mathematical Tripos in 1942 he was assigned to duties with the Ministry of Supply as a scientific officer and he was sent to the Cavendish Laboratory. His remarkable talents in applied mathematics and physics were quickly directed towards military problems requiring these skills and he began to work on a project examining how to penetrate armour. Before this it is doubtful whether he had developed a particular interest in problems concerning elasticity, but the work he began at the Cavendish Laboratory so fascinated him that he continued to have a deep interest in problems of elasticity throughout his career. In 1943 Sneddon married Mary Macgregor. War work often meant that personnel were moved from place to place and soon Sneddon was transferred to work at Fort Halstead in Kent. Here he found himself a colleague with the leading physicist N F Mott and soon they were forming plans for a joint text. When the war ended Sneddon was appointed to a research post in the H H Wills Physical Laboratory at Bristol University where he continued to work with Mott on nuclear physics and also on their book on wave mechanics. Wave Mechanics and its Applications was published in 1948 with Mott and Sneddon as joint authors. The book discussed

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sneddon.html (1 of 4) [2/16/2002 11:32:39 PM]

Sneddon

applications of quantum mechanics rather than studying the theoretical foundations of the topic. The book looks at applications to the electronic structure of atoms including perturbation and variation methods and a study of electron spin. Also considered are interatomic forces and valence, the theory of solids, collision problems, radiation theory and relativistic quantum theory. W H Furry, reviewing the text, wrote:... the physical viewpoints used are interesting and stimulating. Before the book was published Sneddon had returned to Glasgow as a lecturer in physics, or rather natural philosophy as the subject was called in the ancient Scottish Universities at that time. He took up the appointment in 1946 and his outstanding research achievements led to the award of a D.Sc. and also the award of the Kelvin Medal. Munn writes in [2]:Students of that era knew that his youthful appearance and relaxed, friendly manner concealed a formidable intellect. They will also doubtless recall being amazed at the first sight of his unique, immaculate handwriting. Although in the physics department, Sneddon continued to be interest in applied mathematics. His next major text was Fourier Transforms which appeared in 1951. Remarkably the book was reprinted from the 1951 original in 1995, showing what a classic text Sneddon wrote. The original book was dedicated:To the University of Glasgow on the occasion of its fifth centenary, 1451-1951. The book discusses applications of Fourier, Mellin, Laplace and Hankel transforms to the solution of problems in physics and engineering. It is a major text containing around 550 pages and is mainly concerned with applications which involve the solution of ordinary differential equations and boundary value and initial value problems for partial differential equations. The types of physical problems considered include: vibrations of strings, membranes, heavy chains, elastic beams and plates, potential flow, surface waves, slow viscous flow, and heat conduction. More unusual applications are to topics such as the theory of cosmic ray showers. The book concludes with chapters which bring together many results from Sneddon's own papers on boundary value problems in elasticity. G E H Reuther gave this evaluation of the work:The book is distinguished from existing textbooks on operational methods both by its more "applied" flavour and by its much wider scope. It does not confine itself merely to the Laplace transform, and many of the applications are of a more advanced nature than is usual - the later chapters are based almost entirely on work published within the last ten years. The exposition is lucid ... and answers to problems are often evaluated numerically and illustrated by diagrams. The problems are well chosen to illustrate various points of technique in using transform methods ... Before the book appeared in print Sneddon had left Glasgow to take up the chair of mathematics at the University College of North Staffordshire (which later became Keele University). He enjoyed the challenge of building up a department in a new institution and now in a department of mathematics Sneddon made the small shift in attitude required to be an applied mathematician rather than a theoretical physicist. Each of Sneddon's previous two moves had preceded the publication of a major work by him and Sneddon's final move back to the University of Glasgow in 1956 followed a similar pattern. Glasgow established the Simson Chair of Mathematics to which Sneddon was appointed. Although he had been happy at the University College of North Staffordshire still Glasgow held a special place in his affections and in many ways it was a happy homecoming. This time two books were published around http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sneddon.html (2 of 4) [2/16/2002 11:32:39 PM]

Sneddon

the time of his move. The first was Special functions of mathematical physics and chemistry published in the Oliver and Boyd series in 1956. It was aimed at students of applied mathematics, physics, chemistry and engineering who needed to work with the 'special' functions of Legendre, Bessel, Hermite and Laguerre. I [EFR] purchased this book when I was a student in the 1960s. It was, as all the Oliver and Boyd series books, sold at a price a student could afford and it provided a straightforward account of the topic in a short but very clear style. Sneddon's next text Elements of partial differential equations appeared the following year in 1957. It was written with exactly the same basic philosophy as all Sneddon's previous books. He describes his aims in the Preface:The aim of this book is to present the elements of the theory of partial differential equations in a form suitable for the use of students and research workers whose main interest in the subject lies in finding solutions of particular equations rather than in the general theory. The applications of the methods are again the strength of the book which considers the use of partial differential equations in thermodynamics, stochastic processes, and birth and death processes for bacteria. The book deals with, among other topics, Laplace's equation, mixed boundary value problems, the wave equation and the heat equation. In 1960 Sneddon published a joint text with J G Defares, An introduction to the mathematics of medicine and biology. The applications considered in this text are at the forefront of research interests today and show how forward thinking Sneddon was in areas to which to apply his powerful mathematical methods. Another major text which he published was Mixed boundary value problems in potential theory in 1966. This was again a work at the cutting edge of research containing a very complete description of the classical problem of potential theory, namely to determine the electrostatic potential due to a thin circular disk raised to a prescribed potential. Essentially every interesting result which had been published on the topic in the preceding twenty years was discussed with hardly any work of significance being omitted. In 1969 Sneddon published Crack problems in the classical theory of elasticity with M Lowengrub. This book, which examined the problem of the formation and propagation of cracks in elastic bodies, was another masterpiece. Written on a topic on which Sneddon published many papers, it was a comprehensive account of the mathematical analysis of the theoretical distribution of stresses induced in perfectly elastic bodies by the presence of cracks. Another major contribution by Sneddon was his work editing Russian translations of major texts. He began this work around 1960 and was involved with the translation into English of the five volume work by V I Smirnov A course of higher mathematics. He was also involved with the English translation of works by Gelfond and Linnik. By some sort of symmetry many of Sneddon's texts were translated into Russian. Sneddon travelled widely, particularly in North America where he held a number of visiting professorships, but he also made visits to Poland, Russia, Italy and Australia. He received many honours for his work, notably election to the Royal Society of Edinburgh in 1958 and to the Royal Society of London in 1983. He received honorary degrees from a number of universities including Hull, Strathclyde, Warsaw and Heriot-Watt. He was also elected to other academies such as the Polish Academy of Science and appointed a Commander of the Order of Polonia Restituta. He was awarded an OBE in 1969 to recognise his contributions on numerous government committees. He retired from the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sneddon.html (3 of 4) [2/16/2002 11:32:39 PM]

Sneddon

Simson Chair of Mathematics in 1985 but continued as an honorary senior fellow. Other than mathematics Sneddon's main love was music and the arts in Scotland. This was not just a private interest to occupy his leisure time, rather he gave generously of his time on the boards including that of the Scottish National Orchestra, the Citizen's Theatre and he served on the advisory board of Scottish Opera. In [3] Munn sums up his personal qualities with these words:At a personal level Ian will be remembered with affection for his warm personality, his irrepressible, boyish sense of humour - no-one had a greater fund of anecdotes - and his unfailing kindness in helping young people with their careers. Pack, in [4], expresses similar sentiments:Sneddon was a great conversationalist, with a story for every occasion. He knew so many people, from all over the world, that he could always come up with something interesting. He had a lively sense of humour and a warm, kind personality. Article by: J J O'Connor and E F Robertson List of References (4 books/articles)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR December 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Sneddon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sneddon.html (4 of 4) [2/16/2002 11:32:39 PM]

Snell

Willebrord van Roijen Snell Born: 1580 in Leiden, Netherlands Died: 30 Oct 1626 in Leiden, Netherlands

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Willebrord Snell studied law at the University of Leiden but was very interested in mathematics and taught mathematics even while he studied law. From about 1600 he travelled to various European countries, mostly discussing astronomy. In 1602 he went to Paris where his studies continued. He received his degree from Leiden in 1607. Snell's father, Rudolph Snell (1546-1613), was professor of mathematics at Leiden and, in 1604, Willebrord visited Switzerland with his father. In 1613 he succeeded his father as professor of mathematics at the University of Leiden. In 1617 Snell published Eratosthenes Batavus, which contains his methods for measuring the Earth. He proposed the method of triangulation and this work is the foundation of geodesy. Snell also improved the classical method of calculating approximate values of by polygons. Using his method 96 sided polygons give correct to 7 places while the classical method yields only 2 places. van Ceulen's 35 places could be found with polygons of 230 sides rather than 262. Although he discovered the law of refraction, a basis of modern geometric optics, in 1621, he did not publish it and only in 1703 did it become known when Huygens published Snell's result in Dioptrica. Snell also discovered the sine law. Snell studied the loxodrome, the path on the sphere that makes constant angle with the meridians. This appears in Tiphys batavus published in 1624, a work in which he studied navigation.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Snell.html (1 of 2) [2/16/2002 11:32:41 PM]

Snell

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) A Poster of Willebrord Snell

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1600 to 1625

Honours awarded to Willebrord Snell (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Snellius

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Snell.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Snell.html (2 of 2) [2/16/2002 11:32:41 PM]

Snyder

Virgil Snyder Born: 9 Nov 1869 in Dixon, Iowa, USA Died: 4 Jan 1950 in Ithaca, New York, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Virgil Snyder attended Iowa State College from 1886 to 1889 being awarded his Sc.B. in 1889. He then attended Cornell University from 1890 to 1892 when, as was the custom for American mathematicians at that time, he went to Germany. Snyder studied for his doctorate at Göttingen. He was awarded a doctorate in December 1894 for a dissertation Über die linearen Comlexe der Lie'schen Kugelgeometrie written under Klein's supervision. Snyder returned to Cornell University where he was to spend the rest of his career, first as an instructor, then from 1903 as assistant professor, being promoted to full professor in 1910. He retired in 1938 after a career which is summed up by Archibald in [1] as follows:During more than forty years at Cornell University, Professor Snyder has devoted himself whole-heartedly, and with high idealism, to improving the teaching of mathematics, to promoting the welfare of his students and guiding them into research, and to carrying on his own original work in the fields of geometry of the line and sphere, configurations of ruled surfaces, and birational transformations. Snyder was an editor of the Bulletin of the American Mathematical Society from 1904 to 1920. He was vice-president of the American Mathematical Society in 1916 and president of the Society from 1927 to 1928. In [1] Archibald describes his interests as:... a lover of travel ... intensely interested in politics ... favorite recreation is mountain climbing and going on long hikes...

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Snyder.html (1 of 2) [2/16/2002 11:32:43 PM]

Snyder

Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Honours awarded to Virgil Snyder (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1927 - 1928

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Snyder.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Snyder.html (2 of 2) [2/16/2002 11:32:43 PM]

Sobolev

Sergei Lvovich Sobolev Born: 6 Oct 1908 in St Petersburg, Russia Died: 3 Jan 1989 in Leningrad (now St Petersburg), Russia

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sergei L'vovich Sobolev's father, Lev Aleksandrovich Sobolev, was an important layer and barrister. His mother, Nataliya Georgievna, played an important role in Sobolev's upbringing, particularly after the death of Sobolev's father when Sobolev was 14 years old. He studied at the Khar'kov Workers' Technical School preparing to enter the high school which he did in 1922 around the time of his father's death. The high school which he entered was called the 190th School of Leningrad at the time although previously it had been called the Lentovskii High School. In [5] (see also [4]) it is explained that this school :... was founded during the First Russian Revolution by the foremost St Petersburg teachers for pupils who had been excluded from the State Schools and Technical Colleges because of their participation in the revolutionary movement. After graduating from high school in 1925, Sobolev entered the Physics and Mathematics Faculty of Leningrad State University where his talents were quickly spotted by Smirnov who had returned to Leningrad three years earlier. Sobolev became interested in differential equations, a topic which would dominate his research throughout his life, and even at this stage in his career he produced new results which he published. By 1929 Sobolev had completed his university education and he began to teach in a number of different educational establishments. For example his first appointment was in 1929 at the Theoretical Department of the Seismological Institute of the Academy of Sciences. However, in addition, he taught at the Leningrad Electrotechnic Institute in 1930-31.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sobolev.html (1 of 4) [2/16/2002 11:32:45 PM]

Sobolev

In 1932 the Steklov Institute of Physics and Mathematics was divided into separate Departments of Mathematics and of Physics. Vinogradov headed the Mathematics Department and invited Sobolev to join the Department. By this time, however, Sobolev had already ([4] and [5]):... published a number of profound papers in which he put forward a new method for the solution of an important class of partial differential equations. Working with Smirnov, Sobolev studied functionally invariant solutions of the wave equation. These methods allowed them to find closed form solutions to the wave equation describing the oscillations of an elastic medium. The methods also led them to a complete theory of Rayleigh surface waves and Sobolev went on to solve problems on diffraction. Sobolev was honoured for this outstanding work by election as Corresponding Member of the Academy of Sciences of the USSR in 1933. On 28 April 1934, at the general meeting of the Division of Mathematical and Natural Sciences of the Academy of Sciences of the USSR, a decision was taken to split the Departments of the Steklov Institute of Physics and Mathematics, which had been created two years earlier, into two independent Institutes, the Steklov Mathematical Institute and the Lebedev Physical Institute. In the same year the Steklov Mathematical Institute was moved from Leningrad to Moscow and Sobolev went with the new Institute to Moscow. By 1935 Sobolev was head of the Department of the Theory of Differential Equations at the Institute. During the 1930s Sobolev introduced notions which were fundamental in the development of several different areas of mathematics [22]:The study of Sobolev function spaces, which he introduced in the 1930s, immediately became a whole area of functional analysis. Sobolev's notion of generalised function (distribution) turned out to be especially important; with further developments by Schwartz and Gelfand, it became one of the central notions of mathematics. While working in Moscow, Sobolev built on the standard variational method for solving elliptic boundary value problems by introducing these Sobolev function spaces. He gave inequalities on the norms on these spaces which were important in the theory of embedding function spaces. He applied his methods to solve difficult problems in mathematical physics. In 1939 Sobolev was elected a full member of the Academy of Sciences of the USSR. He was only 31 years of age at the time of his election which was a remarkable achievement. It made him the youngest full member of the Academy of Sciences and in fact he remained the youngest member for quite a few years. At the beginning of World War II, the Steklov Mathematical Institute was moved from Moscow to Kazan. In the October of that year Sobolev was appointed as Director of the Institute and in the spring of 1943 he supervised the move of the Institute back to Moscow. Sobolev became one of the first recipients of a Stalin prize (later called a State prize) in the first presentation of these prizes in 1941. His period as Director of the Institute ended in February 1944. A new area of his research involved the study of the motion of a fluid in a rotating vessel. He was led to study a number of new problems which ([4] and [5]):... led him to lay the foundations of the theory of operators in a space with an indefinite metric, and to introduce new ideas in the spectral theory of operators. These ideas in the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sobolev.html (2 of 4) [2/16/2002 11:32:45 PM]

Sobolev

main concern generalised solutions of non-classical boundary value problems. In the early 1950s Sobolev's work turned towards computational mathematics and in 1952 he became head of the first department of computational mathematics in the Soviet Union when he organised the first such department at Moscow State University. However in 1956 he joined with a number of colleagues in proposing ways in which the large areas of Russia in the east could be opened up with educational initiatives. The scheme was to set up a number of Institutes for Scientific research to balance the large number of high quality educational establishments in the east of the Soviet Union. After the plan was approved, Sobolev spent some time in Moscow recruiting staff and organising the establishment of an Institute in Novosibirsk. In [4] and [5] his contribution to the Institute in Novosibirsk is stressed:During the difficult formative years of the Institute Sobolev, by his excellent example, infused his young colleagues with the best habits for scientific work. In ten years, under the leadership of Sobolev, the Institute of Mathematics of the Siberian Branch of the Academy of Sciences of the USSR has become one of the greatest centres for the mathematical sciences of international status. In 1958 Sobolev was part of the Soviet delegation to the International Mathematical Union, the delegation being led by Vinogradov, and Sobolev attended the International Congress at Edinburgh that year and gave an invited address on partial differential equations. From 1960 until 1978 Sobolev, in addition to his work at the Institute of Mathematics of the Siberian Branch of the Academy of Sciences, was a professor at Novosibirsk University. During this period he published his famous text Applications of functional analysis in mathematical physics which appeared in 1962, the English translation being published by the American Mathematical Society in the following year. During the 1960s much of Sobolev's research was directed towards numerical methods, in particular to interpolation. Although interpolation for functions of a single variable was well worked out, the problem of interpolation in many dimensions was largely unsolved. Sobolev applied his theories of generalised functions and of embeddings of function spaces to cubature formulas, the multi-dimensional analogues of quadrature formulas for functions of one variable. Again a major text by Sobolev Introduction to the theory of cubature formulas has been extremely influential in this area. Sobolev received many honours for his fundamental contributions to mathematics. He was elected to many scientific societies, including the Academy of Sciences of the USSR, the Académie des Sciences de France, and the Accademia Nazionale dei Lincei. He was awarded many prizes, including three State Prizes and the 1988 M V Lomonosov Gold Medal from the Academy of Sciences of the USSR. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (26 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1960 to 1970

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sobolev.html (3 of 4) [2/16/2002 11:32:45 PM]

Sobolev

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sobolev.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sobolev.html (4 of 4) [2/16/2002 11:32:45 PM]

Sokhotsky

Yulian-Karl Vasilievich Sokhotsky Born: 2 Feb 1842 in Warsaw, Poland Died: 14 Dec 1927 in Leningrad, USSR (now St Petersburg, Russia)

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sokhotsky entered the Warsaw state gymnasium for primary education in 1850. In 1860 he enrolled in the Physics and Mathematics Faculty at the University of St Petersburg. He was awarded his Master's Degree in 1867 for a dissertation Theory of integrals with applications. In this thesis Sokhotsky discussed the Cauchy integral and the theory of analytic functions, which he called "single-valued". He received a doctorate from St Petersburg in 1873. His doctoral dissertation On definite integrals and functions with application in expansion of series was an early investigation of the theory of singular integral equations. It investigated in detail Cauchy-type integrals which played an important role in boundary value problems in the theory of functions of a complex variable. Sokhotsky became a professor at the University of St Petersburg in 1883. Influenced by Chebyshev, he studied special functions, in particular Jacobi polynomials and Lamé functions. His work is important in the development of the theory of functions, in particular having applications in the theory of hypergeometric series and differential equations. Other topics Sokhotsky studied included Zolotarev's theory of divisibility of algebraic numbers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sokhotsky.html (1 of 2) [2/16/2002 11:32:47 PM]

Sokhotsky

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sokhotsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sokhotsky.html (2 of 2) [2/16/2002 11:32:47 PM]

Sokolov

Yurii Dmitrievich Sokolov Born: 26 May 1896 in Labinskaya Stanitsa (now Labinsk), Russia Died: 2 Feb 1971 in Kiev, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yurii Sokolov was born in Labinskaya Stanitsa, a name of the town whose name means Cossack village. He graduated from Kiev Institute of Peoples Education in 1921 and then he taught in the Spplied Mathematics Division of the Academy. From 1934 he taught at the Institute of Mathematics at the Academy of Sciences of Ukraine. He also taught at several other higher educational institutions in Kiev. Sokolov's main work was on the n-body problem, which he worked on for nearly 50 years. He also worked on functional equations and on such practical problems as the filtration of groundwater. Other work by Sokolov was on celestial mechanics and hydromechanics. One of the topics which will always be associated with Sokolov's name is a method for finding approximate solutions to differential and integral equations. The method which he introduced is now sometimes called the averaging method with functional corrections, sometimes called the Sokolov method. His methods were highly practical and useful in many applications to mathematical physics, but they were also studied with the highest degree of rigour. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sokolov.html (1 of 2) [2/16/2002 11:32:49 PM]

Sokolov

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sokolov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sokolov.html (2 of 2) [2/16/2002 11:32:49 PM]

Somerville

Mary Fairfax Greig Somerville Born: 26 Dec 1780 in Jedburgh, Roxburghshire, Scotland Died: 29 Nov 1872 in Naples, Italy

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mary Somerville was the daughter of William George Fairfax and his second wife Margaret Charters. Mary Fairfax was born in the church manse in Jedburgh, the home of her mother's sister Martha Charters and Martha's husband Thomas Somerville. Mary's father was a naval officer, later Vice-Admiral Sir William George Fairfax, who was at sea at the time of her birth. Mary's mother had visited London from where her husband embarked on a long sea voyage. Margaret Fairfax broke her journey north at Jedburgh where Mary was born. The family home was in Burntisland in the county of Fife, Scotland. You can see pictures of the house in Burntisland. Mary was the fifth of seven children but three died very young. Of the four remaining children, Mary was brought up with her brother who was three years older than she was. A sister was born when Mary was seven, and a second brother when she was ten. The two brothers were given a good education but, in keeping with the ideas of the time, little need was seen to educate girls so Mary's parents saw no need to provide an education for their daughter. As a young child what little education she did receive was from her mother who taught her to read but it was not considered necessary to teach her to write. When Mary was ten years old she was sent to Miss Primrose's boarding school for girls in Musselburgh (a few miles east of Edinburgh on the Firth of Forth). Burntisland and Musselburgh are on opposite sides of the Firth of Forth, Burntisland on the north, Musselburgh on the south. The school in Musselburgh neither gave Mary a happy time nor a good education. Anyway she spent only one year there and, on leaving felt (in her own words) (see [1] or [2]):... like a wild animal escaped out of a cage.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Somerville.html (1 of 5) [2/16/2002 11:32:51 PM]

Somerville

After this Mary returned to her home in Burntisland but she began to educate herself by reading every book that she could find in her home. Far from being encouraged in reading, members of her family such as her aunt criticised her for spending time on this unladylike occupation. In order that she might learn the correct skills for a young lady, Mary was sent to a school in Burntisland where she was taught needlework. However, one member of Mary's family did encourage her with her educational ambitions. When visiting her uncle in Jedburgh Mary told him that she had been teaching herself Latin. Far from being cross, her uncle encouraged her and the two would read Latin before breakfast while Mary stayed in the Jedburgh manse. When Mary was about thirteen, the family rented a house in Edinburgh where they spent the winter months, the summers being spent in Burntisland. Mary balanced her life between the social life expected of a young lady at this time and her own private study. She did learn many skills that were seen appropriate for a young lady. In addition to the needlework mentioned above, she learnt to play the piano and was given lessons in painting from the artist Alexander Nasmyth. In fact it was through Nasmyth that Mary first became interested in mathematics. She overheard him explaining to another pupil that Euclid's Elements formed the basis for understanding perspective in painting, but much more, it was the basis for understanding astronomy and other sciences. This comment was enough to start Mary on the road to study Euclid's Elements which she did with the help of her younger brother's tutor. There was another quite different reason why Mary became interested in studying algebra. She read an article on the subject in a woman's magazine belonging to a friend. Her younger brother's tutor was able to provide Mary with algebra texts and help introduce her to the subject. Mary became so engrossed in mathematics that her parents worried that her health would suffer because of the long hours of study that she put in, usually during the night. Her father believed (as was common at the time) that [1]:... the strain of abstract thought would injure the tender female frame. Social life in Edinburgh was strongly encouraged, however, where Mary enjoyed [1]:... parties, visits, balls, theatres, concerts, and innocent flirtations ... Mary married Samuel Greig in 1804 when she was 24 years old. Her husband was a naval officer who was a distant relation on her mother's side of the family (Samuel Greig's father was a nephew of Mary's maternal grandfather). However Samuel was in the Russian navy and Mary's parents did not allow the marriage to take place until Greig received an appointment in London, for they did not want Mary to go to Russia. Mary and Samuel Greig went to London but Mary found that her husband did not understand her desires to learn. She later wrote (see [1] or [6]):He had a very low opinion of the capacity of my sex, and had neither knowledge of, nor interest in, science of any kind. Samuel Greig died 3 years after the marriage. By this time Mary had given birth to two sons and on the death of her husband she returned to Scotland with them. She now had a circle of friends who strongly encouraged her in her studies of mathematics and science. In particular John Playfair, then professor of natural philosophy at Edinburgh, encouraged her and through him she began a correspondence with William Wallace (Playfair's former pupil) who was then professor of mathematics at the Royal Military College at Great Marlow. In this correspondence they discussed the mathematical problems set in the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Somerville.html (2 of 5) [2/16/2002 11:32:51 PM]

Somerville

Mathematical Repository and in 1811 Mary received a silver medal for her solution to one of these problems. At this time Mary also read Newton's Principia and, at Wallace's suggestion, Laplace's Mécanique Céleste and many other mathematical and astronomical texts. In 1812 Mary Greig married William Somerville who was an inspector of hospitals. William was the son of her aunt Martha and her husband Thomas Somerville in whose manse she had been born. Unlike her first husband, William was interested in science and also supportive of his wife's desire to study. At this time William and Mary lived in Edinburgh and, advised by Wallace, Mary read the most advanced French texts of the day. In addition she studied botany and improved her knowledge of Greek. With her husband she studied geology and they moved in a close circle of friends that included Playfair, Leslie, Sir William Scott, and the physicist David Brewster. In 1814 Mary's oldest daughter from her first marriage died at the age of nine and, in the same year, the only son of her second marriage died as a baby. When William Somerville was appointed as Inspector to the Army Medical Board in 1816, the family moved from Edinburgh to London. Mary's husband was elected to the Royal Society and Mary and William moved in the leading scientific circles of the day. Their friends included George Airy, John Herschel, William Herschel, George Peacock, and Charles Babbage. Mary wrote [2]:We frequently went to see Mr Babbage while he was making his calculating machines. In addition they met with leading European scientists and mathematicians who visited London. In 1817 William and Mary visited Paris and were introduced to the leading scientists there by Biot and Arago (who they had met in London). Mary met Laplace, Poisson, Poinsot, Emile Mathieu, and many others. Returning to London, Mary and William lived in central London which enabled them to continue close contact with their many scientific friends. In 1824 William was appointed as a physician at the Royal Hospital in Chelsea, and the family moved to Chelsea, then on the edge of London. Mary Somerville published her first paper The magnetic properties of the violet rays of the solar spectrum in the Proceedings of the Royal Society in 1826. The paper [12]:... showed ingenuity in original speculation, and attracted much interest at the time, although the theory it propounded was subsequently negatived ... In 1827 Lord Brougham made a request on behalf of the Society for the Diffusion of Useful Knowledge for Mary Somerville to translate Laplace's Méchanique Céleste. However Mary went far beyond a translation, for she explained in detail the mathematics used by Laplace which was unfamiliar to most mathematicians in England at that time. When completed, the work with title The Mechanism of the Heavens was far too large to be published by the Society for the Diffusion of Useful Knowledge and John Herschel recommended its publication to the publisher John Murray. The book appeared in 1831 and was an immediate success both in terms of the number of copies sold and the praise given to it. Also in 1831 James David Forbes, later to become the Principal of the University of St Andrews, was in London and wrote in his notebook his impressions of Mary:Below middle size, fair, countenance not particularly expressive except eyes which are piercing. Short-sighted. Manners the simplest possible. Her conversation very simple and pleasing. Simplicity not showing itself in abstaining from scientific subjects with which she is so well acquainted, but in being ready to talk on them all with the naiveté of a child and

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Somerville.html (3 of 5) [2/16/2002 11:32:51 PM]

Somerville

the utmost apparent unconsciousness of the rarity of such knowledge as she possesses, so that it requires a moment's reflection to be aware that one is hearing something very extraordinary from the mouth of a woman. Mary Somerville spent about a year abroad in 1832-33. Most of the time was spent in Paris where she renewed old friendships with the mathematicians there, and where she worked on her next book The connection of the physical sciences which was published in 1834. Her discussion of a hypothetical planet perturbing Uranus in the sixth edition (1842) of this work led Adams to his investigation and subsequent discovery of Neptune. Another friend of the family was Lady Byron, by this time estranged from her husband Lord Byron, and her daughter Ada Lovelace. Back in London, Mary helped Ada in her study of mathematics and provided strong encouragement to her. Honours now come quickly to Mary Somerville. She was elected to the Royal Astronomical Society in 1835 (at the same time as Caroline Herschel). She was elected to honorary membership of the Société de Physique et d'Histoire Naturelle de Genève in 1834 and, in the same year, to the Royal Irish Academy. Sir Robert Peel, British prime minister from 1834-35 and again from 1841-46, awarded her a civil pension of 200 per annum, during his first period of office. This was increased to 300 in 1837 by William Lamb, 2nd Viscount Melbourne (British prime minister from 1835-41). A letter which Mary wrote to Arago contained information important enough for him to have an extract from the letter published as a paper in Comptes Rendus in 1836. In 1838 William Somerville's health deteriorated and the family went to Italy. (William survived for 22 further years there.) Most of the rest of Mary's life was spent in Italy where she wrote many works which influenced Maxwell. Most important of her later publications was Physical geography which was published in 1848. It was her most successful text and used until the beginning of the 20th century in schools and universities. Many further honours were given to Mary as a result of this publication. She was elected to the American Geographical and Statistical Society in 1857 and the Italian Geographical Society in 1870. Also in 1870 she received the Victoria Gold Medal of the Royal Geographical Society. Mary Somerville was a strong supporter of women's education and women's suffrage. When John Stuart Mill, the British philosopher and economist, organised a massive petition to parliament to give women the right to vote, he had Mary put her signature first on the petition. Somerville College in Oxford was named after her in 1879 because of her strong support for women's education. Many tributes to Mary Somerville sum up her contribution. From [12]:Her grasp of scientific truth in all branches of knowledge, combined with an exceptional power of exposition, made her the most remarkable woman of her generation. Sir David Brewster, inventor of the kaleidoscope, wrote in 1829 (nine years before becoming Principal of the University of St Andrews) that Mary Somerville was:... certainly the most extraordinary woman in Europe - a mathematician of the very first rank with all the gentleness of a woman ... She is also a great natural philosopher and mineralogist. Let us end this biography with Mary Somerville's own words, written late in her life [2]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Somerville.html (4 of 5) [2/16/2002 11:32:51 PM]

Somerville

Sometimes I find [mathematical problems] difficult, but my old obstinacy remains, for if I do not succeed today, I attack them again on the morrow. Article by: J J O'Connor and E F Robertson List of References (12 books/articles)

A Quotation

A Poster of Mary Somerville

Mathematicians born in the same country

Other references in MacTutor

1. 1. 2. Pictures of the house in Burntisland. 3. 4. Somerville's American connections. 5. 6. Somerville's booklist. 7.

Honours awarded to Mary Somerville (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Somerville

Other Web sites

1. Agnes Scott College 2. Virginia Tech 3. Malaspina (an e-text of The Mechanism of the Heavens) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Somerville.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Somerville.html (5 of 5) [2/16/2002 11:32:51 PM]

Sommerfeld

Arnold Johannes Wilhelm Sommerfeld Born: 5 Dec 1868 in Königsberg, Prussia (now Kaliningrad, Russia) Died: 26 April 1951 in Munich, Germany

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Arnold Sommerfeld attended the Gymnasium in Königsberg. Two slightly older pupils at the same school were Minkowski and Wien. Sommerfeld studied at the University of Königsberg where he was taught by Hilbert, Hurwitz and Lindemann. At this time the University of Königsberg was famous for its school of Theoretical Physics which had been founded by Franz Neumann. However Sommerfeld's interests were in mathematics rather than physics. In 1891 Sommerfeld was awarded his doctorate from Königsberg. In 1893 Sommerfeld left Königsberg for Göttingen where he became Klein's assistant. Hurwitz had left Königsberg a year before Sommerfeld and Lindemann left in the same year as Sommerfeld. Two years later Hilbert was to follow Sommerfeld to Göttingen. At Göttingen, the direction of Sommerfeld's research was immediately influenced by Klein who at this time was heavily involved in applying the theory of functions of a complex variable, and other pure mathematics, to a range of physical topics from astronomy to dynamics. Sommerfeld's first work under Klein's supervision was an impressive piece of work on the mathematical theory of diffraction. His work on this topic contains important theory of partial differential equations. Other important work which he undertook while at Göttingen included the study of the propogation of electromagnetic waves in wires and the study of the field produced by a moving electron. From 1897 Sommerfeld taught at Clausthal where he became professor of mathematics at the mining academy. Then, three years later, he became professor of mechanics at the Technische Hochschule of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sommerfeld.html (1 of 3) [2/16/2002 11:32:53 PM]

Sommerfeld

Aachen. In 1897 he began a 13 year study of gyroscopes woking on a 4 volume work jointly with Klein. In 1906 he became professor of theoretical physics at Munich and worked on atomic spectra. He studied the hypothesis that X-rays were waves and proved this by using crystals as three dimensional diffraction gratings. From 1911 his main area of interest became quantum theory. Sommerfeld's work led him to replace the circular orbits of the Niels Bohr atom with elliptical orbits; he also introduced the magnetic quantum number in 1916 and, four years later, the inner quantum number. It was theoretical work attempting to explain the inner quantum number that led to the discovery of electron spin. In the later part of his career, Sommerfeld used statistical mechanics to explain the electronic properties of metals. This replaced an earlier theory due to Lorentz in 1905 based on classical physics. Sommerfeld's approach was to regard electrons in a metal as a degenerate electron gas. He was able to explain features which were unexplained by the earlier classical theory. Sommerfeld had built up a very famous school of theoretical physics at Munich but its thirty years of fame ended with the Nazi rise to power. In 1940 the school closed but by this time Sommerfeld was 71 years old. He survived World War II and eventually in a street accident in Munich. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Honours awarded to Arnold Sommerfeld (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1926

Lunar features

Crater Sommerfeld

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sommerfeld.html (2 of 3) [2/16/2002 11:32:53 PM]

Sommerfeld

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sommerfeld.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sommerfeld.html (3 of 3) [2/16/2002 11:32:53 PM]

Sommerville

Duncan MacLaren Young Sommerville Born: 24 Nov 1879 in Beawar, Rajasthan, India Died: 31 Jan 1934 in Wellington, New Zealand Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Duncan Sommerville was the son of the Rev Dr James Sommerville. He was educated at Perth Academy (less than 50 km from St Andrews) then at the University of St Andrews in Scotland. He was awarded a scholarship in 1899 to allow him to continue his studies at St Andrews. He taught there from 1902 to 1914 being appointed Lecturer in Mathematics in 1905. Turnbull, writing in [3], describes Sommerville in these terms:His scholarly and unobtrusive demeanour as a young lecturer won the admiration of his colleagues and pupils in St Andrews where his teaching left a permanent mark. While he was essentially a geometer he had considerable interests in other sciences, and it is noteworthy that the classes which he chose to attend in his fourth year of study had been Anatomy and Chemistry. Crystallography in particular appealed to him, and doubtless these possible outlets influenced his geometrical concepts and led Sommerville to ponder over space filling figures, and gave an early impetus to thoughts in a field he made particularly his own. He had an original mind, and beneath his outward shyness considerable talents lay concealed: his intellectual grasp of geometry was balanced by a deftness in making models, and on the aesthetic side by an undoubted talent with the brush. In 1915 Sommerville left Scotland for New Zealand to take up a new appointment as Professor of Pure and Applied Mathematics at Victoria College Wellington. In 1919, when the professor of mathematics at Otago University suffered a nervous breakdown, a young student there A C Aitken was left without support and Sommerville began to tutor Aitken with a weekly correspondence. Outside mathematics one of Sommerville's interests was astronomy and he was a founder of the New Zealand Astronomical Society as well as being its first secretary. Sommerville worked on non-euclidean geometry and the history of mathematics. He proved in 1905 that there are eleven Archimedian tilings. His research was described by G Timmus as:... the classification of all types on non-euclidean geometry (including those usually excluded as bizarre), the extension, involving the measurement of generalised angles in higher space, of Euler's Theorem on polyhedra, space filling figures, the classification of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sommerville.html (1 of 2) [2/16/2002 11:32:54 PM]

Sommerville

polytopes (i.e. the generalisation, in higher space, of polyhedra), it is typical that this includes polytopes in non-euclidean space ... In a review of [2] Daniel Coray states:By removing a finiteness condition which is habitually made on the angles formed by the various elements of a pencil (of lines, planes, etc.), Sommerville obtained more general geometries than the usual ones (Euclid, Lobachevsky, Riemann). He classified them into 9 types of plane geometries, 27 in dimension 3, and more generally 3n in dimension n. A number of these geometries have found applications, for instance in physics. In 1911 he published Bibliography of non-Euclidean Geometry, including the Theory of Parallels, the Foundations of Geometry and Space of n Dimensions. There are 1832 references to n-dimensional geometry. Books which Sommerville published were Elements of Non-Euclidean Geometry (1914), Analytic Conics (1924), Introduction to Geometry of n dimensions (1929) and Three Dimensional Geometry (1934). He also wrote 30 papers on combinatorial geometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sommerville.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sommerville.html (2 of 2) [2/16/2002 11:32:54 PM]

Somov

Osip Ivanovich Somov Born: 1 June 1815 in Otrada, Moscow gubernia (now oblast), Russia Died: 26 April 1876 in St Petersburg, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Osip Somov attended secondary school in Moscow and then entered Moscow University to study mathematics and physics. He graduated in 1835 and, from 1839 taught at Moscow Commercial College. From 1841 Somov attended the University of St Petersburg where he was awarded a doctorate and was appointed professor of applied mathematics. He taught at the University of St Petersburg for 25 years. Somov originated a geometrical approach to theoretical mechanics in Russia. He studied the rotation of a solid body about a point, studying examples arising from the work of Euler, Poinsot, Lagrange and Poisson. Other topics Somov studied included elliptic functions and their application to mechanics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Somov.html (1 of 2) [2/16/2002 11:32:56 PM]

Somov

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Somov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Somov.html (2 of 2) [2/16/2002 11:32:56 PM]

Sonin

Nikolay Yakovlevich Sonin Born: 22 Feb 1849 in Tula, Russia Died: 27 Feb 1915 in Petrograd (now St Petersburg), Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nikolay Sonin attended Moscow University studying mathematics and physics there from 1865 to 1869. He obtained a Master's Degree with a thesis submitted in 1871, then he taught at the University of Warsaw where he obtained a doctorate in 1874. He was appointed to a chair in the University of Warsaw in 1876. In 1894 Sonin moved to St Petersburg where he taught at the University for Women. Sonin worked on special functions, in particular cylindrical functions. He also worked on the EulerMaclaurin summation formula. Other topics Sonin studied included Bernoulli polynomials (Jacob Bernoulli) and approximate computation of definite integrals, continuing Chebyshev's work on numerical integration. Together with A A Markov, Sonin prepared a two volume edition of Chebyshev's works in French and Russian. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sonin.html (1 of 2) [2/16/2002 11:32:58 PM]

Sonin

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sonin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sonin.html (2 of 2) [2/16/2002 11:32:58 PM]

Spanier

Edwin Henry Spanier Born: 8 Aug 1921 in Washington, D.C., USA Died: 11 Oct 1996 in Scottsdale, Arizona, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edwin Spanier attended the University of Minnesota, graduating in 1941. The Second World War meant that Spanier spent three years doing army service before studying for his doctorate. This army service was in the U.S. Signal Corps. After his three years of army service, Spanier studied for his doctorate at the University of Michigan. Spanier's doctoral supervisor was Norman Steenrod and under his supervision Spanier wrote a thesis on algebraic topology for which he was awarded his doctorate in 1947. The following year Spanier spent as a research fellow at the Institute for Advanced Study at Princeton. Then in 1948 he was appointed to the faculty of the University of Chicago. During the time he held a post at Chicago, Spanier spent the year 1952-53 in Paris supported by a Guggenheim Fellowship, and the year 1958-59 as a member of the Institute for Advanced Study. Then in 1959 Spanier was appointed as professor of mathematics at Berkeley. At Berkeley, Spanier built up a strong group working in geometry and topology by several appointments of topologists to the faculty of Berkeley and also by attracting many top topologists to spend periods as visitors at Berkeley. From the time of his doctoral work until around the time of the publication of his classic teext algebraic topology in 1966, Spanier work almost exclusively on algebraic topology. His first important paper on Borsuk's cohomotopy groups was published in the Annals of Mathematics in 1949. This work gave an algebraic classification of maps from polyhedra to spheres. Chern was appointed professor of geometry at Chicago in 1949, the year following Spanier's appointment, and the two began the study of homology groups of fibre spaces with their joint paper The homology structure of sphere bundles in 1950. Spanier http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Spanier.html (1 of 3) [2/16/2002 11:33:00 PM]

Spanier

began joint work with Henry Whitehead and in a series of papers they introduced the method of duality in homotopy theory. The range of his work is described in [1]:In all, Spanier published moer than forty papers in algebraic topology, contributing to most to most of the major research areas in the field, including cohomology operations, obstruction theory, homotopy theory, imbeddability of polyhedra in Euclidean spaces, and topology of function spaces. Many of his results are now standard tools in all fields that untilize global geometrical reasoning. These include not only various subjects in pure mathematics, but also diverse areas in applied mathematics, including computer science, mathematical physics, economic models, and game theory. Interestingly, one of Spanier's theories, now called Alexander-Spanier homology, is currently being applied to analyse differential equations - a return to Poincaré's original use of algebraic topology. We have suggested that his work on algebraic topology went on until around the time that his famous book was published in 1966. However, he began to collaborate on work on formal languages from about 1961 onwards. This work is of major importance in theoretical computer science, and also has applications in other areas such as the theory of dynamical systems. Spanier had an outstanding reputation as a lecturer and writer, his lectures and publications being [1]:... characterised by unusual lucidity and precision and an even rarer naturalness and simplicity. No matter how complex the subject, at the end the reader feels the theorems are the right ones, the hypotheses natural, and the methods as simple as possible. Spanier returned to algebraic topology for the publications in the last years of his life. In fact his final publication returned to the topic of his first, namely the the axioms which Eilenberg and Steenrod proposed for homology theory. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Spanier.html (2 of 3) [2/16/2002 11:33:00 PM]

Spanier

JOC/EFR October 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Spanier.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Spanier.html (3 of 3) [2/16/2002 11:33:00 PM]

Spence

William Spence Born: 1777 in Scotland Died: 22 May 1815 in Glasgow, Scotland Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Spence was educated in Glasgow as a manufacturer but nevertheless devoted himself to mathematics, working in isolation in Greenock. A work of 1809 on Logarithmic Transcentents showed him as one of the first British mathematicians to be familiar with the work of Lagrange and Arbogast. He published on algebraic and differential equations (1814) and other manuscripts were edited and published in 1820 by John Herschel. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Spence.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Spence.html [2/16/2002 11:33:01 PM]

Sporus

Sporus of Nicaea Born: about 240 in (possibly) Nicaea (now Iznik), Bithynia (now Turkey) Died: about 300 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sporus was taught by Philon of Gadara. In his turn, Sporus taught Pappus of Alexandria, or perhaps was an older pupil at the same time as Pappus, and we know of him through Pappus's writings and the writings of Eutocius. Sporus worked mainly on the classical problems of squaring the circle and duplicating the cube. His solution of the problem of duplicating the cube is similar to that of Diocles and in fact Pappus also followed a similar construction. However, they avoid using the cissoid but instead rotate a ruler about a point until certain intercepts are equal. Sporus also used approximations which are early examples of integration. Not only did Sporus work on squaring the circle and duplicating the cube but he also constructively criticised others work in these areas. One of his contributions, which is described by Pappus, was to criticise the method of squaring the circle using the quadratrix of Hippias. He uses an argument based on the fact that to be able to draw the quadratrix using Hippias's construction, one needs to know the ratio of a radius of a circle to its circumference and being able to construct this ratio is equivalent to being able to square the circle. There seems little doubt that Sporus's criticism is valid. Sporus also criticised Archimedes for not producing a more accurate approximation of however supports Archimedes, writing (in Heath's translation, see for example [2]):-

. Eutocius

[Archimedes] object in this book was to find an appropriate [approximation of ] suitable for use in daily life. Hence we cannot regard as appropriate the censure of Sporus of Nicaea, who seems to charge Archimedes with having failed to determine with accuracy the length of the straight line which is equal to the circumference of the circle., to judge by his passage in his Keria where Sporus observes that his own teacher, meaning Philon of Gadara, reduced the matter to more accurate numerical expression than Archimedes did... Sporus also wrote on the size of the Sun and on comets. Sporus's writings and teaching clearly had a large impact on Pappus who describes him as having a high reputation. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sporus.html (1 of 2) [2/16/2002 11:33:03 PM]

Sporus

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sporus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sporus.html (2 of 2) [2/16/2002 11:33:03 PM]

Spottiswoode

William Spottiswoode Born: 11 Jan 1825 in London, England Died: 27 June 1883 in London, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Spottiswoode's father was a member of the printing firm of Eyre and Spottiswoode, the Queen's printers, and he was related to John Spottiswood who was archbishop of St Andrews. William attended school in Laleham, then went to Eton College, one of the most prestigious schools in England situated on the Thames near London. From Eton he went to another top school in Harrow School, another prestigious school in Greater London. From Harrow, Spottiswoode was awarded a Lyon Scholarship to attend Balliol College, Oxford which he entered in 1842. Three years later he graduated with a First Class degree in mathematics. In 1846 and 1847 he was awarded mathematics scholarships at Balliol College where he became a lecturer in mathematics. In 1846 his father died and Spottiswoode became Queen's Printer. In the following year his first mathematical publication appeared Meditationes Analyticae. Herbert Rix, writing in [1], describes Spottiswoode's mathematical contributions:His mathematical work was described as 'the incarnation of symmetry'. Besides supplying new proofs by elegant methods of known theorems, he did abundance of important original work. His series of memoirs on the contact of curves and surfaces, contributed to the 'Philosophical Transactions' of 1862 and subsequent years, mainly gave him his high rank as a mathematician. He was also the author in 1851 of the first elementary treatise on determinants, and to his treatise much of the rapid development of that subject is attributable. In 1853 Spottiswoode was elected a Fellow of the Royal Society of London. Spottiswoode was appointed http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Spottiswoode.html (1 of 3) [2/16/2002 11:33:05 PM]

Spottiswoode

president of the mathematical section of the British Association in 1865. Around 1870 there were major changes to the direction of his research. This was a time when he received high office in a number of societies, being president of the London Mathematical Society from 1870 to 1872 and, in 1871, being elected treasure of the Royal Society of London. Spottiswoode's research changed to physical topics, and from 1871 he studied the polarisation of light and later he studied electrical discharge in rarefied gases. In 1878 Spottiswoode was elected president of the Royal Society of London, and in the same year he was president of the British Association for its Dublin meeting. At the Dublin meeting he gave his presidential address on the growth of mechanised invention applied to mathematics. Spottiswoode published 99 papers and several books. His interests however were not confined to mathematics and physics since he was also a leading expert on European languages and on oriental languages. He died of typhoid while still president of the Royal Society. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles)

A Quotation

Mathematicians born in the same country Honours awarded to William Spottiswoode (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1853

London Maths Society President

1870 - 1872

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Spottiswoode.html (2 of 3) [2/16/2002 11:33:05 PM]

Spottiswoode

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Spottiswoode.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Spottiswoode.html (3 of 3) [2/16/2002 11:33:05 PM]

Sridhara

Sridhara Born: about 870 in possibly Bengal, India Died: about 930 in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sridhara is now believed to have lived in the ninth and tenth centuries. However, there has been much dispute over his date and in different works the dates of the life of Sridhara have been placed from the seventh century to the eleventh century. The best present estimate is that he wrote around 900 AD, a date which is deduced from seeing which other pieces of mathematics he was familiar with and also seeing which later mathematicians were familiar with his work. We do know that Sridhara was a Hindu but little else is known. Two theories exist concerning his birthplace which are far apart. Some historians give Bengal as the place of his birth while other historians believe that Sridhara was born in southern India. Sridhara is known as the author of two mathematical treatises, namely the Trisatika (sometimes called the Patiganitasara ) and the Patiganita. However at least three other works have been attributed to him, namely the Bijaganita , Navasati , and Brhatpati. Information about these books was given the works of Bhaskara II (writing around 1150), Makkibhatta (writing in 1377), and Raghavabhatta (writing in 1493). We give details below of Sridhara's rule for solving quadratic equations as given by Bhaskara II. There is another mathematical treatise Ganitapancavimsi which some historians believe was written by Sridhara. Hayashi in [7], however, argues that Sridhara is unlikely to have been the author of this work in its present form. The Patiganita is written in verse form. The book begins by giving tables of monetary and metrological units. Following this algorithms are given for carrying out the elementary arithmetical operations, squaring, cubing, and square and cube root extraction, carried out with natural numbers. Through the whole book Sridhara gives methods to solve problems in terse rules in verse form which was the typical style of Indian texts at this time. All the algorithms to carry out arithmetical operations are presented in this way and no proofs are given. Indeed there is no suggestion that Sridhara realised that proofs are in any way necessary. Often after stating a rule Sridhara gives one or more numerical examples, but he does not give solutions to these example nor does he even give answers in this work. After giving the rules for computing with natural numbers, Sridhara gives rules for operating with rational fractions. He gives a wide variety of applications including problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns. Some of the examples are decidedly non-trivial and one has to consider this as a really advanced work. Other topics covered by the author include the rule for calculating the number of combinations of n things taken m at a time. There are sections of the book devoted to arithmetic and geometric progressions, including

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sridhara.html (1 of 3) [2/16/2002 11:33:07 PM]

Sridhara

progressions with a fractional numbers of terms, and formulas for the sum of certain finite series are given. The book ends by giving rules, some of which are only approximate, for the areas of a some plane polygons. In fact the text breaks off at this point but it certainly was not the end of the book which is missing in the only copy of the work which has survived. We do know something of the missing part, however, for the Patiganitasara is a summary of the Patiganita including the missing portion. In [7] Shukla examines Sridhara's method for finding rational solutions of Nx2 1 = y2, 1 - Nx2 = y2, Nx2 C = y2, and C - Nx2 = y2 which Sridhara gives in the Patiganita. Shukla states that the rules given there are different from those given by other Hindu mathematicians. Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation. Unfortunately, as we indicated above, the original is lost and we have to rely on a quotation of Sridhara's rule from Bhaskara II:Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root. To see what this means take ax2 + bx = c. Multiply both sides by 4a to get 4a2x2 + 4abx = 4ac then add b2 to both sides to get 4a2x2 + 4abx + b2 = 4ac + b2 and, taking the square root 2ax + b = (4ac + b2). There is no suggestion that Sridhara took two values when he took the square root. Article by: J J O'Connor and E F Robertson List of References (9 books/articles) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Other references in MacTutor

Chronology: 900 to 1100

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sridhara.html (2 of 3) [2/16/2002 11:33:07 PM]

Sridhara

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sridhara.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sridhara.html (3 of 3) [2/16/2002 11:33:07 PM]

Sripati

Sripati Born: 1019 in (probably) Rohinikhanda, Maharashtra, India Died: 1066 in India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Sripati's father was Nagadeva (sometimes written as Namadeva) and Nagadeva's father, Sripati's paternal grandfather, was Kesava. Sripati was a follower of the teaching of Lalla writing on astrology, astronomy and mathematics. His mathematical work was undertaken with applications to astronomy in mind, for example a study of spheres. His work on astronomy was undertaken to provide a basis for his astrology. Sripati was the most prominent Indian mathematicians of the 11th century. Among Sripati's works are: Dhikotidakarana written in 1039, a work of twenty verses on solar and lunar eclipses; Dhruvamanasa written in 1056, a work of 105 verses on calculating planetary longitudes, eclipses and planetary transits; Siddhantasekhara a major work on astronomy in 19 chapters; and Ganitatilaka an incomplete arithmetical treatise in 125 verses based on a work by Sridhara. The titles of Chapters 13, 14, and 15 of the Siddhantasekhara are Arithmetic, Algebra and On the Sphere. Chapter 13 consists of 55 verses on arithmetic, mensuration, and shadow reckoning. It is probable that the lost portion of the arithmetic treatise Ganitatilaka consisted essentially of verses 19-55 of this chapter. The 37 verses of Chapter 14 on algebra state various rules of algebra without proof. These are given in verbal form without algebraic symbols. In verses 3, 4 and 5 of this chapter Sripati gave the rules of signs for addition, subtraction, multiplication, division, square, square root, cube and cube root of positive and negative quantities. His work on equations in this chapter contains the rule for solving a quadratic equation and, more impressively, he gives the identity: (x + y) = [(x + (x2 - y)]/2 + [(x - (x2 - y)]/2) Other mathematics included in Sripati's work includes, in particular, rules for the solution of simultaneous indeterminate equations of the first degree that are similar to those given by Brahmagupta Sripati obtained more fame in astrology than in other areas and it is fair to say that he considered this to be his most important contributions. He wrote the Jyotisaratnamala which was an astrology text in twenty chapters based on the Jyotisaratnakosa of Lalla. Sripati wrote a commentary on this work in Marathi and it is one of the oldest works to have survived that is written in that language. Marathi is the oldest of the regional languages in Indo-Aryan, dating from about 1000. Another work on astrology written by Sripati is the Jatakapaddhati or Sripatipaddhati which is in eight chapters and is [1]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sripati.html (1 of 2) [2/16/2002 11:33:08 PM]

Sripati

... one of the fundamental textbooks for later Indian genethlialogy, contributing an impressive elaboration to the computation of the strengths of the planets and astrological places. It was enormously popular, as the large number of manuscripts, commentaries, and imitations attests. Now genethlialogy was the science of casting nativities and it was the earliest branch of astrology which claimed to be able to predict the course of a person's life based on the positions of the planets and of the signs of the zodiac at the moment the person was born or conceived. There is one other work on astrology the Daivajnavallabha which some historians claim was written by Sripati while other claim that it is the work of Varahamihira. As yet nobody has come up with a definite case to show which of these two is the author, or even whether the author is another astrologer. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sripati.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sripati.html (2 of 2) [2/16/2002 11:33:08 PM]

Stackel

Paul Gustav Stäckel Born: 20 Aug 1862 in Berlin, Germany Died: 12 Dec 1919 in Heidelberg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Stäckel studied at Berlin and obtained his doctorate in 1885 working under Kronecker and Weierstrass. He taught in Berlin, Königsberg, Kiel, Hannover, Karlsruhe and Heidelberg. Stäckel worked on differential equations and applied these to analytic mechanics. He was interested in the existence of solutions of differential equations. Later in life he became interested in set theory and prime numbers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stackel.html (1 of 2) [2/16/2002 11:33:10 PM]

Stackel

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stackel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stackel.html (2 of 2) [2/16/2002 11:33:10 PM]

Stampioen

Jan Jansz de Jonge Stampioen Born: 1610 in Rotterdam, Netherlands Died: 1690 in The Hague, Netherlands Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jan Stampioen taught mathematics in Rotterdam, but in 1638 he moved to The Hague to become tutor to Prince William. Two years later Prince William (II) succeeded his father and was to go on to transform the Netherlands into a parliamentary state. While in The Hague Stampioen opened a printing shop in which he printed his own writings on mathematics. Stampioen appended his own treatment of spherical trigonometry to van Schooten's sine tables. In 1633 he challenged Descartes to a public competition and rejected Descartes' solution as not complete. He posed two further challenges under the alias of John Baptista involving the solution of cubics and gave solutions under his own name. A young surveyor Waessenaer solved the first, but Stampioen rejected his solution. The argument was adjudicated by van Schooten who favoured Waessenaer. Descartes also became involved in the argument. In 1644 Stampioen was employed to tutor Huygens and his brother in mathematics. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stampioen.html (1 of 2) [2/16/2002 11:33:11 PM]

Stampioen

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stampioen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stampioen.html (2 of 2) [2/16/2002 11:33:11 PM]

Steenrod

Norman Earl Steenrod Born: 22 April 1910 in Dayton, Ohio, USA Died: 14 Oct 1971 in Princeton, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Norman Steenrod's father was Earl Lindsay Steenrod and his mother was Sarah Rutledge. Both were teachers but neither had any interest in mathematics. They had three children who survived childhood. Earl Steenrod taught mechanical drawing but had astronomy as a hobby and he interested Norman in this exciting subject while he was still a young boy. Sarah Steenrod had music as a hobby and she gave Norman a lifelong interest in this. Norman attended school in Dayton and he was such an outstanding pupil that he was able to complete the twelve school years in only nine. At this stage he was only fifteen years old and so before entering university he worked for two years as a tool designer. This was not an arbitrary occupation to put off two years, rather it was his elder brother's trade and Norman had been taught the job by his brother. The job had a very positive effect on Steenrod's finances too, for it gave him the extra cash needed to pay his university expenses. In 1927 Steenrod enrolled at the University of Miami at Oxford, Ohio. From there he moved to the University of Michigan at Ann Arbor where he attended courses in physics, philosophy and economics. He took just one mathematics course but it was an important one for the future direction of his research, for he took a topology course given by Raymond Wilder who had been a student of Robert Moore. Steenrod graduated from Ann Arbor in 1932 but did not obtain a fellowship to allow him to undertake research. He spent the year 1932-33 back at his home in Dayton. He worked hard on topology problems which Wilder had given him and he made sufficient progress, despite working on his own, that by the end of the year he had written his first paper. Now things got better for Steenrod for the quality of the paper led quickly to offers of fellowships from Harvard, Princeton and Duke. Again he decided to make http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steenrod.html (1 of 3) [2/16/2002 11:33:14 PM]

Steenrod

some money and before taking up the Harvard fellowship, the one he had decided to accept, so he worked at the Flint Chevrolet plant as a die designer using again the skills he had learnt from his brother. Before he had completed his year at Harvard he was offered fellowships for the following year by both Harvard and Duke. Before he had made his decision about which to accept he received another offer, this time from the University of Michigan. He decided to accept this latter offer to work with Wilder. However he discovered that Wilder would not be at Michigan during that year since he had arranged to spend it at Princeton. Wilder and Lefschetz put a strong case to Princeton for them to offer Steenrod a fellowship but even after achieving this it still took all their powers of persuasion to convince Steenrod to take up the Princeton offer. At Princeton Steenrod worked for his doctorate being supervised by Lefschetz. He took only two years before he was awarded his Ph.D. then continued to work at Princeton as an instructor. He married Carolyn Witter in 1938 and they had two children, a girl born in 1942 and a son in 1947. In 1939 Steenrod accepted an appointment to the University of Chicago where he worked for three years, returning to the University of Michigan in 1942. Steenrod spent five years back at Ann Arbor before accepting an offer from Princeton. He moved to Princeton in 1947 and remained on the Faculty there for the rest of his career. After his first research work on point-set topology Steenrod then worked on algebraic topology. He is best known for introducing the Steenrod algebra which came about through his work in classifying by homotopy the maps of a complex into a sphere. He published a paper of fundamental importance on this topic in 1942 in which Steenrod squares are introduced for the first time. Lectures which he gave on this topic at Princeton were written up by David Epstein and published as Cohomology operations in 1962. One of the other major topics of Steenrod's research was fibre bundles. Again Steenrod published a book on the topic which has become a classic. The book The Topology of Fibre Bundles was published in 1951. Chern, reviewing the book, writes:This is the first systematic account of fibre bundles, covering the development of the subject from its birth in 1935 until the most recent period. Most of the essential aspects are treated ... . The book presupposes little knowledge of algebraic topology from the reader. It is divided into three parts. Part I deals with the general theory. ... Perhaps the most important result is the covering homotopy theorem, which has many consequences. Part II is devoted to the homotopy theory of bundles. A clear and concise treatment of the homotopy groups is given, the first one in book form. As one of its most interesting applications, information is obtained on the topological properties of spheres. ... Part III treats the cohomology theory of bundles. ... In 1957 Steenrod was given the honour of being asked to give the American Mathematical Society Colloquium lectures. Rather surprisingly although notes of the lectures circulated for many years they were not published until 1972, after Steenrod's death. The introduction to this book sets the scene:The lectures gave an excellent introduction to a central problem of algebraic topology and its applications, namely the problem of extending continuous functions. In particular, they gave a clear account of some of Steenrod's more important contributions to the subject and covered, as well, the related work of several other mathematicians. Many advances have been made since 1957, but most of them were inspired by the ideas presented in these

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steenrod.html (2 of 3) [2/16/2002 11:33:14 PM]

Steenrod

lectures. This paper is strongly recommended for anyone wanting an introduction to the subject. Finally we mention the important work which Steenrod did on homology theories which appeared in the famous book Foundations of algebraic topology which he wrote with Samuel Eilenberg and was published in 1952. The authors promised to write a second volume of this work but it never came about. Steenrod received many honours for his major contribution to topology, the most important of which was his election to the National Academy of Sciences. James writes in [2]:Algebraic topology underwent a spectacular development in the years following the second world war. From a position of minor importance, as compared as compared with the traditional areas of analysis and algebra, its concepts came to exert a profound influence, and it is now commonplace that a mathematical problem is "solved" by reducing it to a homology-theoretic one. To a great extent the success of this development can be attributed to Steenrod's influence. We noted above Steenrod's interest in music. His other interests outside mathematics included tennis, golf, chess and bridge. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1940 to 1950

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Steenrod.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steenrod.html (3 of 3) [2/16/2002 11:33:14 PM]

Stefan_Josef

Josef Stefan Born: 24 March 1835 in St Peter (near Klagenfurt), Austria Died: 7 Jan 1893 in Vienna, Austria

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Josef Stefanwas born to Slovenian parents in Austria. He graduated in mathematics and physics at the University of Vienna and was appointed lecturer in mathematical physics in 1858 and became a professor there in 1866. In 1866 he bacame director of the Physical Institute at Vienna. He showed empirically, in 1879, that total radiation from a blackbody is proportional to the fourth power of its absolute temperature. Boltzmann, one of his students, showed in 1884 that this Stefan-Boltzmann law could be demonstrated mathematically. Stefan then applied it to determine the approximate temperature of the surface of the Sun. He also did important work on heat conduction in fluids and the kinetic theory of heat. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country Cross-references to History Topics

The quantum age begins

Honours awarded to Josef Stefan (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stefan_Josef.html (1 of 2) [2/16/2002 11:33:16 PM]

Stefan_Josef

Lunar features

Crater Stefan

Other Web sites

1. J Stefan Institute 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stefan_Josef.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stefan_Josef.html (2 of 2) [2/16/2002 11:33:16 PM]

Stefan_Peter

Peter Stefan Born: 1941 in Bratislava, Slovakia Died: 18 June 1978 in Tryfan, Snowdonia, Wales

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Peter Stefan attended school in Bratislava and then attended the university in Prague obtaining his first degree in 1965. Peter obtained a post on there and lectured for 3 years. In 1968 he was invited to attend a conference on Dynamical Systems at the University of Warwick. These major conferences at Warwick had many international visitors throughout an academic year with periods of higher activity. They have been important in the development of many mathematical topics. Peter had been involved in politics in Czechoslovakia, supporting the political movement which hoped to humanise Communist rule by introducing basic civil freedoms, an independent judiciary, and other democratic institutions. During Peter's visit to Warwick the Soviets invaded Czechoslovakia on the night of August 20-21, 1968, installing a Soviet controlled security service. Peter feared that he would be in danger if he returned, and since the reform programme had stopped, he preferred the freedom in Britain. Stefan remained at the University of Warwick where he studied for a Ph.D. which was awarded in 1973. His thesis was on Accessibility and singular foliations and is important in control theory and in the mathematical theory of entropy. However Stefan did not remain at Warwick while working for his doctorate. He spent 1969/70 at Manchester returning for a year to Warwick before being appointed to a lectureship at the University College of North Wales at Bangor, a post he held until his death at the age of 37. He spent one year in Paris during his tenure of the Bangor post, spending 1976/77 at the Institut des Hautes Etudes Scientifique.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stefan_Peter.html (1 of 2) [2/16/2002 11:33:18 PM]

Stefan_Peter

Stefan had a love of freedom and he translated this into a love of climbing after his return from Paris. He was killed in a climbing accident in Snowdonia. He was climbing on his own at the time. In [1] his attitude towards mathematics is summed up as follows:Peter had a strong sense of what was important in mathematics ... Given a first-rate mathematical idea, he made it part of himself. That often required an exhaustive search for the right perspective in mathematical development, in exposition, technical accuracy, and historical viewpoint. His fine taste and judgement shine throughout his work. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Stefan_Peter.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stefan_Peter.html (2 of 2) [2/16/2002 11:33:18 PM]

Steiner

Jakob Steiner Born: 18 March 1796 in Utzenstorf, Switzerland Died: 1 April 1863 in Bern, Switzerland

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jakob Steiner did not learn to read and write until he was 14 and only went to school at the age of 18, against the wishes of his parents. He then studied at the Universities of Heidelberg and Berlin, supporting himself with a very modest income from tutoring. He was an early contributor to Crelle's Journal, the first journal devoted entirely to mathematics founded in 1826. He was appointed to a chair at the University of Berlin in 1834, a post he held until his death. He was one of the greatest contributors to projective geometry. He discovered the Steiner surface which has a double infinity of conic sections on it. The Steiner theorem states that the two pencils by which a conic is projected from two of its points are projectively related. Another famous result is the Poncelet-Steiner theorem which shows that only one given circle and a straight edge are required for Euclidean constructions. He disliked algebra and analysis and believed that calculation replaces thinking while geometry stimulates thinking. He was described by Thomas Hirst as follows: He is a middle-aged man, of pretty stout proportions, has a long intellectual face, with beard and moustache and a fine prominant forehead, hair dark rather inclining to turn grey. The first thing that strikes you on his face is a dash of care and anxiety, almost pain, as if arising from physical suffering he has rheumatism. He never prepares his lectures beforehand. He thus often stumbles or fails to prove what he wishes at the moment, and at every such failure he is sure to make some characteristic remark.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steiner.html (1 of 2) [2/16/2002 11:33:20 PM]

Steiner

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Jakob Steiner

Mathematicians born in the same country

Some pages from publications

The title page of Systematische Entwickelung (1832)

Cross-references to History Topics

1. A comment from Thomas Hirst's diary 2. The development of group theory 3. An overview of the history of mathematics

Cross-references to Famous Curves

Tricuspoid

Other references in MacTutor

1. Minimal paths 2. Chronology: 1820 to 1830 3. Chronology: 1830 to 1840

Other Web sites

1. Clark Kimberling 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Steiner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steiner.html (2 of 2) [2/16/2002 11:33:20 PM]

Steinhaus

Hugo Dyonizy Steinhaus Born: 14 Jan 1887 in Jaslo, Kingdom of Galicia, Austrian Empire (now Poland) Died: 25 Feb 1972 in Wroclaw, Poland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hugo Steinhaus was born in Galicia into a family of Jewish intellectuals. The town of his birth, Jaslo, was in Galicia, about half way between Kraków and Lvov (although a bit nearer Kraków than Lvov). Galicia was attached to Austria in the 1772 partition of Poland. However, by the time Steinhaus was born in Jaslo, Austria had named the region the Kingdom of Galicia and Lodomeria and given it a large degree of administrative autonomy. Steinhaus's uncle was an important person being a politician in the Austrian parliament. Steinhaus studied for one year in Lvov, spent one term in Munich but then spent five years studying mathematics at the University of Göttingen. There he was influenced by an amazingly strong group of mathematicians including Felix Bernstein, Carathéodory, Courant, Herglotz, Hilbert, Klein, Koebe, Landau (although he only arrived in Göttingen after Steinhaus had been there three years), Runge, Toeplitz, and Zermelo. For his doctorate Steinhaus studied under Hilbert's supervision. He was awarded his doctorate, with distinction, for a dissertation Neue Anwendungen des Dirichlet'schen Prinzips in 1911. The main influence on the direction that Steinhaus's research would take was none of the major mathematical figures at Göttingen but rather the influence came from Lebesgue. Steinhaus studied Lebesgue's two major books Leçons sur l'intégration et la recherché des fonctions primitives (1904) and Leçons sur les séries trigonmétriques (1906) around 1912 after completing his doctorate. After military service in the Polish Legion at the beginning of World War I, Steinhaus lived in Kraków. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steinhaus.html (1 of 5) [2/16/2002 11:33:22 PM]

Steinhaus

He relates in [2] how, despite the war in 1916, it was safe to walk in Kraków:During one such walk I overheard the words "Lebesgue measure". I approached the park bench and introduced myself to the two young apprentices of mathematics. They told me they had another companion by the name of Witold Wilkosz, whom they extravagantly praise. The youngsters were Stefan Banach and Otto Nikodym. From then on we would meet on a regular basis, and ... we decided to establish a mathematical society. The mathematical society which Steinhaus proposed was started as the Mathematical Society of Kraków and, shortly after the war ended, it became the Polish Mathematical Society. Steinhaus described the beginnings of the new mathematical society in [2] in a passage which tells us quite a lot about his life in Kraków at the time:As initiator of the idea, I made my room available for meetings and, as the first step in preparations, nailed an oilcloth blackboard to the wall. When the French manager of the boarding house saw what I had done, she was terrified - what was the proprietor going to say? I calmed her down reminding her that the owner of the building was my uncle's brother-in-law, and she forgave my transgression. However, I had made a mistake. Mr L took the position of a traditional, hard-nosed landlord and was unmoved by the lofty goal the blackboard was supposed to serve. The society expanded - it was the first ray of light of this kind in Poland. Also at this time Steinhaus started a collaboration with Banach and their first joint work was completed in 1916. Steinhaus took up an appointment as an assistant at the Jan Kazimierz University in Lvov and, around 1920, he was promoted to Extraordinary Professor. Banach was by this time on the staff at Lvov and the school rapidly grew in importance. Kac, who was a student of Steinhaus in Lvov during the 1930s, described the influence of Lebesgue's work on the Lvov school:The influence of Lebesgue on the Lvov school was very direct. The school, founded ... by Steinhaus and Banach, concentrated mainly on functional analysis and its diverse applications, the general theory of orthogonal series, and probability theory. There is no doubt that none of these theories would have achieved today's level of prominence without an essential understanding of the Lebesgue measure and integral. On the other hand, the ideas of Lebesgue measure and integral found their most striking and fruitful applications there in Lvov. Steinhaus was the main figure in the Lvov School up till 1941. In 1923 he published in Fundamenta Mathematicae the first rigorous account of the theory of tossing coins based on measure theory. In 1925 he was the first to define and discuss the concept of strategy in game theory. Steinhaus published his second joint paper with Banach in 1927 Sur le principe de la condensation des singularités. In 1929, together with Banach, he started a new journal Studia Mathematica and Steinhaus and Banach became the first editors. The editorial policy was:... to focus on research in functional analysis and related topics. Another important publishing venture in which Steinhaus was involved, begun in 1931, was a new series of Mathematical Monographs. The series was set up under the editorship of Steinhaus and Banach from Lvov and Knaster, Kuratowski, Mazurkiewicz, and Sierpinski from Warsaw. An important contribution to the series was a volume written by Steinhaus jointly with Kaczmarz in 1937, The theory of orthogonal series. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steinhaus.html (2 of 5) [2/16/2002 11:33:22 PM]

Steinhaus

Steinhaus is best known for his book Mathematical Snapshots written in 1937. Kac, writing in [7] says:... to understand and appreciate Steinhaus's mathematical style, one must read (or rather look at) snapshots. ... designed to appeal to "the scientist in the child and the child in the scientist" ... it expresses, not always explicitly and at times even unconsciously, what Steinhaus thought mathematics is and should be. To Steinhaus mathematics was a mirror of reality and life much in the same way as poetry is a mirror, and he liked to "play" with numbers, sets, and curves, the way a poet plays with words, phrases, and sounds. Stark [15] describes Steinhaus lectures in Lvov:My class was guided by Professor Steinhaus. It was a very big class, and the analysis lecture was attended by over 220 students squeezed into a smallish and poorly ventilated lecture room, standing in the aisles, and sitting on the window sills. ... His figure, perched high on the podium by a small five by five foot blackboard dominated the crowded room. ... despite Steinhaus's attention to preparation, the lectures were too difficulty for the average student. The mathematicians of the Lvov school did a great deal of mathematical research in the cafés of Lvov. The Scottish Café was the most popular with the mathematicians in general but not with Steinhaus who (according to Ulam):... usually frequented a more genteel tea shop that boasted the best pastry in Poland. This was Ludwik Zalewski's Confectionery at 22 Akademicka Street. It was in the Scottish Café, however, that the famous Scottish Book consisting of open questions posed by the mathematicians working there came into being. Steinhaus, who sometimes joined his colleagues in the Scottish Café, contributed ten problems to the book, including the final one written on 31 May 1941 only days before the Nazi troops entered the town. You can see a picture of the Scottish Café. When the prospect of war was looming in 1938, Steinhaus proposed Lebesgue for an honorary degree from Lvov. Steinhaus joked to Kac that [7]:It will not be a bad record to leave behind, to have had Banach as the first and Lebesgue as the last doctoral candidate. The reception for Lebesgue, after the award of his degree, was held in the Scottish Café but only fifteen mathematicians attended, showing that the school of mathematics in Lvov had shrunk considerably due to the political situation. Steinhaus spent the war years from June 1941 hiding from the Nazis, suffering great hardships, going hungry most of the time but always thinking about mathematics [7]:... even then his sharp restless mind was at work on a multitude of ideas and projects. In 1945 Steinhaus moved to the University of Wroclaw but made many visits to universities in the United States including Notre Dame. Kac in [7] writes:... it was he who, perhaps more than any other individual, helped to raise Polish mathematics from the ashes to which it had been reduced by the Second World War to the position of new strength and respect which it now occupies.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steinhaus.html (3 of 5) [2/16/2002 11:33:22 PM]

Steinhaus

After the end of World War II the Scottish Book, which seems to have been preserved through the war by Steinhaus, was sent by him to Ulam in the United States. The book was translated into English by Ulam and published. Steinhaus, now in the University of Wroclaw, decided that the tradition of the Scottish Book was too good to end. In 1946 he extended the tradition to Wroclaw starting the New Scottish Book. Let us finally examine some of Steinhaus's mathematical contributions which we have not mentioned above. In 1944 Steinhaus proposed the problem of dividing a cake into n pieces so that it is proportional (each person is satisfied with their share) and envy free (each person is satisfied nobody is receiving more than a fair share). For n = 2 the problem is trivial, one person cuts the cake, the other chooses their piece. Steinhaus found a proportional but not envy free solution for n = 3. An envy free solution to Steinhaus's problem for n = 3 was found in 1962 by John H Conway and, independently, by John Selfridge. For general n the problem was solved by Steven Brams and Alan Taylor in 1995. Steinhaus's bibliography, see [10], contains 170 articles. He did important work on functional analysis, but he himself described his greatest discovery in this area as Stefan Banach. Some of Steinhaus's early work was on trigonometric series. He was the first to give some examples which would lead to marked progress in the subject. He gave an example of a trigonometric series which diverged at every point, yet its coefficients tended to zero. He also gave an example of a trigonometric series which converged in one interval but diverged in a second interval. As we have noted above, other contributions by Steinhaus were on orthogonal series, probability theory, real functions and their applications. In particular he is associated with the theory of independent functions, arising from his work in probability theory, and he was the first to make precise the concepts of "independent" and "uniformly distributed". In addition to his famous book Mathematical Snapshots he also wrote the highly acclaimed One Hundred Problems .... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (16 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steinhaus.html (4 of 5) [2/16/2002 11:33:22 PM]

Steinhaus

JOC/EFR February 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Steinhaus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steinhaus.html (5 of 5) [2/16/2002 11:33:22 PM]

Steinitz

Ernst Steinitz Born: 13 June 1871 in Laurahütte, Silesia, Germany (now Huta Laura, Poland) Died: 29 Sept 1928 in Kiel, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernst Steinitz entered the University of Breslau in 1890. He went to Berlin to study mathematics there in 1891 and, after spending two years in Berlin, he returned to Breslau in 1893. In the following year Steinitz submitted his doctoral thesis to Breslau and, the following year, he was appointed Privatdozent at the Technische Hochschule Berlin - Charlottenburg. The offer of a professorship at the Technical College of Breslau saw him return to Breslau in 1910. Ten years later he moved to Kiel where he was appointed to the chair of mathematics at the University of Kiel. Steinitz was a friend of Toeplitz. The direction of his mathematics was also much influenced by Heinrich Weber and by Hensel's results on p-adic numbers in 1899. In [2] interesting results by Steinitz are discussed. These results were given by Steinitz in 1900, when he was a Privatdozent at the Technische Hochschule Berlin - Charlottenburg, at the annual meeting of the Deutsche Mathematiker - Vereinigung in Aachen. In his talk Steinitz introduced an algebra over the ring of integers whose base elements are isomorphism classes of finite abelian groups. Today this is known as the Hall algebra. Steinitz made a number of conjectures which were later proved by Hall. Steinitz is most famous for work which he published in 1910. He gave the first abstract definition of a field in Algebraische Theorie der Körper in that year. Prime fields, separable elements and the degree of transcendence of an extension field are all introduced in this 1910 paper. He proved that every field has an algebraically closed extension field, perhaps his most important single theorem. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steinitz.html (1 of 2) [2/16/2002 11:33:24 PM]

Steinitz

The now standard construction of the rationals as equivalence classes of pairs of integers under the equivalence relation: (a, b) is equivalent to (c, d) if and only if ad = bc was also given by Steinitz in 1910. Steinitz also worked on polyhedra and his manuscript on the topic was edited by Rademacher in 1934 after his death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Ernst Steinitz

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1910 to 1920

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Steinitz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steinitz.html (2 of 2) [2/16/2002 11:33:24 PM]

Steklov

Vladimir Andreevich Steklov Born: 9 Jan 1864 in Nizhny Novgorod (was Gorky from 1932-1990), Russia Died: 30 May 1926 in Gaspra, Crimea, USSR

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vladimir Steklov's father was a clergyman who taught history and Hebrew at the Nizhni Novgorod seminary. His uncle on his mother's side of the family was a famous literary critic. Vladimir Steklov inherited many of these family talents and he himself had considerable literary and musical talents. Had he not made a career from mathematics he could well have made his profession as an opera singer. Steklov entered the Alexander Institute in Nizhni Novgorod at the age of 10 years. He studied there for eight years graduating in 1884. He entered Moscow University but, after one year, he transferred to Kharkov University. There he became a student of Lyapunov and, after a highly successful university career, he graduated in 1887. Steklov remained at Kharkov University working towards becoming a university teacher. In 1891 he was appointed Lecturer in Mechanics and worked towards his Master's Degree. For his Master's thesis Steklov worked on the equations of a solid body moving in an ideal non-viscous fluid. There were four cases to be considered in integrating the equations which arose from this problem, and two of these cases had been solved by Clebsch in 1871. Steklov solved the third case in his thesis, the final case being solved by Steklov's supervisor Lyapunov in 1893. In fact 1893 was the year that Steklov was awarded his Master's Degree. Then, in 1896, Steklov was appointed to an extraordinary professorship of mechanics and continued to work for his doctoral dissertation. For his doctoral dissertation Steklov worked on problems that arose in potential theory, electrostatics and hydromechanics. He reduced problems of this type to boundary-value problems of Dirichlet type using http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steklov.html (1 of 4) [2/16/2002 11:33:26 PM]

Steklov

rigorous mathematical analysis. Steklov was awarded his doctorate in 1902 and, in that year, his supervisor Lyapunov accepted an appointment at St Petersburg. Steklov was appointed to the Chair of Applied Mathematics at Kharkov University which Lyapunov had vacated. Steklov had been an active member of the Kharkov Mathematical Society, serving the Society as secretary in 1891 and deputy chairman in 1899. In 1902, the year he was appointed to the Chair of Applied Mathematics, Steklov became chairman of the Society and held this post until 1906 when he moved to St Petersburg to take up the Chair of Mathematics there. One of the most remarkable facts about Steklov was that he kept a diary for about 20 years, documenting in detail every day the events of his life. We must know more of Steklov's life than almost any other mathematician but the diaries are of interest far beyond the world of mathematics as they record events through a particularly dramatic period of Russian history. Steklov's entry for each day would begin with a record of the weather conditions at 10 a.m. He would record the temperature, atmospheric pressure, the degree of cloud cover, whether there was rain or snow. Then every night he would complete his record of the events of the day. He gave details of letters he had received and letters he had written that day. All those he visited or who visited him are recorded. The final entry was the weather conditions at 3.30 a.m. giving the same details as the 10 a.m. entry. Through the diary we know how well every student that Steklov examined performed in their examination. We know the lectures that he gave on each day. For example on 10 September 1908:Began lecturing at the University on the integration of partial differential equations. There were quite a lot of people. Before the lecture I said a few words about A N Korkin [who died on 8 September 1908] and suggested that the students pay tribute to his memory by standing up, which of course they did. His lecture course on the integration of partial differential equations was to third year students and the lectures went on until April 1909. He records on 25 April 1909:Finished my lectures on integration of equations. For some reason, the students greeted the end of my lecture with applause (were they happy that I had finished?). Steklov had strong political views at a time when political feelings in Russia ran high. He was strongly against the Tsarist Regime and sympathised with progressive students. In fact this made him a very popular teacher at St Petersburg and, coupled with his excellent lecturing skills, brought many students to study at the Department of Mathematics and Physics. Among the students that he taught at St Petersburg were Friedmann, Smirnov and Tamarkin. The outbreak of World War I in 1914 brought an upsurge of patriotic fervour centred on the tsar which was not shared by Steklov . When the German form of the city's name was changed to the Russian name of Petrograd, Steklov wrote in his diary entry of 2 September 1914:St Petersburg has been renamed Petrograd by Imperial Order. Such trifles are all our tyrants can do ... religious processions and extermination of the Russian people by all possible means. Bastards! Well, just you wait. They will get it hot one day! In 1910 Steklov had been elected to the Academy of Sciences. Then in 1919 he became vice-president of the Academy. A P Yushkevich writes in [1]:During the civil war, military conflicts, economic decline, and the early stages of reconstruction, he proved to be a brilliant scientific administrator. For eight years he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steklov.html (2 of 4) [2/16/2002 11:33:26 PM]

Steklov

worked tirelessly to maintain, and later to enlarge, the activity of the Academy and to reorganise it in order to bring science and practical requirements closer together. In 1921 Steklov founded the Institute of Physics and Mathematics and served as its director until his death in 1926. In 1934 the Institute was split into two separate Institutes of Physics and of Mathematics and the Mathematics Institute was named the V A Steklov Mathematical Institute. Steklov made many important contributions to applied mathematics. In addition to the work for his master's thesis and his doctoral thesis referred to above, he reduced problems to boundary value problems of Dirichlet type where Laplace's equation must be solved on a surface. He wrote General Theory of Fundamental Functions in which he examined expansions of functions as series in an infinite system of orthogonal eigenfunctions. In fact the term "Fundamental Functions", which is due to Poincaré, means eigenfunctions in today's terminology. Steklov was not the first to examine series expansions in terms of infinite sets of orthogonal eigenfunctions, of course Fourier had examined a special case of this situation many years before. Steklov, however, produced many papers on this topic which led him to a general theory to replace the special cases examined by others. He studied a generalisation of Parseval's equality for Fourier series to his general setting showing this to be a fundamental property. In all his list of publications contains 154 items. E A Tropp, V Ya Frenkel and A D Chernin, writing in [3], say of Steklov:He was not only an outstanding mathematician but also had an unusually bright personality. ... Steklov has bequeathed to us not only classical works in mathematics and mathematical physics, but also works of some literary merit. These include his book To America and back (1925), based on his impressions of his trip to the United States, as well as his books about Lomonosov and Galileo. Yet his vast literary legacy [his diary] is still waiting to be published and commented upon. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles) Mathematicians born in the same country Honours awarded to Vladimir A Steklov (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Steklov

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steklov.html (3 of 4) [2/16/2002 11:33:26 PM]

Steklov

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Steklov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Steklov.html (4 of 4) [2/16/2002 11:33:26 PM]

Stepanov

Vyacheslaw Vassilievich Stepanov Born: 4 Sept 1889 in Smolensk, Russia Died: 22 July 1950 in Moscow, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vyacheslaw Stepanov entered Moscow University to study mathematics and physics in 1908. He was supervised there by Egorov. He spent some time in Göttingen where he attended lectures by Hilbert and Landau. He returned to Moscow and, much influenced by Egorov and Luzin, he worked on periodic functions and differential equations. Stepanov was appointed as lecturer in Moscow in 1915, then from 1928 he became professor. He was appointed Director of the Research Institute of Mathematics and Mechanics from 1939, a post he held until his death. Harald Bohr had introduced the notion of an almost periodic function and Stepanov constructed and investigated new classes of these functions. In the theory of differential equations he worked on the general theory of dynamical systems studied by G D Birkhoff. In this area Stepanov extended work by Poincaré. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stepanov.html (1 of 2) [2/16/2002 11:33:28 PM]

Stepanov

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stepanov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stepanov.html (2 of 2) [2/16/2002 11:33:28 PM]

Stevin

Simon Stevin Born: 1548 in Bruges, Flanders (now Belgium) Died: 1620 in The Hague, Holland

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Simon Stevin was a Dutch mathematician and engineer who founded the science of hydrostatics by showing that the pressure exerted by a liquid upon a given surface depends on the height of the liquid and the area of the surface. Stevin was a bookkeeper in Antwerp, then a clerk in the tax office at Brugge. After this he moved to Leiden where he first attended the Latin school, then he entered the University of Leiden in 1583 (at the age of 35). While quartermaster in the Dutch army, Stevin invented a way of flooding the lowlands in the path of an invading army by opening selected sluices in dikes. He was an outstanding engineer who built windmills, locks and ports. He advised the Prince Maurice of Nassau on building fortifications for the war against Spain. The author of 11 books, Stevin made significant contributions to trigonometry, geography, fortification, and navigation. In Wereldschrift he defended the sun centred system of Copernicus. Inspired by Archimedes, Stevin wrote important works on mechanics. In his book De Beghinselen der Weeghconst in 1586 appears the theorem of the triangle of forces giving impetus to statics. In 1585 he published De Thiende in which he presented an elementary and thorough account of decimal fractions. Although he did not invent decimals (they had been used by the Arabs and the Chinese long before Stevin's time) he did introduce their use in mathematics. His notation was to be taken up by Clavius and

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stevin.html (1 of 3) [2/16/2002 11:33:30 PM]

Stevin

Napier. Stevin states that the universal introduction of decimal coinage, measures and weights would only be a matter of time. Stevin's notion of a real number was accepted by essentially all later scientists. Particularly important was Stevin's acceptance of negative numbers but he did not accept the 'new' imaginary numbers and this was to hold back their acceptance. Stevin, in his book Stelreghel meaning Algebra, used the notation +, - and . In 1586 (3 years before Galileo) he reported that different weights fell a given distance in the same time. Article by: J J O'Connor and E F Robertson List of References (16 books/articles)

A Quotation

A Poster of Simon Stevin

Mathematicians born in the same country

Some pages from publications

A page from the preface of La Disme (1634) a French translation of De Thiende .

Cross-references to History Topics

An overview of the history of mathematics

Other references in MacTutor

Chronology: 1500 to 1600

Honours awarded to Simon Stevin (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Stevinus

Other Web sites

1. The Galileo Project 2. Amsterdam, Netherlands (An English translation of La Theinde) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stevin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stevin.html (2 of 3) [2/16/2002 11:33:30 PM]

Stevin

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stevin.html (3 of 3) [2/16/2002 11:33:30 PM]

Stewart

Matthew Stewart Born: 15 Jan 1717 in Rothesay, Isle of Bute, Scotland Died: 23 Jan 1785 in Catrine, Ayrshire, Scotland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Matthew Stewart attended school on the Isle of Bute, then in 1734 he began to work under Simson at the University of Glasgow. In 1741 he began to work under Maclaurin at the University of Edinburgh. During this time he continued to correspond with Simson, who was by now a friend rather than a teacher, on Greek geometry. Stewart was licensed to preach at Dunoon in 1744, then one year later he became a minister at Roseneath, Dumbartonshire. He resigned when appointed to Maclaurin's chair at Edinburgh in 1747. Maclaurin had left to serve the troops in the Jacobite rebellion of 1745. Stewart's fame is based on General Theorems (1746), described by Playfair as among the most beautiful, as well as most general, propositions known in the whole compass of geometry. In 1756 Stewart wrote on Kepler's second law of planetary motion using geometrical methods. In 1761 Stewart wrote Tracts Physical and Mathematical describing planetary motion and the perturbation of one planet on another and, two years later, he wrote the supplement The Distance of the Sun from the Earth determined by the Theory of Gravity. This work achieved the rather inaccurate result of 119 million miles (the correct value is approximately 93 million miles). His geometrical methods required too many simplifications to get a better result and this work was criticised by John Landen in 1771. Stewart was elected a Fellow of the Royal Society in 1764. In 1772 his health began to deteriorate and his duties as professor at Edinburgh were taken over by his son Dugald. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stewart.html (1 of 2) [2/16/2002 11:33:31 PM]

Stewart

Honours awarded to Matthew Stewart (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1764

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stewart.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stewart.html (2 of 2) [2/16/2002 11:33:31 PM]

Stewartson

Keith Stewartson Born: 20 Sept 1925 in Barnsley, Yorkshire, England Died: 7 May 1983 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Keith Stewartson was an undergraduate at St Catharine's College, Cambridge during World War II, then after War service involving studying compressible fluid flow problems, he returned to Cambridge to receive a Ph.D. in 1949 for work on boundary layer theory. Keith was appointed lecturer at Bristol in 1949, the same year as his research supervisor at Cambridge Leslie Howarth was appointed to the Chair of Applied Mathematics at Bristol. He spent periods abroad in 1953/54 mainly in North America, where he spent time at the California Institute of Technology, and also in Israel and Australia. Stewartson was appointed to a chair at the University of Durham in 1958, then a chair in University College, London in 1964, a post he held until his death. He spoke on boundary layer theory, in particular flows at high Reynolds numbers, in his inaugural lecture in London. Stewartson studied rotating fluid flows, shear layers, magnetohydrodynamics and flow at both high and low Reynolds numbers. His output was large, with 186 papers. He was quick to recognise the importance of computers. In [6] Riley writes:... he recognised the value of the computer, and the impact that it would make on his subject, at an early stage. For him computational methods were a complement to analytic techniques. They could be used to help verify a delicate analytical structure ..., or even suggest the analytic structure ... Alternatively they could reveal unexpected phenomena which would then be subjected to analysis. Stewartson received many honours. He was elected a Fellow of the Royal Society in 1965. In the previous year he had become one of the founders of the Institute of Mathematics and its Applications.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stewartson.html (1 of 2) [2/16/2002 11:33:33 PM]

Stewartson

In [7] Stewartson is compared with Thomson, Rayleigh and Stokes. Others have compared him with Euler. This is indeed praise but his international domination of his subject clearly makes him worthy of the comparisons. Article by: J J O'Connor and E F Robertson List of References (8 books/articles) Mathematicians born in the same country Honours awarded to Keith Stewartson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1965

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stewartson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stewartson.html (2 of 2) [2/16/2002 11:33:33 PM]

Stieltjes

Thomas Jan Stieltjes Born: 29 Dec 1856 in Zwolle, Overijssel, The Netherlands Died: 31 Dec 1894 in Toulouse, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thomas Stieltjes' father, also named Thomas Stieltjes, had been awarded a doctorate from Leiden University. He became well-known both as a civil engineer and also as a member of the parliament. As an engineer he is famed for the construction of harbours in Rotterdam and is remembered today through a statue, erected by his friends, which stands on the Noordereiland, at the Burgemeester Hoffman Plein, in Rotterdam. In fact the dictionary of biography of famous people from the Netherlands which I [EFR] consulted before writing this biography contained a two page entry concerning Thomas Stieltjes, the father, while there was only a single paragraph on his son, Thomas Stieltjes the mathematician. Thomas Stieltjes senior had seven children and Thomas Stieltjes junior had two brothers and four sisters. Stieltjes started his studies at the Polytechnical School of Delft in 1873 but spent his student years reading Gauss and Jacobi in the library rather than attending lectures. It may have been enjoyable to Stieltjes to read the works of these great mathematicians rather than study the coursework but the consequence was that he failed his examinations. Two further failures in 1875 and 1876 had his father in despair, but having a friend H G van de Sande-Bakhuyzen as director of Leiden Observatory, Thomas's father was able to put in a good word for his son. Stieltjes became assistant at Leiden Observatory in April 1877 but his father, who had done so much to help his son, was not to live long after helping his son to obtain this position for he died in Rotterdam on 23 June 1878. Perhaps the most significant event in Stieltjes' life, as far as mathematics was concerned, occurred on 8 November 1882 when he began a correspondence with Hermite which was to last the rest of his life (which would only be 12 more years during which time they exchanged 432 letters). The original reason that Stieltjes wrote to Hermite concerned his work on celestial mechanics. However the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stieltjes.html (1 of 4) [2/16/2002 11:33:35 PM]

Stieltjes

correspondence turned quickly to mathematics and Stieltjes began to devote all his spare time to mathematical research. It is a great credit to van de Sande-Bakhuyzen, the director of Leiden Observatory, that he responded quickly to Stieltjes' request on 1 January 1883 to stop his observational work to allow him to work more on mathematical topics. The year 1883 was eventful for Stieltjes for he married Elizabeth Intveld in May. Although clearly an important event in Stieltjes' personal life, this was also an important event for Stieltjes as a mathematician for his wife strongly encouraged him to move from astronomical work to mathematics. In September Stieltjes was asked to substitute at the University of Delft for F J van den Berg who had taken ill. From September to December 1883 Stieltjes lectured on analytical geometry and on descriptive geometry. This confirmed what must have been becoming increasing clear in Stieltjes' mind, that mathematics was the only p[ossible career for him. He completed the move on 1 December 1883 when he resigned his post at the observatory. Stieltjes applied for a chair in Groningen and on 15 January 1884 he wrote to Hermite:I have been offered, some days ago, a professorship in analysis (differential and integral calculus) at the University of Groningen. I have accepted this offer and I believe that this position will permit me to become more useful. I owe much, for this position, to the extreme kindness of my old boss Mr Bakhuyzen, the director of the observatory. One of these days, my nomination will become definite. He was to be disappointed, however, for although he was placed first for a chair at Groningen he was not appointed because of his lack of qualifications. He wrote to Hermite on 13 March 1884 saying:The Groningen Faculty has indeed put me in first place for the vacancy, but the Minister has named one of the others. Probably the reason will have been that I had no chance of following the standard path, for I have not received any degree from the University. Although this gives the complete facts that Stieltjes was aware of, the events surrounding this appointment were a little more complex than he realised. In fact the appointing committee had drawn up a list of three names in order of preference in 1883. Top of the list had been Korteweg with Stieltjes in second place and indeed the position was offered to Korteweg who was a professor at the University of Amsterdam. After considering the offer Korteweg decided that he did not wish to leave Amsterdam and move to Groningen so he turned the offer down. At this stage the appointing committee at Groningen drew up a new list putting Stieltjes in first place with Floris de Boer second. However, despite Stieltjes accepting the position at this stage, a Royal Decree was issued on 12 March 1884 appointing de Boer to the chair. As one might imagine Hermite was very disturbed to learn that Stieltjes had been ruled out after an offer had been made to him because of his lack of a degree. In May 1884 Hermite attended celebrations at the University of Edinburgh in Scotland to celebrate the three hundredth aniversary of the founding of the university. There he talked to Bierens de Haan, a professor of mathematics from the Netherlands, and they devised a plan to help Stieltjes by having him proposed for an honorary degree at Leiden University. After de Haan returned, he and van de Sande-Bakhuyzen proposed Stieltjes for an honorary degree in mathematics and astronomy. On 27 May the Senate of Leiden University recorded the following minute:The Rector reported that a request has been received from the Faculty of Mathematics and

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stieltjes.html (2 of 4) [2/16/2002 11:33:35 PM]

Stieltjes

Physics to confer the degree of doctor honoris causa in Mathematics and Astronomy upon Mr T J Stieltjes, a former employee of Leiden Observatory. On behalf of the Faculty, Mr Lorentz explained the merits of Mr Stieltjes and indicated the reasons which led to the proposal. It was decided to reach a conclusion in the next meeting of the Senate. The Senate wrote to Stieltjes regarding the award of the honorary degree but somehow it failed to reach him in time. Stieltjes, on 19 June 1884, replied:To the Senate of Leiden University. The undersigned wishes to thank you for the honourable distinction, conferred upon him by Your College, and to assure you that the distinction is highly appreciated. Due to a regrettable misunderstanding he was not aware of the intention of a public ceremony on last Tuesday June 17 at 3 o'clock. Leiden, June 19, 1884 T J Stieltjes Stieltjes went with his family to Paris in April 1885 and in the same year he was elected to the Royal Academy of Sciences in Amsterdam. He received his doctorate of science in 1886 for a thesis on asymptotic series. In the same year Stieltjes was appointed to the University of Toulouse, being appointed to a chair of differential and integral calculus in Toulouse in 1889. Stieltjes worked on almost all branches of analysis, continued fractions and number theory. He is often called "the father of the analytic theory of continued fractions" for his work in this area. On 18 June 1894 he published Recherches sur les fractions continues in Comptes Rendus de l'Académie des Sciences. A fuller version of this work appeared in the Annales de la Faculté des Sciences de Toulouse. It is a beautifully written work of major importance described in [1] as:... clear, self-contained, almost lyric in style. Stieltjes studied the continued fraction:

If we consider the first n terms of this continued fraction then we obtain the rational function Pn(z)/Qn(z). Stieltjes examined the sequence of rational functions Pn(z)/Qn(z) and the connections between the roots of the polynomials Pn(z) and Qn(z). He is best remembered for the Stieltjes integral which he introduced in Recherches sur les fractions continues while solving the moment problem, that is, given the moments of all orders of a body, find the distribution of its mass. This problem arose in the study of two functions arising as the limits of the sequences P2n(z)/Q2n(z) and P2n+1(z)/Q2n+1(z).

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stieltjes.html (3 of 4) [2/16/2002 11:33:35 PM]

Stieltjes

The first part of Stieltjes' article in the Annales de la Faculté des Sciences de Toulouse covers 120 pages and appeared in 1894. The second part is a forty page article which appeared after Stieltjes' death in 1895. His work on continued fractions had already been awarded the Ormoy Prize of the Académie des Sciences in 1893. Recherches sur les fractions continues is described in [1] as:... a mathematical milestone. The work represents the first general treatment of continued fractions as part of complex analytic function theory; previously, only special cases had been considered. Stieltjes' work is also seen as an important first step towards the theory of Hilbert spaces. Also important is his work on divergent series and discontinuous functions. Stieltjes also contributed to ordinary and partial differential equations, the gamma function, interpolation, and elliptic functions. Stieltjes died on 31 December 1894 and was buried in the cemetery of Terre Cabade in Toulouse on 2 January 1895. His grave has recently been restored. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of Thomas Stieltjes

Mathematicians born in the same country

Other Web sites

1. Leiden, Netherlands 2. Stieltjes Constants 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stieltjes.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stieltjes.html (4 of 4) [2/16/2002 11:33:35 PM]

Stifel

Michael Stifel Born: 1487 in Esslingen, Germany Died: 19 April 1567 in Jena, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Michael Stifel attended the University of Wittenberg where he was awarded an M.A. He made his life in the Church entering the Augustinian monastery at Esslingen. He was ordained in 1511 while at the monastery. However Stifel did not conform correctly to the Catholic faith and he became unhappy with taking money from the poor. He was forced out of the monastery at Esslingen in 1522. He sought refuge with Lutherans and eventually he went to Wittenberg and lived in Luther's own house for a while. In 1523 Luther obtained a position of pastor for Stifel but anti-Lutheran pressure forced him out of a number of positions. In 1528, Luther set him up in a parish at Lochau (now Annaberg). Stifel made the error of predicting the end of the world and, when it was seen that he was wrong, he was arrested and dismissed from his post. In 1535 he went to a parish in Holzdorf and remained there for 12 years. In the religious Schmalkaldic War of 1547, the Lutheran duke Maurice of Saxony and Holy Roman emperor Charles V tried to take a region of Saxony away from Protestant control. Stifel was forced to flee from his parish again. This time Stifel went to Prussia and obtained a parish near Königsberg. During this time he lectured on mathematics and theology at the University of Königsberg. Arguments with colleagues led to him returning to Saxony three years later. In 1559 Stifel obtained a post at the University of Jena, where he lectured on arithmetic and geometry. Stifel's research was on arithmetic and algebra. He invented logarithms independently of Napier using a totally different approach. His most famous work is Arithmetica integra was published in 1544 while he was in Holzdorf. The work contains binomial coefficients and the notation +, -, . Stifel used a clever rearrangement of the letters LEO DECIMVS to "prove" that Leo X was 666, the number of the beast given in the Book of Revelation. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stifel.html (1 of 2) [2/16/2002 11:33:37 PM]

Stifel

List of References (14 books/articles)

A Quotation

Mathematicians born in the same country Some pages from publications

A page from Arithmetica integra (1544) Another page from Arithmetica integra showing the use of + and - signs and his notation for the unknown and its powers

Other references in MacTutor

Chronology: 1500 to 1600

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stifel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stifel.html (2 of 2) [2/16/2002 11:33:37 PM]

Stirling

James Stirling Born: May 1692 in Garden (near Stirling), Scotland Died: 5 Dec 1770 in Edinburgh, Scotland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James Stirling's father was Archibald Stirling and his mother, Archibald Stirling's second wife, was Anna Hamilton. James was their third son and he was born on the family estate at Garden, about 20 km west of the Scottish town of Stirling. The family were strong supporters of the Jacobite cause and this was to have a significant influence on James Stirling's life. The Jacobite cause was that of the Stuart king, James II (of Britain -- James VII of Scotland: Jacobus in Latin), exiled after the Revolution of 1688, and his descendants. Scotland was united to England and Wales in 1707. The Stuarts were Scottish but Roman Catholics and therefore they had only limited support. They did, however, offer an alternative to the British crown with an exiled court in France which had strong support from many such as the Stirling family. When James Stirling was about 17 his father was arrested, imprisoned and accused of high treason because of his Jacobite sympathies. However he was acquitted of the charges. Nothing is known of Stirling's childhood or indeed about his undergraduate years in Scotland. The first definite information that we know is that he travelled to Oxford in the autumn of 1710 in order to matriculate there. Indeed Stirling matriculated at Balliol College Oxford on 18 January 1711 as a Snell Exhibitioner. The terms of the Snell Exhibitions is described in [3]:The Snell Exhibitions to Balliol College were established by the will of an Ayrshire man John Snell (1629?-1679). They were originally intended for Scottish students within Scotland who had not graduated and who would subsequently return to Scotland as priests of the Church of England. Nominations were to be made by the College of Glasgow, one of the requirements of candidates being that they should have spent at least one year at Glasgow. Based on this, together with information from Ramsay (see [4]) who knew Stirling in later life and wrote that he was:bred at the University of Glasgow it is usual to state that indeed Stirling studied at the University of Glasgow (as is done in [1]). However this is not absolutely certain. We know that Ramsay is not always completely reliable. Stirling's name does not appear in the list of students matriculating at Glasgow (not all student's names occur so this is http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stirling.html (1 of 6) [2/16/2002 11:33:39 PM]

Stirling

not very significant). Tweddle [3] notes that a student with the name 'James Stirling' matriculated at the University of Edinburgh on 24 March 1710, did not graduate, and has a signature which is similar to that of the mathematician. Another fact, which is not insignificant, is that Stirling's father was a graduate of Edinburgh. It would be nice to solve this and many other puzzles associated with Stirling's life but they may always remain as puzzles. Stirling was awarded a second scholarship in October 1711, namely the Bishop Warner Exhibition. He should have sworn an oath when matriculating but his Jacobite sympathies would not let him do this and he was excused. Queen Anne died in August 1714 and the German, George I, acceded to the British throne. In 1715 there was the first Jacobite Rebellion, which melted away after the drawn Battle of Sheriffmuir on 13 November 1715. However the concession of allowing Stirling not to swear the oath was withdrawn. He lost his scholarships when he continued to refuse to take the oath. Then he was accused of corresponding with Jacobites who had been involved in planning the rebellion. Life must have been difficult for him at this time and he even appeared at the assizes charged with 'cursing King George' but he was acquitted. Certainly Stirling could now not graduate from Oxford but he remained there for some time. In the minutes of a meeting of the Royal Society of London on 4 April 1717, when Brook Taylor lectured on extracting roots of equations and on logarithms, it is recorded:Mr Stirling of Balliol College Oxford had leave to be present. In 1717 Stirling published his first work Lineae Tertii Ordinis Neutonianae which extends Newton's theory of plane curves of degree 3, adding four new types of curves to the 72 given by Newton. The work was published in Oxford and Newton himself received a copy of the work which is dedicated to the Venetian ambassador Nicholas Tron. Lineae Tertii Ordinis Neutonianae contains other results that Stirling had obtained. There are results on the curve of quickest descent, results on the catenary (in particular relating this problems to that of placing spheres in an arch), and results on orthogonal trajectories. The problem of orthogonal trajectories had been raised by Leibniz and many mathematicians worked on the problem in addition to Stirling, including Johann Bernoulli, Nicolaus(I) Bernoulli, Nicolaus(II) Bernoulli, and Leonard Euler. It is known that Stirling solved the problem early in the year 1716. In 1717 Stirling went to Venice. The Venetian ambassador Tron left London to return to Venice in June 1717 and it is almost certain that Stirling travelled with him. Stirling seems to have been promised a chair of mathematics in Venice but, for some reason that is not known, the appointment fell through. What Stirling did in Venice is also not known but he certainly continued his mathematical research. Stirling certainly was in Venice in 1719 since he submitted a paper Methodus differentialis Newtoniana illustrata to the Royal Society of London from Venice at that time. The paper was received by the Royal Society and reported to their meeting on 18 June 1719. Now Nicolaus(I) Bernoulli occupied the chair at the University of Padua from 1716 until 1722. Stirling must have met Nicolaus(I) Bernoulli and got to know him quite well since, in 1719, he wrote to Newton, again from Venice, offering to act as a go-between. In 1721 Stirling was in Padua and we know that he attended the University of Padua at that time.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stirling.html (2 of 6) [2/16/2002 11:33:39 PM]

Stirling

In 1722 Stirling returned to Glasgow, perhaps around the time that his friend Nicolaus(I) Bernoulli left Padua. There is a story told by Tweedie in [5] that Stirling learned the secrets of the glass industry while in Italy and had to flee for fear of his life since the glass-makers may have tried to assassinate him to prevent their secrets becoming known. It is not clear what he did between that time and late 1724 but it is clear that, at least from 1722, he had the intention of becoming a teacher in London. In August 1722 Maclaurin visited Newton in London and Newton showed him a letter from Stirling in which Stirling wrote that he intended to set himself up as a mathematics teacher in London. Certainly Stirling was friendly with Newton and the letter was almost certainly asking for Newton's help in this venture, help which Newton was giving in telling Maclaurin of Stirling's plans. In late 1724 Stirling travelled to London where he was to remain for 10 years. These were ten years in which Stirling was very active mathematically, corresponding with many mathematicians and enjoying his friendship with Newton. Newton proposed Stirling for a fellowship of the Royal Society of London and, on 3 November 1726, Stirling was elected. Stirling achieved his aim of becoming a teacher in London when he was appointed to William Watt's Academy in Little Tower Street, Covent Garden, London which was [1]:... one of the most successful schools in London; and, although he had to borrow money to pay for the mathematical instruments he needed. The school's prospectus of 1727 lists a course on mechanical and experimental philosophy given by Stirling and others. The syllabus included mechanics, hydrostatics, optics, and astronomy. While in London, Stirling published his most important work Methodus Differentialis in 1730. This book is a treatise on infinite series, summation, interpolation and quadrature. The asymptotic formula for n! for which Stirling is best known appears as Example 2 to Proposition 28 of the Methodus Differentialis. One of the main aims of the book was to consider methods of speeding up the convergence of series. Stirling notes in the Preface that Newton had considered this problem. As an example of the problem he is trying to solve Stirling gives the example of the series 1/[2n(2n-1)] which had been studied by Brouncker in his work on the area under a hyperbola. Stirling writes, in Methodus Differentialis, that:...if anyone would find an accurate value of this series to nine places ... they would require one thousand million of terms; and this series converges much swifter than many others... Many examples of his methods are given, including Leibniz's problem of /4 = 1 - 1/3 + 1/4 - 1/5 + 1/6 - ... and he also gives a theorem to treat convergence of an infinite product. Included in this work on accelerating convergence is a discussion of De Moivre's methods. We mentioned above that he studied interpolation in the Methodus Differentialis. For example he defined the series Tn+1 = nTn with T1 = 1. He then considered T3/2 , between the terms T1 and T2. In today's notation this would be (1/2) and Stirling here is studying the Gamma function. He calculated T3/2 to ten decimal places. In fact T3/2 = (1/2) =

.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stirling.html (3 of 6) [2/16/2002 11:33:39 PM]

Stirling

The book contains other results on the Gamma function and the Hypergeometric function. De Moivre published Miscellanea Analytica in 1730. Stirling wrote to De Moivre pointing out some errors that he had made in a table of logarithms of factorials in the book and also telling De Moivre about Example 2 to Proposition 28 of Methodus Differentialis. De Moivre was able to extend his earlier results using Stirling's ideas and published a Supplement to Miscellanea Analytica a few months later. Clearly Stirling and De Moivre regularly corresponded around this time for in September 1730 Stirling relates the episode and new results of De Moivre in a letter to Cramer. There is another area of Stirling's work that we shall examine, namely his work on gravitation and the figure of the Earth. However, before doing so we will look at a correspondence that Stirling had with Euler since this relates to the work we have just discussed on series. Euler wrote to Stirling on 8 June 1736 from St Petersburg. We quote from his letter where he gives his opinion on Stirling's work (see [7] or [3]):... the more I have learned from your excellent articles, which I have seen here and there in your Transactions, concerning the nature of series, a study in which I have indeed expended much effort, the more I have wished to become acquainted with you in order that I could receive more from you yourself and also submit my own deliberations to your judgement. But before I wrote to you, I searched all over with great eagerness for your excellent book on the method of differences, a review of which I had seen a short time before in the Acta Lipsiensis, until I achieved my desire. Now that I have read through it diligently, I am truly astonished at the great abundance of excellent methods contained in such a small volume, by means of which you show how to sum slowly converging series with ease and how to interpolate progressions which are very difficult to deal with. But especially pleasing to me was proposition XIV of part 1 in which you give a method by which series, whose law of progression is not even established, may be summed with great ease using only the relation of the last terms, certainly this method extends very widely and is of the greatest use. In fact the proof of this proposition, which you seem to have deliberately withheld, caused me enormous difficulty, until at last I succeeded with very great pleasure in deriving it from the preceding results, which is the reason why I have not yet been able to examine in detail all the subsequent propositions In 1735 Stirling returned to Scotland where he was appointed manager of the 'Scotch mining company, Leadhills' in Lanarkshire at a salary of 120 per year. This was a job that Stirling did very well, he [8]:... proved extremely successful as a practical administrator, the condition of the mining company improving vastly owing to his method of employing labour to work the mines. However the work was very demanding. It was two years before he got round to replying to Euler's letter from which we quoted above. In the reply, dated 16 April 1738, and written from Edinburgh he explains why he has not replied sooner (see [7] or [3]):During these last two years I have been involved in a great many business matters which have required me to go frequently to Scotland, and then return to London. And it was on account of these affairs that first of all your letter came late into my hands and then that, even to this very day, there is scarcely time available for reading through your letter with the attention which it deserves. For after deliberations have been interrupted, not to say neglected, for a long time, patience is required before the mind can be brought to think http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stirling.html (4 of 6) [2/16/2002 11:33:39 PM]

Stirling

about the same things once again. In the same letter Stirling offered to put Euler's name forward for election to the Royal Society of London. He did not do that, however, probably again through pressure of work with the mining company and it was not until 1746 that he was proposed by several mathematicians not including Stirling. It appears that Stirling never replied to this second letter from Euler. He wrote to Maclaurin on 26 October 1738 saying that Euler's second letter was:... full of many ingenious things, but it is long and I am not quite master of all the particulars. In 1745 Stirling published a paper on the ventilation of mine shafts. He certainly did not give up mathematics when he took up the post in the mining company, and in [3] there is a discussion of unpublished mathematical work in notebooks of Stirling that were probably written between 1730 and 1745. The year 1745 was the date of the most major of the Jacobite rebellions and Maclaurin played an active role in the defence of Edinburgh against the Jacobites. Charles Edward, the Young Pretender, entered Edinburgh with an army of 2,400 men on 17 September 1745. In 1746 Maclaurin died, partly as a consequence of the battles of the previous year, and Stirling was considered for his chair at Edinburgh. However Stirling's strong support for the Jacobite cause meant that such an appointment was impossible, especially in the year after the rebellion. Stirling was elected to membership of the Royal Academy of Berlin in 1746. In 1753 he resigned from the Royal Society of London as he was in debt to the Society and could no longer afford the annual subscriptions. It cost him 20 to resign. One non-mathematical contribution by Stirling is described in [8] (see also [5]):... he surveyed the Clyde with a view to rendering it navigable by a series of locks, thus taking the first step towards making Glasgow the commercial capital of Scotland. The citizens were not ungrateful, and in 1752 presented him with a silver tea-kettle 'for his service, pains, and trouble'. Finally we must discuss Stirling's second major mathematical contribution, namely his work on the figure of the Earth. On 6 December 1733 Stirling read a paper to the Royal Society of London entitled Twelve propositions concerning the figure of the Earth. The minutes of the Society state:Mr Stirling was ordered thanks, and was desired to communicate his Propositions. Indeed Stirling did submit an extended version of his results which appeared as Of the figure of the Earth, and the variation of gravity on the surface in 1735. In [1] the paper is described:In it he stated, without proof, that the Earth is an oblate spheroid, supporting Newton against the rival Cassinian view. Certainly Stirling was considered that leading British expert on the subject for the next few years by all including Maclaurin and Simpson who went on to make major contributions themselves. As Stirling's unpublished manuscripts show [3], he did go much further than the 1735 paper but probably the pressure of work at the mining company gave him too little time to polish the work. He explains in a letter to Maclaurin, dated 26 October 1738, why he has not published despite pressure to do so:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stirling.html (5 of 6) [2/16/2002 11:33:39 PM]

Stirling

I got a letter this last summer from Mr Machin wholly relating to the figure of the Earth and the new mensuration, he seems to think this a proper time for me to publish my proposition on that subject when everybody is making a noise about it; but I choose rather to stay till the French arrive from the south, which I hear will be very soon. And hitherto I have not been able to reconcile the measurements made in the north to the theory.... In fact the French expedition to Ecuador, referred to by Stirling as 'the south', left in 1735 but did not return until 1744. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles)

Mathematicians born in the same country

Other references in MacTutor

1. Stirling's formula 2. Stirling numbers of the first kind 3. Stirling numbers of the second kind 4. Bell numbers

Honours awarded to James Stirling (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1726

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR Sept1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stirling.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stirling.html (6 of 6) [2/16/2002 11:33:39 PM]

Stokes

George Gabriel Stokes Born: 13 Aug 1819 in Skreen, County Sligo, Ireland Died: 1 Feb 1903 in Cambridge, Cambridgeshire, England

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Stokes' father, Gabriel Stokes, was the Protestant minister of the parish of Skreen in County Sligo. His mother was the daughter of a minister of the church so George Stokes's upbringing was a very religious one. He was the youngest of six children and every one of his three older brothers went on to become a priest. As the priest of the church in Cambridge which Stokes later attended wrote (see [14]):Though he was never narrow in his faith and religious sympathies, he always held fast by the simple evangelical truths he learnt from his father... In [3] the atmosphere in which George grew up is described in words which are more colourful than those which might be used today:The home-life in the Rectory at Skreen was very happy, and the children grew up in the fresh sea-air with well-knit frames and active minds. Great economy was required to meet the educational needs of the large family... It was not only religious teaching, but a wider introduction to education, which Gabriel Stokes was able to give his children. In particular, having studied at Trinity College Dublin, he was able to teach George Latin grammar. Before going to school George was also taught by the clerk in his father's parish in Skreen. Leaving Skreen in 1832, George attended school in Dublin. He spent three years at the Rev R H Wall's school in Hume Street, Dublin but he was not a boarder at the school, living for these three years with his uncle John Stokes. In fact the family finances would not have allowed him a more expensive education, but at this school [3]:He pursued the usual school studies, and attracted the attention of the mathematical master by his solution of geometrical problems.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stokes.html (1 of 6) [2/16/2002 11:33:41 PM]

Stokes

It was during George's three years in Dublin that his father died and this event had, as one would expect, a major effect on the young man. In 1835, at the age of 16, George Stokes moved to England and entered Bristol College in Bristol. The two years which Stokes spent in Bristol at this College were important ones in preparing him for his studies at Cambridge. The Principal of the College, Dr Jerrard, was an Irishman who had attended Cambridge University with William Stokes, one of George's elder brothers. Dr Jerrard was himself a mathematician but Stokes was taught mathematics at Bristol College by Francis Newman (who was the brother of John Henry Newman, later Cardinal Newman, who became the leader of the Oxford Movement in the Church of England which was founded in 1833). Clearly Stokes's talent for mathematics was shown during his studies at Bristol College, for he won mathematics prizes and Dr Jerrard wrote to him (see [3]):I have strongly advised your brother to enter you at Trinity, as I feel convinced that you will in all human probability succeed in obtaining a Fellowship at that College. It was not Trinity, rather Pembroke College, Cambridge, which Stokes entered in 1837. There are slight inconsistencies in what his mathematical background was on entering Cambridge. In the course at Bristol College (according to the College literature) (see [4]):... a student was to become acquainted with the differential and integral calculus and to go on to statics, dynamics, conic sections and the first three sections of Newton's Principia... However, Stokes himself wrote in 1901 (see for example [3]):I entered Pembroke College, Cambridge in 1837. In those days boys coming to the University had not in general read so far in mathematics as is the custom at present; and I had not begun the differential calculus when I entered the College, and had only recently read analytical sections. In Stokes's second year at Cambridge he began to be coached by William Hopkins, a famous Cambridge coach who played a more important role than the lecturers. Stokes wrote [3]:In my second year I began to read with a private tutor, Mr Hopkins, who was celebrated for the very large number of his pupils high places in the University examinations for mathematical honours... Hopkins was to exert a strong influence on the direction of Stokes's mathematical interests. Hopkins [4]:... praised the study of physical astronomy and physical optics, for example, because they revealed mathematics to be 'the only instrument of investigation by which man could possibly have attained to a knowledge of so much of what is perfect and beautiful in the structure of the material universe, and the laws that govern it'. In 1841 Stokes graduated as Senior Wrangler (the top First Class degree) in the Mathematical Tripos and he was the first Smith's prizeman. Pembroke College immediately gave him a Fellowship. He wrote [3]:After taking my degree I continued to reside in College and took private pupils. I thought I would try my hand at original research.... It was William Hopkins who advised Stokes to undertake research into hydrodynamics and indeed this was the area in which Stokes began to work. In addition to Hopkins' advice, Stokes was also inspired to enter this field by the recent work by George Green. Stokes published papers on the motion of incompressible fluids in 1842 and 1843, in particular On the steady motion of incompressible fluids in

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stokes.html (2 of 6) [2/16/2002 11:33:41 PM]

Stokes

1842. After completing the research Stokes discovered that Duhamel had already obtained similar results but, since Duhamel had been working on the distribution of heat in solids, Stokes decided that his results were obtained in a sufficiently different situation to justify him publishing. Stokes then continued his investigations, looking at the situation where he took into account internal friction in fluids in motion. After he had deduced the correct equations of motion Stokes discovered that again he was not the first to obtain the equations since Navier, Poisson and Saint-Venant had already considered the problem. In fact this duplication of results was not entirely an accident, but was rather brought about by the lack of knowledge of the work of continental mathematicians at Cambridge at that time. Again Stokes decided that his results were obtained with sufficiently different assumptions to justify publication and he published On the theories of the internal friction of fluids in motion in 1845. The work also discussed the equilibrium and motion of elastic solids and Stokes used a continuity argument to justify the same equation of motion for elastic solids as for viscous fluids. Perhaps the most important event in the recognition of Stokes as a leading mathematician was his Report on recent researches in hydrodynamics presented to the British Association for the Advancement of Science in 1846. But a study of fluids was certainly not the only area in which he was making major contributions at this time. In 1845 Stokes had published an important work on the aberration of light, the first of a number of important works on this topic. He also used his work on the motion of pendulums in fluids to consider the variation of gravity at different points on the earth, publishing a work on geodesy of major importance On the variation of gravity at the surface of the earth in 1849. In 1849 Stokes was appointed Lucasian Professor of Mathematics at Cambridge. In 1851 Stokes was elected to the Royal Society, awarded the Rumford medal of that Society in 1852, and he was appointed secretary of the Society in 1854. The Lucasian chair paid very poorly so Stokes needed to earn additional money and he did this by accepting an additional position to the Lucasian chair, namely that of Professor of Physics at the Government School of Mines in London. Stokes's work on the motion of pendulums in fluids led to a fundamental paper on hydrodynamics in 1851 when he published his law of viscosity, describing the velocity of a small sphere through a viscous fluid. In addition to several important investigations concerning the wave theory of light, such as a paper on diffraction in 1849. This paper is discussed in detail in [10] in which the authors write:...the results of Stokes are related to the elastic theory of light, and supplement and expand a number of questions, previously studied for the most part in the works of A Cauchy. Stokes's methods for solving diffraction problems, differing considerably from the methods employed by Cauchy, form the basis of the further studies of the mathematical theory of the phenomenon of diffraction. Stokes named and explained the phenomenon of fluorescence in 1852. Stokes's interpretation of this phenomenon, which results from absorption of ultraviolet light and emission of blue light, is based on an elastic aether which vibrates as a consequence of the illuminated molecules. The paper [8] discusses this in detail and is particularly interesting since the author makes full use of Stokes's unpublished notebooks. In 1854 Stokes theorised an explanation of the Fraunhofer lines in the solar spectrum. He suggested these were caused by atoms in the outer layers of the Sun absorbing certain wavelengths. However when Kirchhoff later published this explanation Stokes disclaimed any prior discovery. Stokes's career certainly took a rather different tack in 1857 when his moved from his highly active http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stokes.html (3 of 6) [2/16/2002 11:33:41 PM]

Stokes

theoretical research period into one where he became more involved with administration and experimental work. Certainly his marriage in 1857 was not unconnected with the change in tack and, particularly since it gives us an insight into Stokes's personality, we shall look at the events. Stokes became engaged to marry Mary Susanna Robinson, the daughter of the astronomer at Armagh Observatory in Ireland. In [3] a number of letters from Stokes to Mary Susanna Robinson are given. On 21 January 1857 he wrote of his feelings for her:I was capable of being moved, mathematically, as it were, by the belief that a particular course was right; and I do believe that God put these views in my mind, working by means of that which was in me to supply that which was wanting. Three days later he wrote that she had stopped him becoming an old bachelor:I feel that perhaps my marriage with you would be even the turning-point of my salvation. A further three days later he wrote:You are quite right in saying that it is well not to go brooding over one's own thoughts and feelings, and in a family that is easy, but you don't know what it is to live utterly alone. On the 31 March 1857 he wrote again expressing his feelings in rather mathematical terms:I too feel that I have been thinking too much of late, but in a different way, my head running on divergent series, the discontinuity of arbitrary constants, ... I often thought that you would do me good by keeping me from being too engrossed by those things. These letters clearly did not express the love that Mary hoped to find in them and when Stokes wrote her a 55 page letter (which was possibly deliberately destroyed) about the duty he felt towards her, she came close to calling off the wedding at the last moment. On receiving her letter showing that she was unhappy to go ahead with the marriage Stokes replied:Then it is right that you should even now draw back, nor heed though I should go to the grave a thinking machine unenlivened and uncheered and unwarmed by the happiness of domestic affection. The marriage did go ahead and Stokes certainly turned away from his life of intense mathematical research. It may appear from the above quotations that in fact Stokes was really looking for this change in his life and perhaps he sought marriage partly so that this change in his life-style could come about. At that time, fellows at Cambridge had to be unmarried, and so on his marriage in 1857 Stokes had to give up his fellowship at Pembroke College. However, a change in the rules in 1862 allowed married men to hold fellowships and he was able to take up the fellowship at Pembroke again. Stokes continued as secretary of the Royal Society from his appointment in 1854 until 1885 when he was elected President of the Society. He held the position of President until 1890. He was also president of the Victoria Institute from 1886 until his death in 1903. There were other administrative tasks which he undertook. In 1859 he had written to Thomson saying:I have another iron in the fire now: I have just been appointed an additional secretary of the Cambridge University Commission. P G Tait mentioned this in his criticism of the way that science was organised in Britain [14]:What a comment on things as they are is furnished by the spectacle of genius like that of Stokes wasted on the drudgery of Secretary to the Commissioners for the University of Cambridge; or of a Lecturer in the School of Mines; or the exhausting labour and totally

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stokes.html (4 of 6) [2/16/2002 11:33:41 PM]

Stokes

inadequate remuneration of a Secretary to the Royal Society. Stokes received the Copley medal from the Royal Society of London in 1893 and he was given the highest possible honour by his College when he served as Master of Pembroke College in 1902-3. Stokes's influence is summed up well by Parkinson in [1]:... Stokes was a very important formative influence on subsequent generations of Cambridge men, including Maxwell. With Green, who in turn had influenced him, Stokes followed the work of the French, especially Lagrange, Laplace, Fourier, Poisson and Cauchy. This is seen most clearly in his theoretical studies in optics and hydrodynamics; but it should also be noted that Stokes, even as an undergraduate, experimented incessantly. Yet his interests and investigations extended beyond physics, for his knowledge of chemistry and botany was extensive, and often his work in optics drew him into those fields. One notable omission from his publication list was a treatise on light. This omission was in part due to the change in his research output after 1857 but it was also partly due to not wishing to report upon speculative ideas in a field which was in a rapid state of progress. Stokes's failure to publish a treatise on optics is discussed in detail in [6]. However, he did lecture on optics in his Burnett lectures at the University of Aberdeen in 1891-93 and these lectures were published. Stokes's mathematical and physical papers were published in 5 volumes, the first 3 of which Stokes edited himself in 1880, 1883 and 1891. The last 2 were edited by Sir Joseph Larmor with the work being completed in 1905. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (16 books/articles) A Poster of George Stokes

Mathematicians born in the same country

Cross-references to History Topics

Special relativity

Other references in MacTutor

Chronology: 1840 to 1850

Honours awarded to George Stokes (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1851

Royal Society Copley Medal

Awarded 1893

Lucasian Professor of Mathematics

1849

Lunar features

Crater Stokes

Planetary features

Crater Stokes on Mars

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stokes.html (5 of 6) [2/16/2002 11:33:41 PM]

Stokes

Other Web sites

1. Bob Bruen 2. Dublin, Ireland 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stokes.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stokes.html (6 of 6) [2/16/2002 11:33:41 PM]

Stolz

Otto Stolz Born: 3 May 1842 in Hall (now Solbad Hall in Tirol), Austria Died: 25 Oct 1905 in Innsbruck, Austria

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Otto Stolz was born in Innsbruck and studied in Innsbruck, Vienna, then in Berlin from 1869 to 1871. He also studied at Göttingen before becoming a professor at his home town of Innsbruck. Despite many attempts to get him to accept a chair in Vienna, he remained in his home town building up a strong reputation despite low funding and only a few students but attracting colleagues like Gegenbauer. His topics of research were wide ranging but his most important work was in analysis where he wrote one of the first books on Weierstrass style analysis. He also returned to infinitesimals of the sort condidered by Newton and attempted to make this approach rigorous. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stolz.html (1 of 2) [2/16/2002 11:33:43 PM]

Stolz

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stolz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stolz.html (2 of 2) [2/16/2002 11:33:43 PM]

Stone

Marshall Harvey Stone Born: 8 April 1903 in New York, USA Died: 9 Jan 1989 in Madras, India

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Marshall Stone's father was a distinguished lawyer and the family tradition would have had him follow his father's subject. He studied at Harvard from 1919 to 1922, then was appointed an instructor at Harvard for session 1922/23 to see whether he would enjoy teaching mathematics and whether he would take his mathematical studies further. Indeed he did rapidly decide that he wanted to pursue a career in mathematics and studied for his doctorate under Birkhoff. His doctorate was awarded in 1926 for a thesis entitled Ordinary Linear Homogeneous Differential Equations of Order n and the Related Expansion Problems. By 1925 he was appointed to Columbia University, in 1927 to Harvard. During this period Stone's interests followed very much those of his research supervisor Birkhoff. He published eleven papers on the theory of orthogonal expansions between 1925 and 1928. In these papers a special role is played by expansions in terms of the eigenfunctions of linear differential operators. Although he would return to Harvard again in 1933, Stone first accepted a post at Yale from 1931 to 1933. Back at Harvard in 1933 he was promoted to full professor there in 1937. During these years Stone's research took a number of directions. From 1929 he worked on self-adjoint operators in Hilbert space and included his results in a major publication of a 600 page book Linear transformations in Hilbert space and their applications to analysis. In 1932 he proved results on spectral theory, arising from group theoretical methods in quantum mechanics, which had been conjectured by Weyl. Then in 1934 he published two papers on Boolean http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stone.html (1 of 3) [2/16/2002 11:33:45 PM]

Stone

algebras. He made this study while attempting to understand more deeply the basics underlying his results on spectral theory. One particularly important result proved by Stone during this period was a substantial generalisation of Weierstrass's results on uniform approximation of continuous functions by polynomials. This result is now known as the Stone-Weierstrass theorem. During World War II Stone undertook secret war work and then in 1946 he left Harvard to take up the chairmanship of the mathematics department at the University of Chicago. He did an outstanding job in returning this famous research school to the eminence it had attained earlier by making appointments such as Weil, Chern and MacLane. From 1952 Stone stepped down as head of department in favour of MacLane but he remained at Chicago until he retired in 1968. His interests, which included cooking, are described in [1]:Of all Stone's many interests his love of travel was surely dominant. He began to travel when he was quite young and was on a trip to India when he died. ... Marshall Stone was a man with a very broad outlook and a wide range of interests who seems to have thought rather deeply about a number of issues. ... here was an unusually thoughtful man with a high degree of penetration and insight. ... he seemed well endowed with a quality which I can only describe as wisdom. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles)

A Quotation

A Poster of Marshall Stone

Mathematicians born in the same country

Honours awarded to Marshall Stone (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1943 - 1944

AMS Colloquium Lecturer

1939

AMS Gibbs Lecturer

1956

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stone.html (2 of 3) [2/16/2002 11:33:45 PM]

Stone

JOC/EFR June 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stone.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stone.html (3 of 3) [2/16/2002 11:33:45 PM]

Stott

Alicia Boole Stott Born: 8 June 1860 in Cork, Ireland Died: 17 Dec 1940 in England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alicia Boole was the third daughter of George Boole. George Boole died when Alicia was only four years old and she was was brought up partly in England by her grandmother, partly in Cork by her great-uncle. When she was twelve years old she went to London where she joined her mother and sisters. With no formal education she suprised everyone when, at the age of eighteen, she was introduced to a set of little wooden cubes by her brother-in-law Charles Howard Hinton. Alicia Boole experimented with the cubes and soon developed an amazing feel for four dimensional geometry. She introduced the word 'polytope' to describe a four dimensional convex solid. MacHale, in [3], writes:She found that there were exactly six regular polytopes on four dimensions and that they are bounded by 5, 16 or 600 tetrahedra, 8 cubes, 24 octahedra or 120 dodecahedra. She then produced three-dimensional central cross-sections of all the six regular polytopes by purely Euclidean constructions and synthetic methods for the simple reason that she had never learned any analytic geometry. She made beautiful cardboard models of all these sections.... After taking up secretarial work near Liverpool in 1889 she met and married Walter Stott in 1890. Stott learned of Schoute's work on central sections of the regular polytopes in 1895 and Alicia Stott sent him photographs of her cardboard models. Schoute came to England and worked with Alicia Stott, persuading her to publish her results which she did in two papers published in Amsterdam in 1900 and 1910. The University of Groningen honoured her by inviting her to attend the tercentenary celebrations of the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stott.html (1 of 2) [2/16/2002 11:33:48 PM]

Stott

university and awarding her an honorary doctorate in 1914. In 1930 she was introduced to Coxeter and they worked together on various problems. Alicia Stott made two further important discoveries relating to constructions for polyhedra related to the golden section. Coxeter described his time doing joint work with her saying:The strength and simplicity of her character combined with the diversity of her interests to make her an inspiring friend. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Stott.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Stott.html (2 of 2) [2/16/2002 11:33:48 PM]

Struik

Dirk Jan Struik Born: 30 Sept 1894 in Rotterdam, Netherlands Died: 21 Oct 2000

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Dirk Struik's father was a teacher. Dirk attended the Hogere Bugerschool in The Hague. This was a special type of school which was designed for children of middle class parents who were aspiring to better their status. Struik developed left wing views while at school, being influenced by one of the teachers. The Hogere Bugerschools allowed entry to the university system after passing additional examinations and this Struik did entering the University of Leiden in 1912. Struik commuted from Rotterdam to Leiden University by train, so he never became fully involved with student life. He described his undergraduate years in [8]:No one ever told you which lectures to hear if you wanted to pass your exams; the grapevine took care of that, and you just followed the others who were in the same boat. No deans, provosts, marshals, student advisors, psychologists, and other such academic sages. No fortnightly tests either; there were just two exams in four years ... The range of courses studied by Struik included mathematics and physics. In physics he was taught by Lorentz and de Sitter. Lorentz retired in 1912 when Ehrenfest was appointed to his chair. Struik was strongly influenced by Ehrenfest and attended the weekly seminar which he set up. In 1917, while working on a dissertation, his funds ran out and he left the university to take up a post teaching mathematics in a school in Alkmaar, north of Amsterdam. However, he received a letter from J A Schouten asking if he would like to become his assistant. Ehrenfest had recommended Struik and, after worrying about leaving a teaching post he enjoyed, he decided to accept Schouten's offer and joined him in Delft. Struik decided to change to the topic he was

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Struik.html (1 of 3) [2/16/2002 11:33:50 PM]

Struik

studying with Schouten, tensor analysis, for his doctoral thesis and he presented his dissertation on applications of tensor methods to Riemannian manifolds in 1922. Struik was appointed to a post at the University of Utrecht in 1923. The same year he married a Czech mathematician Ruth Ramler who had obtained a doctorate in mathematics from the University of Prague under the supervision of G Pick and G Kowalewski. However Struik was by this time known for left wing views and this made his academic future look uncertain. In September 1924, funded by a Rockefeller Fellowship, the Struiks spent nine months in Rome. There Dirk worked with Levi-Civita and Ruth worked with Enriques. They had the chance to meet many other mathematicians, Amaldi, Castelnuovo, Volterra and Bianchi who were working in Rome as well as Hadamard, Zariski and Aleksandrov who were visiting Rome. It was while Struik was in Rome that he first became interested in the history of mathematics. However this interest was about to take a new turn, for in June 1925, with an extension of his Rockefeller Fellowship he arrived in Göttingen. Klein, a great mathematical hero of Struik's, died only days after Struik arrived in Göttingen to work with Courant. Courant approached Struik to prepare an edition of Klein's lectures on the history of 19th century mathematics for publication. Another important event at Göttingen for Struik was meeting Norbert Wiener there. While at Göttingen, Struik made full use of the excellent library there to study the Renaissance mathematicians Ries, Rudolff, Apianus, Stifel and Stevin. When his Fellowship ended he returned to Delft but with no prospect of a permanent post. When he received offers from Otto Schmidt to go to Moscow and from Norbert Wiener to visit MIT he had to choose one but it was a hard choice. He decided to visit the USA and left for New York in November 1926. Struik was to work at MIT for the rest of his career. It was a career based on collaboration with Wiener, a continued collaboration with Schouten and an increasing involvement with the history of mathematics. The years of World War II brought changes as Alberts notes in [1]:During the war years much of the normal mathematical research activity at MIT came to a standstill. Some of the professors were involved in research for the military; others, including Struik, carried heavy teaching duties connected with the training of military personnel. Aside from this, Struik spent much of his time pursuing an entirely new research project: to study the origins of American science in their social and economic setting, a subject that had barely been touched on by historians before this time. Even more significantly, the dialectical-materialist approach Struik adopted towards this subject was unprecedented. Struik's Marxist views, however, were bound to lead to trouble in the McCarthy era and indeed this is exactly what happened. At first the McCarthy period was, as Struik put it:... half reminiscent of Nazi Germany, half of Alice in Wonderland. In April 1949 he was accused by an F.B.I. informant. By July 1951 he was charged with being a member of the Communist Party and having taught Marxism. He was brought before the House Un-American Activities Committee and, on legal advice, invoked the Fifth Amendment and refused to answer each of over 200 questions that were put to him. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Struik.html (2 of 3) [2/16/2002 11:33:50 PM]

Struik

Struik was later indicted on charges and bail set at $1000 which was put up by friends who supported him. He was suspended from MIT, with salary, while he was indicted. Since there was no real evidence against Struik the case was not brought, but on the other hand it was not dropped until 1955. During this period Struik concentrated on historical projects, having been prevented from teaching. By late 1955 Struik was reinstated in his teaching duties and held these until 1960 when, at the age of 65, he had to retire. MIT refused him an Emeritus position and his attempts to find positions in other universities in the United States failed. He eventually accepted invitations from Puerto Rico, Costa Rica and Utrecht. He turned his attention to a number of topics of special interest to him, in particular to promoting the history of the sciences, especially mathematics, in Latin America. In [12] a former student at MIT described Struik's teaching:He taught mathematics not as some esoteric mystery, but as practical common sense. And yet, at the same time he gave us a glimpse of the sheer beauty of it. It was at this time that I understood Edna St Vincent Millay's line "Euclid alone has looked on beauty bare". Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Struik.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Struik.html (3 of 3) [2/16/2002 11:33:50 PM]

Study

Eduard Study Born: 23 March 1862 in Coburg, Germany Died: 6 Jan 1930 in Bonn, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eduard Study was the son of a teacher. He studied mathematics and sciences at the universities of Jena, Strasbourg, Leipzig and Munich from 1880. He obtained his doctorate from the University of Munich in 1884. In 1885 Study was appointed as a lecturer in mathematics at the University of Leipzig where he was greatly influenced by Gordan, but in general he was largely self taught in mathematics and this was to show in his very individual approach. After three years at Leipzig, Study moved to the University of Marburg. In 1893 he visited the United States where he taught at several universities but he was mainly based at Johns Hopkins University. On his return to Germany in 1894 he was appointed extraordinary professor of mathematics at Göttingen. Again Study was to move after three years, this time to a full professorship at Greifswald. In 1904 he made his final move when he accepted the chair at the University of Bonn which had been left vacant on the death of Lipschitz in October 1903. Study held the chair at Bonn until he retired in 1927. Study became a leader in the geometry of complex numbers. He reformulated, independently of Severi, the fundamental principles of enumerative geometry due to Schubert. He also worked on invariant theory helping to develop a symbolic notation. In 1923 he published important work on real and complex algebras of low dimension publishing these results. Study's contribution is summarised by W Burau in [1] as follows:... Study demonstrated what he considered to be a thorough treatment of a problem. ... With

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Study.html (1 of 2) [2/16/2002 11:33:52 PM]

Study

Corrado Segre, Study was one of the leading pioneers in the geometry of complex numbers. ... Adept in the methods of invariant theory ... Study, employing the identities of the theory, sought to demonstrate that geometric theorems are independent of coordinates. ... Study was the first to investigate systematically all algebras possessing up to four generators over R and C. Other areas which Study worked on were straight lines in elliptic space, with his student at Bonn J L Coolidge, and he simplified the method of differential operators. In 1903 he published Geometrie der Dynamen which considered euclidean kinematics and the mechanics of rigid bodies. In [1] the impact of Geometrie der Dynamen is described :Unfortunately, because of its awkward style and surfeit of new concepts, this work has never found the public it merits. Study remained in Bonn after his retirement and died of cancer three years later. One final fact about Study is of interest. He had always been interested in biology from his student days and one of the ways that he continued this interest through his life was by having an impressive collection of butterflies. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1920 to 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Study.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Study.html (2 of 2) [2/16/2002 11:33:52 PM]

Sturm

Jacques Charles François Sturm Born: 22 Sept 1803 in Geneva, Switzerland Died: 18 Dec 1855 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles-François Sturm's father was Jean-Henri Sturm whose family came from Strasbourg to settle in Geneva about 50 years before Charles-François's birth. Jean-Henri Sturm was a teacher of arithmetic who had married Jeanne-Louise-Henriette Gremay. Charles-François's parents gave him a good education and at school he showed great promise, particularly in Greek and Latin poetry for which he had a remarkable talent. Sturm came from a Protestant family and, in order to learn German, he attended the local Lutheran church where sermons were preached in that language. When Sturm was sixteen years old his father died and he changed tack in his academic studies, leaving the humanities and taking up the study of mathematics. He was taught mathematics at Geneva Academy by Simon Lhuilier in 1821 and immediately Lhuilier recognised the mathematical genius in Sturm. However, Lhuilier was over seventy years of age and close to retiring at this time so it was his successor Jean-Jacques Schaub who inspired Sturm. Now Schaub did more than teach Sturm mathematics for he supported him financially at the Academy. Sturm's family had been left in considerable financial difficulties on the death of his father so the financial assistance allowed Sturm to continue with his education. At the Academy Sturm's best friend was Daniel Colladon and the friendship would have a marked influence on Sturm's early research career. After leaving the Academy, Sturm was appointed as a tutor to the youngest son of Mme de Staël at the Châteaux de Coppet close to Geneva. He took up his appointment in May 1823 and found that it left him plenty of free time to devote to his own studies. He used his time well and began to write articles on geometry which were published in Gergonne's Annales de mathématique pures et appliquées. Before the end of 1823 the family moved from the château to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sturm.html (1 of 4) [2/16/2002 11:33:54 PM]

Sturm

spend six months in Paris and Sturm, as tutor, naturally accompanied them. In Paris he was introduced into the scientific circles by the family. Sturm wrote to his friend Colladon (see [1]):As for M Arago, I have two or three times been among the group of scientists he invites to his house every Thursday, and there I have seen the leading scientists, Laplace, Poisson, Fourier, Gay-Lussac, Ampère, etc. ... I often attend the meetings of the Institute that take place every Monday. This was clearly an extremely fortunate opportunity for Sturm. Although he returned to the château in May 1824 he left after six further months to devote himself to scientific research. The Paris Academy had set a prize topic on the compressibility of water and Sturm, with his friend Colladon, decided to begin experiments on Lake Geneva with the aim of putting in an entry for the prize. The experiments were not a great success since they did not yield the expected results and Colladon received a serious injury to his hand while conducting the experiments. In December 1825 Sturm and Colladon went to Paris to take courses in mathematics and physics and also to collect further instruments to repeat their experiments. The Paris contacts that Sturm had made proved useful for he lived at Arago's house for a while as tutor to his son. He was also given the use of Ampère's laboratory. The time was very fruitful for Sturm who attended lectures by Ampère, Gay-Lussac, Cauchy, and Lacroix. Fourier suggested projects for both Sturm and Colladon, recognising that Colladon was essentially a physicist while Sturm was a mathematician. Despite completing their paper for the Grand Prix of the Académie des Sciences they did not win the prize; in fact none of the submissions was deemed good enough and the same topic was set again. By this time Sturm and Colladon were both working as assistants to Fourier. Colladon made further experiments on Lake Geneva and after revising their joint memoir they successfully won the prize. The value of the prize was enough to allowed Sturm and Colladon to continue their research in Paris. This point marked the end of their successful collaboration and the two embarked on different research projects. Sturm's theoretical work in mathematical physics involved the study of caustic curves, and poles and polars of conic sections. One of Sturm's most famous papers Mémoire sur la résolution des équations numériques was published in 1829. It considered the problem of determining the number of real roots of an equation on a given interval. The problem was a famous one with a long history having been considered by Descartes, Rolle, Lagrange and Fourier. The first to give a complete solution was Cauchy but his method was cumbersome and impractical. Sturm achieved fame with his paper which, using ideas of Fourier, gave a simple solution. Hermite wrote:Sturm's theorem had the good fortune of immediately becoming a classic and of finding a place in teaching that it will hold forever. His demonstration, which utilises only the most elementary considerations, is a rare example of simplicity and elegance. Strangely although the theorem quickly became a classic it was soon relegated to history and, contrary to what Hermite believed, vanished from textbooks. As the title indicated, two events in the history of the algebraic theorem of Sturm are examined in [5]. The author describes how Tarski showed in 1940 that Sturm's method of proof could be used in mathematical logic to prove the completeness of elementary

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sturm.html (2 of 4) [2/16/2002 11:33:54 PM]

Sturm

algebra and geometry. The 1829 paper was not the last of Sturm's work on this algebraic equations and in [12] Sinaceur:... seeks to determine the mutual influence between A-L Cauchy's and Ch-F Sturm's research from 1829 to around 1840 on the roots of algebraic equations. Paris was not an easy place for a foreigner and Protestant to obtain a post at this time and, despite his fame from the 1829 paper, he was not appointed. The revolution of July 1830 changed the political climate and after this Arago succeeded in getting Sturm appointed as professor of mathematics in the Collège Rollin. He became a French citizen in 1833 and was elected to the Académie des Sciences in 1836. These were the years during which he published some important results on differential equations. Sturm became interested in obtaining results on specific differential equations which occurred in Poisson's theory of heat. Liouville was also working on differential equations derived from the theory of heat. Papers of 1836-1837 by Sturm and Liouville on differential equations involved expansions of functions in series and is today well-known as the Sturm-Liouville problem, an eigenvalue problem in second order differential equations. He worked at the Ecole Polytechnique in Paris from 1838 where he became a professor of analysis and mechanics in 1840. In the same year he succeeded Poisson in the chair of mechanics in the Faculté des Sciences, Paris. For around ten years he gave excellent lectures but his wish to give his students the best possible courses meant that he gave a great deal of his time to preparing his lecture courses on differential and integral calculus and on rational mechanics. These courses became the widely used texts Cours d'analyse de l'Ecole Polytechnique 2 Vol. (1857-63) and Cours de méchanique de l'Ecole Polytechnique 2 Vol. (1861) both published posthumously. His time for research was now limited but he still made important contributions undertaking research on infinitesimal geometry, projective geometry and the differential geometry of curves and surfaces. He also did important work on geometrical optics. From 1851 his health began to fail and despite brave attempts to overcome the problem and return to teaching (which he managed to do for a while) he died after a long illness.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) A Poster of Charles-François Sturm Cross-references to History Topics

Mathematicians born in the same country 1. Doubling the cube 2. Trisecting an angle 3. Matrices and determinants

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sturm.html (3 of 4) [2/16/2002 11:33:54 PM]

Sturm

Honours awarded to Charles-François Sturm (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1840

Royal Society Copley Medal

Awarded 1840

Commemorated on the Eiffel Tower Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sturm.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sturm.html (4 of 4) [2/16/2002 11:33:54 PM]

Sturm_Rudolf

Friedrich Otto Rudolf Sturm Born: 6 Jan 1841 in Breslau, Germany (now Wroclaw, Poland) Died: 12 April 1919 in Breslau, Germany (now Wroclaw, Poland) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Rudolf Sturm's father was a businessman. After attending the St Maria Magdalena Gymnasium in Breslau he entered Breslau University in October 1859 to study mathematics and physics. There he was taught by Schroeter who encouraged him to study synthetic geometry. He was awarded a doctorate by Breslau in 1863 for a dissertation entitled De superficiebus tertii ordinis disquisitiones geometricae . In this work he studied third degree surfaces in their projective representations and proved theorems which had been stated, but not proved, by Steiner. After the award of his doctorate he taught at Breslau as an assistant. He continued to work on surfaces and, in 1864, he shared with Cremona the Steiner prize of the Berlin Academy for his investigations of surfaces. In 1866 he became a science teacher in Bromberg, which is the German name for the city now called Bydgoszcz in northern Poland. The year after he took up the post in Bromberg he published Synthetische untersuchungen über Flächen which collected together his prize winning results and his other work in the area. In 1872 Sturm was appointed assistant professor at the Technical College in Darmstadt where he taught descriptive geometry and graphic statics. In order to provide a good teaching book for his students, Sturm published a textbook Elemente der darstellenden Geometrie on descriptive geometry and graphical statics for his students in 1874. He became an ordinary professor at Münster in 1878, then he returned to Breslau in 1892 where he again held an ordinary professorship. He remained in this post until his death. Sturm wrote extensively on geometry and, other than the teaching textbook on descriptive geometry and graphical statics which we mentioned above and one other teaching text Maxima und Minima in der elementaren Geometrie which he published in 1910, all his work was on synthetic geometry. He wrote a three volume work on line geometry published between 1892 and 1896, and a four volume work on projective geometry, algebraic geometry and Schubert's enumerative geometry the first two volumes of which he published in 1908 and the second two volumes in 1909. These two multi-volume works collect together most of his life's research. Let us first comment on the three volume work, which was the biggest treatise ever to be written on line geometry. There of course a tension in this topic between the totally geometric approach and the algebraic approach. Sturm was a staunch advocate of the former approach even when tackling topics http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sturm_Rudolf.html (1 of 2) [2/16/2002 11:33:56 PM]

Sturm_Rudolf

where the algebraic approach would have been markedly simpler. I [EFR] certainly remember examination questions which one of my lecturers would set asking "without using such and such a theorem, prove that ...". It always seemed to me to be asking one to do mathematics with one hand tied behind your back and to a certain extent that is the effect of Sturm's approach. However, he would have argued that his approach was consistent through this large piece of mathematics and indeed this was the case. It is also true that much of what he treats in the text is particularly well suited to his approach, while only a relatively small amount of the material could have been handled more simply with algebraic methods. Burau writes in [1] that:... in the first two volumes Sturm treated linear complexes, congruences, and the simplest ruled surfaces up to tetrahedral complexes, all of which can be particularly well handled in a purely geometric fashion. He did not systematically investigate the remaining quadratic complexes until volume three, where the difficulties of his approach - as compared with an algebraic treatment - place many demands on the reader. Sturm's four volume work contains over 1800 pages. It examines geometric relationships, in particular transformations such as Cremona transformations. The work in some respects represents the crowning achievement of synthetic geometry developed in Sturm's style. The subject had little in the way of opportunities for further progress and as a consequence, despite Sturm supervising quite a few doctoral students, nevertheless he did not build a school to continue to develop his mathematical ideas. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sturm_Rudolf.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sturm_Rudolf.html (2 of 2) [2/16/2002 11:33:56 PM]

Subbotin

Mikhail Fedorovich Subbotin Born: 29 June 1893 in Ostrolenka (now Ostroleka), Lomzhinsk guberniya, Russia (now Poland) Died: 26 Dec 1966 in Leningrad, USSR (now St Petersburg, Russia) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Mikhail Fedorovich Subbotin's father, Fedor Subbotin, was an army officer. Subbotin entered the Faculty of Physics and Mathematics at Warsaw University in 1910. At this time the University of Warsaw was a Russian university and Warsaw was in "Vistula Land", as it was called, under Russian control. Russia had implemented a policy to have "Vistula Land" dominated by Russian culture. In a policy implemented between 1869 and 1874, all secondary schooling was in Russian. Warsaw only had a Russian language university after the Polish University of Warsaw was closed by the Russian regime in 1869. At the Russian University of Warsaw, Subbotin was awarded the Copernicus Scholarship after a competition run by the department. Although mainly interested in mathematics at this stage, he did begin to find an interest in astronomy when he worked as a calculator for the university observatory. Subbotin graduated from the University of Warsaw in 1914 and began his career with an appointment as a junior astronomer at the university. World War I began in 1914 and, by the end of the year, Russian forces controlled almost all of Galicia. However, the Central Powers (Germany and Austria- Hungary) recaptured Galicia and large parts of Congress Poland. The Russian University of Warsaw was evacuated to Rostov-on-Don in 1915 and Subbotin went with the rest of the university. A German governor general was installed in Warsaw and a new Kingdom of Poland was declared on 5 November 1916 but by this time Subbotin was working for his Master's Degree in Rostov-on-Don. In 1917 Subbotin submitted his thesis and was awarded his Master's Degree in Rostov-on-Don. Despite his early appointment as a junior astronomer, Subbotin's work at this time was firmly in the area of mathematics. He had already published two papers prior to submitting his Master's thesis, one was On the determination of singular points of analytic functions while the second was published in France and was on singular points of certain differential equations. From the evacuated university in Rostov-on-Don, Subbotin moved to the Polytechnic Institute in Novocherkassk. First he was appointed there as an assistant, but soon was made a Privatdozent, then promoted to professor of mathematics. In 1921, while progressing well in his career in Novocherkassk, Subbotin received an invitation to work at the Main Russian Astrophysical Laboratory. He accepted the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Subbotin.html (1 of 3) [2/16/2002 11:33:57 PM]

Subbotin

offer and left Novocherkassk in 1922 to take up a position as Director of the Russian Astrophysical Laboratory in Tashkent. The Russian Astrophysical Laboratory in Tashkent had been created out of the Tashkent Observatory and, in 1925, the Tashkent Observatory became independent again with Subbotin as the first Director of the newly re-established Observatory. From 1930 he worked in astronomy and celestial mechanics at Leningrad (St Petersburg) University being appointed as head of the astronomy department there. He held a variety of posts such as Chairman of the Department of Celestial Mechanics (1935-44), Head of the Theoretical Section of Pulkovo Observatory (1931-34), and Head of Leningrad University Observatory (1934-39). World War II began near the end of 1939 and, in June 1941, Germany invaded the Soviet Union. By September 1941 German armies and the armies of their ally Finland were close to Leningrad. The whole population of the city took part in its defence building antitank fortifications round the city while 200,000 troops of the Russian army defended the city. By November 1941 Leningrad was almost completely cut off. The siege took a huge toll on the population with 650,000 dying in 1942 from starvation, disease, or more directly from German artillery. In 1942 one million of the children, the sick, and the elderly were evacuated from the city. Subbotin nearly died of hunger in the siege but, seriously ill, he was evacuated in February 1942 just in time to save his life. It is worth noting that Leningrad held out and by January 1944 the German troops were driven back and the siege lifted. Subbotin had been taken to Sverdlovsk in February 1942 to recover his health. The University of Leningrad was re-established at Saratov and by the end of 1942 Subbotin was in Saratov as the Director of the Leningrad Astronomical Institute. Subbotin recommended that the Institute become the Institute of Theoretical Astronomy of the USSR Academy of Sciences in 1943 and his recommendation was followed. When the situation allowed, the Institute was returned to Leningrad with Subbotin as its head. Subbotin's early work was in the theory of functions and probability. Later he worked in celestial mechanics producing new methods of calculating orbits from three observations based on solving the Euler-Lambert equations. His astronomy work used his mathematical skills and [1]:... Subbotin not only showed the possibility of improving the convergence of the trigonometric series by which the behaviour of perturbing forces is represented, but also gave an expression for determining Laplace coefficients and presented formulas for computing the coefficients of the necessary members of the trigonometric series. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Subbotin.html (2 of 3) [2/16/2002 11:33:57 PM]

Subbotin

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Subbotin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Subbotin.html (3 of 3) [2/16/2002 11:33:57 PM]

Suetuna

Zyoiti Suetuna Born: Nov 1898 in Kunisaki, South Island, Japan Died: Aug 1970 in Tokyo, Japan Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

After being educated in his home town, Zyoiti Suetuna moved to Tokyo in 1916 when he began to study at the First High School. Certainly his talents had been recognised at his local school for the First High School was only designed for the most talented pupils to finish their school education before entering university. At the First High School Suetuna's mathematical talents became clear to his teachers and in 1919 he entered Tokyo University to take a degree in mathematics. This was an exciting period to study at Tokyo University for Takagi published his famous paper on class field theory in 1920. When Suetuna was in his final undergraduate year his studies were supervised by Takagi and this inspired Suetuna to work on number theory. He graduated from Tokyo University in 1922 and was appointed as a lecturer at Kyushu University. While he was there he read Landau's two volume work on the distribution of the primes and then began to read the papers of Hardy and Littlewood which were being published at that time. In particular he read Hardy and Littlewood's paper The approximate functional equation in the theory of the zeta function with applications to the divisor problems of Dirichlet and Piltz which appeared in the Proceedings of the London Mathematical Society. This paper formed the basis for the first sixteen of Suetuna's papers which were on L-functions (generalisations of the zeta function) and appeared from 1924 up to 1931. Suetuna taught for two years at Kyushu University, being promoted to associate professor during that period, then returned to Tokyo University in 1924. In 1927 Suetuna went to study in Europe, in particular spending two years at Göttingen with Landau's school. While there he became particularly interested in the work on the foundations of mathematics which was being studied by Hilbert's school. He attended lectures by Bernays on the foundations of mathematics during his stay in Göttingen. Another active research group in Göttingen at this time was the algebra group led by Emmy Noether. The style of mathematical research carried out by this group with its lively discussion seminars impressed Suetuna so much that he went on to introduce this research style to Japan on his return. Now 1927, the year Suetuna went to Europe, was the one in which Artin published his general reciprocity law which in some sense completed the proofs of the ideas Takagi had introduced in 1920. Suetuna was fascinated by this result of Artin and he went to Hamburg in 1929 to study with him. Hasse was also proving important results in this area and Suetuna collaborated with Hasse visiting him in Halle in 1929 to complete work on their joint paper A general divisor problem.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Suetuna.html (1 of 3) [2/16/2002 11:33:59 PM]

Suetuna

The European trip had been extremely productive for Suetuna and when he returned to Tokyo University in 1931 he had gained greatly from the experience. One purpose of his visit had been to learn probability and statistics, for these topics did not have active researchers in Japan at this time and Tokyo University had sent him to Germany to gain expertise in these areas. Suetuna did not spent a great deal of time attending lectures on probability and statistics, preferring to work on his own research topics. However he did read books on probability and statistics while in Germany and by the time he returned to Japan he had gained considerable expertise in probability and statistics despite concentrating on his research topics in algebra and number theory. Back in Japan Suetuna introduced a weekly seminar on the Göttingen style where new research results could be discussed. He continued to publish research papers on topics related to Artin's 1927 paper and he also wrote several books: one on algebra and number theory, one on analytic number theory, and one on probability. All were based on lecture courses he gave, the probability book being based on a ten lecture course he was invited to give at Hokkaidu University in 1940. The Analytical theory of numbers was originally published in the form of lecture notes but in 1950 a revised edition was published which incorporated recent developments of the theory. Ikehara, reviewing the book , writes:This book, based mainly on the Riemann zeta-functions and L-functions, is a unique exposition of the analytical theory of numbers in a modern sense as can be seen from the chapter headings: I) Riemann's zeta-functions; II) Hecke's L-functions; III) Dirichlet's L-functions; and IV) Artin's L-series. Suetuna was appointed to Takagi's chair in 1936 when Takagi retired; Suetuna had been promoted to full professor in the previous year. The Second World War disrupted life in Japan and in particular it essentially ended Suetuna's research career. After the War he published some articles on the foundations of mathematics and mathematical philosophy which were areas of interest for him. He was particularly interested in Buddhism and this strongly influenced his philosophical thinking. The paper [2] details his publications consisting of thirty works on number theory, twelve on the foundations of mathematics and eight on Buddhist philosophy. These eight publications make Suetuna an important figure in Buddhist philosophy and probably mean that he is more famous for that topic than he is for mathematics. Suetuna was elected to the Japan Academy in 1947. he retired from his chair in Tokyo University in 1959 and was appointed as Director of the Institute of Statistical Mathematics. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Suetuna.html (2 of 3) [2/16/2002 11:33:59 PM]

Suetuna

Glossary index

Quotations index

Mathematicians of the day JOC/EFR September 2001

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Suetuna.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Suetuna.html (3 of 3) [2/16/2002 11:33:59 PM]

Suter

Heinrich Suter Born: 4 Jan 1848 in Hedingen, Zurich Canton, Switzerland Died: 17 March 1922 in Dornach, Switzerland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Heinrich Suter studied at the University of Zurich where he was taught by Christoffel, Reye, Geiser and Wolf. He then studied at Berlin under Kronecker, Kummer and Weierstrass. Suter obtained a doctorate from Zurich in 1871 on the history of mathematics. Suter taught mathematics and physics in Swiss schools and became the leading expert on Muslim mathematics. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Suter.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Suter.html [2/16/2002 11:34:00 PM]

Suvorov

Georgii Dmitrievic Suvorov Born: 19 May 1919 in Saratov, Russia Died: 12 Oct 1984 in Donetske, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georgii Suvorov was born in Saratov, in western Russia on the Volga River. He studied at Tomsk University, the first Siberian university which had been founded in 1888, graduating in 1941. He taught at Tomsk University until 1965 when he was appointed to head a department in the Institute of Applied Mathematics and Mechanics of the Academy of Sciences of the Ukraine in Donetske. Suvorov made major contributions to the theory of functions. He worked, in particular, on the theory of topological and metric mappings on 2-dimensional space. Another area which Suvorov worked on was the theory of conformal mappings and quasi-formal mappings. His results in this area, mostly from the late 1960s when he was at Donetske, are of particular significance. He extended Lavrentev's results in this area, in particular Lavrentev's stability and differentiability theorems, to more general classes of transformations. On of the many innovations in Suvorov's work was new methods which he introduced to help in the understanding of metric properties of mappings with bounded Dirichlet integral. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Suvorov.html (1 of 2) [2/16/2002 11:34:02 PM]

Suvorov

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Suvorov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Suvorov.html (2 of 2) [2/16/2002 11:34:02 PM]

Swain

Lorna Mary Swain Born: 22 March 1891 in Hampstead, London, England Died: 8 May 1936 in Cambridge, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Lorna Swain was educated at South Hampstead High School in London. From there she won a scholarship to study at Newnham College, Cambridge and she began her studies in 1910. In 1913 she graduated with a First Class honours degree in mathematics and was appointed to an assistant lecturer post at Newnham College with the start of the appointment delayed for a year to allow her to spend a year in Göttingen. However 1914 was not the best of year to begin research in Germany. Shortly after she arrived there war broke out and she hurriedly returned to England. Her mathematical books and papers fared less well and remained interned for several years before eventually being returned. Swain's mathematical interests were in fluid dynamics so, as Göttingen was out of the question, she went to Manchester to work with Horace Lamb. While in Manchester she published her first paper, a joint paper with Lamb. As planned, she returned to Newnham College in 1915 to take up her lecturing post. However World War I again affected her career since she now undertook war work at the Royal Aircraft Establishment. The war work she undertook was related to her expertise in fluids, and she worked on problem of vibration of propellers of aircraft. Her work was written up jointly with H A Webb and appeared as a Report of the Advisory Committee for Aeronautics in 1919. In 1920 Swain was appointed Director of Studies at Newnham College. With heavy teaching and administrative responsibilities she had less time to undertake research but she did publish an important work on motion through a viscous fluid in 1923 in the Proceedings of the Royal Society. Kennedy describes her commitment to teaching and education in [1] as follows:It was one of her guiding principles to do everything she could to encourage the study of mathematics not only in her own College but generally among women. For this reason she was an active member of the Mathematical Association and served on their committee to investigate the teaching of applied mathematics. ... She felt that school work could foster or spoil a gift for mathematics and that it was of the first importance that girls should have good teaching. Her own work, she thought, could contribute to the supply of teachers... all her students know how carefully she arranged the work for each individual, giving extra

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Swain.html (1 of 2) [2/16/2002 11:34:04 PM]

Swain

help herself when she thought it necessary. In 1926 Lorna Swain was promoted to College Lecturer and she then had the opportunity to give advanced courses on hydromechanics and dynamics. A years study leave in 1928-29 allowed her to spend the year at Göttingen that the war had robbed her of earlier. This was a time when she was able to undertake concentrated research and another important work On the turbulent wake behind a body of revolution was published in 1929. Lorna Swain is described in [1] as:... unassuming, reserved in manner, often silent, a singularly faithful friend. Her former pupils were sure of her interest and encouragement in their careers, of a warm and sincere welcome when they visited her, of prompt and valuable advice when they consulted her. Her death at a young age took away someone at the height of their teaching and research powers:Her wisdom and influence were increasing, her powers of mind were still expanding, when she was overtaken by illness and death. It would be idle to guess what more she might have done if she had lived; it is certain she contributed worthily to mathematical education and to mathematical thought. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Other Web sites

Agnes Scott College

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Swain.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Swain.html (2 of 2) [2/16/2002 11:34:04 PM]

Sylow

Peter Ludwig Mejdell Sylow Born: 12 Dec 1832 in Christiania (now Oslo), Norway Died: 7 Sept 1918 in Christiania (now Oslo), Norway

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ludwig Sylow studied at Christiania University and won a mathematics contest in 1853. He then took the high school teacher examination in 1856 and, as no university post was available, taught in the town of Frederikshald from 1858 to 1898. Sylow continued his mathematical studies however (see [2]) at first working on elliptic functions in the tradition of Abel and Jacobi, inspired by the professor of pure mathematics Ole Jacob Broch. Finding Abel's papers on the solvability of algebraic equations by radicals more interesting, Sylow was led from them (by the professor in applied mathematics, Carl Bjerknes) to Galois. In 1861 Sylow obtained a scholarship to travel and visited Berlin and Paris. In Paris he attended lectures by Chasles on the theory of conics, by Liouville on rational mechanics and by Duhamel on the theory of limits. He says, in the report he wrote at the end of the scholarship, that he also:made myself acquainted with newer works, particularly in the theory of equations. In Berlin he had useful discussions with Kronecker but was unable to attend courses by Weierstrass who was ill at the time. In 1862 Sylow lectured at the University of Christiania, substituting for Broch. In his lectures Sylow explained Abel's and Galois's work on algebraic equations. A summary of these lectures is presented in [2]. It is worth noting that although he had not proved 'Sylow's theorems' at this time (he published them 10 years later) he did pose a question about them. After proving Cauchy's theorem that a group of order

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sylow.html (1 of 3) [2/16/2002 11:34:05 PM]

Sylow

divisible by a prime p has a subgroup of order p, Sylow asks whether it can be generalised to powers of p. Between 1873 and 1881 Sylow and Lie prepared an edition of Abel's complete work. Lie said that most of the work was done by Sylow. However Sylow's fame rests on one 10 page paper published in 1872. In this paper Théorèmes sur les groupes de substitutions which Sylow published in Mathematische Annalen Volume 5 (pages 584 to 594) appear the three Sylow theorems. Cauchy had already proved that a group whose order is divisible by a prime p has an element of order p. Sylow proved what is perhaps the most profound result in the theory of finite groups. If pn is the largest power of the prime p to divide the order of a group G then i. G has subgroups of order pn, ii. G has 1 + kp such subgroups, iii. any two such subgroups are conjugate. Almost all work on finite groups uses Sylow's theorems. Sylow became an editor of Acta Mathematica and, in 1894, he was awarded an honorary doctorate from the university of Copenhagen. Lie had a special chair created for Sylow at Christiania University and Sylow taught at the university from 1898. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) A Poster of Ludwig Sylow

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1870 to 1880

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sylow.html (2 of 3) [2/16/2002 11:34:05 PM]

Sylow

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Sylow.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sylow.html (3 of 3) [2/16/2002 11:34:05 PM]

Sylvester

James Joseph Sylvester Born: 3 Sept 1814 in London, England Died: 15 March 1897 in London, England

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

James Joseph Sylvester attended two primary schools in London, then his secondary schooling was at the Royal Institution in Liverpool. In 1833 he became a student at St John's College, Cambridge. Two other famous mathematicians took the tripos examination in the same year as Sylvester, namely Duncan Gregory and George Green. Sylvester came second, Green who was 20 years older than the other two came fourth with Duncan Gregory fifth. (The mathematician who came first did little work of importance after graduating: this was not at uncommon in the 'speed test' which the tripos was at that time.) At this time it was necessary for a student to sign a religious oath to the Church of England before graduating and Sylvester, being Jewish, refused to take the oath necessary so could not graduate. For the same reason he was not eligible for a Smith's prize nor for a Fellowship. From 1838 Sylvester taught physics for three years at the University of London, one of the few places which did not bar him because of his religion. His former teacher De Morgan was one of his colleagues. At the age of 27 he was appointed to a chair in the University of Virginia but he resigned after a few months. A student who had been reading a newspaper in one of Sylvester's lectures insulted him and Sylvester struck him with a sword stick. The student collapsed in shock and Sylvester believed (wrongly) that he had killed him. He fled to New York boarding the first available ship back to England. On his return Sylvester worked as an actuary and lawyer but gave mathematics tuition. His pupils included Florence Nightingale. By good fortune Cayley was also a lawyer and both worked at the courts http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html (1 of 4) [2/16/2002 11:34:07 PM]

Sylvester

of Lincoln's Inn in London. Cayley and Sylvester discussed mathematics as they walked around the courts and, although very different in temperament, they became life long friends. Sylvester tried hard to return to being a professional mathematician and he applied for a lectureship in geometry at Gresham College, London in 1854 but he was not appointed. Another failed application was for the chair in mathematics at the Royal Military Academy at Woolwich, but, after the successful applicant died within a few months of being appointed, Sylvester became professor of mathematics at Woolwich. Sylvester did important work on matrix theory, a topic in which he became interested during walks with Cayley while they were at the courts of Lincoln's Inn. In 1851 he discovered the discriminant of a cubic equation and first used the name 'discriminant' for such expressions of quadratic equations and those of higher order. In particular he used matrix theory to study higher dimensional geometry. He also contributed to the creation of the theory of elementary divisors of lambda matrices. De Morgan was the first president of the London Mathematical Society. Sylvester became the second president of that society. He was the first recipient of the gold medal which the Society awarded to honour De Morgan. Being at a military academy Sylvester had to retire at age 55. At first it looked as though he might give up mathematics since he published his only book at this time and it was on poetry. Clearly Sylvester was proud of this work, entitled The Laws of Verse, since after this he sometimes signed himself "J. J. Sylvester, author of The Laws of Verse". For three years Sylvester appears to have done no mathematical research but then Chebyshev visited London and the two discussed mechanical linkages which can be used to draw straight lines. After working on this topic Sylvester lectured on it at an evening lecture at the Royal Institution entitled On recent discoveries in mechanical conversion of motion. One mathematician in the audience at this lecture was Kempe and he became absorbed by this topic. Kempe and Sylvester worked jointly on linkages and made important discoveries. In 1877 Sylvester accepted a chair at Johns Hopkins University and founded in 1878 the American Journal of Mathematics, the first mathematical journal in the USA. When Smith died in 1883 Sylvester, although 68 years old at this time, was appointed to the Savilian chair of Geometry at Oxford. However Sylvester only liked to lecture on his own research and this was not well liked at Oxford where students wanted only to do well in examinations. In 1892, at the age of 78, Oxford appointed a deputy professor in his place and Sylvester, by this time partially blind and suffering from loss of memory, returned to London where he spent his last years at the Athenaeum Club. Macfarlane [14] describes Sylvester in the following way: Sylvester was fiery and passionate ... Sylvester never wrote a paper without foot-notes, appendices, supplements, and the alterations and corrections in his proofs were such that the printers found their task well-nigh impossible. ... Sylvester satisfied the popular idea of a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html (2 of 4) [2/16/2002 11:34:07 PM]

Sylvester

mathematician as one lost in reflection, and high above mundane affairs. ... Sylvester was an orator, and if not a poet, he at least prided himself on his poetry. One of Sylvester's students at Johns Hopkins University describes his teaching there ...the substance of his lectures had to consist largely of his own work, and, as a rule, of work hot from the forge. The consequence was that a continuous and systematic presentation of any extensive body of doctrine already completed was not to be expected from him. Any unsolved difficulty, any suggested extension, such would have been passed by with a mention by other lecturers, became inevitably with him the occasion of a digression which was sure to consume many weeks, if indeed it did not take him away from the original object permanently. Nearly all of the important memoirs which he published, while in Baltimore, arose in this way. We who attended his lectures may be said to have seen these memoirs in the making. The following quote, from Thomas Hirst, tells us something about Sylvester's personality: On Monday having received a letter from Sylvester I went to see him at the Athenaeum Club. ... He was, moreover, excessively friendly, wished we lived together, asked me to go live with him at Woolwich and so forth. In short he was eccentrically affectionate. Sylvester sent the following puzzle to the Educational Times. It tells us of one of his hobbies as well as his interest in puzzles: I have a large number of stamps to the value of 5 pence and 17 pence only. What is the largest denomination which I cannot make up with a combination of these two different values. [The answer is 63 pence. Can you prove this!] Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (19 books/articles)

Some Quotations (12)

A Poster of James J Sylvester

Mathematicians born in the same country

Some pages from publications

A page from Sylvester's on a New Class of Theorems (1850) containing the first use of the word matrix .

Cross-references to History Topics

1. A comment from Thomas Hirst's diary 2. Matrices and determinants

Cross-references to Famous Curves Other references in MacTutor

Bicorn 1. Chronology: 1850 to 1860 2. Chronology: 1870 to 1880

Honours awarded to James J Sylvester (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1839

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html (3 of 4) [2/16/2002 11:34:07 PM]

Sylvester

Royal Society Copley Medal

Awarded 1880

Royal Society Royal Medal

Awarded 1861

London Maths Society President

1866 - 1868

LMS De Morgan Medal

Awarded 1887

Lunar features

Crater Sylvester

Savilian Professor of Geometry

1883

Other Web sites

1. The Prime Pages (The Prime Number Theorem) 2. Amsterdam, Netherlands (Some details of linkages) 3. A Ranicki (Some quotations) 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Sylvester.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Sylvester.html (4 of 4) [2/16/2002 11:34:07 PM]

Synge

John Lighton Synge Born: 23 March 1897 in Dublin, Ireland Died: 30 March 1995 in Dublin, Ireland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Synge's father Edward Synge could trace his family back to the fifteenth century. The origin of the name "Synge" is described in the introduction to General relativity : papers in honour of J L Synge :The name Synge is said to have originated with Henry VIII, who commanded a favourite choirboy to "Synge, Millington, synge". John Synge's mother, Ellen Price, was of Scottish origin, being descended from the Scottish Stuarts, in particular a Sir William Stuart who went to Ireland in the early seventeenth century. After being educated at St Andrews College in Dublin, Synge entered Trinity College, Dublin in 1915. There his achievements were quite remarkable, winning a Foundation Scholarship at the end of his first year. This scholarship was normally won by students in their third year of study. He was awarded his M.A. in 1919 in Mathematics and Experimental Physics. He received the Large Gold Medal for his outstanding work. After graduating, Synge was appointed to a lectureship in mathematics at Trinity College, but he only held the post for a short time, leaving for Canada in 1920. from 1920 to 1925 Synge was an assistant professor of mathematics at the University of Toronto. Then in 1925 he returned to Trinity College, Dublin where he was elected to a fellowship and also appointed to the chair of Natural Philosophy (the older name for Physics). In 1930 Synge headed back to North America and again was appointed to the University of Toronto, this time as Professor of Applied Mathematics. He spent some time in 1939 at Princeton University and, in 1941, he was a visiting professor at Brown University. In 1943 Synge was appointed as Chairman of the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Synge.html (1 of 4) [2/16/2002 11:34:10 PM]

Synge

Mathematics Department of Ohio State University, then, three years later, as Head of the Mathematics Department of the Carnegie Institute of Technology in Pittsburg. He also spent a short time as ballistic mathematician in the US Air Force during 1944-45. In 1948 he returned to Ireland to take up the post of Senior Professor in the School of Theoretical Physics of the Dublin Institute for Advanced Studies. This School, which had been set up when the Dublin Institute for Advanced Studies had been founded in 1940, had several outstanding members including Erwin Schrödinger who was also a Senior Professor. P S Florides [3] describes the range of topics that Synge worked on:Professor Synge made outstanding contributions to widely varied fields: classical mechanics, geometrical mechanics and geometrical optics, gas dynamics, hydrodynamics, elasticity, electrical networks, mathematical methods, differential geometry and, above all, Einstein's theory of relativity. Synge's most important contributions to theoretical physics were, in many respects, the product of his great skill and insight as a geometer [3]:He felt just as much at home in the ordinary three dimensional Euclidean space as in the four dimensional space-time of relativity. In an astonishing paper in the Proceedings of the Royal Irish Academy (... 1950) he was able, for the first time, to penetrate and explore in detail the region inside the Schwarzschild radius (what we now call a black hole). At a time when many relativists thought that it did not even make sense to talk about this region, this work is very remarkable indeed. Synge retired in 1972 and during his time at the Dublin Institute for Advanced Studies about 12% of all workers in relativity theory studied there. Professor Herman Bondi, who gave the first J L Synge Public Lecture in 1992, remarked:Every one of the other 88% has been deeply influenced by his geometric vision and the clarity of his expression. During his long scientific career, Synge published over 200 papers and 11 books. Three of his books were aimed at providing an understanding of science to a wider audience and, with such a fine expository style, he was well suited to write such works. As Florides says in [3]:Every single book and every single paper is a remarkable work of art. Synge's classic, written in 1956, had a large influence [1]:It is a remarkable fact that hardly a single space-time diagram is to be found in the standard texts on relativity before Synge's own presentation in 1956. To many, the 1956 book came as a revelation. Here was a royal road to relativity which did not involve precarious juggling with factors of (1 - v2/c2). The pervasive influence of the book can be traced in much of the best research in the last fifteen years [this written in 1971], with its new emphasis on invariant geometrical characterisation. The preface to the 1956 work contains the delightful thought:Splitting hairs in an ivory tower is not to everyone's liking and no doubt many a relativist looks forward to the day when governments will ask his advice on important questions. Synge's skill at communication was not only a gift that he had for writing, but also a gift in lecturing. His son-in-law Douglas Dryer wrote:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Synge.html (2 of 4) [2/16/2002 11:34:10 PM]

Synge

Those who have attended J. L.'s lectures well remember the effectiveness with which they were delivered. J. L. was by no means inattentive to achieving such effectiveness. Once, upon opening a public lecture he posed the question, if I recall correctly, as to whether a scientific lecture is to be compared with a theatrical performance, a religious sermon or a circus. Those who were fortunate enough to have attended Synge lectures say that he combined all three. Synge received many honours for his work. The greatest of these was probably his election as a fellow of the Royal Society of London in 1943. He was also elected a fellow of the Royal Society of Canada and was president of the Royal Irish Academy from 1961 to 1964. He is described in [3] as:... a kind and generous man. He encouraged and inspired several generations of students who will always remember him with gratitude, fondness and the deepest respect. As to hobbies, Synge was a keen cyclist, was passionately interested in sailing and painted some very fine compositions. In particular he painted a picture representing Schrödinger held in the hand of God contemplating unified field theory. Synge was a member of a highly talented family. We mention three other members of the family. First John Millington Synge (1871-1909), John Lighton Synge's uncle, who was a playwright and a powerful poetic dramatist who portrayed the primitive life of the Aran Islands and the western Irish coast. Secondly there was Richard Laurence Millington Synge (1914-1994), Nobel laureate for Chemistry in 1952, who shared the Prize for Chemistry with A J P Martin for their development of partition chromatography, a method used to separate mixtures of closely related chemicals such as amino acids for identification. Thirdly we mention Cathleen Synge Morawetz, Professor of Mathematics at the New York Courant Institute, who is John Lighton Synge's daughter. She was the first woman to hold the Directorship of the Courant Institute and she was President of the American Mathematical Society in 1995-96. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

A Quotation

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Synge.html (3 of 4) [2/16/2002 11:34:10 PM]

Search Form

Synge

Glossary index

Quotations index

Mathematicians of the day JOC/EFR June 1998

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Synge.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Synge.html (4 of 4) [2/16/2002 11:34:10 PM]

Szasz

Otto Szász Born: 11 Dec 1884 in Hungary Died: 19 Dec 1952 in Cincinnati, Ohio, USA

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Otto Szász studied at the University of Budapest but he went from there to Göttingen, Munich and Paris where he greatly broadened his education. He received his doctorate in 1911 from the University of Budapest and was appointed onto the staff there as a privatdozent. From 1914 he lectured in Frankfurt, first as a privatdozent and then later as a professor, being a colleague of Bieberbach for 6 years. In 1933 the Nazis came to power in Germany and Szász was forced out of his chair at Frankfurt. It was a period of extraordinary hardship for many mathematicians in Germany, Poland, Hungary and countries around them. Many mathematicians, like Szász, lost their positions. Others suffered far greater hardships; some were sent to concentration camps, many were murdered. However, there was a positive side effect which changed the course of mathematical research in the United States, for many of these top quality European mathematicians went there and, to the country's credit, it did an exceptional job in accepting them and helping them to find positions. Szász, like many others, left Germany and emigrated to the United States in 1933. His initial position was at the Massachusetts Institute of Technology. His next post was at Brown University and then, in 1936, he was appointed to the Faculty of the University of Cincinnati where he spent the rest of his career. He did make a research visit spending a year at the Institute of Numerical Analysis at the University of California in Los Angeles but he seemed content to devote the rest of his life to teaching, research, and his students in Cincinnati [1]:His steady preoccupation with mathematics, his erudition and broad knowledge of classical and contemporary literature and his perseverance in dealing with open problems of a number of varied fields have secured a firm place for him in the mathematical life of Hungary, Germany and of the United States.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Szasz.html (1 of 3) [2/16/2002 11:34:12 PM]

Szasz

Szász's main work was in real analysis, particularly Fourier series. In fact his most notable research was done before he emigrated to the United States, but this was not too surprising since he was in his fiftieth year when he emigrated. His most important contributions are probably between 1915 and 1930 when he made a series of remarkable contributions to a number of different areas. Some of his earliest work was on continued fractions in which he studied certain convergence questions. Perron, influenced by Pringsheim at Munich, had published an important work on continued fractions Die Lehre von den Kettenbrüchen in 1913. Szász generalised some of Perron's results and also, in 1915, published a paper proving one of Perron's conjectures. A few years before Szász began his mathematical researches, Sergi Bernstein had made major contributions to the theory of approximation. Sergi Bernstein stated a problem concerning the completeness of a certain set of powers on an interval and, although Szász did not solve this problem, he did make contributions which were themselves important in the development of the theory of approximations. Other work by Szász made major contributions to questions posed by Landau on the maximum modulus of the partial sums of a power series. He also studied problems on power series related to work of Frigyes Riesz. In fact Szász worked on problems associated with both Riesz brothers, and he gave a very simple proof a theorem by Marcel Riesz on rational functions with given bounds on the unit circle. Some of Szász's contributions to Fourier series related to results proved by Sergi Bernstein, Hardy, Littlewood and Fejér. All these results were achieved before 1933 and many of the mathematicians we have mentioned, such as Perron, Pringsheim, Landau, and Fejér were all Szász's personal friends. His major contribution during the years from 1915 to 1930 was recognised by the Hungarian Mathematical and Physical Society in 1939 when they awarded him their Julius König prize. We must not give the impression, however, that Szász did not make any research contributions after emigrating to the United States. His work in this later period was mainly on [1]:... Tauberian theorems, various methods of summability, Gibbs phenomenon, etc. The following tribute to Szász is made by Szegö in [1]:His students and friends will always warmly remember him as a man of gentle, unassuming and quiet personality. His life and energy were dedicated to the promotion of simple and beautiful problems of mathematics, in particular of the classical analysis. His nearly one hundred and thirty mathematical papers remain a living document of his efforts. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Szasz.html (2 of 3) [2/16/2002 11:34:12 PM]

Szasz

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Szasz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Szasz.html (3 of 3) [2/16/2002 11:34:12 PM]

Szego

Gábor Szegö Born: 20 Jan 1895 in Kunhegyes, Hungary Died: 7 Aug 1985 in Palo Alto, California USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gábor Szegö was born in Kunhegyes, a small town in Hungary about 120 km southeast of Budapest. His undergraduate studies were undertaken in Budapest. After attending university in Budapest, Szegö went to Berlin where he studied under, among others, Frobenius, Schwarz, Knopp and Schottky, and Göttingen where he studied with Hilbert, Landau and Haar. He returned to Hungary where he worked under Féjer and Kurschak. He acted as a coach to the young von Neumann. He enlisted in the Austro-Hungarian cavalry in the First World War and spent some time in the Air force where he met von Mises. In 1921 he moved to Berlin where he became a friend of Schur and worked with von Mises and Schmidt. He cooperated with Pólya in bringing out a joint Problem Book: Aufgaben und Lehrsätze aus der Analysis, vols I and II (Problems and Theorems in Analysis ) (1925) which has since gone through many editions and which has had an enormous impact on later generations of mathematicians. Pólya wrote of their collaboration (see [2]):It was a wonderful time; we worked with enthusiasm and concentration. We had similar backgrounds. We were both influenced, like all young Hungarian mathematicians of that time, by Lipót Fejér. We were both readers of the same well-directed Hungarian Mathematical Journal for high school students that stressed problem solving. We were interested in the same kinds of questions, in the same topics; but one of us knew more about

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Szego.html (1 of 2) [2/16/2002 11:34:13 PM]

Szego

one topic, and the other more about some other topic. It was a fine collaboration. The book Aufgaben und Lehrsätze aus der Analysis, the result of our cooperation, is my best work and also the best work of Gábor Szegö. In 1926 he moved to Königsberg to succeed Knopp as professor. He stayed there until 1934 when the pressure on him as a Jew forced him to move to the USA where he found a post at Washington University in St Louis, Missouri. In 1938 he moved to Stanford where he remained for the rest of his working life. Szegö's most important work was in the area of extremal problems and Toeplitz matrices. This work led him to introduce the notion of the Szegö reproducing kernel. From these beginnings he moved to prove a number of limit theorems, now known as the Szegö limit theorem, the strong Szegö limit theorem and Szegö's orthogonal polynomials and on the unit circle. He produced over 130 research articles as well as several influential books. In addition to the books he wrote with Pólya, described above, Szegö wrote research monographs on his own work. Orthogonal Ploynomials appeared in 1939 and was published by the American Mathematical Society. It has proved highly successful, running to four editions and many reprints over the years. In a collaboration with Ulf Grenander, Szegö wrote Toeplitz forms and their applications which was published in 1958. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Gábor Szegö

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Szego.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Szego.html (2 of 2) [2/16/2002 11:34:13 PM]

Tacquet

Andrea Tacquet Born: 23 June 1612 in Antwerp, Belgium Died: 22 Dec 1660 in Antwerp, Belgium Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Andrea Tacquet was educated at the Jesuit College in Antwerp, entering the Jesuit Order in 1629, then studied mathematics, logic and physics at Louvain from 1631 until 1635. During the last two of these years he was a student of Gregory of Saint-Vincent. After this Tacquet spent a while teaching Greek and poetry at Bruges. From 1640 Tacquet studied theology at Louvain while at the same time he taught mathematics there. From 1644 he taught mathematics at the college of Louvain, then at Antwerp from 1645 until 1649 and from 1655 until 1660, spending the years 1649 to 1655 back at Louvain. He had been ordained in 1646. Tacquet's most important work Cylindricorum et Annularium, on cylinders and rings, followed the approach of Valerio. Tacquet wrote many good elementary texts written as mathematics textbooks for Jesuit colleges. His Elementa geometriae was his most popular teaching work. He also wrote the textbook Astronomia. His books had a considerable effect on Pascal. Tacquet's work was sent to Huygens and a correspondence between the two resulted. He also corresponded with Frans van Schooten. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Andrea Tacquet (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Tacquet

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tacquet.html (1 of 2) [2/16/2002 11:34:15 PM]

Tacquet

Other Web sites

1. Jesuit Scientists 2. The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tacquet.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tacquet.html (2 of 2) [2/16/2002 11:34:15 PM]

Tait

Peter Guthrie Tait Born: 28 April 1831 in Dalkeith, Midlothian, Scotland Died: 4 July 1901 in Edinburgh, Scotland

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

P G Tait's father was John Tait and his mother was Mary Ronaldson. John Tait was a secretary to Walter Francis Scott, the fifth duke of Buccleuch. Peter had two sisters and he began his schooling in the Grammar School in Dalkeith. However, when he was six years old his father died and Peter, with his two sisters and his mother, moved to Edinburgh to live with an uncle John Ronaldson. An Edinburgh banker, John Ronaldson was nevertheless interested in science, in particular in astronomy, geology and with the newly invented photography. He soon interested his young nephew Peter in these subjects and it is fair to say that Peter's interest in science was a direct consequence of his uncle's enthusiasm for the sciences. When the family moved to Edinburgh Peter, of course, had to leave his school in Dalkeith. He next attended a private school in Circus Place Edinburgh, then in 1841, when he was ten years old, he entered Edinburgh Academy. Lewis Campbell, who later became the professor of Greek at the University of St Andrews, and James Clerk Maxwell were one year above Tait at the Academy. In fact Maxwell was slightly younger than Tait so the difference of one year certainly did not reflect their respective ages. Tait was top of his class in each one of his six years at Edinburgh Academy. His early interests, however, were not in science but rather in classics. By his fourth year at the Academy mathematics had become his real love and that was the subject in which he really excelled. In 1846 he was placed first in the mathematics section of the Edinburgh Academical Club Prize which was no mean achievement given that he beat Lewis Campbell, who was placed second, and Maxwell who was placed third. In 1847, Tait's final year at Edinburgh Academy, Maxwell had his revenge since he was placed first for the Edinburgh Academical Club Prize with Tait second.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tait.html (1 of 7) [2/16/2002 11:34:17 PM]

Tait

At the age of 16, in November 1847, Tait entered the University of Edinburgh. Maxwell entered Edinburgh University at the same time at Tait and together they attended the second mathematics class taught by Kelland and the natural philosophy (physics) class taught by James David Forbes. Tait remained at Edinburgh University for only one year before entering Peterhouse, Cambridge in 1848. There he was tutored by William Hopkins through what was a remarkable undergraduate career. In January 1852, at the age of twenty, he graduated as senior Wrangler in the Mathematical Tripos. This means that he was placed first among the First Class degrees in mathematics awarded by Cambridge in that year. He was also the first Smith's prizeman. Maxwell followed Tait to Peterhouse in 1850 but transferred to Trinity where he believed that it was easier to obtain a fellowship. Another fellow student and friend of Tait's was William Steele who was in the same year as Tait and graduated as Second Wrangler. Tait won a Fellowship at Peterhouse and, in addition to coaching undergraduates for the Tripos, he began to collaborate with Steele in writing a text Dynamics of a particle. Tragically Steele died before much progress had been made with writing the book but Tait continued with the project and generously published the book under their joint authorship despite having written most of it himself. It was published in 1856. In September 1854 Tait took up an appointment as professor of mathematics at Queen's College, Belfast. A number of the colleagues and friends he made in Belfast were to have a very significant effect on Tait's career. One of these was Thomas Andrews and the two collaborated in experiments to determine the density of ozone and also the affects of passing electrical discharge through oxygen and other gasses. Tait had not been involved in experimental work up to this time and it is certainly due to the influence of Andrews that he added this interest to his growing range of skills. This research carried out with Andrews took Tait towards chemistry and this was a subject he retained an interest in through his career. Another friendship of real significance was that with Hamilton. Tait had read Hamilton's Lectures on quaternions in 1853 while he was still at Cambridge but although the topic fascinated him he was more taken up with physical applications of mathematics at the time and did not pursue the topic at that stage. Then in July 1858 Tait read a paper by Helmholtz in Crelle's Journal on the motion of a perfect fluid. Helmholtz's paper Uber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen began by decomposing the motion of a perfect fluid into translation, rotation and deformation. Tait saw that using quaternions he could express the fluid velocity as a "vector function". It was the physical insight which Hamilton's quaternion differential calculus then gave which impressed Tait and he began to work hard developing a physical theory. Tait began to correspond with Hamilton in August 1858 and, in reply to Hamilton's question as to how he had stated to work with quaternions, Tait wrote to Hamilton on 7 December 1858 (see for example [5]):... it was only in August last that I suddenly bethought me of certain formulas I had admired years ago on page 610 of your Lectures - and I thought (and still think) likely to serve my purpose exactly. (The matter which more immediately suggested this to me was a paper by Helmholtz in Crelle's Journal (Vol. LX) which I as reading in July last as soon as we received it ... The title (in German) I forget - but a manuscript translation of my own which I now have beside me is headed "Vortex motion" ... ). If Tait's friendship with Hamilton was to prove important for his future research, then other friendships which Tait formed were important in his family life. Two of his friends at Peterhouse were sons of the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tait.html (2 of 7) [2/16/2002 11:34:17 PM]

Tait

Rev James Porter and through them Tait met their sister, Margaret Archer Porter, whom he married in Belfast on 13 October 1857. The Chair of Natural Philosophy at the University of Edinburgh became vacant in 1859, J D Forbes having moved to the University of St Andrews. Tait was a candidate for the chair but so was Maxwell who had been forced to seek another post when Marischal College and King's College in Aberdeen combined. Routh, who had been First Wrangler at Cambridge in Maxwell's year, was also a candidate but the real competition was always going to be between Tait and Maxwell. Tait won despite Maxwell's outstanding scientific achievements. When the Edinburgh paper, the Courant, reported the result it noted that Tait had been chosen in preference to Maxwell since:... there is another quality which is desirable in a Professor in a University like ours and that is the power of oral exposition proceeding on the supposition of imperfect knowledge or even total ignorance on the part of pupils. The claim that Tait was the better person to teach poorly qualified pupils was certainly a fair one and, of course, Tait's personality meant that he made a stronger impression on the appointing committee rather than the much more reserved Maxwell. By the time he arrived in Edinburgh in 1860 Tait was making strong contributions in applying Hamilton's quaternions. In the year he took up the chair of Natural Philosophy at Edinburgh he published Quaternion investigations connected with electro-dynamics and magnetism in which he reworked Helmholtz's hydrodynamic- electromagnetic analogy in the language of quaternions. As Epple writes in [9]:... not only quaternion analysis profited from acquiring a new physical meaning. Quaternionic formulas also helped to grasp physical situations which could be described in terms of fluid motion more easily. By 1863 when he published Note on a quaternion transformation in the Proceedings of the Royal Society of Edinburgh, Tait claimed that:... the next grand extensions of mathematical physics will, in all likelihood, be furnished by quaternions. Hamilton died in 1865 and Tait took over the crusade to give quaternions a leading role in mathematical physics. Among the many contributions he made to the topic we should mention his two important texts Elementary Treatise on Quaternions (1867), and Introduction to Quaternions (1873). Maxwell was impressed by Tait's many works on physical applications of quaternions and wrote in a letter to William Thomson in 1871:You should let the world know that the true source of mathematical methods as applicable to physics is to be found in the Proceedings of the Royal Society of Edinburgh. The volumesurface- and line- integrals of vectors and quaternions and their properties as in the course of being worked out by Tait is worth all that is going on in other seats of learning. Despite his intense work on quaternions, Tait was involved in many other activities. In 1862 he had published joint work with James A Wanklyn on electricity produced during evaporation and during effervescence. Three years later he published a paper on the motion of iron filings on a vibrating plate which was subjected to a magnetic field. In 1866 he started a joint project with the physicist Balfour Stewart on heating a disk which was rapidly rotating in a vacuum. This was a topic Tait came back to on http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tait.html (3 of 7) [2/16/2002 11:34:17 PM]

Tait

several occasions throughout his career. Then in 1867 he published, in addition to the treatise on quaternions, his translation of Helmholtz's 1858 article and also the Treatise on Natural Philosophy for which he may be best known. In 1861 Tait had been working on a text on mathematical physics. His friend William Thomson (later Lord Kelvin):... to my great delight offered to join. The two intended to write a two volume work and Treatise on Natural Philosophy (1867) was to be the first of the volumes. However the second volume was never written. The treatise, known as 'T & T', was written mainly by Tait who seemed to find time despite his numerous other activities, while Thomson found that his many other activities prevented him finding as much time as Tait to work on the book. Hamilton Dickson writes in [8]:The work was epoch-making, and created a revolution in scientific development. For the first time 'T & T', as the authors called themselves, traced to Newton the concept of the 'conservation of energy' which was just then obtaining recognition among physicists, and they showed once and for all that 'energy' was the fundamental physical entity and that its 'conservation' was its predominant and all-controlling property. We have already detailed major achievements for Tait dated 1867 but there is one further event of that year which we should mention. Helmholtz, in his 1858 paper, described the theoretical behaviour of vortex rings. He claimed that two interacting rings would change size and velocity as they interacted but would retain their ring shape. Tait verified Helmholtz' theoretical claims with experiments with smoke rings in 1867. He used two boxes each with a rubber diaphragm which shot out white smoke rings when the diaphragm was struck. Thomson wrote to Helmholtz on 22 January 1867:... a few days ago Tait showed me in Edinburgh a magnificent way of producing [vortex rings]. We sometimes can make one ring shoot through another, illustrating perfectly your description; when one ring passes near another, each is much disturbed, and is seen to be in a state of violent vibration for a few seconds, till it settles again into its circular form. ... The vibrations make a beautiful subject for mathematical work. These experiments were to have a major influence on Thomson who saw the permanence of form as a possible explanation for atoms and therefore explain the way that the different elements could be built. Tait was not convinced by Thomson's idea at first, rightly so of course since, although a beautiful idea, it is quite wrong. The idea led Tait, Thomson and Maxwell to begin to work on knot theory since the basic building blocks, in Thomson's vortex atom theory, would be the rings knotted in three dimensions. By Helmholtz' theory of a perfect fluid, these knotted rings, although they could be distorted, would retain the 'same knot' as a circular knotted piece of string that can be moved around yet the form of the knot remains an invariant. Tait, Thomson and Maxwell exchanged letters in which they invented many topological ideas as they looked at knots. Soon they discovered Listing's 1847 contributions to knot theory. Tait, although at first unconvinced by Thomson's vortex atom theory, began to include the theory in his lecture courses at Edinburgh in the early 1870s and he gave popular lectures describing the theory. In 1876 Tait began an intense study of knots, attempting to classify them. He published seven papers on knots in the Proceedings of the Royal Society of Edinburgh in the academic year 1876-77. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tait.html (4 of 7) [2/16/2002 11:34:17 PM]

Tait

Tait considered alternating knots, namely those which when traversing the projection in 2-dimensional space the crossings go alternately over and under. He labelled the n crossings of such a knot A, B, C, ... and then the knot would be described by the sequence of crossings of length 2n where each of A, B, C, ... occurred exactly twice when the knot was traversed. There were then two basic problems to solve. Firstly which sequences of the above type corresponded to a knot, and secondly how could it be determined when two knots described by such sequences were the same. You can see more details of this in our article on Topology and Scottish mathematical physics. Without any rigorous theory, which would have been well beyond nineteenth century mathematics, Tait began to classify knots using his mathematical and geometrical intuition. By 1877 he had classified all knots with seven crossings but he stopped there. One of the problems he considered after that was the colouring of graphs since he claimed to have a correct proof of the four colour theorem. His proof is fallacious and, sadly, he did not relate colouring of graphs to the knot theory he had considered a few years earlier. Another topic which he had worked on over a number of years was the results of the Challenger expedition on deep sea temperatures. In 1881 Tait published an important paper on the topic in which he showed how to correct the temperature readings because of the high pressures on the thermometers. He returned to the topic of knots in his address to the Edinburgh Mathematical Society in 1883:We find that it becomes a mere question of skilled labour to draw all the possible knots having any assigned number of crossings. The requisite labour increases with extreme rapidity as the number of crossings is increased. ... I have not been able to find time to carry out this process further than the knots with seven crossings. ... It is greatly desired that someone, with the requisite leisure, should try to extend this list, if possible up to 11 ... Kirkman read the text of Tait's address and began to work on classifying knots with more than seven crossings. He sent Tait his results on knot projections with up to nine crossings in May 1884 but he had not looked at the problem of deciding which of the projections led to equivalent knots. Tait worked on this side of the problem and, considering only alternating knots, solved the equivalence problems within a few weeks. Tait seemed to know how to tell whether two knots were equivalent without rigorous methods. He states this quite clearly in the paper he wrote tabulating the knots where he says that his methods have:... the disadvantage of being to a greater or less extent tentative. Not that the rules laid down ... leave any room for mere guessing, but they are too complex to be always completely kept in view. Thus we cannot be absolutely certain that by means of such processes we have obtained all the essentially different forms which the definition we employ comprehends. Despite the problems Tait knew exactly what he was doing for, remarkably, his tables are correct. When Kirkman sent him all knot projections with 10 crossings in January 1885 again Tait found all inequivalent knots. The tables were printed in September 1885 and again they are completely correct. By then he had received from Kirkman 1581 knot projections with 11 crossings and this time Tait felt that he did not have the time to solve the equivalence problem for these. It would be quite impossible in an article of this length to cover all the topics which Tait worked on. Knott [5] lists 365 papers and 22 books written by Tait. We will mention two final topics which he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tait.html (5 of 7) [2/16/2002 11:34:17 PM]

Tait

worked on after ending his work on knots. Thomson suggested that he work on the kinetic theory of gasses and between 1886 and 1892 Tait published more than 20 papers on the topic. In this work he gave what Thomson considered the first proof of the Waterston-Maxwell equipartition theorem. Tait also wrote a classic paper on the trajectory of golf balls (1896). The subject of golf was one of great interest to Tait. Of his four sons, the third was Frederick Gutherie Tait. He became the leading amateur golfer in 1893 and won the Open Golf Championship in 1896 and again in 1898. Freddie Tait, as he was known in the golfing world, has a street in St Andrews named after him which is not far from my [EFR] home. Freddie was a military man in the Black Watch. He gave up his golf when he volunteered to serve in the Boer War in 1899. He was wounded at Magersfontein on 19 December 1899 and killed during fighting at Koodoosberg on 7 February 1900. A deeply religious man, Tait wrote, with the physicist Balfour Stewart, The Unseen Universe (1875):... to overthrow materialism by a purely scientific argument. Because of the public demand, he wrote a sequel Paradoxical Philosophy (1878). We have painted a very positive picture of Tait in the details we have given above. This is right for he deserves no less. However there was another side to his character which we should mention. He became involved in many arguments with his fellow scientists and at least twice engaged in very public arguments. Tait was prone to let his heart rule his head in such situations and he often came of worse in the scientific debate. One of his disputes was with Heaviside and Gibbs whose vector methods he argued vigorously against over a long period. Certainly Tait came off worst in this arguments, perhaps his heart was too set on quaternionic methods to allow his head to see the importance of the ideas of Heaviside and Gibbs. Another bitter dispute was with Clausius and Tyndall. Tait was patriotic to the extent that he would let such considerations prejudice his view of science. The dispute began over who was the first to propose the equivalence of work and heat. Tait and Tyndall began an argument over whether Joule or Julius Robert Mayer von Mayer had priority. Tait wrote a highly prejudiced account of the history of thermodynamics which was stupidly pro-British and Tyndall were right to be offended. Hopkins stumbled into the controversy when Tyndall had asked him to send him all von Mayer's papers but then he was as pro-German as Tait was pro-British when he published an article in 1868 stating that not only did von Mayer have priority but so did the German nation. A more bitter dispute between Tait and Clausius began in 1872 when Maxwell published his Theory of Heat. Clausius stated that the British were trying to claim more than they deserved for the theory of heat which, given Tait's writing, was a fair comment. Maxwell, however, had over a number of years fully recognised Clausius' contribution unlike Tait with his prejudiced approach. Of course Tait's patriotism also meant that he was a devoted supporter of the Royal Society of Edinburgh which he served faithfully from the time he was elected a Fellow shortly after being appointed to the chair in Edinburgh. He served the Society as General Secretary for 22 years from 1879 until 1901. He won the Gunning Victoria Jubilee Prize and twice the Keith prize from that Society. Although never elected a Fellow of the Royal Society of London, he did have the distinction of receiving that Society's Royal Medal in 1886. Other honours given to Tait included the award of honorary degrees by the University of Glasgow and the University of Ireland, as well as being elected to honorary membership of the academies of Denmark, Holland, Sweden and Ireland. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tait.html (6 of 7) [2/16/2002 11:34:17 PM]

Tait

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles)

A Quotation

Mathematicians born in the same country Cross-references to History Topics

1. A visit to Maxwell's house. 2. the four colour theorem 3. Abstract linear spaces 4. Chrystal and the RSE 5. Topology and Scottish mathematical physics

Honours awarded to P G Tait (Click a link below for the full list of mathematicians honoured in this way) Royal Society Royal Medal

Awarded 1886

Fellow of the Royal Society of Edinburgh Honorary Fellow of the Edinburgh Maths Society

Elected 1883

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR December 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Tait.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tait.html (7 of 7) [2/16/2002 11:34:17 PM]

Takagi

Teiji Takagi Born: 21 April 1875 in Kazuya Village (near Gifu), Japan Died: 29 Feb 1960 in Tokyo, Japan

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Teiji Takagi was born in a rural area of Gifu Prefecture in central Japan. His father was an accountant on a farm in this mountainous region and Teiji was brought up on the farm on which his father worked. His mother was a devoted Buddhist and she took Teiji, when he was a young child, with her when she went to the temple. Teiji soon showed himself to be a childhood prodigy by quickly learning to recite the prayers. He attended primary school in Kazuya Village before going to middle school in Gifu entering this second stage of his education in 1886. At that time there were no mathematics texts written in Japanese so the pupils studying mathematics had to use English texts. Takagi studied Algebra for beginners by Todhunter and Geometry by Wilson. In 1891 Takagi began the third stage of his schooling which he took at the Third High School in Kyoto. There were, at that time, eight academies and the brightest pupils went to the one corresponding to the area in which they lived in order to prepare for a university education. Takagi therefore, after showing great talents at middle school, made the natural progression to Kyoto where he studied for three years. In 1894 he graduated for the Third High School and entered Tokyo University, the only university in Japan at that time. At Tokyo University Takagi took courses on calculus and analytic geometry. However he learnt more advanced mathematics by reading books rather than from lecture courses which he attended. He learnt about algebraic curves from George Salmon's book and he also studied Serret's Algèbre Supérieure. He eagerly read Heinrich Weber's Algebra text when it arrived in Japan and by 1898 Takagi had published his first paper. The paper shows a remarkably modern approach to algebra, very surprising for someone who had learnt most of his mathematics from textbooks. The paper begins:In looking back at the history of branches of mathematics, we see that they start with special

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Takagi.html (1 of 4) [2/16/2002 11:34:19 PM]

Takagi

and concrete beginnings and proceed by generalisation as they advance. This is manifested, for instance, in the theory of groups, which is one of the most important fields of present day mathematics, and is related to various other branches. It started as the theory of permutation groups, but now the general theory of groups does not suppose that elements of groups should be permutations. As Cayley has remarked, one has only to suppose that composition of elements satisfies certain laws ... We ... hope that the reader has understood that the essential point in algebra does not lie in the nature of the elements (which are not necessarily numbers) but in the way elements are composed. The paper also in noteworthy for containing an abstract definition of a field. Takagi graduated from Tokyo University in 1897, and in the following year he was chosen as one of twelve students from Japan to study abroad. He sailed to Germany where he studied courses given by Fuchs, Frobenius and Schwarz at Berlin University. However, to his surprise, he discovered that he already knew most of the mathematics in these courses from the books that he had read back in Japan. He then read Hilbert's Zahlbericht, a report on algebraic number theory which had been published in 1897. Takagi wrote to Hilbert who arranged accommodation for him in Göttingen in a house in which he himself had previously lived. In [8] Takagi wrote:At the time when I studied in Germany, Göttingen was perhaps the only place in the world where research in algebraic number theory was going on. Thus, when I told Hilbert that I wanted to study this theory, he did not seem to believe me immediately ... If Takagi expected Hilbert to be actively engaged in algebraic number theory then he would have been disappointed. Hilbert had left this topic immediately after writing the Zahlbericht and by the time Takagi reached Göttingen he was engaged in studying the foundations of geometry and then integral equations. Although Hilbert was not directly involved with Takagi's research, the topic he worked on was certainly one that Hilbert considered of the utmost importance for it was a special case of what became Hilbert's 12th problem in his Paris lecture of 1900. In 1901 Takagi left Göttingen and returned to Japan where he was appointed as Assistant Professor in Algebra in the Department of Mathematics at Tokyo University. He married Toshi Tani in 1902 and they had three sons and five daughters. He completed a doctorate in Tokyo in 1903 presenting a thesis based on work he had undertaken in Göttingen. He was promoted to full professor in Tokyo University in 1904; he held this post until he retired in 1936. On his return to Tokyo in 1903 Takagi proved a conjecture on abelian extensions of imaginary number fields made by Kronecker. Kronecker described this conjecture as:... the dearest dream of youth. Although Takagi was enthusiastic about research he did not continue to develop the work that he had begun in his thesis. He began to write textbooks, which of course were important for the development of Japanese mathematics both at school and university level. The first of these texts was A new course of arithmetic published in 1904. The 500 page work developed real numbers via Dedekind cuts. it was the first of many texts that Takagi wrote: between 1904 and 1911 he wrote 13 texts, but many were multi-volume works so the total number of volumes amounted to 20. In [8] Takagi describes how he was led to begin research again:I am by nature someone who needs stimulus in order to work. there are now quite a number of Japanese mathematicians, but in those days, we had few colleagues. I did not have a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Takagi.html (2 of 4) [2/16/2002 11:34:19 PM]

Takagi

heavy workload so you might imagine that I did research on class field theory in those carefree days, but it was not so. The First World war started in 1914 and this gave me a stimulus, rather a negative stimulus. No scientific information reached Europe for four years. some said this would be the end of Japanese science while newspaper articles wrote of their sympathy for Japanese professors losing their jobs. This made me realise the obvious truth that every researcher had to be independent. Possibly I would have done no research for myself had it not been for World War I. Takagi then went on in [8] to describe the ideas that relaunched his research career:Concerning class field theory, I confess that I was misled by Hilbert. Hilbert considered only unramified class fields. from the standpoint of the theory of algebraic functions which are defined by Riemann surfaces, it is natural to limit consideration to the unramified case ... after the end of scientific exchange between Japan and Europe ... I was freed from that idea and suspected that every abelian extension might be a class field if the latter is not limited to the unramified case. I thought at first that this could not be true. were it false the idea should contain an error and I tried my best to find this error. At that time I almost suffered from a nervous breakdown. I dreamt often that I had resolved the question. I woke up and tried to remember my reasoning but in vain. I tried my utmost to find a counterexample to the conjecture which seemed all too perfect. finally I made my theory confirming this conjecture, but I could not rid myself of the doubt that it might contain an error which would invalidate the whole theory. I badly lacked colleagues who could check my work. Takagi spoke of his work on class field theory, building on Heinrich Weber's work, at the International Congress of Mathematicians in Strasbourg in 1920. While in Europe he visited Hecke and Blaschke in Hamburg. He wrote his most important paper in 1920 which introduced the Takagi class-field theory generalising Hilbert's class field. In 1922 Siegel persuaded Artin to read this paper and its significance was realised. It became the framework of algebraic number theory. Hasse included Takagi's theory in his treatise on class field theory a few years later. In 1925 Hilbert wrote to Takagi in Japan asking if his paper could be published in Mathematische Annalen. Around this time other mathematicians working in the same area as Takagi started to be appointed to Tokyo University and at last he had the mathematical colleagues he had longed for. Iyanaga, the author of [6], became Takagi's student in 1926. He describes his teaching style [6]:[Takagi gave] his lectures without prepared papers, showing however, traits of spirit from time to time with sharp critical remarks, sometimes mixed with jokes. He spoke rather slowly in a low voice and almost never repeated the same thing; he wrote very neatly on the blackboard but the colour of his chalk was rather light; the speed of flow of his lecture was quite rapid and the students had to listen with great attention. Soon honours began to be given to Takagi for his outstanding work. He was honoured by Czechoslovakia, the university of Oslo, and the National Research Council of Japan. Fueter was President of the International Congress of Mathematicians at Zurich in 1932 and Takagi was appointed Vice-President. He served on the committee to award the first Fields' Medals for the 1936 Congress. In 1936 Takagi retired but continued publishing books and papers. His two most important books from

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Takagi.html (3 of 4) [2/16/2002 11:34:19 PM]

Takagi

this time are Introduction to analysis (1938), Algebraic number theory (1948) and an important work on the history of mathematics in the 19th century. Takagi continued to live in Tokyo after he retired until 1945 when his house was destroyed by bombing near the end of World War II. He returned to the village of his birth, coming back to Tokyo in 1947 to live with his eldest son. His wife died of cancer in 1952 and Takagi himself died at the age of 88 in the hospital of Tokyo university. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) A Poster of Teiji Takagi

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1920 to 1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Takagi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Takagi.html (4 of 4) [2/16/2002 11:34:19 PM]

Talbot

William Henry Fox Talbot Born: 11 Feb 1800 in Melbury Abbas, Dorset, England Died: 17 Sept 1877 in Lacock Abbey (near Chippenham), Wiltshire, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henry Fox Talbot studied at Cambridge and wrote papers on elliptic integrals, building on work of Euler, Legendre, Jacobi and Abel. For this work he was elected a Fellow of the Royal Society in 1831. In addition to his mathematical work, Talbot also published on astronomy and physics. He gave the Bakerian lecture to the Royal Society in 1837 with the title Further observations on the optical phenomena of crystals and he received the Royal Medal from the Royal Society in 1838. In 1833 Talbot was elected to parliament but retired one year later. He was a close friend of John Herschel and together they studied light. An interest in chemistry, together with his interest in light, took him into photography and he is best remembered for his pioneering work in this area. In 1844 he published Pencil of nature the first photographically illustrated book. He was also interested in archaeology and was one of the first to translate the cuneiform writing from Nineveh. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Talbot.html (1 of 2) [2/16/2002 11:34:21 PM]

Talbot

Mathematicians born in the same country Honours awarded to Henry Fox Talbot (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1831

Royal Society Royal Medal

Awarded 1838

Royal Society Bakerian lecturer

1837

Fellow of the Royal Society of Edinburgh Lunar features

Crater Talbot

Other Web sites

1. Fox Talbot Museum 2. University of Glasgow, UK 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Talbot.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Talbot.html (2 of 2) [2/16/2002 11:34:21 PM]

Taniyama

Yutaka Taniyama Born: 12 Nov 1927 in Kisai (north of Tokyo), Japan Died: 17 Nov 1958 in Tokyo, Japan

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yutaka Taniyama graduated from the University of Tokyo in 1953. He remained there as a 'special research student', then as an associate professor. His interests were in algebraic number theory. He wrote Modern number theory (1957) in Japanese, jointly with G Shimura. Although they planned an English version, they lost enthusiasm and never found the time to write it before Taniyama's death. However they probably give the reason themselves in the 1957 preface:We find it difficult to claim that the theory is presented in a completely satisfactory form. In any case, it may be said, we are allowed in the course of progress to climb to a certain height in order to look back at our tracks, and then to take a view of our destination. Taniyama's fame is mainly due to two problems posed by him at the symposium on Algebraic Number Theory held in Tokyo in 1955. (His meeting with Weil at this symposium was to have a major influence on Taniyama's work.) These problems form the basis of a conjecture : every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field. This conjecture proved to be a major factor in the proof of Fermat's Last Theorem by Wiles. With seemingly a great future in front of him, both in mathematics and his life (he was planning marriage) he took his own life. In a note he left he took great care to describe exactly where he had reached in the calculus and linear algebra courses he was teaching and to appologise to his colleagues for the trouble his death would cause them. As to the reason for taking his life he says Until yesterday I had no definite intention of killing myself. ... I don't quite understand it myself, but it is not the result of a particular incident, nor of a specific matter. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taniyama.html (1 of 2) [2/16/2002 11:34:23 PM]

Taniyama

About a month later the girl who he was planning to marry also committed suicide. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Cross-references to History Topics

Fermat's last theorem

Other references in MacTutor

Chronology: 1950 to 1960

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Taniyama.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taniyama.html (2 of 2) [2/16/2002 11:34:23 PM]

Tannery_Jules

Jules Tannery Born: 24 March 1848 in Mantes-sur-Seine, France Died: 11 Dec 1910 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jules Tannery's parents were Delphin Tannery, who was an engineer working for the French railways, and Opportune Perrier. Delphin and Opportune Tannery had three children, the eldest being a daughter while the second oldest was Paul Tannery and Jules was the youngest. Delphin Tannery's job with the railways involved him moving around the country depending on where railways were being constructed. The family moved from Mantes to Redon in Ille-et-Vilaine in the Brittany region of northwestern France. He later worked at Mondeville close to Caen. Jules was brought up in a deeply Christian family. His parents Delphin and Opportune Tannery, who were Roman Catholics, gave him his early education, but he was also greatly influenced at home by his elder brother Paul Tannery who had a passion for classics and philosophy. Jules attended schools in several towns as his parents moved because of his father's job, but he completed his secondary education at the Lycée at Caen. At Caen Jules proved himself to be a truly outstanding pupil, winning prizes and delighting his parents who were very keen to see their sons achieve great things with their education. Completing his secondary education in 1866, Tannery sat the entrance examinations in science for both the Ecole Normale Supérieur and the Ecole Polytechnique. Having attained the highest possible grade in both examinations, he had the choice of the two schools and, perhaps rather surprisingly, chose the Ecole Normale Supérieur. Graduating in 1869 and placed top of among all the graduates in that year, Tannery became a mathematics teacher at the Lycée in Rennes, then in 1871 he moved to teach the Lycée at Caen. This

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tannery_Jules.html (1 of 3) [2/16/2002 11:34:25 PM]

Tannery_Jules

proved to be a difficult time for him, for he had been given a very strict Christian upbringing, but had became deeply interested in the culture of the ancient scholars. In particular he admired the ideas of Lucretius, a Latin poet and philosopher who lived in the first century BC. Lucretius's ideas on ethical and logical doctrines caused Tannery to undergo a religious crisis as he found himself drawn towards the philosophy of this pagan writer. In 1872 Tannery returned to Paris and began teaching at the Ecole Normale Supérieur. He had left the school without studying for his doctorate and, now that he had returned there, Hermite encouraged him to undertake research in mathematics. He was awarded his doctorate in 1874 for his thesis Propriétés des intégrales des équations différentielle linéaires à coefficients variables. Tannery was an outstanding teacher of mathematics and he taught at a number of different establishments. He taught higher mathematics at the Lycée Saint-Loius, and taught at the Sorbonne. Then in 1881 he was appointed Maître de Conférences at the Ecole Normale Supérieur, and soon after this to the Ecole Normale for women in Sèvres. From 1884 he was an adviser of studies at the Ecole Normale and, from 1903, Professor of differential and integral calculus at the Faculty of Science in Paris. Speziali, writes in [1]:... Tannery played an important role in the pedagogical reforms in France at the beginning of the twentieth century. Through his lectures and supervisory duties at the Ecole Normale this gifted teacher gave valuable guidance to many students and inspired a number of them to seek careers in science (for example, Paul Painlevé, Jules Drach, and Emile Borel). His main contributions were to the history and philosophy of mathematics. He also wrote some excellent books with a large impact on younger mathematicians. Speziali, writes in [1]:Tannery possessed considerable gifts as a writer. the pure and elegant style of the poems he composed in his free hours clearly bears the stamp of a classic sensibility. his vast culture, nobility of character, and innate sense of a rationally grounded morality are reflected in his Pensées, a collection of his thoughts on friendship, the arts, and beauty. often they exhibit a very refined sense of humour. Tannery worked on Galois' notes and letters. On June 13, 1909, a plaque was placed on Galois's birthplace at Bourg-la-Reine, and Jules made an eloquent speech of dedication. It was published in the Bulletin des Sciences Mathématiques (1909). It is worth noting that Tannery had been an editor of the Bulletin des Sciences Mathématiques since 1876 and he continued in that role until his death. He worked on this major project with Darboux, Hoüel and Emile Picard. Tannery made an impressive contribution to the Bulletin, writing large numbers of reviews. For example he wrote over 200 book reviews for the Bulletin in the period between 1905 and 1910. Emile Picard writes in [2] about both Jules Tannery and his brother Paul Tannery:They were extremely close all their lives. Of very different natures, the two braothers complement each other ... Jules's philosophy ... did not free him from intellectual anxiety. his outlook was less universal than his brother's, but also more profound. He had both the subtle mind of the metaphysician and the penetrating insight of the disillusioned moralist. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tannery_Jules.html (2 of 3) [2/16/2002 11:34:25 PM]

Tannery_Jules

List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tannery_Jules.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tannery_Jules.html (3 of 3) [2/16/2002 11:34:25 PM]

Tannery_Paul

Paul Tannery Born: 20 Dec 1843 in Mantes-la-Jolie, Yvelines, France Died: 27 Nov 1904 in Pantin, Seine-St Denis, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Tannery was a brother of Jules Tannery whose biography is also given in this archive. Since Paul Tannery is famous as an historian of mathematics it is particularly fitting that we should give details of his life and work in this archive dedicated to the history of mathematics. Paul Tannery's parents were Delphin Tannery, who was an engineer working for the French railways, and Opportune Perrier. Delphin and Opportune Tannery had three children, the eldest being a daughter while the second oldest was Paul and Jules Tannery was the youngest. Delphin Tannery's job with the railways involved him moving around the country depending on where railways were being constructed. The family moved from Mantes to Redon in Ille-et-Vilaine in the Brittany region of northwestern France. He later worked at Mondeville close to Caen. Paul was brought up in a deeply Christian family. His parents Delphin and Opportune Tannery, who were Roman Catholics, gave him his early education, then he attended schools in several towns as his parents moved because of his father's job. First he attended school a private school in Mantes, then the Lycée in Le Mans finally completing his secondary education at the Lycée at Caen. Although he was enrolled for a science course at Caen, his interests included classics and philosophy. His parents who were very keen to see their sons achieve great things with their education so they were delighted when, after completing his secondary education in 1860, Tannery sat the entrance examinations in science for the Ecole Polytechnique. Having attained one of the highest possible grades in both examination he entered the school, taking a wide range of courses in science, mathematics, technology and classical subjects such as Hebrew. He graduated from the Ecole Polytechnique in 1863 and entered

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tannery_Paul.html (1 of 4) [2/16/2002 11:34:27 PM]

Tannery_Paul

the Ecole d'Application des Tabacs. He would spend the rest of his working life involved with the tobacco industry. It was in this period after he graduated the Tannery became interested in the positivist philosophy of Comte. Auguste Comte, the French philosopher known as the founder of sociology and of positivism, published his philosophy in a work entitled Cours de philosophie positive in six volumes between 1830 and 1842. It presented a philosophy designed to produce a system of government making full use of technology in a way appropriate to modern industrial society. Tannery was much influenced by Comte's philosophy and it would influence his work throughout his life. Tannery worked from 1865 to 1867 in the state tobacco factory at Lille before moving to Paris to an administrative post in the administration of the state tobacco industry. The Franco-German War began on 19 July 1870. Tannery, a staunch French patriot, served in the French army as an artillery captain. The war went badly for France and on 19 September 1870 the Germans began a siege of Paris. Tannery was present during the siege, and was dismayed when Paris surrendered on 28 January 1871. The Treaty of Frankfurt, signed on 10 May 1871 was an humiliation for France, with Germany annexing Alsace and half of Lorraine. Tannery refused to acknowledge the terms of this treaty. The war having ended, Tannery returned to his administrative post but he became very interested in mathematics being particularly influenced by his brother Jules Tannery who by this time was teaching mathematics. Sent to Périgord to help with the construction of buildings, Tannery became ill. This in many ways worked in his favour since he used the time during which he was recovering to study more deeply, in particular learning ancient languages. Sent to Bordeaux in 1874, again to help with construction work, Tannery found there friends and university facilities which allowed him to spend all his free time working on the history of mathematics in ancient cultures. He had all the necessary skills to make a major contribution to this field and indeed his first publication, occurred around this time. Taton writes in [1] that he became:... a fairly regular contributor to about fifteen French and foreign periodicals. He published hundreds of memoirs, articles, notes, and reviews while pursuing a brilliant career in the state tobacco administration. Although many other historians of science have been obliged to conduct their research concurrently with their professional activities, none of them seems to have produced a body of work comparable to Tannery's in scope and importance. In 1877 Tannery made the rather strange move of requesting a move to Le Havre. This was a strange move since he was immediately deprived of the academic environment of Bordeaux. However, several trips abroad saw his became friends with other notable historians of science such as Heiberg, Zeuthen and Moritz Cantor. He carried on a correspondence with the mathematicians he had met after his return. For example the paper [5] describes three letters he exchanged with Zeuthen dealing with questions in the Greek theory of conic sections and the significance of certain constructions by means of "neusis." Tannery married Marie-Alexandrine Prisset in 1881. She was 13 years younger than he was and, had relatively little education but, after Tannery's death, she would undertake the difficult task of preparing a complete publication of his works with remarkable skill. By 1883 Tannery was finding the academic isolation of Le Havre too much to bear and he applied for a transfer to Paris. This was granted and, although he did not remain there for very long, it proved a period in which his work on the history of Greek geometry flourished. From 1886 until 1888 Tannery was in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tannery_Paul.html (2 of 4) [2/16/2002 11:34:27 PM]

Tannery_Paul

Tonneins and, deprived of library and other resources, he was unable to carry out any serious work on the history of mathematics except editorial work. In January 1888 he returned to Bordeaux as director of the tobacco factory and was able to slot into the intellectual life he had lived there earlier. From 1890 until 1893 he was back at the headquarters of the administration of the state tobacco industry in Paris, and then in 1893 he made his final move to be director of a tobacco factory at Pantin near Paris. A rather strange, and sad, episode occurred near the end of his life. The chair of history of science at the Collège de France became vacant in 1903. Tannery was named as the first choice to fill the post by the two consultative bodies. So certain was Tannery that his appointment after that was merely a formality, that he started work on writing his inaugural lecture.. The minister of education chose not to follow the recommendations of the consultative bodies, however, and appointed a philosopher with little experience in the history of science. It seems that the government did not want a Roman Catholic for the post and also preferred the course proposed by the other candidate on contemporary science rather than Tannery's proposed course on the history of science. It did not escape the notice of Tannery, his friends, and his international colleagues that a course on the history of science seemed appropriate for a professor of the history of science. All their protests were in vain and, as Taton writes in [1]:... there is no doubt that the "scandal of 1903" did great damage to the development of the history of science in France. He died of cancer of the pancreas at the age of sixty while still at the height of his intellectual powers. His main contributions were to the history of Greek mathematics and to the philosophy of mathematics. He published a history of Greek science in 1887, a history of Greek geometry in the same year, and a history of ancient astronomy in 1893. Tannery did work of great importance as an editor of famous mathematics texts. He edited the work of Fermat in three volumes (jointly with C Henry) between 1891 and 1896. In addition he edited the work of Diophantus in two volumes (1893-95). He was an editor of the twelve volume complete works of Descartes Oeuvres de Descartes (1897-1913). Tannery became so skilled in using Greek numerals in his historical work that he believed that they had certain advantages over our present system. Taton, himself a famous historian of mathematics, sums up Tannery's work in [1]:Perhaps its most notable characteristic is an unwavering concern for rigour and precision. the detailed studies that constituted the bulk of his output were, in Tannery's view, only a necessary stage in the elaboration on much broader syntheses that would ultimately lead to a comprehensive history of science that he himself could openly initiate. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tannery_Paul.html (3 of 4) [2/16/2002 11:34:27 PM]

Tannery_Paul

Cross-references to History Topics

1. Squaring the circle 2. Doubling the cube 3. How do we know about Greek mathematicians?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tannery_Paul.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tannery_Paul.html (4 of 4) [2/16/2002 11:34:27 PM]

Tarry

Gaston Tarry Born: 27 Sept 1843 in Villefranche de Panat, Aveyron, France Died: 21 June 1913

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gaston Tarry studied mathematics at secondary school in Paris and joined the civil service. He spent his whole career working in Algeria. He was interested in geometry and published numerous articles in various journals from 1882 until his death. He did extensive work on magic squares and on number theory. He is best known for his work on Euler's 36 Officer Problem, proving that two orthogonal Latin squares of order 6 did not exist. He also published an algorithm for exploring mazes which is called after him. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tarry.html (1 of 2) [2/16/2002 11:34:29 PM]

Tarry

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tarry.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tarry.html (2 of 2) [2/16/2002 11:34:29 PM]

Tarski

Alfred Tarski Born: 14 Jan 1902 in Warsaw, Russian Empire (now Poland) Died: 26 Oct 1983 in Berkeley, California, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alfred Tarski taught at the University of Warsaw, Harvard University, and then joined the staff at University of California at Berkeley in 1942. He was appointed professor of mathematics there in 1949 becoming research professor at the Miller Institute of Basic Research in Science in 1958-1960. Formal scientific languages can be subjected to more thorough study by the semantic method that he developed. He worked on model theory, mathematical decision problems and with universal algebra. He produced axioms for 'logical consequence', worked on deductive systems, the algebra of logic and the theory of definability. Tarski wrote more than ten books in different areas of mathematics, and his teaching influenced many young mathematicians. His work includes Geometry (1935), A decision method for elementary algebra and geometry (1948), Undecidable theories (1953), Logic, semantics, metamathematics (1956). An important paper in 1924 with Banach investigated the equivalence of geometric figures by finite decompositions. Group theorists study 'Tarski monsters', infinite groups whose existence seems intuitively impossible. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles)

A Quotation

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tarski.html (1 of 2) [2/16/2002 11:34:31 PM]

Tarski

A Poster of Alfred Tarski

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1920 to 1930

Honours awarded to Alfred Tarski (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1952

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tarski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tarski.html (2 of 2) [2/16/2002 11:34:31 PM]

Tartaglia

Niccolo Fontana Tartaglia Born: 1499 in Brescia, Republic of Venice (now Italy) Died: 13 Dec 1557 in Venice, Republic of Venice (now Italy)

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Niccolo Fontana known as Tartaglia, was born in Brescia in 1499, the son of a humble mail rider. He was nearly killed as a teenager, when in 1512 the French captured his home town and put it to the sword. Amidst the general slaughter, the twelve year old boy was dealt horrific facial sabre wounds that cut his jaw and palate and he was left for dead. His mother's tender care ensured that the youngster did survive, but in later life Niccolo always wore a beard to camouflage his disfiguring scars and he could only speak with difficulty, hence his nickname Tartaglia, or stammerer. Tartaglia was self taught in mathematics but, having an extraordinary ability, was able to earn his living teaching at Verona and Venice. As a lowly mathematics teacher in Venice, Tartaglia gradually acquired a reputation as a promising mathematician by participating successfully in a large number of debates. The first person known to have solved cubic equations algebraically was del Ferro but he told nobody of his achievement. On his deathbed, however, del Ferro passed on the secret to his (rather poor) student Fior. Fior began to boast that he was able to solve cubics and a challenge between him and Tartaglia was arranged in 1535. Each man was to submit thirty questions for the other to complete. Fior was supremely confident that his ability to solve cubics would be enough to defeat Tartaglia but because negative numbers were not used there was more than one type of cubic equation and Fior had only been shown by del Ferro how to solve one type. Tartaglia submitted a variety of different questions, exposing Fior as an, at best, mediocre mathematician. Fior, on the other hand, offered Tartaglia thirty opportunities to solve the cosa and cube problem since he believed that he would be unable to solve this type. However, in the early hours of 13 February 1535, inspiration came to Tartaglia and he discovered the method to solve both types of cubic. Tartaglia now knowing the method to solve the cosa and cube problems, quickly http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tartaglia.html (1 of 4) [2/16/2002 11:34:33 PM]

Tartaglia

solved all thirty of Fior's problems in less than two hours. As Fior had made little headway with Tartaglia's questions, it was obvious to all who the winner was. At this point that Cardan enters the story. As public lecturer of mathematics at the Piatti Foundation in Milan, he was aware of the cosa and cube problems, but, until the contest, he had taken Pacioli at his word and assumed as Pacioli stated in the Suma published in 1494, that solutions were impossible. Cardan was greatly intrigued when he learned of the contest and immediately set to work on trying to discover Tartaglia's method for himself, but was unsuccessful. A few years later, in 1539 he contacted Tartaglia, through an intermediary, requesting that the method could be included in a book he was publishing that year. Tartaglia declined this opportunity, stating his intention to publish his formula in a book of his own that he was going to write at a later date. Cardan, accepting this, then asked to be shown the method, promising, if he was, to keep it secret. Tartaglia, however, refused. An incensed Cardan now wrote to Tartaglia directly, expressing his bitterness, challenging him to a debate but, at the same time, hinting that he had been discussing Tartaglia's brilliance with the governor of the emperor's army in Milan, Alfonso d'Avalos, the Marchese del Vasto, who one of Cardan's powerful sponsors. On receipt of this letter, Tartaglia radically revised his attitude, realising that acquaintance with the influential Milanese governor could be very rewarding and could provide a way out of the modest teacher's job he then held, and into a lucrative job at the Milanese court. He wrote back to Cardan in friendly terms, angling for an introduction to the Signor Marchese. Cardan was delighted at Tartaglia's new approach, and, inviting him to his house, assured Tartaglia that he would arrange a meeting with d'Avalos. So, in March 1539, Tartaglia left Venice and travelled to Milan. To Tartaglia's dismay, the governor was temporarily absent from Milan but Cardan attended to his guest's every need and soon the conversation turned the cosa and cube problem. Tartaglia, after much persuasion, agreed to tell Cardan his method, if the Cardan would swear never to reveal it and furthermore, to only ever write it down in code so that on his death, nobody would discover the secret from his papers. This Cardan readily agreed to, and Tartaglia divulged his formula in a poem, to help protect the secret, should the paper fall into the wrong hands. Anxious now to leave Cardan's house, he obtained from his host, a letter of introduction to the Marchese and left to seek him out. Instead though, he turned back for Venice, wondering if his decision to part with his formula had been a mistake. By the time he reached Venice, Tartaglia was sure he had made a mistake in trusting Cardan and began to feel very angry that he had been induced to reveal his secret formula. Cardan published two mathematical books that year and, as soon as he could get copies, Tartaglia checked to make sure his formula was not included. Though he felt a little happier to find that the formula was not included in the texts, when Cardan wrote to him in a friendly manner Tartaglia rebuffed his offer of continued friendship and mercilessly ridiculed his books on the merest trivialities. Based on Tartaglia's formula, Cardan and Ferrari, his assistant, made remarkable progress finding proofs of all cases of the cubic and, even more impressively, solving the quartic equation. Tartaglia made no move to publish his formula, despite the fact that, by now, it had become well known that such a method existed. Tartaglia probably wished to keep his formula in reserve for any upcoming debates.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tartaglia.html (2 of 4) [2/16/2002 11:34:33 PM]

Tartaglia

Cardan and Ferrari travelled to Bologna and learnt from della Nave that del Ferro and not Tartaglia had been the first to solve the cubic equation. Cardan felt that although he had sworn not to reveal Tartaglia's method surely nothing prevented him from publishing del Ferro's formula. In 1545 Cardan published Artis magnae sive de regulis algebraicis liber unus or Ars magna as it is more commonly known which contained solutions to the cubic and quartic equations and all of the additional work he had completed on Tartaglia's formula. Del Ferro and Tartaglia are fully credited with their discoveries, as is Ferrari, and the whole story written down in the text. Tartaglia was furious when he discovered that Cardan had disregarded his oath and his intense dislike of Cardan turned into a pathological hatred. The following year Tartaglia published a book, New Problems and Inventions which clearly stated his side of the story and his belief that Cardan had acted in extreme bad faith. For good measure, he added a few malicious personal insults directed against Cardan. Ars Magna had clearly established Cardan as the world's leading mathematician and he was not much damaged by Tartaglia's venomous attacks. Ferrari, however, wrote to Tartaglia, berating him mercilessly and challenged him to a public debate. Tartaglia was extremely reluctant to dispute with Ferrari, still a relatively unknown mathematician, against whom even a victory would do little material good. A debate with Cardan, on the other hand, held great appeal for Tartaglia. Not only did he hate him but Cardan was a leading figure in the mathematical, medical and literary worlds, and even to enter a debate with him would greatly enhance Tartaglia's standing. For all the brilliance of his discovery of the solution to the cosa and cube problem, Tartaglia was still a relatively poor mathematics teacher in Venice. So Tartaglia replied to Ferrari, trying to bring Cardan into the debate. Cardan, however, had no intention of debating with Tartaglia. Ferrari and Tartaglia wrote fruitlessly to each other for about a year, trading the most offensive personal insults but achieving little in the way of resolving the dispute. Suddenly in 1548, Tartaglia received an impressive offer of a lectureship in his home town, Brescia. To clearly establish his credential for the post, Tartaglia was asked to journey to Milan and take part in the contest with Ferrari. On 10 August 1548 the contest took place in the Church in the Garden of the Frati Zoccolanti. Tartaglia was vastly experienced in such debates and expected to win. However, by the end of the first day, it was clear that things were not going Tartaglia's way. Ferrari clearly understood the cubic and quartic equations more thoroughly and Tartaglia decided that he would leave Milan that night and thus leave the contest unresolved. With Tartaglia departing ignominiously, victory was left to Ferrari. Tartaglia suffered as a result of the contest. After giving his lectures for a year in Brescia, he was informed that his stipend was not going to honoured. Even after numerous lawsuits, Tartaglia could not get any payment and returned, seriously out of pocket, to his previous job in Venice, nursing a huge resentment of Cardan. The defeat in Milan would appear to be responsible for Tartaglia's non-payment. Tartaglia is now remembered in that the name of the formula for solving the cubic has been named the Cardan-Tartaglia formula. However, Tartaglia did contribute to mathematics in a number of other ways. Fairly early in his career, before he became involved in the arguments about the cubic equation, he wrote Nova Scientia (1537) on the application of mathematics to artillery fire. In the work he described new ballistic methods and instruments, including the first firing tables.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tartaglia.html (3 of 4) [2/16/2002 11:34:33 PM]

Tartaglia

Tartaglia also wrote a popular arithmetic text and was the first Italian translator and publisher of Euclid's Elements in 1543. In 1546 he published Quesiti et Inventioni diverse de Nicolo Tartalea referred to above. He also published Latin editions of Archimedes's works. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (20 books/articles)

A Quotation

A Poster of Niccolo Fontana Tartaglia

Mathematicians born in the same country

Cross-references to History Topics

1. Quadratic, cubic and quartic equations 2. Mathematical games and recreations 3. An overview of the history of mathematics

Other references in MacTutor

Chronology: 1500 to 1600

Other Web sites

1. The Catholic Encyclopedia 2. The Galileo Project 3. Karen H Parshall 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tartaglia.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tartaglia.html (4 of 4) [2/16/2002 11:34:33 PM]

Tauber

Alfred Tauber Born: 5 Nov 1866 in Pressburg (now Bratislava), Slovakia Died: 1942 in Theresienstadt, Germany (now Terezin, Czech Republic) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alfred Tauber worked in Vienna. His research was on function theory and potential theory. He obtained important results on divergent series and the name Tauberian Theorems was coined by Hardy and Littlewood. Further major results in this area were obtained by Norbert Wiener. Of lesser importance is Tauber's work on differential equations and the gamma function. The date of his death is unknown. He was sent by the Nazis to Theresienstadt concentration camp on June 28 1942. Just after Tauber arrived the entire non-Jewish population of 3700 of Theresienstadt was evacuated and he was one of 53000 inhabitants of the camp. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tauber.html (1 of 2) [2/16/2002 11:34:35 PM]

Tauber

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tauber.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tauber.html (2 of 2) [2/16/2002 11:34:35 PM]

Taurinus

Franz Adolph Taurinus Born: 15 Nov 1794 in Bad König, German Died: 13 Feb 1874 in Cologne, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Franz Adolph Taurinus's father was a court official working for the counts of Erbach- Schöneberg. His mother was Luise Juliane Schweikart who had a younger brother F K Schweikart, only fourteen years of age when Taurinus was born. F K Schweikart went on to become professor of law at the University of Königsberg and it is fair to say that Taurinus's uncle Schweikart played a major role in influencing his ideas and academic achievements. Following his uncle's academic discipline, Taurinus studied law at Heidelberg, Giessen and Göttingen. From 1822 he lived in Cologne as a man of independent means able to devote himself entirely to research without students to teach. Two years before he went to live in Cologne his uncle became professor of law at the University of Königsberg. the two corresponded on mathematical topics and, largely due to Schweikart's influence, he began to investigate the problem of parallel lines and Euclid's fifth postulate:That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles. In 1697 Girolamo Saccheri assumed the fifth postulate is false and attempted to derive a contradiction. Of course, although he did not intend it to be so, he was then studying non-euclidean geometry. In 1766 Lambert followed a similar line to Saccheri. Lambert noticed that, in this new geometry where the sum of the angles of a triangle was less than 180q, the angle sum of a triangle increased as the area of the triangle decreased. Schweikart himself is famed for investigating this new geometry which he called astral geometry. This is described in [3]. Taurinus not only corresponded on mathematical topics with his uncle but he also corresponded with Gauss about his ideas on geometry. At first Taurinus tried to prove that Euclidean geometry was the only geometry but, in 1826, he accepted the lack of contradiction in other geometries. He published Theorie der Parallellinien in Cologne in 1825 and in the following year he published Geometriae prima elementa also in Cologne. In this last mentioned publication Taurinus accepts that a third system of geometry exists in which the sum of the angles of a triangle is less than 180q. He called this geometry "logarithmic-spherical geometry" and he recognised the lack of a contradiction in this geometry as meaning that it was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taurinus.html (1 of 2) [2/16/2002 11:34:36 PM]

Taurinus

internally consistent. He had developed a non-euclidean trigonometry which he applied to a number of elementary problems. Taurinus came up with the important idea that elliptic geometry could be realised on the surface of a sphere, an idea taken up by Riemann. He also realised that there were an infinite number of non-euclidean geometries and this, Taurinus claimed, was highly significant. It showed that euclidean geometry held a unique dominating role. This is an interesting sideways move since his original aim had been to prove that euclidean geometry was the unique geometry. Finding that this was not so, he still wanted to demonstrate that euclidean geometry was "the" geometry. Haas writes in [1]:Taurinus' works on the problem of parallel lines. like those of his uncle, Schweikart, represent a middle stage in the historical development of this problem between the efforts of Saccheri and Lambert, on the one hand, and those of Gauss, Lobachevsky, and Bolyai, on the other. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Some pages from publications

A letter from Gauss to Taurinus discussing the possibility of non-Euclidean geometry.

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Taurinus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taurinus.html (2 of 2) [2/16/2002 11:34:36 PM]

Taussky-Todd

Olga Taussky-Todd Born: 30 Aug 1906 in Olmütz, Austro-Hungarian Empire (now Olomouc, Czech Republic) Died: 7 Oct 1995 in Pasadena, California, USA

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

When Olga Taussky (she became Taussky-Todd after marrying Jack Todd) was three years old the family moved to Vienna. Then in 1916 the family moved again, this time to Linz. Olga's father died while she was still at school so it was hard to fund her through university. However she entered the University of Vienna and, after initially studying chemistry, took up mathematics. At Vienna Olga was taught by Furtwängler, Hahn, Wirtinger, Menger, Helly and others, while she was a fellow student of Gödel. She wrote her thesis, just as class field theory was being developed, on algebraic number fields. She was awarded a doctorate in 1930. Hahn recommended Taussky to Courant and, in 1931 she was appointed as assistant at Göttingen. Working with Wilhelm Magnus and Helmut Ulm, she edited the first volume of Hilbert's complete works on number theory. While in Göttingen she edited Artin's lectures in class field theory (1932), assisted Emmy Noether in her class field theory and Courant in his differential equations course. In 1932-1933 Taussky tutored in Vienna, then she spent a year at Bryn Mawr before taking up a research fellowship from Girton College, Cambridge in 1935. Hardy helped her obtain a teaching post in a London college in 1937 where she soon met Jack (John Todd). They were married the following year. During World War II Olga and Jack moved from place to place. While teaching near Oxford she http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taussky-Todd.html (1 of 3) [2/16/2002 11:34:38 PM]

Taussky-Todd

supervised Hanna Neumann's D.Phil. thesis on combinatorial group theory. In 1947 Olga and Jack when to the USA and worked at the National Bureau of Standards' National Applied Mathematics Laboratory. Here she worked with computers and has been described as a computer pioneer ... who provided significant contributions to solutions of problems associated with applications of computers. In 1957 Olga and Jack both accepted appointments at the California Institute of Technology. She wrote:After many years of work mostly with applied mathematics, I was in the beginning rather uncertain about the teaching. But again it was the students who came to my assistance [as in the college in London]. It was clear to them that I had much mathematics to give them and they forced it out of me. Olga's honours and work is described in [5] as follows:Olga Taussky-Todd was a distinguished and prolific mathematician who wrote about 300 papers. Throughout her life she received many honors and distinctions, most notably the Cross of Honor, the highest recognition of contributions given by her native Austria. Olga's best-known and most influential work was in the field of matrix theory, though she also made important contributions to number theory. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Other Web sites

1. Agnes Scott College 2. AMS (An electronic version of the AMS obituary) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taussky-Todd.html (2 of 3) [2/16/2002 11:34:38 PM]

Taussky-Todd

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Taussky-Todd.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taussky-Todd.html (3 of 3) [2/16/2002 11:34:38 PM]

Taylor

Brook Taylor Born: 18 Aug 1685 in Edmonton, Middlesex, England Died: 29 Dec 1731 in Somerset House, London, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Brook Taylor's father was John Taylor and his mother was Olivia Tempest. John Taylor was the son of Natheniel Taylor who was recorder of Colchester and a member representing Bedfordshire in Oliver Cromwell's Assembly, while Olivia Tempest was the daughter of Sir John Tempest. Brook was, therefore, born into a family which was on the fringes of the nobility and certainly they were fairly wealthy. Taylor was brought up in a household where his father ruled as a strict disciplinarian, yet he was a man of culture with interests in painting and music. Although John Taylor had some negative influences on his son, he also had some positive ones, particularly giving his son a love of music and painting. Brook Taylor grew up not only to be an accomplished musician and painter, but he applied his mathematical skills to both these areas later in his life. As Taylor's family were well off they could afford to have private tutors for their son and in fact this home education was all that Brook enjoyed before entering St John's College Cambridge on 3 April 1703. By this time he had a good grounding in classics and mathematics. At Cambridge Taylor became highly involved with mathematics. He graduated with an LL.B. in 1709 but by this time he had already written his first important mathematics paper (in 1708) although it would not be published until 1714. We know something of the details of Taylor thoughts on various mathematical problems from letters he exchanged with Machin and Keill beginning in his undergraduate years. In 1712 Taylor was elected to the Royal Society. This was on the 3 April, and clearly it was an election based more on the expertise which Machin, Keill and others knew that Taylor had, rather than on his

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taylor.html (1 of 5) [2/16/2002 11:34:40 PM]

Taylor

published results. For example Taylor wrote to Machin in 1712 providing a solution to a problem concerning Kepler's second law of planetary motion. Also in 1712 Taylor was appointed to the committee set up to adjudicate on whether the claim of Newton or of Leibniz to have invented the calculus was correct. The paper we referred to above as being written in 1708 was published in the Philosophical Transactions of the Royal Society in 1714. The paper gives a solution to the problem of the centre of oscillation of a body, and it resulted in a priority dispute with Johann Bernoulli. We shall say a little more below about disputes between Taylor and Johann Bernoulli. Returning to the paper, it is a mechanics paper which rests heavily on Newton's approach to the differential calculus. The year 1714 also marks the year in which Taylor was elected Secretary to the Royal Society. It was a position which Taylor held from 14 January of that year until 21 October 1718 when he resigned, partly for health reasons, partly due to his lack of interest in the rather demanding position. The period during which Taylor was Secretary to the Royal Society does mark what must be considered his most mathematically productive time. Two books which appeared in 1715, Methodus incrementorum directa et inversa and Linear Perspective are extremely important in the history of mathematics. Second editions would appear in 1717 and 1719 respectively. We discuss the content of these works in some detail below. Taylor made several visits to France. These were made partly for health reasons and partly to visit the friends he had made there. He met Pierre Rémond de Montmort and corresponded with him on various mathematical topics after his return. In particular they discussed infinite series and probability. Taylor also corresponded with de Moivre on probability and at times there was a three-way discussion going on between these mathematicians. Between 1712 and 1724 Taylor published thirteen articles on topics as diverse as describing experiments in capillary action, magnetism and thermometers. He gave an account of an experiment to discover the law of magnetic attraction (1715) and an improved method for approximating the roots of an equation by giving a new method for computing logarithms (1717). His life, however, suffered a series of personal tragedies beginning around 1721. In that year he married Miss Brydges from Wallington in Surrey. Although she was from a good family, it was not a family with money and Taylor's father strongly objected to the marriage. The result was that relations between Taylor and his father broke down and there was no contact between father and son until 1723. It was in that year that Taylor's wife died in childbirth. The child, which would have been their first, also died. After the tragedy of losing his wife and child, Taylor returned to live with his father and relations between the two were repaired. Two years later, in 1725, Taylor married again to Sabetta Sawbridge from Olantigh in Kent. This marriage had the approval of Taylor's father who died four years later on 4 April 1729. Taylor inherited his father's estate of Bifons but further tragedy was to strike when his second wife Sabetta died in childbirth in the following year. On this occasion the child, a daughter Elizabeth, did survive. Taylor added to mathematics a new branch now called the "calculus of finite differences", invented integration by parts, and discovered the celebrated series known as Taylor's expansion. These ideas appear in his book Methodus incrementorum directa et inversa of 1715 referred to above. In fact the first mention by Taylor of a version of what is today called Taylor's Theorem appears in a letter which he http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taylor.html (2 of 5) [2/16/2002 11:34:40 PM]

Taylor

wrote to Machin on 26 July 1712. In this letter Taylor explains carefully where he got the idea from. It was, wrote Taylor, due to a comment that Machin made in Child's Coffeehouse when he had commented on using "Sir Isaac Newton's series" to solve Kepler's problem, and also using "Dr Halley's method of extracting roots" of polynomial equations. There are, in fact, two versions of Taylor's Theorem given in the 1715 paper which to a modern reader look equivalent but which, the author of [8] argues convincingly, were differently motivated. Taylor initially derived the version which occurs as Proposition 11 as a generalisation of Halley's method of approximating roots of the Kepler equation, but soon discovered that it was a consequence of the Bernoulli series. This is the version which was inspired by the Coffeehouse conversation described above. The second version occurs as Corollary 2 to Proposition 7 and was thought of as a method of expanding solutions of fluxional equations in infinite series. We must not give the impression that this result was one which Taylor was the first to discover. James Gregory, Newton, Leibniz, Johann Bernoulli and de Moivre had all discovered variants of Taylor's Theorem. Gregory, for example, knew that arctan x = x - x3/3 + x5/5 - x7/7 + ... and his methods are discussed in [13]. The differences in Newton's ideas of Taylor series and those of Gregory are discussed in [15]. All of these mathematicians had made their discoveries independently, and Taylor's work was also independent of that of the others. The importance of Taylor's Theorem remained unrecognised until 1772 when Lagrange proclaimed it the basic principle of the differential calculus. The term "Taylor's series" seems to have used for the first time by Lhuilier in 1786. There are other important ideas which are contained in the Methodus incrementorum directa et inversa of 1715 which were not recognised as important at the time. These include singular solutions to differential equations, a change of variables formula, and a way of relating the derivative of a function to the derivative of the inverse function. Also contained is a discussion on vibrating strings, an interest which almost certainly come from Taylor's early love of music. Taylor, in his studies of vibrating strings was not attempting to establish equations of motion, but was considering the oscillation of a flexible string in terms of the isochrony of the pendulum. He tried to find the shape of the vibrating string and the length of the isochronous pendulum rather than to find its equations of motion. Further discussion of these ideas is given in [14]. Taylor also devised the basic principles of perspective in Linear Perspective (1715). The second edition has a different title, being called New principles of linear perspective. The work gives first general treatment of vanishing points. Taylor had a highly mathematical approach to the subject and made no concessions to artists who should have found the ideas of fundamental importance to them. At times it is very difficult for even a mathematician to understand Taylor's results. The phrase "linear perspective" was invented by Taylor in this work and he defined the vanishing point of a line, not parallel to the plane of the picture, as the point where a line through the eye parallel to the given line intersects the plane of the picture. He also defined the vanishing line to a given plane, not parallel to the plane of the picture, as the intersection of the plane through the eye parallel to the given plane. He did not invent the terms vanishing point and vanishing line, but he was one of the first to stress their importance. The main theorem in Taylor's theory of linear perspective is that the projection of a straight line not parallel to the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taylor.html (3 of 5) [2/16/2002 11:34:40 PM]

Taylor

plane of the picture passes through its intersection and its vanishing point. There is also the interesting inverse problem which is to find the position of the eye in order to see the picture from the viewpoint that the artist intended. Taylor was not the first to discuss this inverse problem but he did make innovative contributions to the theory of such perspective problems. One could certainly consider this work as laying the foundations for the theory of descriptive and projective geometry. Taylor challenged the "non-English mathematicians" to integrate a certain differential. One has to see this challenge as part of the argument between the Newtonians and the Leibnitzians. Conte in [7] discusses the answers given by Johann Bernoulli and Giulio Fagnano to Taylor's challenge. We mentioned above the arguments between Johann Bernoulli and Taylor. Taylor, although he did not win all the arguments, could certainly dispute with Johann Bernoulli on fairly equal terms. Jones describes these arguments in [1]:Their debates in journals occasionally included rather heated phrases and, at one time, a wager of fifty guineas. When Bernoulli suggested in a private letter that they couch their debate in more gentlemanly terms, Taylor replied that he meant to sound sharp and to "show an indignation". Jones also explains in [1] that Taylor was a mathematician of far greater depth than many have given him credit for:A study of Brook Taylor's life and work reveals that his contribution to the development of mathematics was substantially greater than the attachment of his name to one theorem would suggest. His work was concise and hard to follow. The surprising number of major concepts that he touched upon, initially developed, but failed to elaborate further leads one to regret that health, family concerns and sadness, or other unassessable factors, including wealth and parental dominance, restricted the mathematically productive portion of his relatively short life. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (15 books/articles) A Poster of Brook Taylor

Mathematicians born in the same country

Some pages from publications

The title page from New Principles of Linear Perspective (1719) and some diagrams from it.

Cross-references to History Topics

1. The rise of calculus 2. Mathematics in St Andrews to 1700 3. Mathematical games and recreations 4. An overview of Indian mathematics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taylor.html (4 of 5) [2/16/2002 11:34:40 PM]

Taylor

Other references in MacTutor

1. 2. 3. 4.

Taylor series series for cosine series for sine Chronology: 1700 to 1720

Honours awarded to Brook Taylor (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1712

Lunar features

Crater Taylor

Other Web sites

1. Rouse Ball 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Taylor.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taylor.html (5 of 5) [2/16/2002 11:34:40 PM]

Taylor_Geoffrey

Geoffrey Ingram Taylor Born: 7 March 1886 in St John's Wood, London, England Died: 27 June 1975 in Cambridge, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Geoffrey Taylor was a grandson of George Boole and Alicia Stott was his aunt. He attended school in Hampstead, and there he began to find his love of science. At the age of 11 he attended a series of children's Christmas lectures on The principles of the electric telegraph and these made a strong impression on him. He was introduced to William Thomson at one of these lectures and Lord Kelvin told him he had been friendly with Geoffrey Taylor's grandfather George Boole. In 1899 Taylor went to University College School and in 1905 he won a scholarship to study at Trinity College, Cambridge. There he read mathematics, attending lectures by Whitehead, Whittaker and Hardy. After taking part I of the mathematics tripos he moved towards physics taking part II of the physics tripos. He then won a scholarship to undertake research at Trinity College. One of his first pieces of research was a theoretical study of shock waves where he extended work by Thomson. This work won him a Smith's Prize. In 1910 he was elected to a Fellowship at Trinity College. The following year he was appointed to a meteorology post and his work on turbulence in the atmosphere led to his publication Turbulent motion in fluids which won the Adams Prize at Cambridge in 1915. The outbreak of World War I saw Taylor offer his services and he was sent to Farnborough to use his scientific skills in the design and operation of aeroplanes. Here he worked on the stress on propeller shafts. This led him to think about the limiting strengths of materials and this influenced some of his later projects.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taylor_Geoffrey.html (1 of 3) [2/16/2002 11:34:43 PM]

Taylor_Geoffrey

After World War I Taylor returned to a lectureship at Cambridge. One of the topics he worked an at this stage was an application of turbulent flow to oceanography. He also worked on the problem of bodies passing through a rotating fluid. In 1923 Taylor was appointed to a Royal Society research professorship. This enabled him to stop teaching which he had been doing for the previous four years. As Batchelor writes in [2]:He was not a natural lecturer and not much interested in teaching... At this stage Taylor made a great many fundamental steps in the study of fluids. This period is described in [2]:His investigations in the mechanics of fluids and solids covered an extraordinary wide range, and most of them exhibited the originality and insight for which he was now becoming famous. He undertook research on the deformation of crystalline materials, work which led from his World War I work at Farnborough. Among the many topics he studied was also another major contribution to turbulent flow, where he introduced a new approach through a statistical study of velocity fluctuations. During World War II Taylor again worked on applications of his expertise to military problems such as the propagation of blast waves, studying both waves in the air and underwater explosions. Taylor continued his research after the end of the War, taking the opportunity to complete some more thorough investigations into problems which pressure of finding solutions had prevented him from taking further previously. He retired in 1952 but he continued his work at Cambridge with little evidence that his status had in any way changed until 1972. In that year he suffered a stroke from which he only partially recovered. During his last three year he suffered the frustrations of wanting to get back to scientific work although his physical condition would not allow it. Taylor received many honours during his life. He was elected a Fellow of the Royal Society in 1919, winning its Royal Medal in 1933 and its Copley Medal in 1944:... for his many contributions to aeodynamics, hydrodynamics, and the structure of metals, which have had a profound influence on the advance of physical science and its applications. Also in 1944 he was knighted and appointed to the Order of Merit in 1969. He was elected to membership of academic societies in many countries including the United States, France, Italy, Sweden, The Netherlands, India, Poland and the USSR. He received honorary degrees from more than a dozen universities throughout the world and over twenty Medals for his outstanding contributions to applied mathematics. He published over 250 papers in his long career on applied mathematics, mathematical physics and engineering. His contribution is summed up in [2] as follows:Taylor's work is of the greatest importance to the mechanics of fluids and solids and to their application in meteorology, oceanography, aeronautics, metal physics, mechanical engineering and chemical engineering. The nature of his thinking was like that of Stokes, Kelvin and Rayleigh, although he got more from experiments than any one of these three. He had the rare honour of seeing his scientific papers, some previously unpublished, gathered together and published in four thick volumes during his lifetime. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taylor_Geoffrey.html (2 of 3) [2/16/2002 11:34:43 PM]

Taylor_Geoffrey

List of References (10 books/articles) Mathematicians born in the same country Honours awarded to Geoffrey Taylor (Click a link below for the full list of mathematicians honoured in this way) Royal Society Copley Medal

Awarded 1944

Royal Society Bakerian lecturer

1923

LMS De Morgan Medal

Awarded 1956

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Taylor_Geoffrey.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Taylor_Geoffrey.html (3 of 3) [2/16/2002 11:34:43 PM]

Teichmuller

Paul Julius Oswald Teichmüller Born: 18 June 1913 in Nordhausen im Harz, Germany Died: 11 Sept 1943 in Dnieper region, USSR

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oswald Teichmüller studied at Göttingen under Hasse, writing a thesis on linear operators on Hilbert spaces, then in Berlin under Bieberbach. A member of the Nazi party, he played a major role in getting the students at Göttingen to dismiss Landau. Teichmüller's main contribution is in the area of geometric function theory. He wrote 34 papers in the space of about 6 years, 21 being published in Deutsche Mathematik, the journal for German style mathematics founded by Bieberbach. He introduced quasi-conformal mappings and differential geometric methods into complex analysis. Teichmüller joined the army in 1939, fighting at first in Norway but he was last heard of on the date given above for his death, involved in heavy fighting along the Dnieper. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) A Poster of Oswald Teichmüller

Previous

Mathematicians born in the same country

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Teichmuller.html (1 of 2) [2/16/2002 11:34:44 PM]

Teichmuller

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Teichmuller.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Teichmuller.html (2 of 2) [2/16/2002 11:34:44 PM]

Temple

George Frederick James Temple Born: 2 Dec 1901 in London, England Died: 30 Jan 1992 in Quarr Abbey, Isle of Wight, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Temple was educated in London. In 1918 he entered Birkbeck College as an evening student (the College had previously been known as London Mechanic's Institution and in 1920 was recognised as a school of the University of London for evening and part-time students). George became a part-time research assistant in physics there the following year. He received his B.Sc. in 1922, then two years later he was appointed Demonstrator in Mathematics at Imperial College, London. Whitehead had been the main reason for Temple's move to Imperial as he had been interested in papers in relativity which Temple had published. However Whitehead left the chair at Imperial as Temple arrived and Chapman was appointed to fill the chair. Chapman obtained a scholarship for Temple to undertake further research and he spent a year at Imperial working on quantum theory before going to Cambridge where he worked with Eddington. Temple returned to Imperial as a Reader in 1930, then two years later was appointed to the chair of mathematics at King's College London. During World War II he worked at RAF Farnborough where his work earned him a CBE. During his time in Farnborough he worked on aerodynamics and supersonic fluid flow. After 1945 Temple returned to King's College. He advised the Minister of Civil Aviation on air traffic control during 1948-50. In 1953 Temple moved to the Sedleian chair at Oxford to succeed Chapman. Temple worked on a wide variety of topics. Relativity theory, aerodynamics and quantum mechanics have been mentioned above but he also worked on analysis contributing to the study of the Lebesgue

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Temple.html (1 of 3) [2/16/2002 11:34:46 PM]

Temple

integral. He wrote seven books, two on quantum theory An introduction to quantum theory (1931) and The general principles of quantum theory (1934). His other books included An introduction to fluid dynamics (1958) and The structure of the Lebesgue integration theory (1971). In 1981 (at the age of 80) he published a book on the history of mathematics. This book 100 years of mathematics (1981) took him ten years to write and deals with, in his own words:those branches of mathematics in which I had been personally involved. He declared that it was his last mathematics book, and entered the Benedictine Order as a monk. He was ordained in 1983 and entered Quarr Abbey on the Isle of Wight. However he could not stop doing mathematics and when he died he left a manuscript on the foundations of mathematics. He claims:The purpose of this investigation is to carry out the primary part of Hilbert's programme, i.e. to establish the consistency of set theory, abstract arithmetic and propositional logic and the method used is to construct a new and fundamental theory from which these theories can be deduced. Temple was elected a Fellow of the Royal Society in 1943 and, in 1970, he was awarded the Sylvester Medal of the Society:... in recognition of his many distinguished contributions to applied mathematics, especially in his work on distribution theory. He was President of the London Mathematical Society during the period 1951-53. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Honours awarded to George Temple (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1943

Royal Society Sylvester Medal

Awarded 1970

London Maths Society President

1951 - 1953

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Temple.html (2 of 3) [2/16/2002 11:34:46 PM]

Temple

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Temple.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Temple.html (3 of 3) [2/16/2002 11:34:46 PM]

Tetens

Johannes Nikolaus Tetens Born: 16 Sept 1736 in Tetenbüll, South Schleswig (now Germany) Died: 17 Aug 1807 in Copenhagen, Denmark

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johannes Tetens was professor of physics at Bützow University (1760-1765) then was made director of the Bützow Pädagogium. He was appointed professor of philosophy at the University of Kiel in 1776 and he was later appointed to the chair of mathematics at Kiel. Tetens' major work was in philosophy and his theories were to have a major influence on Kant. Tetens wrote Philosophische Versuche über die menschliche Natur und ihre Entwickelung (1777) which is an investigation of the origin and structure of knowledge. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tetens.html (1 of 2) [2/16/2002 11:34:49 PM]

Tetens

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tetens.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tetens.html (2 of 2) [2/16/2002 11:34:49 PM]

Thabit

Al-Sabi Thabit ibn Qurra al-Harrani Born: 826 in Harran, Mesopotamia (now Turkey) Died: 18 Feb 901 in Baghdad, (now in Iraq)

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thabit ibn Qurra was a native of Harran and a member of the Sabian sect. The Sabian religious sect were star worshippers from Harran often confused with the Mandaeans (as they are in [1]). Of course being worshipers of the stars meant that there was strong motivation for the study of astronomy and the sect produced many quality astronomers and mathematicians. The sect, with strong Greek connections, had in earlier times adopted Greek culture, and it was common for members to speak Greek although after the conquest of the Sabians by Islam, they became Arabic speakers. There was another language spoken in southeastern Turkey, namely Syriac, which was based on the East Aramaic dialect of Edessa. This language was Thabit ibn Qurra's native language, but he was fluent in both Greek and Arabic. Some accounts say that Thabit was a money changer as a young man. This is quite possible but some historians do not agree. Certainly he inherited a large family fortune and must have come from a family of high standing in the community. Muhammad ibn Musa ibn Shakir, who visited Harran, was impressed at Thabit's knowledge of languages and, realising the young man's potential, persuaded him to go to Baghdad and take lessons in mathematics from him and his brothers (the Banu Musa). In Baghdad Thabit received mathematical training and also training in medicine, which was common for scholars of that time. He returned to Harran but his liberal philosophies led to a religious court appearance when he had to recant his 'heresies'. To escape further persecution he left Harran and was appointed court astronomer in Baghdad. There Thabit's patron was the Caliph, al-Mu'tadid, one of the greatest of the 'Abbasid caliphs. At this time there were many patrons who employed talented scientists to translate Greek text into Arabic and Thabit, with his great skills in languages as well as great mathematical skills, translated and revised many of the important Greek works. The two earliest translations of Euclid's Elements were made by

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thabit.html (1 of 4) [2/16/2002 11:34:51 PM]

Thabit

al-Hajjaj. These are lost except for some fragments. There are, however, numerous manuscript versions of the third translation into Arabic which was made by Hunayn ibn Ishaq and revised by Thabit. Knowledge today of the complex story of the Arabic translations of Euclid's Elements indicates that all later Arabic versions develop from this revision by Thabit. In fact many Greek texts survive today only because of this industry in bringing Greek learning to the Arab world. However we must not think that the mathematicians such as Thabit were mere preservers of Greek knowledge. Far from it, Thabit was a brilliant scholar who made many important mathematical discoveries. Although Thabit contributed to a number of areas the most important of his work was in mathematics where he [1]:... played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics. We shall examine in more detail Thabit's work in these areas, in particular his work in number theory on amicable numbers. Suppose that, in modern notation, S(n) denotes the sum of the aliquot parts of n, that is the sum of its proper quotients. Perfect numbers are those numbers n with S(n) = n while m and n are amicable if S(n) = m, and S(m) = n. In Book on the determination of amicable numbers Thabit claims that Pythagoras began the study of perfect and amicable numbers. This claim, probably first made by Iamblichus in his biography of Pythagoras written in the third century AD where he gave the amicable numbers 220 and 284, is almost certainly false. However Thabit then states quite correctly that although Euclid and Nicomachus studied perfect numbers, and Euclid gave a rule for determining them ([6] or [7]):... neither of these authors either mentioned or showed interest in [amicable numbers]. Thabit continues ([6] or [7]):Since the matter of [amicable numbers] has occurred to my mind, and since I have derived a proof for them, I did not wish to write the rule without proving it perfectly because they have been neglected by [Euclid and Nicomachus]. I shall therefore prove it after introducing the necessary lemmas. After giving nine lemmas Thabit states and proves his theorem: for n > 1, let pn = 3.2n-1 and qn = 9.22n-1-1. If pn-1, pn, and qn are prime numbers, then a = 2n pn-1 pn and b = 2nqn are amicable numbers while a is abundant and b is deficient. Note that an abundant number n satisfies S(n) > n, and a deficient number n satisfies S(n) < n. More details are given in [9] where the authors conjecture how Thabit might have discovered the rule. In [13] Hogendijk shows that Thabit was probably the first to discover the pair of amicable numbers 17296, 18416. Another important aspect of Thabit's work was his book on the composition of ratios. In this Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thabit.html (2 of 4) [2/16/2002 11:34:51 PM]

Thabit

arithmetic could be applied. The authors of [22] and [23] stress that by introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalisation of the number concept. Thabit generalised Pythagoras's theorem to an arbitrary triangle (as did Pappus). He also discussed parabolas, angle trisection and magic squares. Thabit's work on parabolas and paraboliods is of particular importance since it is one of the steps taken towards the discovery of the integral calculus. An important consideration here is whether Thabit was familiar with the methods of Archimedes. Most authors (see for example [29]) believe that although Thabit was familiar with Archimedes' results on the quadrature of the parabola, he did not have either of Archimedes' two treatises on the topic. In fact Thabit effectively computed the integral of x and [1]:The computation is based essentially on the application of upper and lower integral sums, and the proof is done by the method of exhaustion: there, for the first time, the segment of integration is divided into unequal parts. Thabit also wrote on astronomy, writing Concerning the Motion of the Eighth Sphere. He believed (wrongly) that the motion of the equinoxes oscillates. He also published observations of the Sun. In fact eight complete treatises by Thabit on astronomy have survived and the article [20] describes these. The author of [20] writes:When we consider this body of work in the context of the beginnings of the scientific movement in ninth-century Baghdad, we see that Thabit played a very important role in the establishment of astronomy as an exact science (method, topics and program), which developed along three lines: the theorisation of the relation between observation and theory, the 'mathematisation' of astronomy, and the focus on the conflicting relationship between 'mathematical' astronomy and 'physical' astronomy. An important work Kitab fi'l-qarastun (The book on the beam balance) by Thabit is on mechanics. It was translated into Latin by Gherard of Cremona and became a popular work on mechanics. In this work Thabit proves the principle of equilibrium of levers. He demonstrates that two equal loads, balancing a third, can be replaced by their sum placed at a point halfway between the two without destroying the equilibrium. After giving a generalisation Thabit then considers the case of equally distributed continuous loads and finds the conditions for the equilibrium of a heavy beam. Of course Archimedes considered a theory of centres of gravity, but in [14] the author argues that Thabit's work is not based on Archimedes' theory. Finally we should comment on Thabit's work on philosophy and other topics. Thabit had a student Abu Musa Isa ibn Usayyid who was a Christian from Iraq. Ibn Usayyid asked various questions of his teacher Thabit and a manuscript exists of the answers given by Thabit, this manuscript being discussed in [21]. Thabit's concept of number follows that of Plato and he argues that numbers exist, whether someone knows them or not, and they are separate from numerable things. In other respects Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments. Thabit also wrote on [1]:... logic, psychology, ethics, the classification of sciences, the grammar of the Syriac

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thabit.html (3 of 4) [2/16/2002 11:34:51 PM]

Thabit

language, politics, the symbolism of Plato's Republic ... religion and the customs of the Sabians. This archive contains information on other members of Thabit's family. His son, Sinan ibn Thabit, and his grandson Ibrahim ibn Sinan ibn Thabit, both were eminent scholars who contributed to the development of mathematics. Neither, however, reached the mathematical heights of Thabit. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (29 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Perfect numbers 2. Arabic mathematics : forgotten brilliance? 3. How do we know about Greek mathematics?

Other references in MacTutor

Chronology: 500 to 900

Honours awarded to Thabit (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Thebit

Other Web sites

1. Muslim scientists 2. Muslims online 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thabit.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thabit.html (4 of 4) [2/16/2002 11:34:51 PM]

Thales

Thales of Miletus Born: about 624 BC in Miletus, Asia Minor (now Turkey) Died: about 547 BC in Miletus, Asia Minor (now Turkey)

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thales of Miletus was the son of Examyes and Cleobuline. His parents are said by some to be from Miletus but others report that they were Phoenicians. J Longrigg writes in [1]:But the majority opinion considered him a true Milesian by descent, and of a distinguished family. Thales seems to be the first known Greek philosopher, scientist and mathematician although his occupation was that of an engineer. He is believed to have been the teacher of Anaximander (611 BC 545 BC) and he was the first natural philosopher in the Milesian School. However, none of his writing survives so it is difficult to determine his views or to be certain about his mathematical discoveries. Indeed it is unclear whether he wrote any works at all and if he did they were certainly lost by the time of Aristotle who did not have access to any writings of Thales. On the other hand there are claims that he wrote a book on navigation but these are based on little evidence. In the book on navigation it is suggested that he used the constellation Ursa Minor, which he defined, as an important feature in his navigation techniques. Even if the book is fictitious, it is quite probable that Thales did indeed define the constellation Ursa Minor. Proclus, the last major Greek philosopher, who lived around 450 AD, wrote:[Thales] first went to Egypt and thence introduced this study [geometry] into Greece. He discovered many propositions himself, and instructed his successors in the principles underlying many others, his method of attacking problems had greater generality in some cases and was more in the nature of simple inspection and observation in other cases. There is a difficulty in writing about Thales and others from a similar period. Although there are http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thales.html (1 of 5) [2/16/2002 11:34:53 PM]

Thales

numerous references to Thales which would enable us to reconstruct quite a number of details, the sources must be treated with care since it was the habit of the time to credit famous men with discoveries they did not make. Partly this was as a result of the legendary status that men like Thales achieved, and partly it was the result of scientists with relatively little history behind their subjects trying to increase the status of their topic with giving it an historical background. Certainly Thales was a figure of enormous prestige, being the only philosopher before Socrates to be among the Seven Sages. Plutarch, writing of these Seven Sages, says that (see [8]):[Thales] was apparently the only one of these whose wisdom stepped, in speculation, beyond the limits of practical utility, the rest acquired the reputation of wisdom in politics. This comment by Plutarch should not be seen as saying that Thales did not function as a politician. Indeed he did. He persuaded the separate states of Ionia to form a federation with a capital at Teos. He dissuaded his compatriots from accepting an alliance with Croesus and, as a result, saved the city. It is reported that Thales predicted an eclipse of the Sun in 585 BC. The cycle of about 19 years for eclipses of the Moon was well known at this time but the cycle for eclipses of the Sun was harder to spot since eclipses were visible at different places on Earth. Thales's prediction of the 585 BC eclipse was probably a guess based on the knowledge that an eclipse around that time was possible. The claims that Thales used the Babylonian saros, a cycle of length 18 years 10 days 8 hours, to predict the eclipse has been shown by Neugebauer to be highly unlikely since Neugebauer shows in [11] that the saros was an invention of Halley. Neugebauer wrote [11]:... there exists no cycle for solar eclipses visible at a given place: all modern cycles concern the earth as a whole. No Babylonian theory for predicting a solar eclipse existed at 600 BC, as one can see from the very unsatisfactory situation 400 years later, nor did the Babylonians ever develop any theory which took the influence of geographical latitude into account. After the eclipse on 28 May, 585 BC Herodotus wrote:... day was all of a sudden changed into night. This event had been foretold by Thales, the Milesian, who forewarned the Ionians of it, fixing for it the very year in which it took place. The Medes and Lydians, when they observed the change, ceased fighting, and were alike anxious to have terms of peace agreed on. Longrigg in [1] even doubts that Thales predicted the eclipse by guessing, writing:... a more likely explanation seems to be simply that Thales happened to be the savant around at the time when this striking astronomical phenomenon occurred and the assumption was made that as a savant he must have been able to predict it. There are several accounts of how Thales measured the height of pyramids. Diogenes Laertius writing in the second century AD quotes Hieronymus, a pupil of Aristotle [6] (or see [8]):Hieronymus says that [Thales] even succeeded in measuring the pyramids by observation of the length of their shadow at the moment when our shadows are equal to our own height. This appears to contain no subtle geometrical knowledge, merely an empirical observation that at the instant when the length of the shadow of one object coincides with its height, then the same will be true for all other objects. A similar statement is made by Pliny (see [8]):Thales discovered how to obtain the height of pyramids and all other similar objects, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thales.html (2 of 5) [2/16/2002 11:34:53 PM]

Thales

namely, by measuring the shadow of the object at the time when a body and its shadow are equal in length. Plutarch however recounts the story in a form which, if accurate, would mean that Thales was getting close to the idea of similar triangles:... without trouble or the assistance of any instrument [he] merely set up a stick at the extremity of the shadow cast by the pyramid and, having thus made two triangles by the impact of the sun's rays, ... showed that the pyramid has to the stick the same ratio which the shadow [of the pyramid] has to the shadow [of the stick] Of course Thales could have used these geometrical methods for solving practical problems, having merely observed the properties and having no appreciation of what it means to prove a geometrical theorem. This is in line with the views of Russell who writes of Thales contributions to mathematics in [12]:Thales is said to have travelled in Egypt, and to have thence brought to the Greeks the science of geometry. What Egyptians knew of geometry was mainly rules of thumb, and there is no reason to believe that Thales arrived at deductive proofs, such as later Greeks discovered. On the other hand B L van der Waerden [16] claims that Thales put geometry on a logical footing and was well aware of the notion of proving a geometrical theorem. However, although there is much evidence to suggest that Thales made some fundamental contributions to geometry, it is easy to interpret his contributions in the light of our own knowledge, thereby believing that Thales had a fuller appreciation of geometry than he could possibly have achieved. In many textbooks on the history of mathematics Thales is credited with five theorems of elementary geometry:(i) A circle is bisected by any diameter. (ii) The base angles of an isosceles triangle are equal. (iii) The angles between two intersecting straight lines are equal. (iv) Two triangles are congruent if they have two angles and one side equal. (v) An angle in a semicircle is a right angle. What is the basis for these claims? Proclus, writing around 450 AD, is the basis for the first four of these claims, in the third and fourth cases quoting the work History of Geometry by Eudemus of Rhodes, who was a pupil of Aristotle, as his source. The History of Geometry by Eudemus is now lost but there is no reason to doubt Proclus. The fifth theorem is believed to be due to Thales because of a passage from Diogenes Laertius book Lives of eminent philosophers written in the second century AD [6]:Pamphile says that Thales, who learnt geometry from the Egyptians, was the first to describe on a circle a triangle which shall be right-angled, and that he sacrificed an ox (on the strength of the discovery). Others, however, including Apollodorus the calculator, say that it was Pythagoras. A deeper examination of the sources, however, shows that, even if they are accurate, we may be crediting Thales with too much. For example Proclus uses a word meaning something closer to 'similar' rather than 'equal- in describing (ii). It is quite likely that Thales did not even have a way of measuring angles so 'equal- angles would have not been a concept he would have understood precisely. He may have claimed no more than "The base angles of an isosceles triangle look similar". The theorem (iv) was attributed to Thales by Eudemus for less than completely convincing reasons. Proclus writes (see [8]):-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thales.html (3 of 5) [2/16/2002 11:34:53 PM]

Thales

[Eudemus] says that the method by which Thales showed how to find the distances of ships from the shore necessarily involves the use of this theorem. Heath in [8] gives three different methods which Thales might have used to calculate the distance to a ship at sea. The method which he thinks it most likely that Thales used was to have an instrument consisting of two sticks nailed into a cross so that they could be rotated about the nail. An observer then went to the top of a tower, positioned one stick vertically (using say a plumb line) and then rotating the second stick about the nail until it point at the ship. Then the observer rotates the instrument, keeping it fixed and vertical, until the movable stick points at a suitable point on the land. The distance of this point from the base of the tower is equal to the distance to the ship. Although theorem (iv) underlies this application, it would have been quite possible for Thales to devise such a method without appreciating anything of 'congruent triangles'. As a final comment on these five theorems, there are conflicting stories regarding theorem (iv) as Diogenes Laertius himself is aware. Also even Pamphile cannot be taken as an authority since she lived in the first century AD, long after the time of Thales. Others have attributed the story about the sacrifice of an ox to Pythagoras on discovering Pythagoras's theorem. Certainly there is much confusion, and little certainty. Our knowledge of the philosophy of Thales is due to Aristotle who wrote in his Metaphysics :Thales of Miletus taught that 'all things are water'. This, as Brumbaugh writes [5]:...may seem an unpromising beginning for science and philosophy as we know them today; but, against the background of mythology from which it arose, it was revolutionary. Sambursky writes in [15]:It was Thales who first conceived the principle of explaining the multitude of phenomena by a small number of hypotheses for all the various manifestations of matter. Thales believed that the Earth floats on water and all things come to be from water. For him the Earth was a flat disc floating on an infinite ocean. It has also been claimed that Thales explained earthquakes from the fact that the Earth floats on water. Again the importance of Thales' idea is that he is the first recorded person who tried to explain such phenomena by rational rather than by supernatural means. It is interesting that Thales has both stories told about his great practical skills and also about him being an unworldly dreamer. Aristotle, for example, relates a story of how Thales used his skills to deduce that the next season's olive crop would be a very large one. He therefore bought all the olive presses and then was able to make a fortune when the bumper olive crop did indeed arrive. On the other hand Plato tells a story of how one night Thales was gazing at the sky as he walked and fell into a ditch. A pretty servant girl lifted him out and said to him "How do you expect to understand what is going on up in the sky if you do not even see what is at your feet". As Brumbaugh says, perhaps this is the first absent-minded professor joke in the West! The bust of Thales shown above is in the Capitoline Museum in Rome, but is not contemporary with Thales and is unlikely to bear any resemblance to him Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thales.html (4 of 5) [2/16/2002 11:34:53 PM]

Thales

Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles)

Some Quotations (2)

A Poster of Thales of Miletus

Mathematicians born in the same country

Cross-references to History Topics

1. Greek Astronomy 2. The trigonometric functions

Other references in MacTutor

1. The result about angles subtended in a circle 2. Chronology: 30000BC to 500BC

Honours awarded to Thales of Miletus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Thales

Other Web sites

1. G Don Allen 2. Internet Encyclopedia of Philosophy 3. Drury College 4. S M Cohen 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thales.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thales.html (5 of 5) [2/16/2002 11:34:53 PM]

Theaetetus

Theaetetus of Athens Born: about 417 BC in Athens, Greece Died: about 369 BC in Athens, Greece Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Most of what we know of Theaetetus's life comes from the writing of Plato. It is clear that Plato held Theaetetus in the highest regard and he wrote two dialogues which had Theaetetus as the principal character, one of the dialogues being Theaetetus while the other is the Sophist. In Theaetetus a discussion between Socrates, Theaetetus and his teacher Theodorus of Cyrene is recorded. This conversation took place in 399 BC and Theaetetus is described as a youth at the time. This allows us to give a fairly accurate date for Theaetetus's birth (although some have claimed that the Greek word could describe a man of up to 21 years old). Again from Plato we learn that Theaetetus's father, Euphronius of Sunium, was a wealthy man and left a large fortune. However, the money was squandered by the trustees of the will but despite this Theaetetus was generous to all around him. In appearance Theaetetus had a snub nose and protruding eyes but he is described by Plato as having a beautiful mind and he is also described as being the perfect gentleman. Theodorus said that of all his pupils [1]:... he had never found one so marvellously gifted. There are two references to a 'Theaetetus' in the Suda Lexicon (a work of a 10th century Greek lexicographer). The first states (see for example [1]):Theaetetus, of Athens, astronomer, philosopher, disciple of Socrates, taught at Heraclea. He was the first to construct the so-called five solids. He lived after the Peloponnesian war. The Peloponnesian War was fought between Athens and Sparta from 431 BC to 404 BC so the dates here are consistent since Theaetetus would be 13 years old when the War ended so saying the he 'lived after the Peloponnesian war' is reasonable. The second reference in the Suda Lexicon states (see for example [1]):Theaetetus, of Heraclea in Pontus, philosopher and pupil of Plato. Of course it is unclear whether these refer to the same person or to two different people. There are many historians of mathematics who believe that these refer to the same person. Bulmer-Thomas in [1], however, thinks that Allman's explanation in [5] is the most likely. According to this theory the second Theaetetus was the son of the first. If this is so then he would have been born when Theaetetus of Athens

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theaetetus.html (1 of 4) [2/16/2002 11:34:55 PM]

Theaetetus

was teaching in Heraclea and would have been sent by his father to Athens to be educated at the Academy there under Plato. Theaetetus took part in the battle between Athens and Corinth in 369 BC. After acquitting himself with distinction in the battle, he was wounded and carried back to Athens. As a result of the wounds that he received in the battle, Theaetetus contracted dysentery and died in Athens. Theaetetus made very important contributions to mathematics and despite none of his writing having survived we do know a great deal about his contribution. Book X and Book XIII of Euclid's Elements are almost certainly a description of Theaetetus's work. This means that it was Theaetetus's work on irrational lengths which is described in the Book X, thought by many to be the finest work of the Elements. Pappus wrote in the introduction to his commentary to Book X of Euclid's Elements (see for example [1]):The aim of Book X of Euclid's treatise on the "Elements" is to investigate the commensurable and the incommensurable, the rational and irrational continuous quantities. This science has its origin in the school of Pythagoras, but underwent an important development in the hands of the Athenian, Theaetetus, who is justly admired for his natural aptitude in this as in other branches of mathematics. One of the most gifted of men, he patiently pursued the investigation of truth contained in these branches of science ... and was in my opinion the chief means of establishing exact distinctions and irrefutable proofs with respect to the above mentioned quantities. Pappus tells us, therefore, that Theaetetus was inspired by the work of Theodorus to work on incommensurables and that he made major contributions to the theory. In Heath's translation, see for example [3], (we repeat in a slightly different form part of the above quotation by Pappus) the theory of irrationals:... was considerably developed by Theaetetus the Athenian, who gave proof, in this part of mathematics as in others, of ability which has been justly admired. ... As for the exact distinctions of the above-named magnitudes and the rigorous demonstrations of the propositions to which this theory gives rise, I believe that they were chiefly established by this mathematician. For Theaetetus had distinguished square roots commensurable in length from those which are incommensurable, and who divided the more generally known irrational lines according to the different means, assigning the medial line to geometry, the binomial to arithmetic and the apotome to harmony, as stated by Eudemus... B L van der Waerden argues in [4] that Book X of the Elements is entirely the work of Theaetetus. He writes:Has the same Theaetetus who studied the medial, the binomial, and the apotome, also defined and investigated the ten other irrationalities, or were they introduced later on? It seems to me that all this is the work of one mathematician. For the study of the 13 irrationalities is a unit. The same fundamental idea prevails throughout the book, the same methods of proof are applied in all cases. ... Hence the entire book is the work of Theaetetus. However, as Bulmer-Thomas points out in [1], van der Waerden's argument only holds up if we assume that Euclid has not done a lots of work in unifying the methods and giving a consistent approach to the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theaetetus.html (2 of 4) [2/16/2002 11:34:55 PM]

Theaetetus

work of Book X. Bulmer-Thomas prefers the conjecture that although Book X is based on Theaetetus's work there is much due to Euclid presented there too. Plato, writing in his work Theaetetus, has Theaetetus describe how he came to generalise Theodorus's proof that 3, 5, ..., 17 were irrational (see for example [3]):The idea occurred to the two of us [Theaetetus and the younger Socrates], seeing that these square roots appeared to be unlimited in multitude, to try to arrive at one collective term by which we could designate all these roots. ... We divided number in general into two classes. The number which can be expressed as equal multiplied by equal we likened to a square in form, and we called it square ... The intermediate number, such as three, five, and any number which cannot be expressed as equal multiplied by equal ... we likened to an oblong figure and called it an oblong number... In [10], however, Paiow argues that Theodorus had a general method and only presented the particular cases for pedagogical reasons. If his arguments are valid then, of course, Theaetetus would not be the first to prove the general result. Theaetetus is also thought to be the author of the theory of proportion which appears in Eudoxus's work. Theaetetus was the first to study the octahedron and the icosahedron and it is believed that Book XIII of Euclid's Elements is based on his work. A comment (thought to be due to Geminus) states [3]:... the five so-called Platonic figures which, however, do not belong to Plato, three of the five being due to the Pythagoreans, namely the cube, the pyramid, and the dodecahedron, while the octahedron and the icosahedron are due to Theaetetus. We quoted from Pappus above where he described Theaetetus's work on the medial, the binomial, and the apotome. Given two magnitudes a, b the medial is ab, the binomial is a + b, and the apotome is a - b. It is easy to see that the medial and the binomial are closely related to the geometric mean and the arithmetic mean respectively. What is much less clear is how the apotome is a - b is related to the modern harmonic mean. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) Mathematicians born in the same country Honours awarded to Theaetetus (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Theaetetus

Other Web sites

Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theaetetus.html (3 of 4) [2/16/2002 11:34:55 PM]

Theaetetus

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Theaetetus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theaetetus.html (4 of 4) [2/16/2002 11:34:55 PM]

Theodorus

Theodorus of Cyrene Born: 465 BC in Cyrene (now Shahhat, Libya) Died: 398 BC in Cyrene (now Shahhat, Libya) Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato, teaching him mathematics, and also the tutor of Theaetetus. Plato travelled to and from Egypt and on such occasions he spent time with Theodorus in Cyrene. Theodorus, however, did not spend his whole life in Cyrene for he was certainly in Athens at a time when Socrates was alive. Theodorus, in addition to his work in mathematics, was [5]:... distinguished ... in astronomy, arithmetic, music and all educational subjects. A member of the society of Pythagoras, Theodorus was one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness. Our knowledge of Theodorus comes through Plato who wrote about him in his work Theaetetus. Theodorus is remembered by mathematicians for his contribution to the development of irrational numbers and it is this aspect of his work which Plato describes (see for example [5]):[Theodorus] was proving to us a certain thing about square roots, I mean the side (i.e. root) of a square of three square units and of five square units, that these roots are not commensurable in length with the unit length, and he went on in this way, taking all the separate cases up to the root of seventeen square units, at which point, for some reason, he stopped. Our whole knowledge of Theodorus's mathematical achievements are given by this passage from Plato. Yet there are points of interest which immediately arise. The first point is that Plato does not credit Theodorus with a proof that the square root of two was irrational. This must be because 2 was proved irrational before Theodorus worked on the problem, some claim this was proved by Pythagoras himself. There is no doubt that Theodorus would have constructed lines of length 3, 5 etc. using Pythagoras's theorem. It is also clear that Theodorus had no general result here, for Plato goes on to describe how Theodorus's results inspired Theaetetus and Socrates to look at generalisations:The idea occurred to the two of us (Theaetetus and Socrates), seeing that these square roots appeared to be unlimited in multitude, to try to arrive at one collective term by which we could designate all these roots.... http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html (1 of 3) [2/16/2002 11:34:57 PM]

Theodorus

So the question which naturally comes next is how did Theodorus prove that 3, 5, ..., 17 were irrational without giving a proof which would clearly prove that any non-square number was irrational. The usual proof that 2 is irrational, namely the one which supposes that 2 = p/q where p/q is a rational in its lowest terms and derives a contradiction by showing that p and q are both even, would have been known to Theodorus. This proof generalises easily (for a modern mathematicians thinking in terms of numbers rather than lengths) to show n is irrational for any non-square n. It is almost impossible to conceive that Theodorus would have used this proof on each of 3, 5, ..., 17 without obtaining a general theorem long before he got to 17. An interesting proposal was made by Zeuthen in 1915. He suggested that Theodorus may have used the result which would later appear in Euclid's Elements namely:If, when the lesser of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable. Heath [5] illustrates the use of this result to show that 5 is irrational. Start with 1 and 5. 5/1 = 2 +( 5-2) 1/( 5-2) = 4 + ( 5-2)2 ( 5-2)/( 5-2)2 = 1/( 5-2) = 4 + ( 5-2)2 ....... The process now clearly fails to terminate since the ratio 1 : ( 5-2) is the same as ( 5-2) : ( 5-2)2. Heath [5] gives a geometric version of this, starting with a right-angled triangle with sides 1, 2 and 5 which may be close to the method that Theodorus used. However there is little chance to do more than guess at Theodorus's method. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 500BC to 1AD

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html (2 of 3) [2/16/2002 11:34:57 PM]

Theodorus

Mathematicians of the day JOC/EFR January 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Theodorus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html (3 of 3) [2/16/2002 11:34:57 PM]

Theodosius

Theodosius of Bithynia Born: about 160 BC in Bithynia, Anatolia (now Turkey) Died: about 90 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Theodosius of Bithynia was for a long time thought to have been born in Tripolis. The reason for this comes from an error in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Theodosius was a (see for example [1]):... philosopher [who] wrote "Sphaerics" in three books, a commentary on the chapter of Theudas, two books "On Days and Nights", a commentary on the "Method of Archimedes", Descriptions of Houses in three books, "Skeptical Chapters", "Astrological works", "On Habitations". Theodosius wrote verses on the spring and other types of works. He was from Tripolis. We are indeed interested in this article in the Theodosius who wrote Sphaerics in three books. He must have lived close to 100 BC. Yet Theudas who is next referred to in the Suda was a sceptic philosopher of the second century AD so we see immediately that there is an error. It would now appear that, apart from the work on Theudas and Skeptical Chapters which was almost certainly written by the same person (one assumes another author by the name of Theodosius), the rest of the entry is correct except for the final two sentences "Theodosius wrote verses on the spring and other types of works. He was from Tripolis." which must refer to what one has to assume is yet a third author called Theodosius. So Theodosius was the author of Sphaerics, a book on the geometry of the sphere, written to provide a mathematical background for astronomy. It is thought that Sphaerics is based on some pre-Euclidean textbook which is now lost. It is conjectured, on rather little evidence one would have to say, that Eudoxus wrote this earlier text. There seems to be no way in which the speculation on this point can ever be settled. Sphaerics contains no trigonometry although it is likely that Hipparchus introduced spherical trigonometry before Sphaerics was written (although, one has to assume, after the book on which Sphaerics is based, which would certainly be the case if this earlier book was written by Eudoxus). Sphaerics was written to supplement Euclid's Elements in particular to make up for the lack of results on the geometry of the sphere in Euclid's work. Theodosius defines a sphere to be a solid figure with the property that any point on its surface is at a constant distance from a fixed point (the centre of the sphere). He gives theorems which generalise those

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theodosius.html (1 of 3) [2/16/2002 11:34:58 PM]

Theodosius

given by Euclid in Book III of the Elements for the circle. The second book of Theodosius's work considers touching circles on a sphere. It then goes on to consider geometry results which are relevant to astronomy and these continue to be studied through Book III. Heath writes [2]:It is evident that Theodosius was simply a laborious compiler, and that there was practically nothing original in his work. Perhaps it is worth remarking that despite our comment above that the work contains no trigonometry, there are some results which we could easily interpret in trigonometrical terms. For example Theodosius proves that for a spherical triangle with angles A, B, C (C a right angle) and sides a, b, c where side a is opposite angle A, etc. then tan a = sin b tan A. Neugebauer, in [3], is highly critical of the Sphaerics calling it dull and pedantic only surviving because it was used as a textbook. More specifically Neugebauer writes:Theodosius comes nowhere near recognising the fundamental importance of the great-circle triangle and his theorems rarely go beyond the geometrically obvious in the relations between a few special great circles and their parallels, without ever mentioning that one is dealing with configurations of interest to astronomy. Two other works by Theodosius have survived in the original Greek. These are On habitations containing 12 theorems and On days and nights. The first of these explains the views of the universe due to the rotation of the Earth and, in particular, it considers how the view is affected by the different places on the Earth in which people live. Theodosius considers the length of the night and day at various points on the earth and claims that the day lasts for seven months at the north pole and the night for five months. On moving south one reaches the circle where at the summer solstice the day is 30 days long. Of course this is very strange and partly explained by Theodosius's definition of night as period of darkness and day as a period of light. Theodosius considered that it was 'day' if the sun was less than 15 below the horizon for then no stars were visible and he seemed to fail to understand that in the polar regions the sun can move almost parallel to the horizon. The other work On days and nights is in two books, the first of which has 13 propositions, the second 19 propositions, which give conditions on the lengths of the night and day depending on the location of the observer. Theodosius also considers the two possibilities, that the length of the year is a rational multiple of the length of the day and that it is an irrational multiple. Neugebauer [3] makes some interesting comments on the diagrams in ancient texts and how they may have been totally changed by both early editors and even by modern editors. Referring to Theodosius's On days and nights he says that:... errors occur in diagrams. Letters are easily misplaced or sometimes an arc may be missing but by and large figures are well drawn. In many cases the extant diagrams show an axial symmetry which is not wrong but which is not required by the theorem or proof in question. Such symmetries detract ... from the general validity of the proposition. It is impossible to tell if such symmetrisations, caused either by the greater simplicity of construction or its aesthetic appeal, belong to the archetype or are copyist or editorial "improvements". http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theodosius.html (2 of 3) [2/16/2002 11:34:58 PM]

Theodosius

Of the other works mentioned in the Suda we have no reason to doubt that Theodosius wrote a commentary on the Method by Archimedes but there is no other evidence to prove whether this is correct or not. Theodosius is also reported to have invented a sundial suitable for all regions but nothing is known about it. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Theodosius.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theodosius.html (3 of 3) [2/16/2002 11:34:58 PM]

Theon

Theon of Alexandria Born: about 335 in (possibly) Alexandria, Egypt Died: about 405 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Theon of Alexandria worked in Alexandria as a teacher of mathematics and astronomy. We know from his own writings that he observed a solar eclipse on 16 June 364 at Alexandria and a lunar eclipse, again in Alexandria, on 25 November 364. We also know that he made a list of Roman consuls which he continued to make until 372. There is a reference in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Theon of Alexandria lived under the Emperor Theodosius I (who reigned from 379 to 395). These dates are therefore consistent. The Suda also states that Theon was a member of the Museum. which was an institute for higher education set up in Alexandria in 300 BC. Again this is possible, but the Museum certainly did not exist much beyond the time of Theon if indeed it existed in his time. On balance it seems reasonable to accept that he was one of its last members. Theon was the father of Hypatia and it certainly seems to be the case that he died before she was murdered in 415. There does not seem to be any other evidence which would let us give a more accurate guess of the dates of his birth and death other than these few indications of times when he was certainly working. Theon is famed for his commentaries on many works such as Ptolemy's Almagest and the works of Euclid. These commentaries were written for his students and some are even thought to be lecture notes taken by students at his lectures. On one work he gave two commentaries and in the preface to the second he explains that he is giving a more elementary account for the majority of his students are unable to understand geometrical proofs. This again confirms that the vigour had gone out of his teaching establishment and indeed the poor quality of students it seemed to be attracting could have been a telling factor in the closure of the Museum (if as we commented above the Suda is right in giving that as his institution). Theon was a competent but unoriginal mathematician. Theon's version of Euclid's Elements (with textual changes and some additions) is thought to have been written with the assistance of his daughter Hypatia and was the only Greek text of the Elements known, until an earlier one was discovered in the Vatican in the late 19th century. However, now that the Vatican manuscript has been discovered it is possible to see exactly the changes that Theon made in his version. The approach that Theon makes appears to make is to try to improve the earlier manuscript rather than to

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theon.html (1 of 3) [2/16/2002 11:35:00 PM]

Theon

try to reproduce an accurate reproduction with comments. So he corrected mistakes which he spotted in the mathematics, but unfortunately not all the points that he fails to understand are mistakes, some are perfectly correct. Theon also tried to standardise the way that Euclid writes, so when Theon came across an expression which was somewhat different from the norm, he replaced it by the standard form of expression. On the positive side, however, Theon amplified Euclid's text whenever he thought that an argument was overly brief, sometimes adding propositions to make the text more easily read by beginners. In this he was successful, so much so in fact that his became the standard edition and almost all earlier editions have been lost. Heath writes of Theon's edition of the Elements [2]:.. while making only inconsiderable additions to the content of the "Elements", he endeavoured to remove difficulties that might be felt by learners in studying the book, as a modern editor might do in editing a classical text-book for use in schools; and there is no doubt that his edition was approved by his pupils at Alexandria for whom it was written, as well as by later Greeks who used it almost exclusively... Theon also produced commentaries on other works of Euclid. Certainly he produced a commentary on Euclid's Optics and on his Data. Theon's commentary on the Data is written at a relatively advanced level and in it Theon tends to shorten Euclid's proofs rather than to amplify them. The Optics on the other hand is elementary and written in a totally different style and some historians conjecture that it is really a set of lecture notes by one of Theon's students. Many times the manuscript contains a phrase such as "he said" and it is thought that a student is indeed writing down what "Theon said". The Catoptrica is a rather different case for here we have a work which on the face of it claims to be written by Euclid. This however is impossible since the contents are a mixture of work dating from Euclid's time together with work which is much later than Euclid's time. The style and elementary nature of the work make authorship by Theon a distinct possibility. If this is the case then again he is writing for his weak students. Theon also wrote extensive commentaries on the astronomical works of Ptolemy, both on the Almagest and the Handy tables. Again his daughter Hypatia assisted him in the commentary on the Almagest and this is Theon's most major piece of work. In the preface to his commentary on the Almagest Theon writes that his intention is to improve on previous commentators (see for example [1]):... who claim that they will only omit the more obvious points, but in fact prove to have omitted the most difficult. However, as Toomer points out in [1], this is exactly what Theon himself goes on to do. Theon wrote two commentaries on Ptolemy's Handy Tables. The small commentary only explains how to use the tables while the large commentary explains their construction. The larger commentary has been published recently by Tihon in [5] and [6]. Although Theon certainly wrote the small commentary after the larger one, since he refers to the larger commentary in the preface to the smaller. However, Tihon discovered that the oldest manuscript which has been preserved, a Vatican manuscript dating from the 9th century, suggests that Theon never completed the text of his large commentary. This Vatican manuscript is made from an earlier copy of Theon's text which was being used in the year 463 in Apamea http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theon.html (2 of 3) [2/16/2002 11:35:00 PM]

Theon

in Syria. As to Theon's commentary on Ptolemy's Syntaxis Heath writes [2]:This commentary is not calculated to give us a very high opinion of Theon's mathematical calibre, but it is valuable for several historical notices that it gives, and we are indebted to it for a useful account of the Greek method of operating with sexagesimal fractions, which is illustrated with examples of multiplication, division, and the extraction of the square root of a non-number by way of approximation. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. How do we know about Greek mathematics? 2. Babylonian numerals

Other references in MacTutor

Chronology: 1AD to 500

Honours awarded to Theon of Alexandria (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Theon Junior

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Theon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theon.html (3 of 3) [2/16/2002 11:35:00 PM]

Theon_of_Smyrna

Theon of Smyrna Born: about 70 Died: about 135 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Little is known of Theon of Smyrna's life. He was called 'the old Theon' by Theon of Alexandria and 'Theon the mathematician' by Ptolemy. The date of his birth is little better than a guess, but we do have some firm data about dates in his life. We know that he was making astronomical observations of Mercury and Venus between 127 and 132 since Ptolemy lists four observations which Theon made in 127, 129, 130 and 132. From these observations Theon made estimates of the greatest angular distance that Mercury and Venus can reach from the Sun. The style of his bust, dedicated by his son 'Theon the priest', gives us the date of his death to within 10 years and it is placed within the period 130-140 (hence our midpoint guess of 135). Theon's most important work is Expositio rerum mathematicarum ad legendum Platonem utilium. This work is a handbook for philosophy students to show how prime numbers, geometrical numbers such as squares, progressions, music and astronomy are interrelated. Its rather curious title means that it was intended as an introduction to a study of the works of Plato, but this is rather fanciful. As Huxley writes in [1]:... the book has little to offer the specialist student of Plato's mathematics. It is, rather, a handbook for philosophy students, written to illustrate how arithmetic, geometry, stereometry, music, and astronomy are interrelated. The most important feature of the work is the wide range of citations of earlier sources. Its worst feature is its lack of originality. Heath writes [2]:Theon's work is a curious medley, valuable, not intrinsically, but for the numerous historical notices which it contains. In the introduction Theon gives his reason for writing the work:Everyone would agree that he could not understand the mathematical arguments used by Plato unless he were practised in this science... One who had become skilled in all geometry and all music and astronomy would be reckoned most happy on making acquaintance with the writings of Plato, but this cannot be come by easily or readily, for it calls for a very great deal of application from youth upwards. In order that those who have failed to become practised in these studies, but aim at a knowledge of his writings, should not wholly fail in their desires, I shall make a summary and concise sketch of the mathematical theorems http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theon_of_Smyrna.html (1 of 3) [2/16/2002 11:35:02 PM]

Theon_of_Smyrna

which are specially necessary for readers of Plato.... The work begins with a collection of theorems which Theon says will be useful for the study of arithmetic, music, geometry, and astronomy in Plato. However his coverage of geometry is none too good and later in the book he makes an excuse for this saying that anyone who reads his book, or the works of Plato, will have already studied elementary geometry. In the section on numbers Theon adopts a Pythagorean approach, writing about odd numbers, even numbers, prime numbers, composite numbers, square numbers, oblong numbers, triangular numbers, polygonal numbers, circular numbers, spherical numbers, solid numbers with three factors, pyramidal numbers, perfect numbers, deficient numbers and abundant numbers. The best section of Expositio rerum mathematicarum is the astronomy section which teaches that the Earth is spherical, that mountains are negligible in height compared with the Earth etc. It includes knowledge of conjunctions, eclipses, occultations and transits. However, Neugebauer writes in [3]:It is clear that Theon's treatise does not pretend to make original contributions to astronomy. Unfortunately it is also clear that Theon has not fully digested the material he is presenting to his readers. Theon also wrote commentaries on the main authorities of mathematics and astronomy. In particular he wrote an important work on Ptolemy and another on Plato's Republic which he refers to himself in work which survives. Whether his work on the ancestry of Plato is a separate work or a section of one of his commentaries on Plato's work, it is impossible to say. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Greek Astronomy 2. Doubling the cube

Honours awarded to Theon of Smyrna (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Theon Senior

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theon_of_Smyrna.html (2 of 3) [2/16/2002 11:35:02 PM]

Theon_of_Smyrna

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Theon_of_Smyrna.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Theon_of_Smyrna.html (3 of 3) [2/16/2002 11:35:02 PM]

Thiele

Thorvald Nicolai Thiele Born: 24 Dec 1838 in Copenhagen, Denmark Died: 26 Sept 1910 in Copenhagen, Denmark

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Thorvald Thiele taught in Copenhagen as well as being chief actuary for an insurance company. With an interest in astronomy, he studied orbits and the three body problem. In 1895 Thiele showed that singularities in the motion of such a system could often be removed by a sutable transformation allowing the motion of bodies after a collision to be studied. He is remembered for having an interpolation formula named after him, the formula being used to obtain a rational function which agrees with a given function at any number of given points. He introduced cumulants (under the name of "half-invariants") in 1889, 1897, 1899, about 30 years before their rediscovery and exploitation by R. A. Fisher. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Other Web sites

Theseus

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thiele.html (1 of 2) [2/16/2002 11:35:04 PM]

Thiele

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thiele.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thiele.html (2 of 2) [2/16/2002 11:35:04 PM]

Thom

René Thom Born: 2 Sept 1923 in Montbéliard, Doubs, France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

René Thom is known for his development of catastrophe theory, a mathematical treatment of continuous action producing a discontinuous result. From 1931 Thom attended the Primary School in Montbéliard, the town of his birth in which his parents were shopkeepers. It was at this primary school that Thom first showed his academic potential winning a scholarship. He attended Collège Cuvier at Montbéliard and received his baccalaureate in elementary mathematics from Besançon in 1940. However, his life was about to be disrupted by World War II. Thom's parents sent him and his brother south to avoid the conflict although they themselves remained in Montbéliard. Thom and his brother eventually reached Switzerland. He writes in [6]:The surprising warmth with which we were welcomed there, all those people offering food and drink at the roadside, still fills me with emotion. After helping with the harvest near Romont, Thom returned to France being taken to Lyon where he lived with a friend of his mother. While in Lyon he continued his education, receiving his baccalaureate in philosophy in June 1941. After this he returned to his parents home in Montbéliard but was soon in Paris again to continue his education. Thom attended the Lycée Saint-Louis in Paris and applied to enter the Ecole Normale Supérieure but failed to gain entrance in 1942. Determined to take advantage of a university education at the Ecole Normale Supérieure he applied again in 1943 and this time he was [6]:... successful (but not brilliantly so!). At Ecole Normale Supérieure times were difficult as Paris was occupied by the German forces. However, mathematically it was an exciting time for Thom who was to be strongly influenced by Henri Cartan and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thom.html (1 of 3) [2/16/2002 11:35:06 PM]

Thom

the Bourbaki approach to mathematics. World War II ended while Thom was still studying at the Ecole Normale Supérieure and [6]:... the last year, after the 'victory', was a year of opening, bringing with it the impression of once more living life to the full. Of this rebirth I can recall a sensation of freedom that I found hard to control. In 1946 Thom moved to Strasbourg so that he could continue to work with Henri Cartan. There he was influenced by others including Ehresmann and Koszul. His doctorate, supervised by Henri Cartan, was awarded in 1951 for a thesis entitled Fibre spaces in spheres and Steenrod squares. The work of the thesis was carried out in Strasbourg but Thom presented it to Paris. The foundations of the theory of cobordism, for which Thom later received a Fields Medal, already appear in his doctoral thesis. Thom was awarded a fellowship to allow him to travel to the United States in 1951 and he relates in [6] how this enabled him to meet Einstein, Weyl, Steenrod and attend the seminars of Calabi and Kodaira. Thom returned to France and taught at Grenoble in 1953-54, then at Strasbourg from 1954 until 1963. He was appointed a professor in 1957. In 1964 he moved to the Institut des Hautes Etudes Scientifique at Bures-sur-Yvette. However this prompted a change in direction as he explains in [6]:Relations with my colleague Grothendieck were less agreeable for me. His technical superiority was crushing. His seminar attracted the whole of Parisian mathematics, whereas I had nothing new to offer. That made me leave the strictly mathematical world and tackle more general notions, like the theory of morphogenesis, a subject which interested me more and led me towards a very general form of 'philosophical' biology. Thom's theory is an attempt to describe, in a way that is impossible using differential calculus, those situations in which gradually changing forces lead to so-called catastrophes, or abrupt changes. The theory has widespread application in the physical and biological sciences and in the social sciences. Presented by Thom in Structural Stability and Morphogenesis (1972), the theory has since been developed by many mathematicians. However, writing in [6], Thom explains why the theory which was marked by enormous popular success has fallen from favour:It is a fact that catastrophe theory is dead. But one could say that it died of its own success. It was brought down by the extension from analytical (or algebraic) models to models that were only smooth. For as soon as it became clear that the theory did not permit quantitative prediction, all good minds ... decided it was of no value. When it comes down to it, this extension resulted from B Malgrange's extension of the preparation theorem. Thom's earlier work had made him well known before he worked on catastrophe theory. His work on topology, in particular on characteristic classes, cobordism theory and the Thom transversality theorem led to his being awarded a Fields Medal in 1958. However, Thom feels that in some sense he did not deserve the honour [6]:... I have the impression that work was done just a little while later that was greater in depth and sagacity than mine and whose authors were quite as deserving, if not more so, of the medal (such as my co-medallist Klaus Roth). I am thinking too of Barry Mazur's demonstration of the Schönflies conjecture: Every sphere Sn-1 in Rn with regular boundary is the boundary of an n-ball. Not to mention the discovery by Milnor of exotic spheres.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thom.html (2 of 3) [2/16/2002 11:35:06 PM]

Thom

Hopf, who awarded the Fields Medal to Thom in Edinburgh, pointed in his presentation address to the importance of Thom's theory:... his basic ideas, the grand simplicity of which I have talked of, are of a very geometric and intuitive nature. These ideas have significantly enriched mathematics, and everything seems to indicate that the impact of Thom's ideas - whether they find their expression in the already known or in forthcoming works - is not exhausted by far. Thom was awarded the Grand Prix Scientifique de la Ville de Paris in 1974. He was made an honorary member of the London Mathematical Society in 1990. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles)

Some Quotations (4)

A Poster of René Thom

Mathematicians born in the same country

Other references in MacTutor

1. Chronology: 1950 to 1960 2. Chronology: 1970 to 1980

Honours awarded to René Thom (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1958

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thom.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thom.html (3 of 3) [2/16/2002 11:35:06 PM]

Thomae

Johannes Karl Thomae Born: 11 Dec 1840 in Germany Died: 2 March 1921

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johannes Thomae taught at Göttingen, Halle, Freiburg and Jena. Cantor had discovered that the points in n-dimensional space could be put in 1-1 correspondence with the line. In a letter of 1877 to Dedekind he said I see it but I don't believe it. This was published in 1878 but since the correspondence was not continuous many attempts to prove the invariance of dimension using continuity were made. Thomae was the first to attempt a general proof of the invariance of dimension but it was not satisfactory since the necessary topological tools had not been developed at this time. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomae.html (1 of 2) [2/16/2002 11:35:08 PM]

Thomae

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thomae.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomae.html (2 of 2) [2/16/2002 11:35:08 PM]

Thomason

Robert Wayne Thomason Born: 5 Nov 1952 in Tulsa, Oklahoma, USA Died: Nov 1995 in Paris, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bob Thomason entered Michigan State University in 1971 to read for a mathematics degree. He graduated in 1973 and then went to Princeton University to study for his doctorate. His supervisor at Princeton was John Moore and he wrote a dissertation on category theory in which he produced results which were to become fundamental tools in topology. After graduating with his doctorate in June 1977 Thomason was appointed as Moore instructor at the Massachusetts Institute of Technology. This was a two year appointment and Thomason remained there until he took up a three year post as Dickson assistant professor at the University of Chicago in 1979. Things did not go smoothly for Thomason, however, at the University of Chicago. While working on conjectures of Quillen-Lichtenbaum connecting K-theory to étale cohomology Thomason produced what was first thought to be a remarkable proof. However, an error was found in 1980 and [4]:... Thomason began to feel uncomfortable about the scepticism expressed by others. Perceiving this as persecution, he resigned from his position at Chicago in June 1980. During the two years after Thomason resigned from Chicago he spent some time at the Massachusetts Institute of Technology and one year at the Institute for Advanced Study at Princeton. This was a mathematically profitable time for him and he was able to complete the results that had gone wrong while he was at Chicago along with some other major pieces of work. Thomason's next post was at Johns Hopkins University where he was appointed in 1983. During the six years he spent there he produced a series of outstanding papers solving, among others, problems arising from Grothendieck's work in his paper with Berthelot and Illusie Théorie des Intersections et Théorème http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomason.html (1 of 3) [2/16/2002 11:35:10 PM]

Thomason

de Riemann-Roch (1971). For example in a 1983 paper he found a partial solution of Grothendieck's absolute cohomological purity conjecture. During his time at Johns Hopkins University, Thomason was awarded a Sloan Fellowship which enabled him to spend the year 1987 at Rutgers University. Thomason spent three years working on the problems of Grothendieck referred to above. In [4] a most remarkable story is told regarding this work:While at Rutgers, he put everything in place except for one step ... On January 22 1988, he had a dream in which his recently deceased friend Thomas Trobaugh told him how to solve the final step.... Awaking with a start, he worked out the argument for the missing step. In gratitude, he listed his friend as a coauthor of the resulting paper. The importance of this work was recognised when Thomason was chosen to give an address at the International Congress of Mathematicians in Kyoto in 1990. In October 1989 Thomason was appointed to a post in Max Karoubi's laboratory at the University of Paris VII. He continued to produce outstanding work, publishing six papers before his death in 1995. He had health problems which were to lead to his death [4]:Bob had diabetes and always had to strictly control what he ate. This made going to restaurants with Bob an awkward affair, because he would not eat something until he was sure it had no nutritional content. Early in November 1995 [some accounts say late October], just before his 43rd birthday, he went into diabetic shock and died in has apartment in Paris. Thomason first important results concerned a proof that all infinite loop space machines produce equivalent output. Working with J P May he wrote a paper The uniqueness of infinite loop space machines. Quoting from the paper:An infinite loop space machine is a functor which constructs spectra out of ... space level data. There are many such machines known ...; they are given by such widely disparate topological constructions that it is far from obvious that they turn out equivalent spectra when fed the same data. The purpose of this paper is to prove that all machines which satisfy certain reasonable properties do in fact turn out equivalent spectra. Thomason then developed material which he had studied for his doctorate considering the homotopy theory of the category of small categories and the homotopy theory of the category of small symmetric monoidal categories. We have already mentioned Thomason's results on the conjectures of Quillen-Lichtenbaum connecting K-theory to étale cohomology which he achieved during 1980-83. A reviewer of his paper Algebraic K-theory and étale cohomology (1985) wrote:This paper is one of the most important papers in algebraic K-theory since a paper by D Quillen ... in 1972. Versions of the paper have circulated in manuscript form for several years and it is good to see it appear finally. The paper is important both for the results proved, and for the techniques used. The author pushes the applications of stable homotopy and homotopical algebra to algebraic K-theory and algebraic geometry further than anyone else and his methods have exerted considerable influence on other workers in the field. Thomason's work during the next three years was on equivariant algebraic K-theory. He worked on the algebraic K-theory of algebraic group actions on schemes. He produced a theorem published in 1988 said http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomason.html (2 of 3) [2/16/2002 11:35:10 PM]

Thomason

to be:... one of the most important and powerful tools in algebraic K-theory. It is also one of the most sought-after. Much work has been done in attempts to extend Quillen's localization theorem to more general contexts, but none has even begun to approach the complete generality achieved in this paper. A 1990 he contributed a paper (the one written after his dream with Thomas Trobaugh as coauthor) to The Grothendieck Festschrift which is described as a 'landmark paper'. Thomason solved a problem in 1995 which had been posed by Grothendieck. The problem concerned lifting a homotopy structure. He lectured on this result in Genova only about three weeks before his death. He never wrote these results up for publication. Weibel paints a picture of Thomason in [4]:Like many of his colleagues, Bob Thomason hated to waste energy on trivial matters, like fashion. He made the decision early in life to dress only in black clothing, thus simplifying that portion of his life. With his pointed goatee, he looked like a beat poet to outsiders, but mathematicians knew him as one of the greatest talents of his generation. Few have had the simultaneous grasp of topology, algebraic geometry and K-theory that Thomason did. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thomason.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomason.html (3 of 3) [2/16/2002 11:35:10 PM]

Thompson_D'Arcy

D'Arcy Wentworth Thompson Born: 2 May 1860 in Edinburgh, Scotland Died: 21 June 1948 in St Andrews, Fife, Scotland

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

D'Arcy Thompson's father was appointed Professor of Greek at Queen's College (now University College) Galway when D'Arcy was 3 years old but he returned to Edinburgh at the age of 10 to attend Edinburgh Academy. He won the prize for Classics, Greek Testament, Mathematics and Modern Languages in his final year at school. After starting a medicine course at Edinburgh University he changed to study science at Cambridge. He graduated with a B.A. in Zoology in 1883 but he would not obtain his doctorate until 29 years later in 1923. In 1884 D'Arcy Thompson was appointed Professor of Biology in Dundee (incorporated as part of the University of St Andrews in 1897) and then in 1917 he was appointed to the Chair of Natural History in St Andrews. He was to hold a chair for 64 years, a record which will not now be broken. D'Arcy combined skills in a way that made him unique. He was a Greek scholar, a naturalist and a mathematician. He was the first biomathematician although he followed in the tradition of another great natural historian with mathematical skills, namely Buffon. His understanding of mathematics was of the modern subject but based on the firm foundations of an understanding of Greek mathematics. Although he was to write around 300 scientific articles and books all D'Arcy's various skills came together in his most famous book On Growth and Form (1917). This book assumes that all science and learning are one, and attempts to reduce biological phenomena to mathematics. He claimed that all animals and plants could only be understood in terms of pure mathematics. The shell of Nautilus (shown in the picture above) and the hexagonal cells of the bee's honeycomb related to logarithmic spirals and minimal areas. D'Arcy related such things to the Greek work on http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thompson_D'Arcy.html (1 of 3) [2/16/2002 11:35:12 PM]

Thompson_D'Arcy

approximating

, 2 and Euclid's Elements.

D'Arcy Thompson was also a very fine writer and he said The little gift I have for writing English which I possess, and try and cultivate and use, is, speaking honestly and seriously, the one thing I am a bit proud and vain of - the one and only thing. At the end of Growth and Form he wrote ... something of the use and beauty of mathematics I think I am able to understand. I know that in the study of material things number, order, and position are the threefold clue to exact knowledge: and that these three, in the mathematician's hands, furnish the first outlines for a sketch of the Universe. D'Arcy Thompson was elected a Fellow of the Royal Society of London in 1916. He was awarded the Darwin Medal of the Society in 1946:... in recognition of his outstanding contributions to the development of biology. He also received recognition for his mathematics, being made an honorary member of the Edinburgh Mathematical Society in 1933. Article by: J J O'Connor and E F Robertson List of References (7 books/articles)

Some Quotations (3)

A Poster of D'Arcy Thompson

Mathematicians born in the same country

Other references in MacTutor

Using a computer to visualise change in biological organisms

Honours awarded to D'Arcy Thompson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1916

Fellow of the Royal Society of Edinburgh Honorary Fellow of the Edinburgh Maths Society

Elected 1933 1. John Casti

Other Web sites

2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thompson_D'Arcy.html (2 of 3) [2/16/2002 11:35:12 PM]

Thompson_D'Arcy

Glossary index

Quotations index

Mathematicians of the day JOC/EFR November 1997

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thompson_D'Arcy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thompson_D'Arcy.html (3 of 3) [2/16/2002 11:35:12 PM]

Thompson_John

John Griggs Thompson Born: 13 Oct 1932 in Ottawa, Kansas, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Thompson studied at Yale University, receiving his B.A. in 1955. He went to the University of Chicago to undertake research and completed his doctorate in 1959. His doctoral thesis, entitled A Proof that a Finite Group with a Fixed-Point-Free Automorphism of Prime Order is Nilpotent was supervised by MacLane. In fact his doctoral thesis solved one of the conjectures of Frobenius which had remained unsolved for around 60 years. Thompson's thesis, as is clear from its title, proved Frobenius's conjecture that a finite group with an automorphism which does not fix any group element is necessarily nilpotent. The solution of Frobenius's conjecture was not done by simply pushing the existing techniques further than others had done; rather it was achieved by introducing many highly original ideas which were to lead to many developments in group theory. Thompson was an assistant at Harvard University in 1961-62, then, in 1962, he was appointed professor at the University of Chicago. In 1968 Thompson accepted a fellowship at University College, Cambridge in England. He was appointed Rouse Ball Professor of Pure Mathematics at Cambridge in 1970. It is no coincidence that starting at the time of Thompson's thesis, group theory leapt into prominence as the mathematical topic which was attracting most attention and which was undergoing the most rapid development. The reason was that suddenly progress began to be made on one of the main problems of finite group theory, namely the classification of finite simple groups. Every finite group can be viewed as built from a finite collection of finite simple groups. The finite simple groups are therefore the building blocks from which finite groups are built. To classify finite groups therefore reduces to two problems, namely the classification of finite simple groups and the solution of the extension problem, that is the problem of how to fit the building blocks together. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thompson_John.html (1 of 4) [2/16/2002 11:35:14 PM]

Thompson_John

Early contributions were made by Galois, Jordan and Emile Mathieu. Claude Chevalley showed in 1955 that the Lie groups have finite analogues which are finite simple groups. M Suzuki in 1960 discovered new infinite families of finite simple groups. These were discovered by him independently of Chevalley's theory but then it was noticed that they were indeed twisted Chevalley groups. An automorphims had been missed in the original working out of Chevalley's theory which is why the Suzuki groups were only discovered some time after. Thompson, working with Walter Feit, proved in 1963 that all nonabelian finite simple groups were of even order. They published this result in Solvability of Groups of Odd Order a 250 page paper which appeared in the Pacific Journal of Mathematics 13 (1963), 775-1029. Despite the importance of the paper several journals declined to publish it because of its length. The paper consists of one whole part of Volume 13 of the Pacific Journal. This result stunned the world of mathematics but it also led mathematicians to believe that a classification of finite simple groups might prove possible. Both Thompson and Feit received the The Frank Nelson Cole Prize in 1965 when the thirteenth award was made to them for this their joint paper. Another major early step by Thompson towards the classification of finite simple groups was his classification of those finite simple groups in which every soluble subgroup has a soluble normaliser. Thompson was awarded a Fields Medal for his work at the International Congress of Mathematicians in Nice in 1970. Brauer, speaking of Thompson's work at the Congress, first spoke of the 'odd order paper':The first paper I have to mention is a joint paper by Walter Feit and John Thompson and, of course, Feit's part in it should not be overlooked. Here, the authors proved a famous conjecture, to the effect that all non-cyclic finite simple groups have even order. I am not sure who was the first to observe this. Fifty years ago [1920] this was already referred to as a very old conjecture. While it was usually mentioned in courses on algebra, it is only fair to say that nobody ever did anything about it, simply because nobody had any idea how to get started. It was not even clear that the whole problem made sense. Was the role of the prime 2 simply a little accident; did 2 play an entirely exceptional role, or were there properties of other prime divisors of the group order which bore at least some resemblance to those of 2? It was only after the Feit-Thompson paper that one could be sure that the whole question was a reasonable one. Brauer went on to speak of Thompson's subsequent work:Thompson's work which has now been honoured by the Fields medal is a sequel to this first paper. In it he determined the minimal simple finite groups, this is to say, the simple groups whose proper subgroups are solvable. Actually, a more general problem is solved. It suffices to assume that only certain subgroups, the so-called local subgroups, are solvable. These are the normalizers of the subgroups of prime power order ... These results are the first substantial results achieved concerning simple groups. A number of important corollaries show that one is now able to answer questions on finite groups which were completely out of reach before. I mention one: a finite group is solvable if and only if every subgroup generated by two elements is solvable. The nonabelian finite simple groups fall into a small number of infinite series and 26 sporadic groups. During the 1970s Thompson contributed to the understanding of these groups. Brauer, in a personal

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thompson_John.html (2 of 4) [2/16/2002 11:35:14 PM]

Thompson_John

comment at the end of [3] predicted this:On reaches a point in life where one wonders what one still expects of life, what one would still like to see happen. This applies to mathematics too. I have passed the point I have mentioned. I like to say that I would like to see the solution of the problem of the finite simple groups and the part I expect Thompson's work to play in it. Quite generally I would like to see to what further heights Thompson's future work will take him. John Thompson's interests after 1970 became broader and over the 1970s he also made major contributions to coding theory. His work on coding theory was to lay the foundation for the solution of a long standing problem, namely the fact that there is no finite plane of order 10. During the 1980s much of Thompson's work was on the problem of which finite groups could occur as Galois groups. Work in this area was started by Hilbert with his proof of the irreducibility theorem, and the authors of [4] state that:Thompson's work may well be the most important advance since Hilbert's time. In 1989 Thompson was one of the five main speakers at the Groups St Andrews meeting. He gave a series on lectures on Galois groups at that meeting. The picture of Thompson shown here was taken in St Andrews during the conference. Thompson has received many awards for his outstanding contributions to mathematics. In addition to the Cole Prize from the American Mathematical Society and the Fields Medal in 1970 described above, he was awarded the Senior Berwick Prize from the London Mathematical Society in 1982, the Sylvester Medal from the Royal Society in 1985 and he received the Wolf Prize and the Poincaré Prize in 1992. He was elected to the National Academy of Sciences in the United States in 1971 and the Royal Society of London in 1979. He was awarded the National Medal of Science in 2000. Among the honorary degrees that he has received are ones from Yale University (1980), the University of Chicago (1985) and the University of Oxford (1987). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1960 to 1970

Honours awarded to John Thompson (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1970

Fellow of the Royal Society

Elected 1979

Royal Society Sylvester Medal

Awarded 1985

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thompson_John.html (3 of 4) [2/16/2002 11:35:14 PM]

Thompson_John

AMS Cole Prize winner

1965

AMS Colloquium Lecturer

1974

LMS Berwick Prize winner

1982

Other Web sites

1. AMS (Classification of finite simple groups) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thompson_John.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thompson_John.html (4 of 4) [2/16/2002 11:35:14 PM]

Thomson

William Thomson (Lord Kelvin) Born: 26 June 1824 in Belfast, Ireland Died: 17 Dec 1907 in Netherhall (near Largs), Ayrshire, Scotland

Click the picture above to see nine larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Thomson's father, James Thomson, had originally intended to become a minister of the Presbyterian Church but had opted for an academic career as a mathematician. William's mother died when he was six years old and from that time he was brought up by his father. James Thomson was the professor of engineering in Belfast at the time of William's birth and, when William was eight years old, his father James was appointed to the chair of mathematics at the University of Glasgow. James Thomson was a dominant father who brought his family up in a strict Presbyterian fashion. William's sister wrote of this childhood [3]:Our father read to us regularly every Sunday morning some chapters in the Old Testament, and in the evening some in the New. However, despite his father being very strict he had a very close relationship with William. It was from his father that William learn mathematics and at a very young age he became an accomplished mathematician with knowledge of the latest developments in the subject. William attended Glasgow University from the age of 10. This early age is not quite as unusual as one would think, for at that time the universities in Scotland to some extent competed with the schools for the most able junior pupils. Thomson began what we would consider university level work in 1838 when he was 14 years old. In the session 1838-39 he studied astronomy and chemistry. The following year he took natural philosophy courses (today called physics) which included a study of heat, electricity and magnetism. His Essay on the Figure of the Earth won him a gold medal from the University of Glasgow when he was 15 years old and it was a truly remarkable work containing important ideas which Thomson returned to throughout his life.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomson.html (1 of 7) [2/16/2002 11:35:16 PM]

Thomson

At the end of session 1839-40 Thomson read Fourier's The Analytical Theory of Heat a work on the application of abstract mathematics to heat flow. He later wrote [8]:I took Fourier out of the University Library; and in a fortnight I had mastered it - gone right through it. In fact there was a strong interest among the lecturers in Glasgow at that time in the French mathematical approach to physical science. In particular the works of Lagrange, Laplace, Legendre, Fresnel and Fourier were treated with "reverence" to use a word which Thomson himself would later use to describe the attitude that his lecturers had towards these French mathematicians. In fact Thomson also read Laplace's Méchanique céleste in session 1839-40 and visited Paris during this session. Wilson, writing in [9], describes Thomson's undergraduate years in Glasgow as follows:... from 1838 to 1841, William appears to have become thoroughly familiar with the phenomena of heat, electricity, and magnetism. Meikleham [the professor of natural philosophy] evidently encouraged something of a unified view of these branches of natural philosophy. Not only did his professors put him in touch with much modern experimental and mathematical research, but they also articulated the ideal of mathematising physical theory, even though none of them was himself a master of that craft. In 1841 Thomson entered Cambridge and in the same year his first paper was published. This paper Fourier's expansions of functions in trigonometrical series was written to defend Fourier's mathematics against criticism from the professor of mathematics at the university of Edinburgh. A more important paper On the uniform motion of heat and its connection with the mathematical theory of electricity was published in 1842 while Thomson was studying for the mathematical tripos examinations at Cambridge. At Cambridge Thomson was coached by William Hopkins, a famous Cambridge coach who played a more important role than the lecturers. Despite the efforts of Babbage, Peacock and Herschel to introduce the new French mathematics into Cambridge, the style of the mathematical tripos taken by Thomson still left much to be desired. Herschel and Babbage had conducted some experiments on magnetism in 1825, developing methods introduced by Arago, but nothing on heat, electricity or magnetism had entered the syllabus of the tripos. Thomson took the final part of the mathematical tripos exams in 1845. He graduated with a BA and he was second wrangler (second place in the list of those obtaining a first class degree). Further examinations saw him become first Smith's prizeman and he was elected a fellow of Peterhouse. Also in 1845 Thomson read George Green's work which was to have a major influence on the direction of his research. His interest in the French approach, and advice from his father, meant that after taking his degree Thomson went to Paris. There he worked in the physical laboratory of Henri-Victor Regnault and he was soon taking part in deep discussions with Biot, Cauchy, Liouville, Dumas, and Sturm. Perhaps the most profitable discussions that Thomson had were with Liouville. It was at Liouville's request that Thomson began to try to bring together the ideas of Faraday, Coulomb and Poisson on electrical theory. Ideas of 'action at a distance' or properties of the 'ether', and ideas of an 'electrical fluid' were difficult to unify. There were problems of whether or not an 'electrical fluid' was an actual physical entity with the properties of a fluid. Thomson was led to study the whole methodology of a physical science, distinguishing 'physical' parts of a theory from 'mathematical' parts.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomson.html (2 of 7) [2/16/2002 11:35:16 PM]

Thomson

In 1846 the chair of natural philosophy at Glasgow became vacant. Thomson's father used his influence in the University to help his son become the leading candidate for the post. Thomson returned from Paris to Glasgow and, in 1846, was unanimously elected professor of natural philosophy at the University. In 1847-49 he collaborated with Stokes on hydrodynamical studies, which Thomson applied to electrical and atomic theory. This collaboration with Stokes was to last for over fifty years with frequent letters on scientific matters being exchanged. Many of these letters have survived, for example copies of 407 letters from Thomson to Stokes and 249 letters from Stokes to Thomson have been published. Many of these letters discuss the mathematical similarities in the theory of heat and the theory of fluids. For example Stokes wrote to Thomson in 1847 (see for example [9]):What an intimate relation there is between the mathematical considerations which are applicable to heat, fluid motion, and attraction. The thermodynamical studies of Thomson led him to propose an absolute scale of temperature in 1848. The absolute scale that he proposed was based on his studies of the theory of heat, in particular the theory proposed by Sadi Carnot and later developed by Clapeyron. The Kelvin absolute temperature scale, as it is now known, was precisely defined much later after conservation of energy had become better understood. It derives its name from the title, Baron Kelvin of Largs, that Thomson received from the British government in 1892, and named after Thomson because of his proposal in this 1848 paper. Thomson's work on heat, and its shortcomings, is described fully in [11]. The author summarises his conclusions:... Thomson published between 1849 and 1852 three influential papers on the theory of heat. However, historians of science have already called attention to Thomson's difficulties in reconciling a principle formulated by James Prescott Joule with another principle formulated by Nicolas Leonard Sadi Carnot, and to errors Thomson made in his calculations. In the meantime, Rudolf Julius Emmanuel Clausius reconciled the two principles, and in 1854 he derived an expression for Carnot's principle. ... it was fundamental to Clausius' reasoning that he took to its ultimate consequence the meaning of Carnot's principle as a 'recovery' condition. Thomson somehow let the meaning of 'recovery' escape him. Therefrom came his troubles. He seems to have been so obsessed by his initial difficulties that he put the emphasis on irreversibility and on conservation of energy, missing 'all the rest'. In 1852 Thomson observed what is now called the Joule-Thomson effect, namely the decrease in temperature of a gas when it expands in a vacuum. Joule's ideas on heat were to change Thomson's views over the years. Thomson came to believe in a dynamical theory of heat and, in 1872, he wrote about how his views were led towards that approach (see for example [1]):... [before 1847] I did not ... know that motion is the very essence of what has hitherto been called matter. At the 1847 meeting of the British Association in Oxford, I learned from Joule the dynamical theory of heat, and was forced to abandon at once many, and gradually from year to year all other, statical preconceptions regarding the ultimate causes of apparently statical phenomena. The dynamical theory of heat led Thomson to also think of a dynamical theory for electricity and magnetism. In 1856 he sent a paper on this subject to the Royal Society of London entitled Dynamical illustrations of the magnetic and helicoidal rotary effects of transparent bodies on polarised light. He

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomson.html (3 of 7) [2/16/2002 11:35:16 PM]

Thomson

explained his notion of electricity in these words a few years later (see [1]):... we can conceive that electricity itself is to be understood as not an accident, but an essence of matter. Whatever electricity is, it seems quite certain that electricity in motion is heat; and that a certain alignment of exes of revolution in this motion is magnetism... This work by Thomson in 1856 on electricity and magnetism is important for it was these ideas which led Maxwell to develop his remarkable new theory of electromagnetism. One might think that Thomson would have eagerly supported Maxwell's theory which his own work had helped to create, but this was not so. Thomson had ideas of his own which he hoped would lead to a unifying theory, and his ideas took him further and further from accepting those of Maxwell. On vortex motion which Thomson published in 1867, set out his ideas. The paper begins:The mathematical work of the present paper has been performed to illustrate the hypothesis that space is continuously occupied by an incompressible frictionless fluid acted on by no force, and that material phenomena of every kind depend solely on motions created in the fluid. However, Thomson's initial hope that his theory could explain electromagnetism, light, gravity, and chemical processes slowly faded. In [18] the author argues for the importance of Thomson's electromagnetic work despite its eventual failure:We wish to stress Thomson's often under-estimated merits in the theory of the electromagnetic field. W Thomson was the first who tried to treat mathematically Faraday's conception of lines of force, and he introduced J C Maxwell to the problems of the electromagnetic field not only by his works, but also by his personal initiative. The author of the biography of Thomson [23], puts forward the view that during the first half of Thomson 's career he seemed incapable of being wrong while during the second half of his career he seemed incapable of being right. This seems too extreme a view but Thomson's refusal to accept atoms, his opposition to Darwin's theories, his incorrect speculations as to the age of the Earth and the Sun, and his opposition to Rutherford's ideas of radioactivity, certainly put him on the losing side of many arguments later in his career. Having studied some of Thomson's research contributions, let us comment on the innovations he introduced into teaching at the University of Glasgow. He introduced laboratory work into the degree courses, keeping this part of the work distinct from the mathematical side. He encouraged the best students by offering prizes. Some prizes were awarded to the best student, a vote being organised among the students to determine the recipient. There were also prizes which Thomson gave to the student that he considered most deserving. Not only did Thomson take a unified view of the physical world in his research, but he carried this into his teaching. One of his students, who attended Thomson's 1859-60 lectures, wrote (see [9]):His impulse was to correlate phenomena and arrive at the principle underlying them, and this gave him a certain impatience with branches of science which were still in the observational stage, and not yet come under mechanical laws. Hence the most brilliant and weighty part of his course was at the end, when he summed up his teaching and generalised energy, and the correlation of the physical forces... Another of Thomson's famous pieces of work was his joint project with Tait to produce their famous text Treatise on Natural Philosophy which they began working on in the early 1860s. They worked by http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomson.html (4 of 7) [2/16/2002 11:35:16 PM]

Thomson

posting a notebook back and forward to each other on this huge project which Thomson envisaged as covering all physical theories. Many volumes were intended, but only the first two were ever written which cover kinematics and dynamics. These were remarkable volumes which became the standard texts for many generations of scientists. Thomson achieved his greatest fame through an event that we have still to discuss. He was always greatly interested in the improvement of physical instrumentation, and Thomson designed and implemented many new devices, including the mirror-galvanometer that was used in the first successful sustained telegraph transmissions in transatlantic submarine cable. Thomson had joined a group of industrialist in the mid 1850s on a project to lay a submarine cable between Ireland and Newfoundland. He played several roles in this project, being on the board of directors and also being an advisor on theoretical electrical matters. The electrician who was in charge of the practical side of the operation was E O W Whitehouse, who insisted on using his own system against Thomson's advice. The cable was successfully laid in 1858, an attempt having failed the previous year when the cable broke. After initial difficulties with transmitting a signal, there was a sudden marked improvement and Whitehouse claimed success for his system. However it was soon discovered that he had substituted Thomson's mirror-galvanometer for his own instruments and there was a furious row between Whitehouse, Thomson and the other directors. Thomson's instruments were fully used for the third attempt at laying a cable in 1865 and this proved highly successful with rapid transmission of signals possible. For his work on the transatlantic cable Thomson was knighted in 1866. As well as fame, his participation in the telegraph cable project led to a large personal fortune brought about by his cable patents and consulting. He was able to buy a 126-ton yacht (the Lalla Rookh) as well as a fine house with surrounding estate. The Glasgow Herald proudly claimed the success of the cable [8]:Is Professor Thomson, the distinguished electrician, without whose inspiring genius this great business had not been so easily achieved, not a Glasgow man? And were the principal electrical instruments employed in testing and working the cable not manufactured by Mr White, the optician of this city, though under Professor Thomson's directions? Thomson published more than 600 papers. He was elected to the Royal Society of London in 1851, received its Royal Medal in 1856, received its Copley Medal in 1883 and served as its president from 1890 to 1895. In addition to his activities with the Royal Society of London, as one would expect of such an eminent Scottish professor, he served the Royal Society of Edinburgh over many years. He served three terms as president of this Society, first from 1873 to 1878, for the second time from 1886 to 1890, and for the third time from 1895 until his death in 1907. Thomson served as president of yet a third society when he was elected as president of the British Association for the Advancement of Science in 1871. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles)

Some Quotations (8)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomson.html (5 of 7) [2/16/2002 11:35:16 PM]

Thomson

A Poster of William Thomson (Lord Kelvin)

Mathematicians born in the same country 1. A comment from Thomas Hirst's diary

Cross-references to History Topics

2. Special relativity 3. Chrystal and the RSE Other references in MacTutor

Chronology: 1840 to 1850

Honours awarded to William Thomson (Lord Kelvin) (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1851

Royal Society Copley Medal

Awarded 1883

Royal Society Royal Medal

Awarded 1856

Royal Society Bakerian lecturer

1856

Fellow of the Royal Society of Edinburgh London Maths Society President

1898 - 1900

Honorary Fellow of the Edinburgh Maths Society

Elected 1883

Lunar features

Crater Thomson and Promontorium Kelvin and Rupes Kelvin 1. L Zapato (Some of Kelvin's papers and addresses) 2. University of Glasgow

Other Web sites

3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomson.html (6 of 7) [2/16/2002 11:35:16 PM]

Thomson

JOC/EFR July 1999

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Thomson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thomson.html (7 of 7) [2/16/2002 11:35:16 PM]

Thue

Axel Thue Born: 19 Feb 1863 in Tönsberg, Norway Died: 7 March 1922 in Oslo, Norway

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Axel Thue was the son of Niels Thue and Nicoline Cathinka Eger. He studied at Voss's school in Oslo, where he showed a great interest in physics, completing his studies there in 1883. He then enrolled at the University of Oslo graduating in 1889. He went to Leipzig in 1890 and spent a year studying under Lie. However, [1]:... his works do not reveal Lie's influence, probably because of Thue's inability to follow anyone else's line of thought. He also spent a while in Berlin where he attended lectures by Helmholtz, Fuchs and Kronecker. Back in Olso, Thue held a scholarship in mathematics from 1891 to 1894. On 6 July 1894 he married Lucie Collett Lund who was ten years younger than Thue. Then Thue was appointed to Trondheim Technical College where he worked from 1894 until 1903. He was appointed as professor of applied mathematics at Oslo University in 1903. He held this post until his death in 1922. In 1909 he produced an important paper, published in Crelle's Journal, on algebraic numbers showing that, for example, y3 - 2x2 = 1 cannot be satisfied by infinitely many pairs of integers. His work was extended by Siegel in 1920 and again by Klaus Roth in 1958. Landau, in 1922, described Thue's work as:... the most important discovery in elementary number theory that I know. Thue's Theorem states that:If f(x, y) is a homogeneous polynomial with integer coefficients, irreducible in the rationals http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thue.html (1 of 2) [2/16/2002 11:35:18 PM]

Thue

and of degree > 2 and c is a non-zero integer then f(x, y) = c has only a finite number of integer solutions. His contributions to the theory of Diophantine equations are discussed in [3]. In fact Thue wrote 35 papers on number theory, mostly on the theory of Diophantine equations, and these are reproduced in [2]. Another famous contribution made by Thue was his 1910 paper on the word problem for finitely presented semigroups. If this work seems a little strange for a professor of applied mathematics then some quotes from Thue will clarify where he stood the issue of applications. He wrote a many articles in series between 1906 and 1912 and he wrote in one of them:For the development of the logical sciences it will be important to find wide fields for the speculative treatment of difficult problems, without regard to eventual applications. Another quote from Thue on applied mathematics (see for example [1]) is:The further removed from usefulness or practical application, the more important. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Axel Thue

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thue.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thue.html (2 of 2) [2/16/2002 11:35:18 PM]

Thurston

William Paul Thurston Born: 30 Oct 1946 in Washington, D.C., USA

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Bill Thurston studied at New College, Sarasota, Florida. He received his B.S. from there in 1967 and moved to the University of California at Berkeley to undertake research under Morris Hirsch's and Stephen Smale's supervision. He was awarded his doctorate in 1972 for a thesis entitled Foliations of 3-manifolds which are circle bundles. This work showed the existence of compact leaves in foliations of 3-dimensional manifolds. After completing his Ph.D., Thurston spent the academic year 1972-73 at the Institute for Advanced Study at Princeton. Then, in 1973, he was appointed an assistant professor of mathematics at Massachusetts Institute of Technology. In 1974 he was appointed professor of mathematics at Princeton University. Throughout this period Thurston worked on foliations. Lawson ([5]) sums up this work:It is evident that Thurston's contributions to the field of foliations are of considerable depth. However, what sets them apart is their marvellous originality. This is also true of his subsequent work on Teichmüller space and the theory of 3-manifolds. In [8] Wall describes Thurston's contributions which led to him being awarded a Fields Medal in 1982. In fact the1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Lectures on the work of Thurston which led to his receiving the Medal were made at the 1983 International Congress. Wall, giving that address, said:Thurston has fantastic geometric insight and vision: his ideas have completely

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thurston.html (1 of 3) [2/16/2002 11:35:20 PM]

Thurston

revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplay between analysis, topology and geometry. Wall [8] goes on to describe Thurston's work in more detail:The central new idea is that a very large class of closed 3-manifolds should carry a hyperbolic structure - be the quotient of hyperbolic space by a discrete group of isometries, or equivalently, carry a metric of constant negative curvature. Although this is a natural analogue of the situation for 2-manifolds, where such a result is given by Riemann's uniformisation theorem, it is much less plausible - even counter-intuitive - in the 3-dimensional situation. Kleinian groups, which are discrete isometry groups of hyperbolic 3-space, were first studied by Poincaré and a fundamental finiteness theorem was proved by Ahlfors. Thurston's work on Kleinian groups yielded many new results and established a well known conjecture. Sullivan describes this geometrical work in [6], giving the following summary:Thurston's results are surprising and beautiful. The method is a new level of geometrical analysis - in the sense of powerful geometrical estimation on the one hand, and spatial visualisation and imagination on the other, which are truly remarkable. Thurston's work is summarised by Wall [8]:Thurston's work has had an enormous influence on 3-dimensional topology. This area has a strong tradition of 'bare hands' techniques and relatively little interaction with other subjects. Direct arguments remain essential, but 3-dimensional topology has now firmly rejoined the main stream of mathematics. Thurston has received many honours in addition to the Fields Medal. He held a Alfred P Sloan Foundation Fellowship in 1974-75. In 1976 his work on foliations led to his being awarded the Oswald Veblen Geometry Prize of the American Mathematical Society. In 1979 he was awarded the Alan T Waterman Award, being the second mathematician to receive such an award (the first being Fefferman in 1976). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles)

A Quotation

Mathematicians born in the same country Other references in MacTutor

Chronology: 1970 to 1980

Honours awarded to Bill Thurston (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1982

AMS Colloquium Lecturer

1989

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thurston.html (2 of 3) [2/16/2002 11:35:20 PM]

Thurston

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thurston.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thurston.html (3 of 3) [2/16/2002 11:35:20 PM]

Thymaridas

Thymaridas of Paros Born: about 400 BC in Paros, Greece Died: about 350 BC Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

We are told a little about Thymaridas' life. He was apparently a rich man but, for some reason we are not told about, he fell into poverty. Thestor of Poseidonia sailed to Paros to help him with money specially collected for his benefit. Thymaridas was a Pythagorean and a number theorist who wrote on prime numbers. Iamblichus tells us that Thymaridas called a prime number rectilinear since it can only be represented one-dimensionally. Non-primes such as 6 are represented by rectangles of sides 2 and 3. We are also told that he called 'one' a 'limiting quantity' or a 'limit of fewness'. Thymaridas also gave methods for solving simultaneous linear equations which became known as the 'flower of Thymaridas'. For the n equations in n unknowns x + x1 + x2 + ... + xn-1 = S x + x1 = a1 x +x2 = a2 . . . x + xn-1 = an-1 then Thymaridas gives the solution x = [(a1 + a2 + .... + an-1) - S]/(n - 2). He also shows how certain other types of equations can be put into this form. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thymaridas.html (1 of 2) [2/16/2002 11:35:22 PM]

Thymaridas

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Thymaridas.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Thymaridas.html (2 of 2) [2/16/2002 11:35:22 PM]

Tibbon

Jacob ben Machir ibn Tibbon Born: 1236 in Marseilles, Spain (now France) Died: 1312 in Montpellier, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jacob ben Tibbon is also known as Prophatius. He was a distinugished Jewish medical man who worked in the medical faculty of the University of Monpellier. He was the grandson of Samuel ben Judah ibn Tibbon who is famed as a translator. Jacob ben Tibbon is himself known as a translator as well as a mathematician and an astronomer. He translated into Hebrew many Arabic versions of Greek works, Euclid's Elements, Ptolemy's Almagest as well as certain Arabic works by al-Ghazali and others. He wrote Jacob's Quadrant in which he describes a quadrant of his own invention. This work contains a table of 11 fixed stars which are to be used in the construction of the instrument. Jacob ben Tibbon also wrote Luhot (Tables) a book of astronomical tables giving ascensions of certain stars at Paris. These tables are mentioned in Dante's Divine Comedy. The Italian astronomer Andalo di Negro wrote Canones Super Almanach Profatii in 1323 which dealt with Jacob ben Tibbon's tables in Luhot. Another work by Jacob ben Tibbon was Almanach Perpetuum which, as the title indicates, was a work on the almanac. Tibbon's work was used by Copernicus in forming his theories. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tibbon.html (1 of 2) [2/16/2002 11:35:23 PM]

Tibbon

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tibbon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tibbon.html (2 of 2) [2/16/2002 11:35:23 PM]

Tietze

Heinrich Franz Friedrich Tietze Born: 31 Aug 1880 in Schleinz (E of Neunkirchen), Austria Died: 17 Feb 1964 in Munich, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Heinrich Tietze's father was Emil Tietze, the Director of the Geological Institute at the University of Vienna, and his mother was Rosa von Hauer, who was the daughter of the geologist Franz Ritter von Hauer. Tietze was a student at the Technische Hochschule in Vienna, starting his studies there in 1898. At Vienna he formed a close friendship with three other students of mathematics, Paul Ehrenfest, Hans Hahn and Gustav Herglotz. They were known as the 'inseparable four'. It was his friend Herglotz who suggested that Tietze spend a year in Munich, and indeed he went there in 1902 to continue his studies. Returning to Vienna, Tietze was supervised during his doctoral studies by Gustav von Escherich and he was awarded his doctorate in 1904. Wirtinger, who had himself studied at Vienna, spent ten years at the University of Innsbruck before returning to a chair at the University of Vienna in 1905. He lectured on algebraic functions and their integrals in his first year back in Vienna, and Tietze attended these lectures and because of them formed an instant liking for topological notions which would from that time on be his main research topic. Tietze submitted his habilitation thesis to Vienna in 1908 and this was on a topological topic considering topological invariants. From 1910 he was an extraordinary professor of mathematics at Brünn (today called Brno), and in 1913 he was promoted to ordinary professor. His career was interrupted, however, in 1914 by the outbreak of World War I. At the start of the war Tietze was called up to serve in the Austrian army. He served for the duration of the war, returning to Brünn when hostilities had ended. The following year, in 1919, he accepted the chair of mathematics at the University of Erlangen. After six years in Erlangen, Tietze accepted a chair at the University of Munich. He remained in Munich for the rest of his life, retiring from his chair in 1950 but continuing his mathematical research almost up to the time of his death at age 83. Of course this means that Tietze spent the difficult years of Nazi control of Germany at Munich and this had many unfortunate consequences. Litten, in [3], gives details of one such difficulty. Caratheodory was a colleague of Tietze's at Munich until he retired in 1938. The quest for a successor took from 1938 until 1944 and resulted in unbelievably complex political considerations. Tietze and his colleagues drew up a short list of three candidates to replace Caratheodory. These were Herglotz, van der Waerden and Siegel. However, all three were

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tietze.html (1 of 3) [2/16/2002 11:35:25 PM]

Tietze

opposed by the Nazi professors at Munich for political reasons. Litten describes the arguments which involved considering the political reliability and the number of Jewish friends of the candidates. Tietze contributed to the foundations of general topology and developed important work on subdivisions of cell complexes. The paper [5] lists six books and 104 papers written by Tietze. The bulk of this work was carried out after he took up the chair at Munich in 1925. Fundamental groups were introduced by Poincaré in 1895 and, in 1908, Tietze recognised that from the abelianised fundamental group of a space all the earlier invariants could be calculated. In that 1908 paper, Tietze produced a finite presentation for the fundamental group and invented the now well-known Tietze transformations to show that fundamental groups are topological invariants. The now famous Tietze transformations change one presentation of a finitely presented group to another presentation without changing the group which is defined by the presentation. It is possible to go from any given finite presentation of a group to any other using Tietze transformations. In the same 1908 paper Tietze gives the first reference to the isomorphism problem for groups, namely: if two groups are defined by finite presentations, is there an algorithm to decide whether they are isomorphic or not? Of course Tietze gives the problem in the context of fundamental groups of topological spaces. It is probably fair to say that von Dyck initiated the study of combinatorial group theory but then Tietze made the first major step forward. Among the topics in topology which Tietze worked on were knot theory, Jordan curves and continuous mappings of areas. Tietze also wrote on map colouring and wrote a well known book Famous Problems of Mathematics. Seebach writes [1]:It shows his gift for representing even difficult mathematical questions in a very clear and impressive manner for interested people. Topics outside topology which Tietze worked on included ruler and compass constructions, continued fractions, partitions, the distribution of prime numbers, and differential geometry. Tietze received many honours for his contributions. In particular he was elected a member of the Bavarian Academy of Sciences and served two terms (1934-42 and 1946-51) as Secretary to the Mathematics and Natural Sciences Division. He was also elected to the Austrian Academy of Sciences in 1959. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles)

A Quotation

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tietze.html (2 of 3) [2/16/2002 11:35:25 PM]

Tietze

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tietze.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tietze.html (3 of 3) [2/16/2002 11:35:25 PM]

Tikhonov

Andrei Nikolaevich Tikhonov Born: 30 Oct 1906 in Gzhatska, Smolensk, Russia Died: 1993

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Like most Russian mathematicians there are different ways to transliterate Andrei Nikolaevich Tikhonov's name into the Roman alphabet. The most common way, other than Andrei Nikolaevich Tikhonov, is to write it as Andrey Nikolayevich Tychonoff. Andrei Nikolaevich Tikhonov attended secondary school as a day pupil and entered the Moscow University in 1922, the year in which he completed his school education. His studied in the Mathematics Department of the Faculty of Mathematics and Physics at Moscow University and made remarkable progress, having his first paper published in 1925 while he was still in the middle of his undergraduate course. This first work was related to results of Aleksandrov and Urysohn on conditions for a topological space to be metrisable. However he did not stop there and continued his investigations in topology. By 1926 he had discovered the topological construction which is today named after him, the Tikhonov topology defined on the product of topological spaces. Aleksandrov, recalling in [4] how he failed to appreciate the significance of Tikhonov's ideas at the time he proposed them, remembered:... very well with what mistrust he met Tikhonov's proposed definition. How was it possible that a topology introduced by means of such enormous neighbourhoods, which are only distinguished from the whole space by a finite number of the coordinates, could catch any of the essntial characteristics of a topological product? Tikhonov certainly had given the right definition and this idea, which was counterintuitive to even as great a topologist as Aleksandrov, allowed Tikhonov to go on and prove such important topological

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tikhonov.html (1 of 4) [2/16/2002 11:35:27 PM]

Tikhonov

results as the product of any set of compact topological spaces is compact. Few mathematicians have gained a worldwide reputation before they even start their research careers but this was essentially how it was for Tikhonov. His results on the Tikhonov topology of products were achieved before he graduated in 1927. With this impressive record he became a research student at Moscow University in 1927. It might be thought that someone who had clearly such an intuitive grasp of topological ideas would be only too pleased to use his talents in that area. Tikhonov, however, had equal talents for other areas of mathematics. The range of his work is summarised in [3]:We owe to Tikhonov deep and fundamental results in a wide range of topics in modern mathematics. His first-class achievements in topology and functional analysis, in the theory of ordinary and partial differential equations, in the mathematical problems of geophysics and electrodynamics, in computational mathematics and in mathematical physics are all widely known. Tikhonov's scientific work is characterised by magnificent achievements in very abstract fields of so-called pure mathematics, combined with deep investigations into the mathematical disciplines directly connected with practical requirements. In fact Tikhonov's work led from topology to functional analysis with his famous fixed point theorem for continuous maps from convex compact subsets of locally convex topological spaces in 1935. These results are of importance in both topology and functional analysis and were applied by Tikhonov to solve problems in mathematical physics. He defended his habilitation thesis in 1936 on Functional equations of Volterra type and their applications to mathematical physics. The thesis applied an extension of Emile Picard's method of approximating the solution of a differential equation and gave applications to heat conduction, in particular cooling which obeys the law given by Josef Stefan and Boltzmann. After successfully defending his thesis, Tikhonov was appointed as a professor at Moscow University in 1936 and then, three years later, he was elected as a Corresponding Member of the Academy of Sciences of the USSR. Tikhonov's approach to problems in mathematical physics is described in [14]:A characteristic of Tikhonov's research is to combine a concrete theme in natural science with investigations into a fundamental mathematical problem. In discussing some general problem in nature he always knows how to pick out a typical concrete physical problem and to give it a clear mathematical formulation. However, his mathematical investigations are never confined to the solution of a given concrete problem, but serve as the starting point for stating a general mathematical problem that is a broad generalisation of the first problem. The extremely deep investigations of Tikhonov into a number of general problems in mathematical physics grew out of his interest in geophysics and electrodynamics. Thus, his research on the Earth's crust lead to investigations on well-posed Cauchy problems for parabolic equations and to the construction of a method for solving general functional equations of Volterra type. ... Tikhonov's work on mathematical physics continued throughout the 1940s and he was awarded the State Prize for this work in 1953. However, in 1948 he began to study a new type of problem when he considered the behaviour of the solutions of systems of equations with a small parameter in the term with the highest derivative. After a series of fundamental papers introducing the topic, the work was carried http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tikhonov.html (2 of 4) [2/16/2002 11:35:27 PM]

Tikhonov

on by his students. Another area in which Tikhonov made fundamental contributions was that of computational mathematics ([11] and [12]):Under his guidance many algorithms for the solution of various problems of electrodynamics, geophysics, plasma physics, gas dynamics, ... and other branches of the natural sciences were evolved and put into practice. ... One of the most outstanding achievemnets in computational mathematics is the theory of homogeneous difference schemes, which Tikhonov developed in collaboration with Samarskii.... In the 1960s Tikhonov began to produce an important series of papers on ill-posed problems. He defined a class of regularisable ill-posed problems and introduced the concept of a regularising operator which was used in the solution of these problems. Combining his computing skills with solving problems of this type, Tikhonov gave computer implementations of algorithms to compute the operators which he used in the solution of these problems. Tikhonov was awarded the Lenin Prize for his work on ill-posed problems in 1966. In the same year he was elected to full membership of the Academy of Sciences of the USSR. Tikhonov's wide interests throughout mathematics led him to hold a number of different chairs at Moscow University, in particular a chair in the Mathematical Physics Faculty and a chair of Computational Mathematics in the Engineering Mathematics Faculty. He also became dean of the Faculty of Computing and Cybernetics at Moscow University. Tikhonov was appointed as Deputy Director of the Institute of Applied Mathematics of the Academy of Sciences, a position he held for many years. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (25 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1999 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tikhonov.html (3 of 4) [2/16/2002 11:35:27 PM]

Tikhonov

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tikhonov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tikhonov.html (4 of 4) [2/16/2002 11:35:27 PM]

Tilly

Joseph Marie de Tilly Born: 16 Aug 1837 in Ypres, Belgium Died: 4 Aug 1906 in Munich, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Tilly was a military man and became a lieutenant in the artillery. In 1858 he was assigned to teach a mathematics course at the regimental school. Tilly studied the principles of geometry, Euclid's 5th postulate and non-euclidean geometry. In 1860 he achieved results similar to those of Lobachevsky but at this stage he had not heard about Lobachevsky. He learnt about Lobachevsky's work in 1866, then in 1870 he published a work on Lobachevsky space. Tilly was the first to study non-euclidean mechanics, a topic he invented. Tilly corresponded with Jules Hoüel, the only French mathematician interested in these topics at that time. Until this point Tilly had worked in isolation. He also wrote on military science and the history of mathematics in Belgium. There were complaints that Tilly had unduly emphasised the scientific education of future officers. An inspector at the military school declared that Tilly was not allowed to use differentials. Tilly must have carried on with his methods of teaching despite this and he was dismissed from his post and forced into early retirement in 1900. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tilly.html (1 of 2) [2/16/2002 11:35:28 PM]

Tilly

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tilly.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tilly.html (2 of 2) [2/16/2002 11:35:28 PM]

Tinbergen

Jan Tinbergen Born: 12 April 1903 in The Hague, Netherlands Died: 9 June 1994 in Netherlands

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jan Tinbergen's father was a schoolmaster. Jan attended the Hogere Bugerschool in The Hague. This was a special type of school which was designed for children of middle class parents who were aspiring to better their status. These schools allowed entry to the university system after passing additional examinations in Latin and Greek and this Tinbergen did entering the University of Leiden in 1921. He studied mathematics and theoretical physics at the University of Leiden and was most influenced there by Ehrenfest. Writing in the newspaper NRC-Handelsblad in 1987 he said of his teachers at Leiden:To Ehrenfest I owe a great deal. I studied physics at a time when a number of fascinating persons were there together. Ehrenfest would not instruct as such, as he preferred dialogue. Thanks to him I could participate in discussions with Albert Einstein. Also Kamerling Onnes, Lorentz and Zeeman were present. Being a student in the hands of such teachers, you are very fortunate indeed. However, Tinbergen had political interests associated with his left wing views. At university he founded a club for social democratic students and also founded a student newspaper. In fact some of Tinbergen's first publications were articles he wrote for the socialist newspaper Het Volk in which he examined the effects of the economic depression of 1920-22 on unemployment and how the lives of the poor had been affected. After completing his undergraduate degree at Leiden, Tinbergen continued to study at Leiden for his doctorate under Ehrenfest's supervision. His thesis combined mathematics, physics and economics. In the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tinbergen.html (1 of 3) [2/16/2002 11:35:30 PM]

Tinbergen

introduction Tinbergen thanks Ehrenfest for pointing out to him a topic which could allow him to combine mathematical theories with his political interests. The main part of the thesis is mathematical, studying minimisation problems. Then he gives two appendices, one describing applications of the mathematics to physics, the second appendix giving applications to economics. Tinbergen submitted his dissertation in 1929. The importance of this work of Tinbergen was that it was one of the first examples of a new idea in mathematics, namely mathematical modelling. Of course mathematical physics had been studied throughout the history of mathematics but Tinbergen's work saw a new path for the applications of mathematics, where the applications could be to a wide variety of areas. Tinbergen's political views meant that he was unwilling to do military service. He was fortunate in that, only a few years earlier in 1923, legislation had been passed in the Netherlands allowing conscientious objectors to avoid military service. The legislation required that a conscientious objector do government service so Tinbergen joined the Dutch government's Central Bureau of Statistics. From 1929 to 1945 he worked as a statistician with the Bureau of Statistics. His work on economics in his doctorate had been entirely theoretical, but now he had access to large amounts of data on which to test and develop theories. From 1933 to 1973 he was professor of economics at The Netherlands School of Economics, Rotterdam. He was appointed to the board of the scientific bureaux of the Dutch Labour Party and he co-authored the Labour Plan in 1935. This plan was based on Tinbergen's mathematically based principles of economics. Alberts writes in [3]:The assumption that these principles of economic rationality might really be made to work was substantiated by Tinbergen one year afterwards in an academic debate on the possibilities of active economic policy. In his contribution to the debate Tinbergen projected a 'quantitative stylising of the Dutch economy' to isolate the important factors and their effects by means of a set of definitions and equations. This "model" of the Dutch economy, as he called it (written with quotation marks at first), would allow one to throw some data into the "mathematical machinery" which would then predict the results. Tinbergen later developed other econometric models, in particular he constructed an econometric model of the USA. In 1969 he jointly won the first Nobel Prize for Economics for this first ever macroeconomic model. In the late 1930s Tinbergen worked as a scientific advisor for the League of Nations. Then, in 1945, he was appointed as director of the Dutch Central Planning Bureau. His major publications include Statistical Testing of Business Cycles (1938), Econometrics (1942), Economic Policy (1956), and Income Distribution (1975). Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tinbergen.html (2 of 3) [2/16/2002 11:35:30 PM]

Tinbergen

Honours awarded to Jan Tinbergen (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1969

Other Web sites

1. Nobel prizes site (A CV of Tinbergen and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tinbergen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tinbergen.html (3 of 3) [2/16/2002 11:35:30 PM]

Tinseau

D'Amondans Charles de Tinseau Born: 19 April 1748 in Besançon, France Died: 21 March 1822 in Montpellier, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Tinseau entered Ecole Royale du Génie at Mézières in 1769 and graduated as a military engineer in 1771. At Mézières he was a student of Monge who encouraged Tinseau to undertake mathematical research. Tinseau wrote on the theory of surfaces, working out the equation of a tangent plane at a point of a surface. He continued Monge's study of curves of double curvature and ruled surfaces. Two papers were published in 1772, one on infinitesimal geometry and one on astronomy. He only published one further paper but continued to correspond with Monge on mathematical topics. Tinseau campaigned against the French Revolution. He tried to organise uprisings and tried to help the Allied powers against the French armies. He gave the strategic plans of the French armies to the Allies in 1813. He was exiled to England and did not return to France until 1816. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tinseau.html (1 of 2) [2/16/2002 11:35:31 PM]

Tinseau

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tinseau.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tinseau.html (2 of 2) [2/16/2002 11:35:31 PM]

Tisserand

François-Félix Tisserand Born: 13 Jan 1845 in Nuits-St-Georges, Côte-d'Or, France Died: 20 Oct 1896 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Félix Tisserand studied at the Ecole Normale. In his doctoral thesis, written in 1868, Tisserand extended Delaunay's work on the three-body problem on the orbit of the Moon. This work was an outstanding contribution to mathematical astronomy coming quickly after the publication of the second volume of Delaunay's work on lunar theory La Théorie du mouvement de la lune which had been published in 1867. Tisserand was then appointed as an assistant-astronomer at the Paris Observatory. This was an unfortunate time at the Paris Observatory since its Director Le Verrier had become very unpopular following his drive for efficiency and attempts were being made to have him removed. In 1870 Delaunay was appointed as Director of the Paris Observatory to replace Le Verrier. Tisserand showed remarkable abilities in his work at the Observatory and it was clear that he would soon achieve an elevated position in the world of astronomy. In 1873 Tisserand was appointed director of the Toulouse Observatory to succeed Daguin, a post which he held for five years. At the Toulouse Observatory Tisserand appointed two young astronomers, Joseph Perrotin and Guillaume Bigourdan, to help him in his observational work. Neither were experienced in astronomy but Tisserand had made two good choices for he soon trained the young men to become astronomers of outstanding qualities. Tisserand did not spend the whole of his five years as director in Toulouse. In 1874 he went to Japan to make observations of the transit of Venus which occurred that year. A transit of Venus is when the planet passed in front of the disc of the sun as viewed from the Earth and Tisserand took a year to make the journey to Japan, make his observations, and return to France. In order to keep the Toulouse Observatory operating during this lengthy absence Jules Gruey, a teacher at the Toulouse Faculty of Science was appointed temporary director. The experience stood Gruey in good stead for he went on to became the director of the Besançon Observatory. After Tisserand returned to France and took over again as director of the Observatory in Toulouse he had a new instrument installed. However the 83 centimetre telescope installed in 1875 was not a great success since the wooden base was not stable enough to allow photography. Having a telescope which could not be used for photography was rather unfortunate since most astronomical work at that time required that facility. Tisserand decided to undertake a programme of measurement of the separation of binary stars, but he discovered that the micrometer used for such measurements did not work. He decided http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tisserand.html (1 of 2) [2/16/2002 11:35:33 PM]

Tisserand

that he could only use the defective telescope for observing the planets and together with his assistants he made observations of the moons of Jupiter and Saturn. From 1892 until his death he was director of the Paris Observatory. When he arrived to take up the post of director work was being undertaken at the Observatory on the Catalogue photographique de la carte du ciel (the Photographic Catalogue of the Map of the Sky). Tisserand took over the task of completing this major work and arranged for its publication. Tisserand is especially remembered for his four volume textbook which is an update of Laplace's work. He published Traité de mécanique céleste , (Treatise on Celestial Mechanics) in four volumes which appeared between the years 1889 and 1896. Despite being 100 years old this textbook is still sometimes referred to by current writers of celestial mechanics books. In 1874 he was elected to the Academy of Sciences as a corresponding member and was elevated to full membership in 1878. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to Félix Tisserand (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Tisserand

Paris street names

Rue Tisserand (15th Arrondissement)

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tisserand.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tisserand.html (2 of 2) [2/16/2002 11:35:33 PM]

Titchmarsh

Edward Charles Titchmarsh Born: 1 June 1899 in Newbury, Berkshire, England Died: 18 Jan 1963 in Oxford, Oxfordshire, England

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edward Titchmarsh was known as Ted to his contemporaries. His grandfather was a grocer in Royston while Ted's father, Edward Harper, became a Congregational minister in Newbury, Berkshire. Ted was the second of three children born to Edward Harper and Caroline Titchmarsh. Ted's father moved to Sheffield, where again he was the Congregational minister and it was in Sheffield that Ted attended King Edward VII School. He later wrote of his time at this school:The first occasion on which I distinguished myself was when I was in one of the fourth forms. The headmaster for some unknown reason made the whole upper school do an arithmetic paper, the same for all forms. The mathematical specialists in the sixth form came out top, and I came next. ... At this point one had to choose either classical or modern subjects: I was put on the classical side. I learnt enough Latin to pass and enough Greek to fail. it became clear that mathematics was my real subject and I began to specialise in it. In December 1916, when he was seventeen years old, Titchmarsh won an Open Mathematical Scholarship to Balliol College, University of Oxford and he began his studies there in October 1917. By this time, of course, Britain had been involved in World War I for three years and Titchmarsh, by now eighteen years of age, was soon undertaking war service. After just one term at Oxford he joined the Royal Engineers and in August 1918 he was sent to France as a dispatch rider. Titchmarsh was fortunate to have arrived in France at a point when the tide of the war had changed. There had been major offensives by the German armies throughout June and July of 1918 but as Titchmarsh landed in France the Allied forces were making steady advances, driving the German troops out of France. Titchmarsh carried out his duties as a dispatch rider first on horseback and then on a motorcycle. By

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Titchmarsh.html (1 of 4) [2/16/2002 11:35:35 PM]

Titchmarsh

October 1918 the Allies had recovered most of German occupied France and a part of Belgium. German morale collapsed, the German Kaiser William II abdicated on 9 November, and on 11 November the war ended with the signing of the Armistice between Germany and the Allies. However Titchmarsh served with the Royal Engineers for almost a further year before he was able to return to his interrupted studies in Oxford in October 1919. At Oxford he was tutored by J W Russell, and Mary Cartwright (see [4] or [5]) wrote:At Russell's first lecture the room was packed to the doors, and Russell said: "Ah, there's my clever pupil Mr Titchmarsh - he knows it all, he can go away." Russell dictated his lectures word for word and examples were handed out - and then, if necessary, solutions to examples. Some of Titchmarsh's solutions replaced the official ones. At Oxford Titchmarsh soon came under the influence of Hardy and he later wrote:From [Hardy] I learnt what mathematical analysis is, and at his suggestion I devoted myself to research in pure mathematics. Hardy held a class which did not form part of the syllabus. The class met once a week on Monday evenings after dinner and it would begin with a talk and then continue until late in the evening with the members taking part in deep mathematical discussions. Titchmarsh attended this class, as did Mary Cartwright. Of course Titchmarsh and Hardy had a common passion, namely cricket, which must have served Titchmarsh well. His uncle was a professional cricketer and Titchmarsh often played in the regular cricket matches with Hardy. Titchmarsh graduated with a First Class degree in 1922 and won mathematical scholarships for his outstanding work. He did not read for his doctorate but was appointed as a Senior Lecturer at University College London in 1923. This appointment in London did not see Titchmarsh end his association with Oxford. Far from it, he took the examinations for a Prize Fellowship at Magdalen College Oxford also in 1923 and, having won the Fellowship, he held it for seven years. Not only did he lecture in London, where he supervised doctoral students, but he also began publishing high quality research papers on mathematical analysis. During the academic year 1928-29 Hardy was at Princeton, and it was Titchmarsh who took over the supervision of Mary Cartwright who was, at that time, one of Hardy's doctoral students at Oxford. Despite having duties at both London and Oxford, Titchmarsh found time to visit his father who was by this time a Congregational minister in Essex. In 1925 he married Kathleen Blomfield, who was the daughter of his father's Church Secretary. They would have three children, all daughters. Charles Burkill held the chair of pure mathematics at Liverpool from 1924 until 1919 when he took up a lectureship at Peterhouse, Cambridge. Titchmarsh was appointed to Burkill's chair at Liverpool, a post he held for two years before he succeeded to Hardy's Savilian chair at Oxford when Hardy moved to Cambridge. Wilson, in [8] recounts how the appointment came about:By chance, Titchmarsh was visiting Oxford to examine a doctorate and bumped into Ferrar who asked him whether he'd applied for Hardy's vacant Oxford Chair. Titchmarsh said no, but (encouraged by Ferrar) thought that he might. He sent in an application on a single sheet saying that he wished to apply for the geometry Chair but could not undertake to lecture on geometry as Hardy had done. two days later he was appointed and the statutes

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Titchmarsh.html (2 of 4) [2/16/2002 11:35:35 PM]

Titchmarsh

were changed to say that the Savilian Professor of Geometry no longer had to lecture on geometry. Titchmarsh held the Savilian Chair of Geometry at Oxford for 30 years. All Titchmarsh's work is in analysis, in fact he refused to lecture on any other topic. His method of working was to work on a topic until he tired of it, when he would write a book on that topic. He studied Fourier series and Fourier integrals writing Introduction to the Theory of Fourier Integrals (1937). Other topics to which he made major contributions included entire functions of a complex variable and, working with Hardy, integral equations. He also did important work on the Riemann zeta function writing The Zeta-Function of Riemann (1930) which he brought up-to-date as The Theory of the Riemann Zeta-Function (1951) which [1]:... contained practically everything that was known on the subject. His most popular book The theory of functions was published in 1932, and [1]:... a generation of mathematicians learned the theory of analytic functions and Lebesgue integration from it, and also learned (by observation) how to write mathematics. From 1939 Titchmarsh concentrated on the theory of series expansions of eigenfunctions of differential equations, work which helped to resolve problems in quantum mechanics. However [1]:... he saw physics as a source of interesting mathematics problems; but his interest was exclusively in the mathematics, without any regard for its real applicability. His work on this topic occupied him for the last 25 years of his life and he published much of it in Eigenfunction Expansions Associated with Second-Order Differential Equations (1946, 1958). Titchmarsh was elected to the Royal Society in 1931 and received its Sylvester Medal in 1955:... in recognition of his distinguished researches on the Riemann zeta-function, analytic theory of numbers, Fourier analysis and eigen-function expansions. He received many other honours for his important contributions to mathematics. He served as President of the London Mathematical Society in 1945-47 and was awarded its De Morgan Medal in 1953 and its Berwick Prize. The University of Sheffield, from the town where he attended school, awarded him an honorary doctorate in 1953. Sir Michael Atiyah reminisces about Titchmarsh in [3]:He was a scholarly man who sat in his room and wrote beautiful books - impeccable, effectively written textbooks, from which many students have learnt their complex analysis. But he was a man of very few words; his influence was not due to personal contact but through his writing. When I first came to Oxford he was curator of the Mathematical Institute, so as a newly arrived member I had to go to him to get a key to my office. I was duly ushered into his big room, where he was sitting at his desk. I sat down and he handed over the key, and then I expected a word of welcome of some words of advice, but we just sat in silence. After five minutes, I left. Charles Coulson, one of his colleagues, wrote a tribute to Titchmarsh:There were many things about Ted that I have always much admired - his utter humility, which never betrayed anything but the greatest simplicity; his complete integrity ... and his kindness to me when I arrived first; to his students (who worshipped him) and to everyone.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Titchmarsh.html (3 of 4) [2/16/2002 11:35:35 PM]

Titchmarsh

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles)

Some Quotations (5)

Mathematicians born in the same country Honours awarded to Edward Titchmarsh (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1931

Royal Society Sylvester Medal

Awarded 1955

London Maths Society President

1945 - 1947

LMS De Morgan Medal

Awarded 1953

LMS Berwick Prize winner

1956

Savilian Professor of Geometry

1932

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Titchmarsh.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Titchmarsh.html (4 of 4) [2/16/2002 11:35:35 PM]

Todd

John Arthur Todd Born: 23 Aug 1908 in Liverpool, England Died: 22 Dec 1994 in Croydon, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Todd was educated at Liverpool Collegiate School and, having sat the scholarship examination for Cambridge, he was awarded an entrance scholarship to Trinity College. Entering Trinity in October 1925 he achieved distinction in an undergraduate career which ended with his graduation in 1928. After graduation Todd remained at Trinity to study for his doctorate in geometry under H F Baker's supervision. He was supported by a research scholarship and, in 1930, he was awarded the highly prestigious Smith's Prize. Among his fellow students at Cambridge at this time were a number of others studying geometry including P du Val, H S M Coxeter and W L Edge. After an impressive prize winning record Todd hoped for a Research Fellowship at Trinity but he failed in three attempts, on one occasion losing out to Coxeter who won the Fellowship. Disappointed at his failure to win the Fellowship, Todd left Cambridge and accepted Mordell's offer of an assistant lectureship at the University of Manchester in 1931. An important year for Todd's mathematical development was the session 1933-34 which he spent at Princeton on a Rockefeller Scholarship. Lefschetz proved the major influence on Todd during this year in the United States, and this broadened Todd's geometrical interests beyond the classical approach he had learnt under Baker. In until 1937 Todd was appointed a lecturer at Cambridge. Baker had retired from the Lowndean Chair in 1936 and the chair had been filled by Hodge. Todd and Hodge began to change the geometry at Cambridge to areas that were then of great interest internationally. However the outbreak of World War II meant that their reforms had to be delayed. Sadly, by the time peace had returned and progress could http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Todd.html (1 of 3) [2/16/2002 11:35:37 PM]

Todd

recommence, Todd was no longer a young reformer. In 1948 Todd had been elected a Fellow of the Royal Society of London and, given his long association with Trinity College, it seemed inconceivable that he would not be elected a Fellow of the College. Despite strong backing by mathematicians in the College, Todd failed to on a number of occasions to be elected. As Atiyah writes in [1]:Unfortunately, opposition in other quarters, based no doubt on the perception of Todd as socially gauche, was too strong. By 1958 disillusioned with his treatment by Trinity, Todd went to Downing College. He became a reader in 1960 and taught in Cambridge until he retired in 1973. His time at Downing became a happier period in his life as the following extract from the Downing College Magazine indicates:When first he arrived ... he was withdrawn and difficult to get to know; perhaps his essentially donnish and reclusive nature had not been helped in that regard by Trinity's not having embraced its own. But ... he grew into an active and genuinely loved person. To his pupils, we think he seemed a benevolent if mildly austere godfather, complete, or perhaps replete, with pipe and moustache. ... He needed friendship; he found it and gave it as a Fellow. ... Every well-equipped College should have a John Todd and we were privileged to have had him and to have enjoyed his company. Atiyah in [1] divides Todd's mathematical interests into invariant theory, group theory and canonical systems. He writes:In each case there is a major algebraic component but, as represented in Todd's work, the essential insight, interest and emphasis was on the geometric meaning behind the formula. Todd was a superb technician and manipulator of formulae but he also brought to bear a keen appreciation of the underlying geometry. Todd generalised the arithmetic genus and the invariants of the canonical system on an algebraic variety to a system of invariants of every codimension. This work is the origin of the Todd genus and Todd polynomials which were named after him. Todd polynomials, and certain other closely related polynomials, are much studied today and have played a major role in the study and classification of manifolds. In group theory Todd provided, certainly according to Coxeter, the main contribution to their joint work on the Todd-Coxeter procedure which they published in 1936. The procedure, today much used in computer implementations, enumerates the cosets of a subgroup of finite index in a finitely presented group. Coxeter explained to me [EFR] once how Todd used the back of old rolls of wallpaper on which to enumerate cosets which he could do at the rate of about 200 an hour. The Todd-Coxeter procedure became the most fundamental idea in the development of computational group theory yet the authors found difficulty in getting their paper published. Certainly several referees failed to recognise the importance of this new idea. I [EFR] do recall one occasion which saddened me greatly. It was at a British Mathematical Colloquium (Todd attended the Colloquium for each one of its first 25 years) where Todd talked on Mathieu groups, a subject to which he made considerable contributions. Some young research students in the audience heckled Todd, presumably they considered his mathematics old fashioned. After Todd retired, to the surprise of his colleagues, he married. He also seemed to give up his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Todd.html (2 of 3) [2/16/2002 11:35:37 PM]

Todd

mathematical work at that time but he had other interests to occupy his time such as stamp collecting and music. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to John Todd (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1948

London Maths Society President

1957 - 1969

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Todd.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Todd.html (3 of 3) [2/16/2002 11:35:37 PM]

Todhunter

Isaac Todhunter Born: 23 Nov 1820 in Rye, Sussex, England Died: 1 March 1884 in Cambridge, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Isaac Todhunter's father, George Todhunter, was a minister at a church in Rye. He died when Isaac was six years old and this left the family in severe financial difficulties. Isaac's mother moved to Hastings and opened a girls' school. Todhunter was sent to a school in Hastings but did not show any promise even being described as 'unusually backward'. However, after being sent to a new school set by J B Austin from London, Isaac made excellent progress. After leaving school he became an assistant school master at a school in Pecham but attended evening classes at London University where he was taught by De Morgan. He passed the entrance examinations and won a scholarship to study mathematics at London University, where, in addition to De Morgan, he was taught by Sylvester. While taking his degree course Todhunter also worked as a mathematics teacher at a large school in Wimbledon. He obtained a BA from University College London in 1842 and then an MA in 1844 with a prize for the top mark in the examination. He went to St John's College Cambridge, entering the College in 1844 and becoming senior wrangler and Smith's prizeman in 1848. He was elected a fellow of the college in 1849 and taught there for 15 years. He also undertook work as a private tutor and P G Tait and E J Routh were among his pupils. One of his pupils was Leslie Stephen, the father of the author Virginia Woolf, who studied mathematics at Cambridge with Todhunter as his tutor. Stephen gave a delightful description of Todhunter as follows:He lived in a perfect atmosphere of mathematics: his books, all ranged in the neatest order, and covered with uniform brown paper, were mathematical. His talk, to us at any rate, was one round mathematics. Even his chairs and tables strictly limited to the requirements of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Todhunter.html (1 of 4) [2/16/2002 11:35:39 PM]

Todhunter

pupils, and the pattern on his carpet, seemed to breath mathematics. By what mysterious process it was that he accumulated stores of miscellaneous information and knew all about the events of the time (for such I afterwards discovered to be the fact) I have never been able to guess. Probably he imbibed it through the pores of his skin. Still less can I imagine how it came to pass that he published a whole series of excellent mathematical works. He probably wrote them in momentary interstices of time between one pupil's entering his sanctum and another leaving it. In [6] there is a description of Todhunter's interests which suggests that he had more interests outside mathematics than his students realised:Todhunter's life was mainly that of a studious recluse. His sustained industry and methodical distribution of his time enabled him to acquire a wide acquaintance with general and foreign literature; and besides being a sound Latin and Greek scholar, he was familiar with French, German, Spanish, Italian, and also Russian, Hebrew, and Sanscrit. He was well versed in the history of philosophy, and on three occasions acted as examiner for the moral sciences tripos. His habits and tastes were singularly simple; and to a gentle kindly disposition he united a high sense of honour, a warm sympathy with all that was calculated to advance the cause of genuinely scientific study in the university, and considerable humour. Todhunter progressed from fellow to principal mathematical lecturer at St John's College where he resisted all attempts to reduce the central role of Euclid in mathematics courses. Sylvester had said in his British Association address of 1869:I should rejoice to see ... Euclid honourably shelved or buried deeper than ever did plummet sound, out of the schoolboy's reach.... Todhunter replied:Whatever may have produced the dislike to Euclid in the illustrious mathematician..., there is no ground for supposing that he would have been better pleased with the substitutes which are now offered and recommended in its place. Todhunter was always ready to respond to arguments and when Tait said:From the majority of the papers in our few mathematical journals, one would almost be led to fancy that British mathematicians have too much pride to use a simple method, while an unnecessarily complex one be had. Todhunter replied:I take down some of these volumes, and turning over the pages I find article after article by Profs Cayley, Salmon and Sylvester, not to mention many other highly distinguished names. The idea of amending the elaborate essays of these eminent mathematicians seems to me something like the audacity recorded in poetry with which a superhuman hero climbs to the summit of the Indian Olympus and overturns the thrones of Vishnu, Brahma and Siva. In 1864 Todhunter resigned his fellowship at St John's College, which he was forced to do as he wished to marry. His marriage to Louisa Anna Maria Davies took place on 13 August 1864. Todhunter had been elected a fellow of the Royal Society in 1862. He became a founding member of the London Mathematical Society in 1865 along with De Morgan.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Todhunter.html (2 of 4) [2/16/2002 11:35:39 PM]

Todhunter

In 1874 Todhunter was elected as an honorary fellow of St John's College but he was taken ill in 1880 and, from that time on, his health deteriorated. He became progressively more paralysed with an illness which led to his death. Todhunter is best known for his textbooks and his writing on the history of mathematics. Among his textbooks are Analytic Statics (1853), Plane Coordinate Geometry (1855), Examples of Analytic geometry in Three Dimensions (1858). He also wrote some more elementary texts, for example Algebra (1858), Trigonometry (1859), Theory of Equations (1861), Euclid (1862), Mechanics (1867) and Mensuration (1869). Among his books on the history of mathematics are A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace (1865, reprinted 1965) and History of the Mathematical Theories of Attraction (1873). The wide circulation of his books is described in [6]:No mathematical treatises on elementary subjects probably ever attained so wide a circulation; and, being adopted by the Indian government, they were translated into Urdu and other Oriental languages. Todhunter received many awards for his contributions to mathematics. In addition to the fellowship of the Royal Society he served on its Council in 1874, the same year in which he was awarded the Adams Prize for his work Researches on the calculus of variations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

A Quotation

A Poster of Isaac Todhunter

Mathematicians born in the same country

Honours awarded to Isaac Todhunter (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1862

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Todhunter.html (3 of 4) [2/16/2002 11:35:39 PM]

Todhunter

JOC/EFR June 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Todhunter.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Todhunter.html (4 of 4) [2/16/2002 11:35:39 PM]

Toeplitz

Otto Toeplitz Born: 1 Aug 1881 in Breslau, Germany (now Wroclaw, Poland) Died: 15 Feb 1940 in Jerusalem (under the British Mandate at the time)

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Otto Toeplitz came from a Jewish family which contained several teachers of mathematics. Both his father, Emil Toeplitz, and his grandfather, Julius Toeplitz, taught mathematics in a Gymnasium and they also both published mathematics papers. Otto was brought up in Breslau and he attended a Gymnasium in that city. His family background made it natural that he also should study mathematics. After completing his secondary education in Breslau, Toeplitz entered the university there to study mathematics. After graduating, he continued with his studies of algebraic geometry at the University of Breslau, being awarded his doctorate in 1905. In 1906 Toeplitz went to Göttingen. Dieudonné, in [9] describes some of the work that Toeplitz did during his seven years in Göttingen. When he arrived there Hilbert was completing his theory of integral equations. This greatly influenced Toeplitz who began to rework the classical theories of forms on n-dimensional spaces for infinite dimensional spaces. He wrote five papers directly related to Hilbert's spectral theory. Also during this period he published a paper on summation processes and discovered the basic ideas of what are now called the 'Toeplitz operators'. When Toeplitz arrived in Göttingen, Hellinger was a doctoral student there. The two quickly became friends and they would collaborate closely for many years. Hellinger left Göttingen in 1909, four years before Toeplitz, but before that the two were already producing joint papers. It was not until 1913 that Toeplitz was offered a teaching post as extraordinary professor at the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Toeplitz.html (1 of 3) [2/16/2002 11:35:41 PM]

Toeplitz

University of Kiel. He was promoted to ordinary professor at Kiel in 1920. A major joint project with Hellinger to write a major encyclopaedia article on integral equations, which they worked on for many years, was completed during this time and appeared in print in 1927. In 1928 Toeplitz accepted an offer of a chair at the University of Bonn. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. However, there was an exemption clause which, among others, exempted non-Aryans who had been in post before 1914. Toeplitz kept his lecturing post in Bonn in 1933. However Toeplitz was dismissed from his chair by the Nazis in 1935 despite the exemption clause in the Civil Service Law which was simply disregarded after decisions at the Nuremberg party congress in the autumn of 1935. Toeplitz had not instantly left Germany in 1933 when the Nazis came to power. As a proud Jew he started working for the Jewish community from that time. On a local level he united Jewish schoolchildren in Bonn and its vicinity bringing them to a Jewish school which he founded. On a country wide by level he selected gifted students for scholarships which allowed them to study abroad. His daughter Eva Wohl writes [13]:Of course it was very painful for him to realize that some of his colleagues at the university had fallen into the Nazis trap; still there remained a handful of very faithful friends with whom he worked till the day of his emigration. My father was very happy to have the opportunity to emigrate to Palestine (as it then was) in 1939 and to be able to help in the building up of Jerusalem University. He had great plans for modernizing the university but unfortunately he became very ill and died a year after his arrival. Toeplitz worked on infinite linear and quadratic forms. In the 1930's he developed a general theory of infinite dimensional spaces and criticised Banach's work as being too abstract. In a joint paper with Köthe in 1934, Toeplitz introduced, in the context of linear sequence spaces, some important new concepts and theorems. Köthe, in [12], describes the:... sometimes clumsy ways of our thinking ... which led to their discoveries. He also relates how the isomorphism problem for sequence spaces appeared in disguise as a new problem on nuclear (F)-spaces with basis. Toeplitz was also very interested in the history of mathematics. For example he wrote an excellent book on the history of the calculus The Calculus: A Genetic Approach (1963). It was originally published posthumously in German in 1949 edited by G Köthe. An historical topic which interested him deeply was the relation between Greek mathematics and Greek philosophy. He was a frequent visitor to the Frankfurt Mathematics Seminar in the 1920s and 30s, where his friend Hellinger worked from 1914, and there the history of mathematics played a large role. Toeplitz believed [1]:... that only a mathematician of stature is qualified to be a historian of mathematics. It was not only the history of mathematics which interested outside his area of research in mathematics. He wrote a popular book on mathematics in collaboration with H Rademacher. This work, The Enjoyment of Mathematics, has been reprinted many times over the years. Toeplitz was also greatly interested in school mathematics and devoted much time to it.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Toeplitz.html (2 of 3) [2/16/2002 11:35:41 PM]

Toeplitz

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles)

A Quotation

A Poster of Otto Toeplitz

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Toeplitz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Toeplitz.html (3 of 3) [2/16/2002 11:35:41 PM]

Torricelli

Evangelista Torricelli Born: 15 Oct 1608 in Rome, Italy Died: 25 Oct 1647 in Florence, Tuscany (now Italy)

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Evangelista Torricelli entered the Jesuit College of Faenza in 1624. He went to the Collegio Romano in Rome where he showed such talent that he was taught by Castelli at the University of Sapienza. Sapienza was the name of the building which the University of Rome occupied at this time and it gave its name to the University. As well as being taught by Castelli, Torricelli became his secretary and held this post from 1626 to 1632. During the next nine years he served as a secretary to Ciampoli and possibly a number of other professors. Torricelli served as Galileo's secretary from 1641 to 1642 and succeeded him as the court mathematician to Grand Duke Ferdinando II of Tuscany. He held this post until his death living in the ducal palace in Florence. Torricelli was the first man to create a sustained vacuum and to discover the principle of a barometer. In 1643 Torricelli proposed an experiment, later performed by his colleague Vincenzo Viviani, that demonstrated that atmospheric pressure determines the height to which a fluid will rise in a tube inverted over the same liquid. This concept led to the development of the barometer. Torricelli also proved that the flow of liquid through an opening is proportional to the square root of the height of the liquid a result now known as Torricelli's theorem. Torricelli found the length of the arc of the cycloid, the curve traced by a point on the circumference of a rotating circle.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Torricelli.html (1 of 3) [2/16/2002 11:35:43 PM]

Torricelli

He made early use of infinitesimal methods and determined the point in the plane of a triangle so that the sum of its distances from the vertices is a minimum (known as the isogonic centre). Torricelli also studied projectile motion. His only published work, Opera geometrica (1644) included important material on this topic. He was a skilled lens grinder, making telescopes and a type of microscope. In fact he made much money from his skill in lens grinding in the last period of his life in Florence. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (31 books/articles) A Poster of Evangelista Torricelli

Mathematicians born in the same country

Some pages from publications

An appendix on the cycloid in De parabole (1644).

Cross-references to History Topics

The rise of the calculus

Cross-references to Famous Curves

1. Cycloid 2. Equiangular spiral 3. Right Strophoid

Other references in MacTutor

1. Work on Minimal Paths 2. Chronology: 1625 to 1650

Honours awarded to Evangelista Torricelli (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Torricelli

Paris street names

Rue Torricelli (17th Arrondissement)

Other Web sites

1. The Galileo Project 2. Parma, Italy (In Italian) 3. The Catholic Encyclopedia 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Torricelli.html (2 of 3) [2/16/2002 11:35:43 PM]

Torricelli

Glossary index

Quotations index

Mathematicians of the day JOC/EFR December 1996

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Torricelli.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Torricelli.html (3 of 3) [2/16/2002 11:35:43 PM]

Trail

William Trail Born: 1746 in Scotland Died: 1831 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Trail studied at Aberdeen, then he studied under Simson at Glasgow. In 1766 he was appointed to the Chair of Mathematics at Marischal College, Aberdeen preferred to Playfair and Robert Hamilton. He resigned in 1779 on obtaining a preferment in the Irish Church. Trail published Elements of Algebra... and a biography of Simson. He was described as a man of great capacity... entirely extinguished... by the sinecure emoluments of the Irish Church. Article by: J J O'Connor and E F Robertson

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Trail.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Trail.html [2/16/2002 11:35:44 PM]

Tricomi

Francesco Giacomo Tricomi Born: 5 May 1897 in Italy Died: 21 Nov 1978 in Torino, Italy

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Francesco Tricomi studied first at the University of Bologna, then at the University of Naples. His first academic appointment was to lecture at Padova, then he moved to the University of Rome. The University of Florence offered Tricomi a chair of mathematics which he accepted but, after a few years, he moved again to take up the chair of mathematics at Torino. In fact before he moved to Torino he had already published a paper in 1923 which was to become very famous. In this paper he studied the theory of partial differential equations of mixed type, in particular the equation yuxx + uyy = 0, now known as the 'Tricomi equation'. The equation became important in describing an object moving at supersonic speed. Of course there were no supersonic aircraft in 1923 but the equation was to play a major role in later studies of supersonic flight. Tricomi's time in Torino was affected by World War II and his work was interrupted for a time. Then, a few years after the end of the war, he was involved in the Bateman project. In 1946 Bateman died and Erdélyi headed a team, which included Magnus and Tricomi, working at the California Institute of Technology to publish the vast range of material left by Bateman. The team produced 3 volumes of Higher Transcendental Functions and 2 volumes of Tables of Integral Transforms. In 1950 Tricomi returned from the United States to Torino to continue his remarkable research output. Tricomi's autobiography [1] list 300 papers, while a further 46 are listed in [6]. These papers cover a vast range of subjects including singular integrals, differential and integral equations, pseudodifferential operators, functional transforms, special functions, probability theory and its applications to number

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tricomi.html (1 of 3) [2/16/2002 11:35:46 PM]

Tricomi

theory. As well as having the 'Tricomi equation' named after him, there are also special functions called 'Tricomi functions'. Tricomi was an editor of Aequationes Mathematicae from the time the journal was founded until his death. The editors described him as:A forthright man, outspoken opponent of dictatorships of all colours, of sloppiness, of abstraction for abstraction's sake, and of the 'publish or perish' syndrome, his interests went far beyond mathematical research. Here we mention only his interest in the teaching of mathematics and in expository work, which led him to write several excellent textbooks, eventually translated from their original Italian to English, French, German and Russian. The fine books refered to in this quotation were certainly vastly superior to that suggested by the modest comment Tricomi wrote in the preface of one of them:Maybe I have not succeeded to make difficult things easy, but at least I have never made an easy subject difficult. Tricomi had a deep interest in problems concerning the history of mathematics and he published many important articles on this topic. In fact, references to articles on Riemann, Hadamard, Enriques and Fubini written by him appear in this Archive. His influence on mathematics goes well beyond the impressive results of his research. His writings have made an important contribution towards the present development of science. As the editors of Aequationes Mathematicae write:... the problems ... which he has solved and the theories which he has initiated and others have continued to work on, will keep his name alive. His passing away is a great loss to the international mathematical community. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) A Poster of Francesco Tricomi

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tricomi.html (2 of 3) [2/16/2002 11:35:46 PM]

Tricomi

JOC/EFR October 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tricomi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tricomi.html (3 of 3) [2/16/2002 11:35:46 PM]

Troughton

Edward Troughton Born: Oct 1753 in Corney, Cumberland, England Died: 12 June 1835 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edward Troughton's father was a farmer and Edward worked for his father when he was young. He became an apprentice mathematical instrument maker in 1770, working for his brother John Troughton in London. In 1779 he became a partner with his brother and they bought The Sign of the Orrey at 136 Fleet Street, London. Troughton soon established himself as the leading maker of instruments in England. Not only did he make great improvements in the design of existing instruments, but he also invented many new instruments. He began his instrument making career with instruments to aid navigation, for example, he designed the 'pillar' sextant, patented in 1788, the dip sector, the marine barometer and the reflecting circle built in 1796. Other instruments which he designed were for use in surveying. He designed the pyrometer, the mountain barometer and the large theodolites which were used in the American Coast Survey of 1815, and base-line measuring apparatus. In fact these instruments were later used in surveys of Ireland and of India. Troughton's most famous instruments were astronomical ones. He made the Groombridge Transit Circle in 1805 and a six foot Mural Transit Circle in 1810 which was erected at the Observatory in Greenwich in 1812. In 1816 he made a ten-foot Transit Circle. He never produced any telescopes, however, and the reason for this was that he suffered from colour-blindness which was a defect which ran in his family. After his brother John died, Edward ran the business alone until, in 1826, because of failing health due to old age, he took on a new partner William Simms. Simms continued Troughton's internationally known http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Troughton.html (1 of 2) [2/16/2002 11:35:48 PM]

Troughton

business after Troughton died in 1835 at his home in Fleet Street. One of Troughton's most important contributions was a method of dividing a circle. His paper on this, An account of the method of dividing astronomical and other instruments by ocular inspection in the Philosophical Transactions of the Royal Society in 1809 won him the Copley medal of the Royal Society. Troughton was elected a Fellow of the Royal Society in 1810. He was also elected a Fellow of the Royal Society of Edinburgh in 1822. In 1822 he published another work A comparison of the repeating circle of Borda with the altitude and azimuth circle in the Memoirs of the Royal Astronomical Society. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Honours awarded to Edward Troughton (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1810

Royal Society Copley Medal

Awarded 1809

Fellow of the Royal Society of Edinburgh

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Troughton.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Troughton.html (2 of 2) [2/16/2002 11:35:48 PM]

Tschirnhaus

Ehrenfried Walter von Tschirnhaus Born: 10 April 1651 in Kieslingswalde (near Görlitz), Germany Died: 11 Oct 1708 in Dresden, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ehrenfried Tschirnhaus had private lessons in mathematics while still at school. He entered the University of Leiden in 1668 and there he studied mathematics, philosophy and medicine. In 1672 war broke out between Holland and France and Tschirnhaus enlisted in a student force. He did not see active service. Tschirnhaus began a European tour in 1674, visiting England where he met Wallis in Oxford and Collins in London. He also visited Leiden and then Paris where he remained for a while after meeting Leibniz and Huygens. Tschirnhaus worked on the solution of equations and the study of curves. He discovered a transformation which, when applied to an equation of degree n, gave an equation of degree n with no term in xn-1 and xn-2. He also studied catacaustic curves in 1682, these being the envelope of light rays emitted from a point source after reflection from a given curve. There is a sinusoidal spiral named after him. For some time Tschirnhaus had one aim in life and that was to obtain a paid position at the Académie Royal des Sciences in Paris. He was elected a member in 1682 but no pension came with the appointment. Tschirnhaus was a scientist, and among other things, he experimented making porcelain from clay mixed with fusible rock in the 1680s. A factory at Meissen started production of his porcelain in 1710 and the first sales of any consequence of Tschirnhaus's porcelain took place at the Leipzig Fair in 1713. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tschirnhaus.html (1 of 2) [2/16/2002 11:35:50 PM]

Tschirnhaus

In 1706 Sweden invaded and Tschirnhaus was in some trouble. However after the war he was offered the position of Chancellor at the University of Halle but remained on his family estate of Kieslingswald. There was great competition from governments to obtain his porcelain techniques but Tschirnhaus kept them for himself and ended his life deeply in debt. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) A Poster of Ehrenfried Tschirnhaus

Mathematicians born in the same country

Cross-references to History Topics

1. Quadratic, cubic and quartic equations 2. Longitude and the Académie Royale

Cross-references to Famous Curves

Tschirnhaus's cubic curve

Other references in MacTutor

Chronology: 1675 to 1700

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tschirnhaus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tschirnhaus.html (2 of 2) [2/16/2002 11:35:50 PM]

Tsu

Tsu Ch'ung Chi Born: 430 in Fan-yang (now Hopeh), China Died: 501 in China

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Tsu was a Chinese mathematician and astronomer. He gave the rational approximation 355/113 to

which is correct to 6 decimal places. He also proved that

3.1415926 < < 3.1415927 a remarkable result on which it would be nice to have more details but Tsu Ch'ung Chi's book, written with his son, is lost. Tsu's astronomical achievements include the making of a new calendar in 463 which never came into use. Tsu also determined the precise time of the solstice by measuring the length of the Sun's shadow at noon on days near the solstice to reduce errors caused by the fact that it is very difficult to determine the exact time of the solstice. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) A Poster of Tsu Cross-references to History Topics

Mathematicians born in the same country 1. Pi through the ages 2. A chronology of pi

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tsu.html (1 of 2) [2/16/2002 11:35:52 PM]

Tsu

Other references in MacTutor

Chronology: 1AD to 500

Honours awarded to Tsu (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Tsu Chung-Chi

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tsu.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tsu.html (2 of 2) [2/16/2002 11:35:52 PM]

Tukey

John Wilder Tukey Born: 16 June 1915 in New Bedford, Massachusetts, USA Died: 26 July 2000 in New Brunswick, New Jersey, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Tukey's parents recognised that he had great potential while he was still only a child so the arranged for him to be educated at home rather than in school. This was possible since his parents were themselves teachers. His formal education began only when he entered Brown University in Providence, Rhode Island to study mathematics and chemistry. After being awarded his Master's Degree in chemistry from Brown University, Tukey went to Princeton University in 1937 to study for his doctorate in mathematics. He received his doctorate in 1939 for a dissertation Denumerability in topology which was published in 1940 as Convergence and uniformity in topology. He had already had three papers published before his doctorate was awarded and, in after graduating he was appointed as an instructor at Princeton. External events were to play a major role in the direction of Tukey's career since he joined the Fire Control Research office to contribute towards the war effort. The work here involved statistics and Tukey quickly found the work very much to his liking. There were other statisticians in Princeton, also contributing towards the war effort, in particular Wilks and Cochran and Tukey soon began exchanging ideas with these men. When World War II ended in 1945 Wilks, by this time well aware of Tukey's remarkable statistical talent, offered him a post in statistics within the mathematics department at Princeton. However one post was not enough to absorb his energy and, also in 1945, Tukey joined the AT&T Bell Laboratories. Tukey's first major contribution to statistics was with his introduction of modern techniques for the estimation of spectra of time series. E J Hannan, reviewing Tukey's papers on this topic writes:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tukey.html (1 of 3) [2/16/2002 11:35:54 PM]

Tukey

They show a remarkable uniformity of attitude characterised by a realistic recognition of the complexity of the situation, a consequent distrust of asymptotic theory, the use of standard statistical techniques as providing benchmarks rather than (say) precise confidence intervals, continual questioning of assumptions, emphasis on computational aspects, emphasis on ways of presenting the analysis, this presentation in ways familiar to the main users rather than in ways adopted in mathematical treatments, the early recognition of the superior qualities of digital devices for general purposes (as compared to analog devices) and a conspicuous fascination with new words and phrases, some of which have become established. There is, of course, also the introduction of new methods, some of which have proved to be important. These include methods for estimating spectra, spectra of higher moments, complex demodulation, methods for determining the magnitude and sign of initial impulses observed after transmission through a (more or less) fixed linear system and the Fourier analysis of the logarithm of a spectral estimate to discern echoes. In 1965, in a paper with J W Cooley published in the Mathematics of Computation, he introduced the important 'Fast Fourier Transform' algorithm. For many people this will be the work for which is best known. However, it is only a small part of a large number of areas with he made significant contributions. His work on the philosophy of statistics and of research is summarised by A D Gordon to include the following topics:... the usefulness and limitation of mathematical statistics; the importance of having methods of statistical analysis that are robust to violations of the assumptions underlying their use; the need to amass experience of the behaviour of specific methods of analysis in order to provide guidance on their use; the importance of allowing the possibility of data's influencing the choice of method by which they are analysed; the need for statisticians to reject the role of 'guardian of proven truth', and to resist attempts to provide once-for-all solutions and tidy over-unifications of the subject; the iterative nature of data analysis; implications of the increasing power, availability and cheapness of computing facilities; the training of statisticians. Tukey also made substantial contributions to the analysis of variance and the problem of making simultaneous inferences about a set of parameter values from a single experiment. Many of his papers are written with others and one of his co-authors, F Mosteller writes in [2]:John loves to work with others, and many have had the pleasure in participating in his genius. Variety and breadth mark his accomplishments. He works successfully on both large- and small- scale problems and on both practical and theoretical problems. ... He is always eager to respond to new questions, and he gives generously of his time and ideas. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles)

A Quotation

A Poster of John Tukey

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1960 to 1970

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tukey.html (2 of 3) [2/16/2002 11:35:54 PM]

Tukey

Other Web sites

1. John Tukey Memorial Site 2. Guardian, UK (Obituary) 3. New York Times (Obituary) 4. University of Minnesota

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tukey.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tukey.html (3 of 3) [2/16/2002 11:35:54 PM]

Tunstall

Cuthbert Tunstall Born: 1474 in Hackforth, Yorkshire, England Died: 18 Dec 1559 in Lambeth, London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Cuthbert Tunstall was educated at Balliol College, Oxford, then left because of the plague to go to King's Hall, Cambridge. When Henry VIII founded Trinity College 50 years later, King's Hall became part of Trinity. Between 1499 and 1505 Tunstall studied Canon and Roman law at Padua. On his return to England he entered the Church, obtaining the parish of Barmston in Yorkshire. After being rector of a number of different parishes, holding several at one time, he became Bishop of London in 1522, then becoming Bishop of Durham in 1530. A conservative during the Reformation he was imprisoned in 1552 and deprived of his bishopric. Reinstated the following year by Mary Tudor, Tunstall was deprived again in 1559 after refusing to swear the oath of supremacy under Elizabeth. Tunstall wrote the first printed work published in England devoted exclusively to mathematics. It was an arithmetic book De arte supputandi libri quattuor (1522) based on Pacioli's Suma. It makes no claim to originality. Tunstall also has the distinction of having the Grynaeus's edition, being the first printed edition, of Euclid's Elements in Greek dedicated to him. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tunstall.html (1 of 2) [2/16/2002 11:35:56 PM]

Tunstall

List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1500 to 1600

Other Web sites

1. The Catholic Encyclopedia 2. The Galileo Project 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Tunstall.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Tunstall.html (2 of 2) [2/16/2002 11:35:56 PM]

Turan

Paul Turán Born: 28 Aug 1910 in Budapest, Hungary Died: 26 Sept 1976 in Budapest, Hungary

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Paul Turán's Ph.D. was supervised by Féjér, but, being Jewish, he could not obtain a job (even as a school teacher). He spent 32 months in a Nazi labour camp from 1941 to 1944 in Hungary. From 1949 he was professor at Budapest University. Turán's first work was on probabilistic number theory and in 1938 he produced the sum-power method on which he wrote 50 papers during his life. He also worked on extremal graph theory (while in the labour camp) and statistical group theory with Erdös. The reference below, written by Paul Erdös, describes Turán's work on graph theory. A mathematician who served under Turán in Budapest described him as outstanding in analytic number theory. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turan.html (1 of 2) [2/16/2002 11:35:58 PM]

Turan

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Turan.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turan.html (2 of 2) [2/16/2002 11:35:58 PM]

Turing

Alan Mathison Turing Born: 23 June 1912 in London, England Died: 7 June 1954 in Wilmslow, Cheshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alan Turing was born at Paddington, London. His father, Julius Mathison Turing, was a British member of the Indian Civil Service and he was often abroad. Alan's mother, Ethel Sara Stoney, was the daughter of the chief engineer of the Madras railways and Alan's parents had met and married in India. When Alan was about one year old his mother rejoined her husband in India, leaving Alan in England with friends of the family. Alan was sent to school but did not seem to be obtaining any benefit so he was removed from the school after a few months. Next he was sent to Hazlehurst Preparatory School where he seemed to be an average to good pupil in most subjects but was greatly taken up with following his own ideas. He became interested in chess while at this school and he joined the debating society. He completed his Common Entrance Examination in 1926 and then went to Sherborne School. Now 1926 was the year of the general strike and when the strike was in progress Turing cycled 60 miles to the school from his home, not too demanding a task for Turing who later was to become a fine athlete of almost Olympic standard. He found it very difficult to fit into what was expected at this public school, yet his mother had been so determined that he should have a public school education. Many of the most original thinkers have found conventional schooling an almost incomprehensible process and this seems to have been the case for Turing. His genius drove him in his own directions rather than those required by his teachers. He was criticised for his handwriting, struggled at English, and even in mathematics he was too interested with his own ideas to produce solutions to problems using the methods taught by his teachers. Despite producing unconventional answers, Turing did win almost every possible mathematics prize while at Sherborne. In chemistry, a subject which had interested him from a very early age, he carried out

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turing.html (1 of 7) [2/16/2002 11:36:00 PM]

Turing

experiments following his own agenda which did not please his teacher. Turing's headmaster wrote (see for example [5]):If he is to stay at Public School, he must aim at becoming educated. If he is to be solely a Scientific Specialist, he is wasting his time at a Public School. This says far more about the school system that Turing was being subjected to than it does about Turing himself. However, Turing learnt deep mathematics while at school, although his teachers were probably not aware of the studies he was making on his own. He read Einstein's own papers on relativity and he also read about quantum mechanics in Eddington's The nature of the physical world. An event which was to greatly affect Turing throughout his life took place in 1928. He formed a close friendship with Christopher Morcom, a pupil in the year above him at school, and the two worked together on scientific ideas. Perhaps for the first time Turing was able to find someone with whom he could share his thoughts and ideas. However Morcom died in February 1930 and the experience was a shattering one to Turing. He had a premonition of Morcom's death at the very instant that he was taken ill and felt that this was something beyond what science could explain. He wrote later (see for example [5]):It is not difficult to explain these things away - but, I wonder! Despite the difficult school years, Turing entered King's College, Cambridge in 1931 to study mathematics. This was not achieved without difficulty. Turing sat the scholarship examinations in 1929 and won an exhibition, but not a scholarship. Not satisfied with this performance, he took the examinations again in the following year, this time winning a scholarship. In many ways Cambridge was a much easier place for unconventional people like Turing than school had been. He was now much more able to explore his own ideas and he read Russell's Introduction to mathematical philosophy in 1933. At about the same time he read von Neumann's 1932 text on quantum mechanics, a subject he returned to a number of times throughout his life. The year 1933 saw the beginnings of Turing's interest in mathematical logic. He read a paper to the Moral Science Club at Cambridge in December of that year of which the following minute was recorded (see for example [5]):A M Turing read a paper on "Mathematics and logic" . He suggested that a purely logistic view of mathematics was inadequate; and that mathematical propositions possessed a variety of interpretations of which the logistic was merely one. Of course 1933 was also the year of Hitler's rise in Germany and of an anti-war movement in Britain. Turing joined the anti-war movement but he did not drift towards Marxism, nor pacifism, as happened to many. Turing graduated in 1934 then, in the spring of 1935, he attended Max Newman's advanced course on the foundations of mathematics. This course studied Gödel's incompleteness results and Hilbert's question on decidability. In one sense 'decidability' was a simple question, namely given a mathematical proposition could one find an algorithm which would decide if the proposition was true of false. For many propositions it was easy to find such an algorithm. The real difficulty arose in proving that for certain propositions no such algorithm existed. When given an algorithm to solve a problem it was clear that it was indeed an algorithm, yet there was no definition of an algorithm which was rigorous enough to allow one to prove that none existed. Turing began to work on these ideas. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turing.html (2 of 7) [2/16/2002 11:36:00 PM]

Turing

Turing was Elected a fellow of King's College, Cambridge in 1935 for a dissertation On the Gaussian error function which proved fundamental results on probability theory, namely the central limit theorem. Although the central limit theorem had recently been discovered, Turing was not aware of this and discovered it independently. In 1936 Turing was a Smith's Prizeman. Turing's achievements at Cambridge had been on account of his work in probability theory. However, he had been working on the decidability questions since attending Newman's course. In 1936 he published On Computable Numbers, with an application to the Entscheidungsproblem. It is in this paper that Turing introduced an abstract machine, now called a Turing machine, which moved from one state to another using a precise finite set of rules (given by a finite table) and depending on a single symbol it read from a tape. The Turing machine could write a symbol on the tape, or delete a symbol from the tape. Turing wrote [12]:Some of the symbols written down will form the sequences of figures which is the decimal of the real number which is being computed. The others are just rough notes to "assist the memory". It will only be these rough notes which will be liable to erasure. He defined a computable number as real number whose decimal expansion could be produced by a Turing machine starting with a blank tape. He showed that was computable, but since only countably many real numbers are computable, most real numbers are not computable. He then described a number which is not computable and remarks that this seems to be a paradox since he appears to have described in finite terms, a number which cannot be described in finite terms. However, Turing understood the source of the apparent paradox. It is impossible to decide (using another Turing machine) whether a Turing machine with a given table of instructions will output an infinite sequence of numbers. Although this paper contains ideas which have proved of fundamental importance to mathematics and to computer science ever since it appeared, publishing it in the Proceedings of the London Mathematical Society did not prove easy. The reason was that Alonzo Church published An unsolvable problem in elementary number theory in the American Journal of Mathematics in 1936 which also proves that there is no decision procedure for arithmetic. Turing's approach is very different from that of Church but Newman had to argue the case for publication of Turing's paper before the London Mathematical Society would publish it. Turing's revised paper contains a reference to Church's results and the paper, first completed in April 1936, was revised in this way in August 1936 and it appeared in print in 1937. A good feature of the resulting discussions with Church was that Turing became a graduate student at Princeton University in 1936. At Princeton, Turing undertook research under Church's supervision and he returned to England in 1938, having been back in England for the summer vacation in 1937 when he first met Wittgenstein. The major publication which came out of his work at Princeton was Systems of Logic Based on Ordinals which was published in 1939. Newman writes in [12]:This paper is full of interesting suggestions and ideas. ... [it] throws much light on Turing's views on the place of intuition in mathematical proof. Before this paper appeared, Turing published two other papers on rather more conventional mathematical topics. One of these papers discussed methods of approximating Lie groups by finite groups. The other paper proves results on extensions of groups, which were first proved by Reinhold Baer, giving a simpler

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turing.html (3 of 7) [2/16/2002 11:36:00 PM]

Turing

and more unified approach. Perhaps the most remarkable feature of Turing's work on Turing machines was that he was describing a modern computer before technology had reached the point where construction was a realistic proposition. He had proved in his 1936 paper that a universal Turing machine existed [12]:... which can be made to do the work of any special-purpose machine, that is to say to carry out any piece of computing, if a tape bearing suitable "instructions" is inserted into it. Although to Turing a "computer" was a person who carried out a computation, we must see in his description of a universal Turing machine what we today think of as a computer with the tape as the program. While at Princeton Turing had played with the idea of construction a computer. Once back at Cambridge in 1938 he starting to build an analogue mechanical device to investigate the Riemann hypothesis, which many consider today the biggest unsolved problem in mathematics. However, his work would soon take on a new aspect for he was contacted, soon after his return, by the Government Code and Cypher School who asked him to help them in their work on breaking the German Enigma codes. When war was declared in 1939 Turing immediately moved to work full-time at the Government Code and Cypher School at Bletchley Park. Although the work carried out at Bletchley Park was covered by the Official Secrets Act, much has recently become public knowledge. Turing's brilliant ideas in solving codes, and developing computers to assist break them, may have saved more lives of military personnel in the course of the war than any other. It was also a happy time for him [12]:... perhaps the happiest of his life, with full scope for his inventiveness, a mild routine to shape the day, and a congenial set of fellow-workers. Together with another mathematician W G Welchman, Turing developed the Bombe, a machine based on earlier work by Polish mathematicians, which from late 1940 was decoding all messages sent by the Enigma machines of the Luftwaffe. The Enigma machines of the German navy were much harder to break but this was the type of challenge which Turing enjoyed. By the middle of 1941 Turing's statistical approach, together with captured information, had led to the German navy signals being decoded at Bletchley. From November 1942 until March 1943 Turing was in the United States liaising over decoding issues and also on a speech secrecy system. Changes in the way the Germans encoded their messages had meant that Bletchley lost the ability to decode the messages. Turing was not directly involved with the successful breaking of these more complex codes, but his ideas proved of the greatest importance in this work. Turing was awarded the O.B.E. in 1945 for his vital contribution to the war effort. At the end of the war Turing was invited by the National Physical Laboratory in London to design a computer. His report proposing the Automatic Computing Engine (ACE) was submitted in March 1946. Turing's design was at that point an original detailed design and prospectus for a computer in the modern sense. The size of storage he planned for the ACE was regarded by most who considered the report as hopelessly over-ambitious and there were delays in the project being approved. Turing returned to Cambridge for the academic year 1947-48 where his interests ranged over many topics far removed from computers or mathematics, in particular he studied neurology and physiology. He did not forget about computers during this period, however, and he wrote code for programming computers. He had interests outside the academic world too, having taken up athletics seriously after the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turing.html (4 of 7) [2/16/2002 11:36:00 PM]

Turing

end of the war. He was a member of Walton Athletic Club winning their 3 mile and 10 mile championship in record time. He ran in the A.A.A. Marathon in 1947 and was placed fifth. By 1948 Newman was the professor of mathematics at the University of Manchester and offered Turing a readership there. Turing resigned from the National Physical Laboratory to take up the post in Manchester. Newman writes in [12] that in Manchester:... work was beginning on the construction of a computing machine by F C Williams and T Kilburn. The expectation was that Turing would lead the mathematical side of the work, and for a few years he continued to work, first on the design of the subroutines out of which the larger programs for such a machine are built, and then, as this kind of work became standardised, on more general problems of numerical analysis. In 1950 Turing published Computing machinery and intelligence in Mind. It is another remarkable work from his brilliantly inventive mind which seemed to foresee the questions which would arise as computers developed. He studied problems which today lie at the heart of artificial intelligence. It was in this 1950 paper that he proposed the Turing Test which is still today the test people apply in attempting to answer whether a computer can be intelligent. Turing did not forget about questions of decidability which had been the starting point for his brilliant mathematical publications. One of the main problems in the theory of group presentations was the question: given any word in a finitely presented group is there an algorithm to decide if the word is equal to the identity. Post had proved that for semigroups no such algorithm exists. Turing though at first that he had proved the same result for groups but, just before giving a seminar on his proof, he discovered an error. He was able to rescue from his faulty proof the fact that there was a cancellative semigroup with insoluble word problem and he published this result in 1950. Boone used the ideas from this paper by Turing to prove the existence of a group with insoluble word problem in 1957. Turing was elected a Fellow of the Royal Society of London in 1951, mainly for his work on Turing machines in 1936. By 1951 he was working on the application of mathematical theory to biological forms. In 1952 he published the first part of his theoretical study of morphogenesis, the development of pattern and form in living organisms. Turing was arrested for violation of British homosexuality statutes in 1952 when he reported to the police details of a homosexual affair. He had gone to the police because he had been threatened with blackmail. He was tried as a homosexual on 31 March 1952, offering no defence other than that he saw no wrong in his actions. Found guilty he was given the alternatives of prison or oestrogen injections for a year. He accepted the latter and returned to a wide range of academic pursuits. Not only did he press forward with further study of morphogenesis, but he also worked on new ideas in quantum theory, on the representation of elementary particles by spinors, and on relativity theory. Although he was completely open about his sexuality, he had a further unhappiness which he was forbidden to talk about due to the Official Secrets Act. The decoding operation at Bletchley Park became the basis for the new decoding and intelligence work at GCHQ. With the cold war this became an important operation and Turing continued to work for GCHQ, although his Manchester colleagues were totally unaware of this. After his conviction, his security clearance was withdrawn. Worse than that, security officers were now extremely worried that someone with complete knowledge of the work going on at GCHQ was now labelled a security risk. He had many http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turing.html (5 of 7) [2/16/2002 11:36:00 PM]

Turing

foreign colleagues, as any academic would, but the police began to investigate his foreign visitors. A holiday which Turing took in Greece in 1953 caused consternation among the security officers. Turing died of potassium cyanide poisoning while conducting electrolysis experiments. The cyanide was found on a half eaten apple beside him. An inquest concluded that it was self-administered but his mother always maintained that it was an accident. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles)

Some Quotations (4)

A Poster of Alan Turing

Mathematicians born in the same country

Other references in MacTutor

1. Turing as a runner 2. Chronology: 1930 to 1940

Honours awarded to Alan Turing (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1951

Other Web sites

1. Turing archive for the history of computing 2. The Turing home page 3. Virginia Tech 4. Simon Fraser University 5. New York Times (His article on computing machines and intelligence) 6. Stanford Encyclopedia of Philosophy (The Church-Turing thesis) 7. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turing.html (6 of 7) [2/16/2002 11:36:00 PM]

Turing

JOC/EFR July 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Turing.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turing.html (7 of 7) [2/16/2002 11:36:00 PM]

Turnbull

Herbert Westren Turnbull Born: 31 Aug 1885 in Tettenhall, Wolverhampton, England Died: 4 May 1961 in Grasmere, Westmoreland, England

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Herbert Turnbull's father was an H M Inspector of Schools. Certainly Turnbull's father was interested in mathematics and transmitted his enthusiasm for the subject to his young son Herbert. In the Preface to The Mathematical Discoveries of Newton (1945) Turnbull thanks his father for a quotation which:... is taken from a lecture on Newton which he gave to a group of Nottinghamshire miners seventy years ago. This means that Turnbull's father gave the lecture on Newton in the 1870s. Turnbull also dates his own interest in the history of mathematics to his childhood, in particular writing in the same Preface:My own interest in Newton dates from childhood: his mathematical prowess was as well known at the family breakfast table as the batsmanship of W G Grace. Herbert was educated at Sheffield Grammar School. He went up to Cambridge where he had a career of great distinction being placed Second Wrangler and, in 1909, he was the winner of the Smith Prize. After graduating, Turnbull taught at St Catherine's College, Cambridge (1909), and then at the University of Liverpool (1910). He writes in [5] in a letter offering a first appointment at St Andrews to Ledermann:I naturally look back on my start at lecturing in thinking of your present position. My first post outside Cambridge (where I gave one course a term) was at Liverpool and involved at least three sets of new lectures to be prepared, and in one term four sets. Turnbull married Ella Drummond Williamson in 1911. She was the daughter of Canon H D Williamson. Herbert and Ella Turnbull had one child, a son. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turnbull.html (1 of 5) [2/16/2002 11:36:03 PM]

Turnbull

After his year as a lecturer at the University of Liverpool Turnbull taught at the Hong Kong University, becoming master at St Stephen's College in Hong Kong in 1911, and warden of the University Hostel two years later. The University Hostel was run by the Church Mission Society as part of Hong Kong University and Turnbull had duties as a mathematics lecturer at the University in addition to being warden of the hostel. He held his posts in Hong Kong until 1915. On his return to England, Turnbull worked as a school teacher for three years in the leading independent school at Repton in the county of Derbyshire in the north of England. He taught at the famous boys' school of Repton which had a long history, being founded in 1556 in buildings which included a restored Augustinian priory established in 1172. Following this Turnbull became an a School Inspector, entering at this stage the same profession as that of his father. From 1919 to 1926 he was a fellow at St John's College, Oxford holding the Fereday Fellowship during this period. As an undergraduate at Cambridge Turnbull had become fascinated by the topic of invariant theory. He published two papers on classical algebra problems in 1910 and 1911, then a further paper in 1916 after his return to England. In 1919 he published two more papers but his involvement in mathematical research had been necessarily limited by the jobs that he had held over these years. Turnbull was appointed Regius Professor of Mathematics in the United College of St Salvator and St Leonard at the University of St Andrews in 1921. He only had five papers published before being appointed to this chair but his quality was evident to all concerned. He held the Regius Chair until he retired in 1950 when he was succeeded as Regius Professor of Mathematics by Copson. In [3] Ledermann describes an interesting incident which occurred soon after he arrived in St Andrews in 1934. Ledermann writes [3]:[Turnbull] was very kind and most patient when our communications were hindered by my poor command of English. Before he was appointed to the Regius Chair of Mathematics at St Andrews Professor Turnbull had been a missionary in China. He had picked up some of the local language there. One day he said to me: "Walter, I see you have some difficulty expressing yourself. Would it help you if I spoke to you in Chinese?" I thanked him for his offer, but asked him, with apologies, to persevere with English. Turnbull was interested in algebra, particularly invariant theory building on work of Gordan and Clebsch. As Ledermann writes in [2]:It was unfortunate for him that already in the 1920's the fashion in algebraical research had drastically changed, and his original work on invariants did not receive the recognition which it would have found two decades earlier. As to Turnbull's approach to mathematics it was [2]:... concrete and formal in the sense that he sought to solve problems by an effective formalism rather than by a conceptual analysis of the underlying structures. His topics were algebraical, but he was fond of presenting them against a geometrical background. Turnbull was also interested in the history of mathematics. He explains in the Preface to his little book The Great Mathematicians his attitude towards historical study in mathematics. We quote it here for two reasons. Firstly it tells us something of Turnbull's character and attitude towards mathematics. Secondly we relate it because this web archive in run from a server which we have named after Turnbull and in this archive of the history of mathematics we have tried to follow Turnbull's thinking on the subject:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turnbull.html (2 of 5) [2/16/2002 11:36:03 PM]

Turnbull

The usefulness of mathematics in furthering the sciences is commonly acknowledged: but outside the ranks of the experts there is little inquiry into its nature and purpose as a deliberate human activity. Doubtless this is due to the inevitable drawback that mathematical study is saturated with technicalities from beginning to end. Fully conscious of the difficulties in the undertaking, I have written this little book in the hope that it will help to reveal something of the spirit of mathematics, without unduly burdening the reader with intricate symbolism. ... I have tried to show how a mathematician thinks, how his imagination, as well as his reason, leads him to new aspects of the truth. Turnbull published his own historical research into mathematics in the James Gregory Tercentenary Volume (1939). In our library in St Andrews the copy of this volume is inscribed in Turnbull's own hand:D. E. Rutherford with very best wishes from H. W. Turnbull Oct. 1939 In the Preface to this work Turnbull describes how he discovered letters from Collins to James Gregory:... in a bundle of remarkable original documents in the Library of the University of St Andrews ... I first examined the documents at St Andrews in 1932, when it was discovered that Gregory, the original recipient of the letters, had used their blank spaces for recording his own mathematical thoughts. As a result of careful scrutiny it has been established that Gregory made several remarkable and unsuspected discoveries, particularly in the calculus and the theory of numbers, which he never published. He was, for example, employing Taylor and Maclaurin expansions more than forty years in advance of anyone else. Turnbull's major beautifully written works include The Theory of Determinants, Matrices, and Invariants (1928), The Great Mathematicians (1929), Theory of Equations (1939), The Mathematical Discoveries of Newton (1945), and An Introduction to the Theory of Canonical Matrices (1945), which was jointly written with Aitken. The Mathematical Discoveries of Newton arose from two lectures which Turnbull gave on Newton. The first was given at a meeting of the Edinburgh Mathematical Society in December 1942 to commemorate the 300th anniversary of Newton's birth. The second was given at a meeting of Edinburgh University's Mathematical and Physical Society. Turnbull writes:Without going into too much detail I have tried to explain - as far as the work of geniuses can be explained - what led Newton to these discoveries. The positive interest afforded by contemplating the wonderful range covered by his early mathematical work provides an adequate theme for this short study, and makes unnecessary an attempt to deal with the controversies which clouded his later years. It is so typical of Turnbull that he chose to emphasise the extraordinarily positive aspects of Newton's life and work. After he retired in 1950 Turnbull, at the request of the Royal Society, began to work on the Correspondence of Isaac Newton. Two volumes of this important work were published before his death. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turnbull.html (3 of 5) [2/16/2002 11:36:03 PM]

Turnbull

Turnbull received many honours for his work, the most major being his election as a Fellow of the Royal Society in 1932. He was also elected to the Royal Society of Edinburgh, receiving their Keith Medal and Gunning Victoria Jubilee Prize. Outside mathematics Turnbull had several major interests. One of these was music, where he was an excellent pianist, playing in a chamber orchestra. Another of his loves was mountaineering and as a member of the Alpine Club he made many ascents without the help of a guide. Ledermann writes in [2]:Nearer his home opportunities for practice were provided on the cliffs of St Andrews bay. He discovered fourteen ways up "The Maiden Rock". The mastery of the "Rock and Spindle" was not exactly part of the mathematical syllabus, but many a student experienced on this striking formation his first thrill of rock climbing under the guidance of his professor of mathematics. "The Maiden Rock" and the "Rock and Spindle" are volcanic stacks which have survived as the sea has washed away the surrounding material. They are close to the cliffs on the south side of St Andrews bay. Ledermann [4] writes of Turnbull's:... kindness and caring - he was a lovely man. In [2] he writes how Professor and Mrs Turnbull:... extended hospitality to countless students and friends. At these gatherings Mrs Turnbull was a gracious and lively hostess. The inevitable shyness of the younger guests was overcome by drawing room games, but the highlight of the evening, for those who could appreciate it, was the performance on two pianos by Professor and Mrs Turnbull. Their playing, highly musical and exquisitely blended, was a beautiful expression of a harmonious partnership. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles)

Some Quotations (2)

A Poster of Herbert Turnbull

Mathematicians born in the same country

Cross-references to History Topics

Matrices and determinants

Honours awarded to Herbert Turnbull (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1932

Fellow of the Royal Society of Edinburgh Honorary Fellow of the Edinburgh Maths Society

Elected 1954

Other Web sites

Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turnbull.html (4 of 5) [2/16/2002 11:36:03 PM]

Turnbull

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Turnbull.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turnbull.html (5 of 5) [2/16/2002 11:36:03 PM]

Turner

Peter Turner Born: 1586 in London, England Died: 1652 in London, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Peter Turner was educated at Oxford first at St Mary Hall, then at Christ Church. He graduated with a B.A. in 1605. Turner became a fellow of Merton College, Oxford in 1607, holding the fellowship until 1648. In 1620 he succeeded Briggs first to the chair of geometry at Gresham College in London, then, in 1630, to the Savilian chair of Geometry at Oxford. Imprisoned after the battle of Edgehill since he was a royalist, Turner lost both his fellowship and his chair in 1648. He had strongly supported the royalist cause during the Civil War and served in a military capacity from 1641. Turner's quality as a mathematician cannot be judged as he left no mathematical publications but we know he wrote very stylish Latin! One might reasonably ask how someone who left behind no evidence of mathematical ability came to hold two of the major mathematical chairs in England. It appears mainly due to William Laud, archbishop of Canterbury (1633-45) and religious adviser to King Charles I. Laud became president of St John's, Oxford in 1611 and then chancellor in 1629. Laud set up a committee which produced the Laudian statutes, new endowments and new buildings in Oxford. Turner was a highly active member of this committee and so came to Laud's notice. Through Laud, Turner gained the appointments to the two chairs. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Peter Turner (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turner.html (1 of 2) [2/16/2002 11:36:04 PM]

Turner

Savilian Professor of Geometry

1631

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Turner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Turner.html (2 of 2) [2/16/2002 11:36:04 PM]

Uhlenbeck

George Eugene Uhlenbeck Born: 6 Dec 1900 in Batavia, Java (now Jakarta, Indonesia) Died: 31 Oct 1988 in Boulder, Colorado, USA

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Uhlenbeck's ancestors had come first from Germany but then later generations were Dutch and served in the Dutch colonies. His father, Eugenius Marius Uhlenbeck was born in Java in what was then called the Dutch East Indies but is now Indonesia. In 1893 Eugenius Uhlenbeck, who served in the Dutch East Indian Army, married Anne Marie Beeger who was the daughter of a Dutch major general. Two of their children died of malaria at a very young age but four survived. Of the surviving children George was the second child, with an older sister Annie and two younger brothers Willem Jan and Eugenius Marius. When George was six years old the family moved permanently to Holland, setting up home in The Hague. There George attended elementary school followed by high school, or the higher burger school as it was then called. Uhlenbeck said [3]:I was a very dutiful student, very dutiful. I always worked very regularly and I was always very good in class. I was certainly not clear until the last years in high school what I was going to do. However, as with so many people, he was influenced by an excellent school teacher. In his last couple of years at school his physics teacher strongly encouraged him, gave him texts on the differential and integral calculus and suggested that he read undergraduate texts on mathematics and physics. One of the books which made him enthusiastic about physics was Lorentz's Lectures on Physics which he read in the Royal Library in The Hague. Although Uhlenbeck performed extremely well in his final school examinations in July 1918, he was not allowed to enter a university since his studies had not included Greek and Latin. This was not his fault, simply that the school he attended did not offer these topics. He had no choice but to enter the Institute of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Uhlenbeck.html (1 of 5) [2/16/2002 11:36:06 PM]

Uhlenbeck

Technology in Delft. He chose to study chemical engineering there but shortly after, when the rules were changed by the Dutch government so that Greek and Latin were no longer required for university entrance, he left the Institute of Technology. Having spent just one semester studying chemical engineering he began his university study at the University of Leiden. Now Uhlenbeck was doubly glad to be able to make the move, firstly because he had always wanted to study physics and mathematics at university, but also because he felt he did not have the dexterity to undertake a practical subject. At Leiden physics was taught by Ehrenfest, H K Onnes and J P Kuenen while Lorentz, by that time retired from his chair, came in to lecture once a week. Uhlenbeck bought a copy of Boltzmann's Lectures on gas dynamics and he studied this in his own time, continuing the way of learning which he had adopted at school. At first he found it almost impossible to understand but after reading an encyclopaedia article by Ehrenfest it began to make sense. Of the lectures he attended those on the foundations of analysis he found greatly to his liking, finding that the rigour of analysis was particularly pleasing to him. Later in his career Ehrenfest would encourage him to present his mathematics with less rigour. Uhlenbeck's parents did not find it easy to provide the necessary financial support for him at university. To keep down the expense, he travelled every day from The Hague to Leiden, a long and tiring journey. In his second year as an undergraduate Uhlenbeck studied Maxwell's theory which he wrote out in great detail. This impressive piece of work led to him being awarded a scholarship for his third year of study which took the financial burden away from his parents. As he had done at school, Uhlenbeck worked exceptionally hard and performed well in his final examinations in December 1920. After graduating, he began postgraduate studies at Leiden under the supervision of Ehrenfest. In order to support himself financially, however, he had to work. He took a part-time job as a teacher in a girls school in Leiden which gave him sufficient money to allow him to rent a room. At last the daily travelling was over. He settled down to the work for his Master's degree which involved attending lecture courses. Pais writes [5]:Ehrenfest's graduate lectures consisted of a two-year course: Maxwell theory, ending with the theory of electrons and some relativity, one year; and statistical mechanics, ending with atomic structure and quantum theory the other. Uhlenbeck attended these lectures and took additional instruction in mathematics. In September 1922, before obtaining his Master's degree, Uhlenbeck took the opportunity to spend time working in Italy when he accepted an appointment as a private tutor to the son of the Dutch ambassador in Rome. He held this job until June 1925, but he returned to Holland each summer. He completed the work for his Master's degree after one more year and then he was examined and received the degree in September 1923. Shortly after the award of the degree Uhlenbeck returned to Italy and, on Ehrenfest's advice, made contact with Enrico Fermi. This proved to be an important friendship for both men. However despite this contact with scientists, Uhlenbeck's academic interests turned towards history. When his appointment in Italy ended and he had returned to Holland he was close to making a decision to become an historian. Ehrenfest, keen to keep him a scientist, suggested that he work for him for a while and gave Uhlenbeck a two year appointment as his assistant. He succeed Struik who had been awarded a Rockefeller Fellowship. Very soon after taking up the appointment, working with a graduate student Samuel Goudsmit, Uhlenbeck made his most important scientific discovery when he discovered

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Uhlenbeck.html (2 of 5) [2/16/2002 11:36:06 PM]

Uhlenbeck

electron spin. In January 1925 Pauli had proposed that the electron should be given an additional fourth quantum number which was a half integer. This was one of the clues which led Uhlenbeck to arrive at the idea of electron spin. He wrote (see for example []):... it occurred to me that , since (I had learned) each quantum number corresponds to a degree of freedom of the electron, Pauli's fourth quantum number must mean that the electron had an additional degree of freedom -- in other words the electron must be rotating. The concept immediately excited Niels Bohr, Pauli, Einstein, Heisenberg and others interested in quantum theory. Uhlenbeck's work progressed well and particularly important was Oskar Klein's appointment to Leiden. The two quickly became friends and exchanged ideas, particularly on Klein's ideas about five dimensional relativity. Uhlenbeck's doctoral work was written in Copenhagen where he spent two months devoted to intensive writing of his dissertation. It was of fundamental importance in quantum mechanics, systematising statistical notions and expanding on the electron spin ideas which had announced two years earlier. Immediately after being examined on his thesis, Uhlenbeck was appointed to Michigan. He married Else Ophorst in August 1927 then, in the following month, he arrived in Ann Arbor to take up the appointment. He returned to the Netherlands to take up a chair in Utrecht in 1935. The chair had become vacant due to the holder, Kramers, moving to Leiden to take up the chair left vacant there after Ehrenfest committed suicide. He took leave from Utrecht in 1938 to spend a year as visiting professor at Columbia University in New York but after a short time back in Holland he left again for the United States in August 1939 shortly before the outbreak of World War II. From 1943 until 1945 Uhlenbeck worked at MIT as a member of the team working on the development of radar. Then, after the war ended, he returned to Ann Arbor where, after spending 1948-49 at Princeton, he was named Henry Cahart professor at Michigan in 1954. Four years later, in 1960, he moved to the Rockefeller Institute in New York where he remained on the staff until he retired in 1971. As well as fundamental work on quantum mechanics, Uhlenbeck worked on atomic structure and the kinetic theory of matter. He extended Boltzmann's equation to dense gasses and wrote two important papers on Brownian motion. The main topic on which he worked throughout his career was statistical physics. The aim of this topic was to understand the relationship between physics at atomic level and that at macroscopic level. In 1955 he explained the attraction of the area to him:To me the great charms of statistical physics lies in its connection with parts of mathematics with which otherwise one rarely comes in contact. Uhlenbeck had a very significant influence on statistical mechanics and brought an area which was very varied and disjointed into some sort of structured whole. He was always very keen on clarity and a logical approach to all the problems he studied. Cohen, a student of Uhlenbeck's, tells us much about Uhlenbeck's character in [2]. ... [Uhlenbeck] often admonished me that rather than trying to be original, it was much more important to be clear and correct and to summarise critically the present status of a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Uhlenbeck.html (3 of 5) [2/16/2002 11:36:06 PM]

Uhlenbeck

field in the Ehrenfest tradition. He wisely observed that what is often of lasting value is not the first original contribution to a problem, but rather the final clearly and critically written survey. That is certainly what he did in this Brownian motion paper! As to the style of Uhlenbeck's papers, Cohen writes:Uhlenbeck's papers are all relatively short and stand out by their conciseness, precision, and clarity, finely honed to a deeper understanding of a basic problem in statistical physics. They do not contain long formal derivations and are almost all geared to concrete problems. ... they were of a classic nobility, mathematical purity and clarity ... He felt that something really original one did only once - like the electron-spin--the rest of one's time one spent on clarifying the basics. Cohen also comments on the high quality of Uhlenbeck's teaching in []:He was an inspiring teacher. With superbly organised and extremely clear lectures, he laid bare for everyone to see the beautiful structure of statistical mechanics, based on the principles of the founding fathers, Maxwell, Boltzmann, and Gibbs. Thus he transmitted to a younger generation what he conceived to be the essence of the past and the way to the future. In doing so, he educated several generations of physicists in statistical mechanics in a style rare in this century. Uhlenbeck received many honours for his work. These include his appointment at Lorentz professor in 1955, and serving as President of the American Physical Society in 1959. He was awarded the Planck Medal in 1965, the Lorentz Medal of the Royal Dutch Academy of Sciences in 1970, the National Medal of Science from the United States in 1977, and the Woolf Prize in 1979. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Honours awarded to George Uhlenbeck (Click a link below for the full list of mathematicians honoured in this way) AMS Gibbs Lecturer

1950

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Uhlenbeck.html (4 of 5) [2/16/2002 11:36:06 PM]

Uhlenbeck

Glossary index

Quotations index

Mathematicians of the day JOC/EFR May 2001

Poster index Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Uhlenbeck.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Uhlenbeck.html (5 of 5) [2/16/2002 11:36:06 PM]

Uhlenbeck_Karen

Karen Keskulla Uhlenbeck Born: 24 Aug 1942 in Cleveland, Ohio, USA

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karen Uhlenbeck's father was an engineer and her mother was an artist. She grew up in the country, the eldest of four children. Many mathematicians know from early age that mathematics will be their life but this was not so with Karen Uhlenbeck. As a child she was interested in books and this led her to an interest in science. She writes:As a child I read a lot, and I read everything. I'd go to the library and then stay up all night reading. I used to read under the desk in school. ... we lived in the country so there wasn't a whole lot to do. I was particularly interested in reading about science. I was about twelve years old when my father began bringing home Fred Hoyle's books on astrophysics. I found them very interesting. I also remember a little paperback book called "One, Two, Three, (and, in?) Infinity" by George Gamow, and I remember the excitement of understanding this very sophisticated argument that there were two different kinds of infinities. Uhlenbeck entered the University of Michigan with the intention of studying physics but a combination of studying exciting mathematics courses and finding that physics practicals were not a strong point lead her to change to mathematics. She was awarded a B.S. in mathematics in 1964. After graduating from the University of Michigan, Uhlenbeck continued her studies at the Courant Institute in New York. However at this time she married and decided to follow her husband when he went to Harvard. She entered Brandeis University and was awarded a Master's Degree in 1966. She remained at Brandeis to study for her doctorate under Richard Palais' supervision, and was awarded a Ph.D. in 1968. Her first appointment was a one year post in 1968-69 at Massachusetts Institute of Technology. Then another temporary post, this time a two year one as a lecturer at the University of California, Berkeley http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Uhlenbeck_Karen.html (1 of 4) [2/16/2002 11:36:09 PM]

Uhlenbeck_Karen

during 1989-71. She describes her search for a permanent position:I was told, when looking for jobs after my year at MIT and two years at Berkeley, that people did not hire women, that women were supposed to go home and have babies. So the places interested in my husband - MIT, Stanford, and Princeton - were not interested in hiring me. I remember that I was told that there were nepotism rules and that they could not hire me for this reason, although when I called them on this issue years later they did not remember saying these things ... I ended up at the University of Illinois, Champaign-Urbana because they hired me, and my husband came along. In retrospect I realized how remarkably generous he was because he could have been at MIT, Stanford, or Princeton. I hated Champaign-Urbana - I felt out of place mathematically and socially, and it was ugly, bourgeois and flat. After being on the faculty at Urbana-Champaign from 1971 to 1976, she moved to the University of Illinois at Chicago where she was promoted to full professor. At this time she:... became friends with S T Yau, whom I credit with generously establishing my finally and definitively as a mathematician. In 1983 she was awarded a MacArthur Prize Fellowship and moved to a professorship at the University of Chicago. In 1988 Uhlenbeck was appointed Professor in the University of Texas at Austin where she also holds the Sid W Richardson Foundation Regents Chair in Mathematics. Uhlenbeck is a leading expert on partial differential equations and describes her mathematical interests as follows:I work on partial differential equations which were originally derived from the need to describe things like electromagnetism, but have undergone a century of change in which they are used in a much more technical fashion to look at the shapes of space. Mathematicians look at imaginary spaces constructed by scientists examining other problems. I started out my mathematics career by working on Palais' modern formulation of a very useful classical theory, the calculus of variations. I decided Einstein's general relativity was too hard, but managed to learn a lot about geometry of space time. I did some very technical work in partial differential equations, made an unsuccessful pass at shock waves, worked in scale invariant variational problems, made a poor stab at three dimensional manifold topology, learned gauge field theory and then some about applications to four dimensional manifolds, and have recently been working n equations with algebraic infinite symmetries. Uhlenbeck's work provided analytic tools to use instantons as an effective geometric tool. In [1] Simon Donaldson reminisces about the work on the applications of instantons that led him to receive a Fields Medal in 1986. He describes the "bubbling" phenomenon saying:In fact the papers of Uhlenbeck which appeared about that time [1982] contained essentially all the analysis required to put this picture on a firm footing. The papers do not discuss "bubbling" explicitly - perhaps the arguments were supposed to be obvious to experts by analogy with the work of Sacks and Uhlenbeck in the harmonic maps case. In 1988 Uhlenbeck lectured on Instantons and Their Relatives at the Centennial Celebration of the American Mathematical Society. Witten, who gave the next talk on Geometry and quantum field theory at the symposium said:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Uhlenbeck_Karen.html (2 of 4) [2/16/2002 11:36:09 PM]

Uhlenbeck_Karen

In the talk just before mine, Karen Uhlenbeck described some purely mathematical developments that at least roughly might be classified in this area. She described advances in geometry that have been achieved through the study of systems of nonlinear partial differential equations. Among other things, she sketched some aspects of Simon Donaldson's work on the geometry of four-dimensional manifolds, instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory. Two years later, in 1990, Witten received a Fields Medal for his work on topological quantum field theories. At the same International Congress of Mathematicians in Kyoto, Karen Uhlenbeck was a Plenary Speaker. Among the many honours that Uhlenbeck has received for her work one should mention in particular that she was elected a Member of the American Academy of Arts and Sciences in 1985 and a Member of the National Academy of Sciences the following year. She has also served on the editorial boards of many journals; a complete list to date is Journal of Differential Geometry (1979-81), Illinois Journal of Mathematics (1980-86), Communications in Partial Differential Equations (1983- ), Journal of the American Mathematical Society (1986-91), Ergebnisse der Mathematik (1987-90), Journal of Differential Geometry (1988-91), Journal of Mathematical Physics (1989- ), Houston Journal of Mathematics (1991- ), Journal of Knot Theory (1991- ), Calculus of Variations and Partial Differential Equations (1991- ), Communications in Analysis and Geometry (1992- ). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Karen Uhlenbeck

Mathematicians born in the same country

Honours awarded to Karen Uhlenbeck (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer Other Web sites

1985 1. Texas (Karen Uhlenbeck's home page) 2. AMS (Moduli spaces and adiabatic limits)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Uhlenbeck_Karen.html (3 of 4) [2/16/2002 11:36:09 PM]

Uhlenbeck_Karen

Glossary index

Quotations index

Mathematicians of the day JOC/EFR April 1998

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Uhlenbeck_Karen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Uhlenbeck_Karen.html (4 of 4) [2/16/2002 11:36:09 PM]

Ulam

Stanislaw Marcin Ulam Born: 3 April 1909 in Lemberg, Poland, Austrian Empire (now Lvov, Ukraine) Died: 13 May 1984 in Santa Fe, New Mexico, USA

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stan Ulam solved the problem of how to initiate fusion in the hydrogen bomb. He also devised the 'Monte-Carlo method' widely used in solving mathematical problems using statistical sampling. At the age of ten, Ulam entered the gymnasium in Lvov and, about this time, he became interested first in astronomy and then in physics. An uncle gave Ulam a telescope when he was about 12 years old and later Ulam tried to understand Einstein's special theory of relativity. However this required an understanding of mathematics and so, at age 14, he began to study mathematics from books, going well beyond the school level mathematics he was learning. Ulam said ([9] or [10]):... I was sixteen when I really learned calculus all by myself from a book by Kowalevski, a German not to be confused with Sonia Kovalevskaya .... Then I read also about set theory in a book by Sierpinski, and I think I understood that. We had a good professor in high school, Zawirski, who was a lecturer in logic at the university. I talked to him about it then and when I entered the Polytechnic Institute. Now with interests in astronomy, physics and mathematics, Ulam entered the Polytechnic Institute in Lvov. In 1927, his first year at the university, he was taught by Kuratowski who had just been appointed to Lvov. Ulam said ([9] or [10]):[Kuratowski] gave an elementary course on set theory, and I asked some questions, then I talked to him after classes, and he became interested in a young student who evidently was interested in mathematics and had some ideas. I was lucky to solve an unsolved problem

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ulam.html (1 of 4) [2/16/2002 11:36:11 PM]

Ulam

that he proposed. Ulam obtained his Ph.D. from the Polytechnic Institute in Lvov in 1933 where he studied under Banach. He investigated a problem which originated with Lebesgue in 1902 to find a measure on [0,1] with certain properties. Banach in 1929 had solved a related measure problem, but assuming the Generalised Continuum Hypothesis. Ulam, in 1930, strengthened Banach's result by proving it without using the Generalised Continuum Hypothesis. In 1935 Ulam received an invitation from von Neumann to visit the Institute for Advanced Study in Princeton for a few months. Planning to spend three months there he sailed from France to New York. At the Institute for Advanced Study he met G D Birkhoff who invited him to Harvard University. Ulam said ([9] or [10]):... I went back to Poland, but the next fall I returned to Cambridge as a member of the so-called Society of Fellows, a new Harvard institution. ... I started teaching right away: first, elementary courses and then quite advanced courses. I became a lecturer at Harvard in 1940, but every year during that time I commuted between Poland and the United States. In the summers I visited my family and friends and mathematicians. In Poland mathematical life was very intense, the mathematicians saw each other often in cafes such as the Scottish Cafe and the Roma Cafe. We sat there for hours and did mathematics. During the summers I did this again. And then in '39, I actually left Poland about a month before World War II started. You can see a picture of the Scottish Café. In 1940 Ulam was appointed as an assistant professor at the University of Wisconsin. In 1943 Ulam became an American citizen. In that year von Neumann asked him to undertake some very important war work. They agreed to meet ([9] or [10]):... in Chicago in some railroad station to learn a little bit more about it. I went there, and he could not tell me where he was going. There were two guys, sort of guards, looking like gorillas, with him. He discussed with me some mathematics, some interesting physics, and the importance of this work. And that was Los Alamos at the very start. A few months later I came with my wife ... arriving for the first time in a very strange place. He worked on the hydrogen bomb at the Los Alamos National Laboratory in New Mexico. This work is described in [1]:Working with physicist Edward Teller, Ulam solved one major problem encountered in work on the fusion bomb by suggesting that compression was essential to explosion and that shock waves from a fission bomb could produce the compression needed. He further suggested that careful design could focus mechanical shock waves in such a way that they would promote rapid burning of the fusion fuel. Teller suggested that radiation implosion, rather than mechanical shock, be used to compress the thermonuclear fuel. This two-stage radiation implosion design, which became known as the Teller-Ulam configuration, led to the creation of modern thermonuclear weapons. Ulam, with J C Everett, also proposed the 'Orion' plan for nuclear propulsion of space vehicles. Rota [21] describes how Ulam's personality changed in 1946:One morning in 1946, in Los Angeles, Stanislaw Ulam, a newly appointed professor at the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ulam.html (2 of 4) [2/16/2002 11:36:11 PM]

Ulam

University of Southern California, awoke to find himself unable to speak. A few hours later, he underwent a dangerous surgical operation after the diagnosis of encephalitis. ... In time, however, some changes in his personality became obvious to those who knew him. ... his ideas, which he spouted out at odd intervals, were fascinating beyond anything I have witenssed before or since. However, he seemed to studiously avoid going into details. ... he came to lean on his unimpaired imagination for his ideas, and on the [hard work] of others for technical support. ... A crippling technical weakness coupled with an extraordinarily creative imagination is the drama of Stan Ulam. Soon after I met him, I was made to understand that, as far as our conversations went, his drama would be one of the Forbidden Topics. ... But he knew I knew, and I knew he knew I knew. While Ulam was at Los Alamos, he developed the 'Monte-Carlo method' which searched for solutions to mathematical problems using a statistical sampling method with random numbers. It is now widely used in computer implementations of mathematical software. He remained at Los Alamos until 1965 when he was appointed to the chair of mathematics at the University of Colorado. At the time of his death he was professor of biomathematics at the University of Colorado. When asked to sum up his work, he said ([9], [10]):Originally I worked in set theory and some of these problems are still being worked on intensively. It is too technical to describe: measurable cardinals, measure in set theory, abstract measure. Then in topology I had a few results. ... Then I worked a little in ergodic theory. Oxtoby and I solved an old problem and some other problems were solved in other fields later. In general I would say luck plays a part, at least in my case. Also I had luck with tremendously good collaborators in set theory, in group theory, in topology, in mathematical physics, and in other method, which is not a tremendously intellectual achievement but is very useful, a few things like that. Ulam's writing include A collection of mathematical problems (1960), Sets numbers and universes (1974) and Adventures of a Mathematician (1976). He was described by Rota in the following way:Ulam's mind is a repository of thousands of stories, tales, jokes, epigrams, remarks, puzzles, tounge-twisters, footnotes, conclusions, slogans, formulas, diagrams, quotations, limericks, summaries, quips, epitaphs, and headlines. In the course of a normal conversation he simply pulls out of his mind the fifty-odd relevant items, and presents them in linear succession. A second-order memory prevents him from repeating himself too often before the same public. His wife, Françoise Ulam, writing in [4] described Ulam working methods:Ulam ... is almost exculsively a talking man, a verbal person. When not thinking ... what he enjoys most is to talk, to discuss, to argue, to converse, with friends and colleagues. Relying on his phenomenal memory, he carries everything in his head. ... The physical act of taking pen to paper has always been painful for him. His mind and his eyes are the obstacles. His mind, because it works much faster than his fingers...; his eyes because one is very myopic the other very presbyotic. ... From childhood fears, then from http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ulam.html (3 of 4) [2/16/2002 11:36:11 PM]

Ulam

youthful vanity he spurned wearing glasses, until very recently. Thus Ulam has always had a very hard time bringing himself to write anything for publication, either in long hand or with a typewriter. Machines and other mechanical objects have always turned him off. ... How then does he ever produce a written text? Mainly by talking ... Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (23 books/articles)

Some Quotations (7)

A Poster of Stan Ulam

Mathematicians born in the same country

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ulam.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ulam.html (4 of 4) [2/16/2002 11:36:11 PM]

Ulugh_Beg

Ulugh Beg Born: 1393 in Soltaniyeh, Timurid, Persia (now Iran) Died: 27 Oct 1449 in Samarkand, Timurid empire

Click the picture above to see three larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ulugh Beg was the grandson of the conqueror Timur, who is often known as Tamerlane (from Timur-I-Leng meaning Timur the Lame, a title of contempt used by his Persian enemies). Although in this archive we are primarily interested in Ulugh Beg's achievements in mathematics and astronomy, we need to examine the history of the area since it had such a major impact on Ulugh Beg's life. Timur, Ulugh Beg's grandfather, came from the Turkic Barlas tribe which was a Mongol tribe that was living in Transoxania, today essentially Uzbekistan. He united several Turko-Mongol tribes under his leadership and set out on a conquest, with his armies of mounted archers, of the area now occupied by Iran, Iraq, and eastern Turkey. Shortly after his grandson Ulugh Beg was born, Timur invaded India and by 1399 he had taken control of Delhi. Timur continued his conquests by extending his empire to the west from 1399 to 1402, winning victories over the Egyptian Mamluks in Syria and the Ottomans in a battle near Ankara. Timur died in 1405 leading his armies into China. After Timur's death his empire was disputed among his sons. Ulugh Beg's father Shah Rukh was the fourth son of Timur and, by 1407, he had gained overall control of most of the empire, including Iran and Turkistan regaining control of Samarkand. Samarkand had been the capital of Timur's empire but, although his grandson Ulugh Beg had been brought up at Timur's court, he was seldom in that city. When Timur was not on one of his military campaigns he moved with his army from place to place and his court, including his grandson Ulugh Beg, travelled with him. In 1409 Shah Rukh decided to make Herat in Khorasan (today in western Afghanistan) his new capital. Shah Rukh ruled there making it a trading and cultural centre. He founded a library there and became a patron of the arts. However Shah Rukh did not give up Samarkand, rather he decided to give it to his son http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ulugh_Beg.html (1 of 4) [2/16/2002 11:36:13 PM]

Ulugh_Beg

Ulugh Beg who was more interested in making the city a cultural centre than he was in politics or military conquest. Although Ulugh Beg was only sixteen years old when his father put him in control of Samarkand, he became his father's deputy and he became ruler of the Mawaraunnahr region. Ulugh Beg was primarily a scientist, in particular a mathematician and an astronomer. However, he certainly did not neglect the arts, writing poetry and history and studying the Qur'an. In 1417, to push forward the study of astronomy, Ulugh Beg began building a madrasah which is a centre for higher education. The madrasah, fronting the Rigestan Square in Samarkand, was completed in 1420 and Ulugh Beg then began to appoint the best scientists he could find to positions there as lecturers. Ulugh Beg invited al-Kashi to join his madrasah in Samarkand, as well as around sixty other scientists including Qadi Zada. There is little doubt that, other than Ulugh Beg himself, al-Kashi was the leading astronomer and mathematician at Samarkand. Letters which al-Kashi wrote to his father have survived. These were written from Samarkand and give a wonderful description of the scientific life there. The contents of one of these letters has only recently been published, see [5]. In the letters al-Kashi praises the mathematical abilities of Ulugh Beg but of the other scientists in Samarkand, only Qadi Zada earned his respect. Ulugh Beg led scientific meetings where problems in astronomy were freely discussed. Usually these problems were too difficult for all except al-Kashi and the letters confirm that al-Kashi was the closest collaborator of Ulugh Beg at his madrasah in Samarkand. In addition to the madrasah, Ulugh Beg built an observatory at Samarkand, the construction of this beginning in 1428. The Observatory, which was circular in shape, had three levels. It was over 50 metres in diameter and 35 metres high. The director of the Observatory was Ali-Kudschi, a Muslim astronomer. Al-Kashi and other mathematicians and astronomers appointed to the madrasah also worked at Ulugh Beg's Observatory. Among the instruments specially constructed for the Observatory was a quadrant so large that part of the ground had to be removed to allow it to fit in the Observatory. There was also a marble sextant, a triquetram and an armillary sphere. The achievements of the scientists at the Observatory, working there under Ulugh Beg's direction and in collaboration with him, are discussed in detail in [4]. This excellent book records the main achievements which include the following: methods for giving accurate approximate solutions of cubic equations; work with the binomial theorem; Ulugh Beg's accurate tables of sines and tangents correct to eight decimal places; formulas of spherical trigonometry; and of particular importance, Ulugh Beg's Catalogue of the stars, the first comprehensive stellar catalogue since that of Ptolemy. This star catalogue, the Zij-i Sultani, set the standard for such works up to the seventeenth century. Published in 1437, it gives the positions of 992 stars. The catalogue was the results of a combined effort by a number of people working at the Observatory including Ulugh Beg, al-Kashi, and Qadi Zada. As well as tables of observations made at the Observatory, the work contained calendar calculations and results in trigonometry. The trogonometric results include tables of sines and tangents given at 1 intervals. These tables display a high degree of accuracy, being correct to at least 8 decimal places. The calculation is built on an accurate determination of sin 1 which Ulugh Beg solved by showing it to be the solution of a cubic equation which he then solved by numerical methods. He obtained http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ulugh_Beg.html (2 of 4) [2/16/2002 11:36:13 PM]

Ulugh_Beg

sin 1 = 0.017452406437283571 The correct approximation is sin 1 = 0.017452406437283512820 which shows the remarkable accuracy which Ulugh Beg achieved. Observations made at the Observatory brought to light a number of errors in the computations of Ptolemy which had been accepted without question up to that time. Data from his Observatory allowed Ulugh Beg to calculate the length of the year as 365 days 5 hours 49 minutes 15 seconds, a fairly accurate value. He produced data relating to the Sun, the Moon and the planets. His data for the movements of the planets over a year is, like so much of his work, very accurate [1]:... the difference between Ulugh Beg's data and that of modern times relationg to [Saturn, Jupiter, Mars, Venus] falls within the limits of two to five seconds. Ulugh Beg's politics were not up to his science and, after his father's death in 1447, he was unable to retain power despite being an only son. He was eventually put to death at Samarkand at the instigation of his own son 'Abd al-Latif. His tomb was discovered in 1941 in the mausoleum built by Timur in Samarkand. It was discovered that Ulugh Beg had been buired in his clothes which is known to indicate that he was considered a martyr. The injuries inflicted on him were evident when his body was examined [1]:... the third cervical vertebra was severed by a sharp instrument in such a way that the main portion of the body and an arc of that vertebra were cut cleanly; the blow, struck from the left, also cut through the right corner of the lower jaw and its lower edge. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles)

A Quotation

A Poster of Ulugh Beg

Mathematicians born in the same country

Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Other references in MacTutor

1. The Observatory at Samarkand. 2. Chronology: 1300 to 1500

Honours awarded to Ulugh Beg (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Ulugh Beigh

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ulugh_Beg.html (3 of 4) [2/16/2002 11:36:13 PM]

Ulugh_Beg

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1999

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ulugh_Beg.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Ulugh_Beg.html (4 of 4) [2/16/2002 11:36:13 PM]

Upton

Francis Robbins Upton Born: 1852 in Peabody, Massachusetts, USA Died: 10 March 1921 in Orange, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Francis Upton studied at Bowdoin College, Brunswick and then he went to Berlin where he studied under Helmholtz. He returned to the USA and studied at Princeton University. He became the very first student to officially earn, by examination, a graduate degree from Princeton. He received the degree of Master of Science from Priceton in 1877 after being instructed by Cyrus Fogg Brackett. It was Brackett who, in 1889, founded at Princeton the first school of electrical engineering in the United States. After obtaining his Master's degree Upton worked with Thomas Edison on mathematical problems associated with devices such as the incandescent lamp, the watt-hour meter and large dynamos. Although Edison was a genius as an inventor he had no formal education so was unable to translate his intuition into mathematics. For this he relied on Upton who produced mathematical formulations of Edison's ideas. Upton went to live in Menlo Park, New Jersey, since he worked there in the laboratory that Edison had set up in that village in 1876. The laboratory was a remarkable research environment for Upton to undertake his reseach in. Better equiped than most of the leading university laboratories, it possessed the finest equipment available. Galvanometers, storage batteries, induction coils, wire chemicals, photographic equipment and a forge were available. It also had a research library where scientific publications were held. About twenty skilled machinists, in addition to the mathematical physicist Upton, staffed what was the first ever industrial research laboratory. In 1879 the design of the electric light bulb was perfected. Despite several eminent scientists predicting http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Upton.html (1 of 2) [2/16/2002 11:36:15 PM]

Upton

that electric light bulbs in a circuit would never work, a lamp powered by current produced by dynamos was demonstrated on 21 October 1879. It was Upton's home in Menlo Park which was the first private house in the world to be lit by electricty, the lamps being powered by a station in Menlo Park capable of lighting 30 bulbs. Upton became a partner and the general manager of the Edison Lamp Works which was established in 1880. The became part of the General Electric Company in 1892. Princeton now has the Francis Robbins Upton Fellowships named his memory. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Upton.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Upton.html (2 of 2) [2/16/2002 11:36:15 PM]

Urysohn

Pavel Samuilovich Urysohn Born: 3 Feb 1898 in Odessa, Ukraine Died: 17 Aug 1924 in Batz-sur-Mer, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pavel Urysohn is also known as Pavel Uryson. His father was a financier in Odessa, the town in which Pavel Samuilovich was born. He came from a family descended from the sixteenth century Rabbi M Jaffe. It was a well-off family and Urysohn received his secondary education in Moscow at a private school there. In 1915 Urysohn entered the University of Moscow to study physics and in fact he published his first paper in this year. Being interested in physics at this time it is not surprising that this first paper was on a physics topic, and indeed it was, being on Coolidge tube radiation. However his interest in physics soon took second place for after attending lectures by Luzin and Egorov at the University of Moscow he began to concentrate on mathematics. Urysohn graduated in 1919 and continued his studies there working towards his doctorate. The authors of [8] write:Luzin was a dynamic mathematician and it was he who persuaded Urysohn to stay on in order to study for a doctorate during 1919-21. At this stage Urysohn was interested in analysis, in particular integral equations, and this was the topic of his habilitation. He was awarded his habilitation in June 1921 and, following this, became an assistant professor at the University of Moscow. Urysohn soon turned to topology. He was asked two questions by Egorov and it was these which occupied him during the summer of 1921. The first question that Egorov posed was to find a general

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Urysohn.html (1 of 4) [2/16/2002 11:36:17 PM]

Urysohn

intrinsic topological definition of a curve which when restricted to the plane became Cantor's notion of a continuum which is nowhere dense in the plane. The second of Egorov's questions was a similar one but applied to surfaces, again asking for an intrinsic topological definition. Now these were difficult questions which had been around for some time. It was not that Egorov had come up with new questions, rather he was giving the bright young mathematician Urysohn two really difficult problems in the hope that he might come up with new ideas. Egorov was not to be disappointed, for Urysohn attacked the questions with great determination. He did not sit still waiting for inspiration to strike, rather he tried one idea after another to see if it would give him the topological definition of dimension that he was looking for. A holiday with other young Moscow mathematicians to the village of Burkov, on the banks of the river Kalyazmy near to the town of Bolshev, did not stop him trying to find the "right" definition of dimension. Quite the opposite, it was a good chance for him to think in congenial surroundings, and one morning near the end of August he woke up with an idea in his mind which he felt, even before working through the details, was right. Immediately he told his friend Aleksandrov about his inspiration. Of course there was a lot of hard work after the moment of inspiration. During the following year Urysohn worked through the consequences building a whole new area of dimension theory in topology. It was an exciting time for the topologists in Moscow for Urysohn lectured on the topology of continua and often his latest results were presented in the course shortly after he had proved them. He published a series of short notes on this topic during 1922. The complete theory was presented in an article which Lebesgue accepted for publication in the Comptes rendus of the Academy of Sciences in Paris. This gave Urysohn an international platform for his ideas which immediately attracted the interest of mathematicians such as Hilbert. Urysohn published a full version of his dimension theory in Fundamenta mathematica. He wrote a major paper in two parts in 1923 but they did not appear in print until 1925 and 1926. Sadly Urysohn had died before even the first part was published. The paper begins with Urysohn stating his aim which was:To indicate the most general sets that still merit being called "lines" and "surfaces" ... In fact Urysohn set out to do far more in this paper than to answer the two questions that Egorov had posed to him. As Crilly and Johnson write [8]:Not only did he seek definitions of curve and surface, but also definitions of n-dimensional Cantorian manifold and hence of dimension itself. The dimension concept was, in fact, the centre of his attention. Now, although Urysohn did not know of Brouwer's contribution when he worked out the details of his theory of topological dimension, Brouwer had in fact published on that topic in 1913. He had given a global definition, however, and this was in contrast to Urysohn's local definition of dimension. Another important aspect of Urysohn's ideas was the fact that he presented them in the context of compact metric spaces. After the Urysohn's death, Aleksandrov argued that although Urysohn's definition of dimension was given for a metric space, it is, nevertheless, completely equivalent to the definition given by Menger for general topological spaces. Urysohn visited Göttingen in 1923. His reports to the Mathematical Society of Göttingen interested Hilbert and while in Göttingen he learnt of Brouwer's contributions to the area made in the paper of 1913 http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Urysohn.html (2 of 4) [2/16/2002 11:36:17 PM]

Urysohn

to which we referred above. Urysohn spotted an error in Brouwer's paper regarding a definition of dimension while he was studying it in Göttingen and easily constructed a counter-example. He met Brouwer at the annual meeting of the German Mathematical Society in Marburg where both gave lectures and Urysohn mentioned Brouwer's error, and his counter-example, in his talk. It was an occasion which made Brouwer begin to think about topology again, for his interests had turned to intuitionism, the subject of his talk at Marburg. In the summer of 1924 Urysohn set off again with Aleksandrov on a European trip through Germany, Holland and France. Again the two mathematicians visited Hilbert and, by 7 May, they must have left since Hilbert wrote to Urysohn on that day telling him his paper with Aleksandrov was accepted for publication in Mathematische Annalen (see below). This letter, given in [11], also thanks Urysohn for caviar he had given Hilbert, and expresses the hope that Urysohn will visit again the following summer. They then met Hausdorff who was impressed with Urysohn's results. He also wrote a letter to Urysohn which was dated 11 August 1924 (see [11]). The letter discusses Urysohn's metrization theorem and his construction of a universal separable metric space. The construction of a universal metric space, containing an isometric image of any metric space, was one of Urysohn's last results. Like Hilbert, Hausdorff expressed the hope that Urysohn would visit again the following summer. Van Dalen writes in [13] about their final mathematical visit which was to Brouwer:This time [Urysohn and Aleksandrov] visited Brouwer, who was most favourably impressed by the two Russians. He was particularly taken with Urysohn, for whom he developed something like the attachment to a lost son. After this visit the two mathematicians continued their holiday to Brittany where they rented a cottage. Urysohn drowned in rough seas while on one of their regular swims off the coast. Urysohn was not only an "inseparable friend" to Aleksandrov but the two collaborated on important publications such as Zur Theorie der topologischen Räume published in Mathematische Annalen in 1924. Urysohn's main contributions, in addition to the theory of dimension discussed above, are the introduction and investigation of a class of normal surfaces, metrization theorems, and an important existence theorem concerning mapping an arbitrary normed space into a Hilbert space with countable basis. He is remembered particularly for 'Urysohn's lemma' which proves the existence of a certain continuous function taking values 0 and 1 on particular closed subsets. After Urysohn's death Brouwer and Aleksandrov made sure that the mathematics he left was properly dealt with. As van Dalen writes [13]:Brouwer was broken hearted. He decided to look after the scientific estate of Urysohn as a tribute to the genius of the deceased. Together with Aleksandrov he acquitted himself of this task. Crilly and Johnson write [8]:Considering that he only had three years to devote to topology, he made his mark in his chosen field with brilliance and passion. He transformed the subject into a rich domain of modern mathematics. How much more might he there have been, had he not died so young? Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Urysohn.html (3 of 4) [2/16/2002 11:36:17 PM]

Urysohn

Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles) Mathematicians born in the same country Other Web sites

Akron (More pictures and links)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Urysohn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Urysohn.html (4 of 4) [2/16/2002 11:36:17 PM]

Vacca

Giovanni Enrico Eugenio Vacca Born: 18 Nov 1872 in Genoa, Italy Died: 6 Jan 1953 in Rome, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giovanni Vacca went to school in Genoa and at this time he was politically very active. In 1892 he helped Filippo Turati, a young lawyer from Milan, to found the Italian Workers' Party (Partito dei Lavoratori Italiani). The following year the party was renamed the Italian Socialist Party (Partito Socialista Italiano). The party was mainly social democratic. It believed in the class struggle and aimed to win seats in parliament. The party had Marxist views but it expected to slowly influence people towards its views. Vacca entered the University of Genoa to study mathematics. However in 1894 Crispi dissolved the Party and the leaders of the Party, including Vacca were banished from Genoa. The banishment was in 1897 and, fortunately, this allowed Vacca to graduate that year before moving away from Genoa. He moved to Turin where he became Peano's assistant in November 1897. In 1899 Vacca went to Hannover to study the unpublished manuscripts of Leibniz. The following year Vacca attended the First International Congress of Philosophy which was held in Paris in 1900. At the Congress Vacca met Couturat and, the following year, Couturat wrote in the preface of La Logique de Leibniz :Our work on the logic of Leibniz was almost completed (at least we thought so) when we had the pleasure, at the International Congress of Philosophy (August, 1900), of making the acquaintance of Mr Giovanni Vacca, at that time mathematical assistant at the University of Turin, who had examined, the year before, the manuscripts of Leibniz preserved in Hannover, and had extracted from them several formulas of logic inserted in the Formulaire

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vacca.html (1 of 3) [2/16/2002 11:36:20 PM]

Vacca

de Mathématiques of Mr Peano. It was he who revealed to us the importance of the unpublished works of Leibniz, and inspired us with the desire to consult them in turn. In 1903 Vacca published a collection of short works by Leibniz and some of his papers which had not been previously published. However before this, in 1902, Vacca's position as Peano's assistant came to an end and he had returned to Genoa. Despite the dissolution of the Italian Socialist Party in 1894, it had been revived in the late 1890s and won 32 parliamentary seats in 1900. Vacca, on his return to Genoa in 1902, worked for the Party again becoming a member of the Socialist Council and also a member of the national party administration. However Vacca continued his mathematical work and gave a course at the University of Genoa on mathematical logic. In 1904 Vacca returned to Turin and was assistant to Peano for one further year. However Vacca had by this time yet a third interest in addition to mathematics and politics. He had become interested in Chinese as early as 1898 when there had been a Chinese exhibition in Turin and he had some lessons in Chinese from two missionaries who had returned from China. In 1905, this interest in Chinese became the road that Vacca decided to follow. He went to Florence to study the Chinese language at the university. Vailati was teaching in Florence at this time and the two already knew each other from their time in Turin. They shared mathematical interests and certainly Vacca continued his mathematical research and interest in the history of mathematics. In fact Vacca would be one of the editors of Vailati's collected works, published in 1911, two years after Vailati's death. Vacca spent 1907-08 in western China, spending a year in the city of Cheng-tu. After his return to Florence he was awarded his doctorate for Chinese studies in 1910 and, the following year, he was appointed to a post teaching Chinese literature at the University of Rome. In 1922 Vacca succeeded his old professor of History and Geography of East Asia to the chair at Florence. Vacca taught Chinese language and literature there until he retired in 1947. Despite his change of topic in mid career, Vacca continued his Chinese and mathematical studies in parallel. For example in 1928 Peano presented a paper by Vacca on Fermat's method of descent to the Academy of Sciences of Turin. Throughout his career he published around 130 papers, 47 relating to his Chinese interests, 38 on mathematical research and 45 on the history of mathematics. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vacca.html (2 of 3) [2/16/2002 11:36:20 PM]

Vacca

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vacca.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vacca.html (3 of 3) [2/16/2002 11:36:20 PM]

Vailati

Giovanni Vailati Born: 24 April 1863 in Crema, Italy Died: 14 May 1909 in Rome, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giovanni Vailati attended school in Monza and in Lodi. In 1880 he entered the University of Turin to study engineering. Peano was appointed as assistant to D'Ovidio at Turin in 1880 and Vailati was among the first group of students for whom he had to care. Clearly Peano had a major influence on Vailati who would become interested in topics on which Peano worked. Vailati graduated with an engineering degree in 1884. However, probably mainly due to Peano, Vailati realised that mathematics was the subject for him so he continued to study at Turin for his mathematics degree which he was awarded in 1888. Perhaps it tell us something of Vailati's character that his classmates at university called him "the Philosopher". Vailati did not seek employment after graduating but returned to him home in Lode where he studied languages. He did make frequent visits to the University of Turin and maintained contacts with mathematics and the mathematicians at the university. Then in 1892 he was appointed as Peano's second assistant, a position he held until 1895. Vailati, in a letter to a cousin dated 22 December 1892, described his duties:In the days just passed my teaching duties have left me little free time, not so much because of the frequency of the hours of school (nine a week, of which three consist of merely attending the lesson of the principal) as for the necessary preparation and elaboration of the topics to discuss so as to avoid the danger of pulling a boner. After completing his years as Peano's assistant, Vailati became assistant in projective grometry and then one year later he became Volterra's assistant at Turin. In 1899, after spending seven years with the duties http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vailati.html (1 of 3) [2/16/2002 11:36:22 PM]

Vailati

of an assistant at Turin, Vailati decided to take up school teaching. He taught at quite a number of schools over the next few years: Syracuse (1899-1900), Bari (1900-1901), Como (1901-1904) and Florence from 1904. Vailati worked on mathematical logic, working closely with Peano on this topic, and also on the history and methodology of science. Vailati had taught a course at Turin on the history of mechanics in the years 1896-1898 and he published three essays based on this course which earned him considerable fame. However during his lifetime his work in philosophy earned him most recognition. He was a pragmatist with views not far from those of Charles Peirce. Vailati was a member of the organising committee of the First International Congress of Philosophy which was held in Paris in 1900 immediately before the First International Congres of Mathematicians. At the First International Congress of Philosophy, Vailati was appointed to the permanent international commission. He attended the Second International Congress of Philosophy in Rome in 1903 and the Third International Congress of Philosophy in Heidelberg in 1908. In fact this third conference was largely a discussion of pragmatism and this is almost certainly due to Vailati's influence on the committee. Kennedy writes in [1]:Vailati came of a Catholic family, but lost his faith during his early university years. Throughout his life he had affectionate and devoted friends. He never married. His premature death was attributed to heart trouble, complicated by pulmonitis. In [2] Peano describes Vailati:... he was modest, friendly with all, and universally esteemed, both for his learning as also for his personal qualities. The picture of Vailati is from a caricature by A Spadini. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Other Web sites

1. Crema, Italy (A site devoted to Vailati) 2. Crema, Italy (A biography in English).

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vailati.html (2 of 3) [2/16/2002 11:36:22 PM]

Vailati

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vailati.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vailati.html (3 of 3) [2/16/2002 11:36:22 PM]

Valerio

Luca Valerio Born: 1552 in Naples, Italy Died: 17 Jan 1618 in Rome, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Luca Valerio was brought up on the island of Corfu, then he studied at the Collegio Romano. In Rome he was taught mathematics by Clavius. He remained at Collegio Romano after taking his first degree and was awarded a doctorate in philosophy and theology After taking his doctorate Valerio remained in Rome. At first he taught rhetoric and Greek at the Collegio Greco. On a visit to Pisa in 1590 Valerio met Galileo. After his return to Rome he began teaching rhetoric at the University of Sapienza. Sapienza was the name of the building which the University of Rome occupied at this time and it gave its name to the University. About 1600 Valerio, still at Sapienza, began to teach mathematics. Valerio's De centro gravitatis, written in 1604, applied methods of Archimedes to find volumes and centres of gravity of solid bodies. He used interesting early ideas of the quotient of limits. Among his other works was Quadratura parabolae (1606). From 1609 until 1616 Valerio corresponded with Galileo. During this period, in 1611, Valerio obtained a position in the Vatican library in addition to his post at Sapienza. He therefore was closely connected with the top people in the Roman Catholic church. Cardinal Robert Bellarmine, the chief theologian of the Roman Catholic church, issued a decree on 5 March 1616 which declared Copernicanism false and erroneous. Valerio took fright at this, ended his correspondence with Galileo, and resigned from the Accademia dei Lincei which he had entered four years earlier. His resignation was not accepted but he was not allowed to attend further meetings. The articles below indicate that Valerio was:a very withdrawn and isolated person. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Valerio.html (1 of 2) [2/16/2002 11:36:23 PM]

Valerio

List of References (9 books/articles) Mathematicians born in the same country Cross-references to History Topics

The rise of the calculus

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Valerio.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Valerio.html (2 of 2) [2/16/2002 11:36:23 PM]

Vallee_Poussin

Charles Jean Gustave Nicolas Baron de la Vallée Poussin Born: 14 Aug 1866 in Louvain, Belgium Died: 2 March 1962 in Louvain, Belgium

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles De la Vallée Poussin's father was the professor of mineralogy and geology at the University of Louvain for around 40 years. The original family name was Lavallée, a name of French origin. A great-grandfather of Vallée Poussin married into the family of Nicolas Poussin, the leading French artist of the 17th century, and being an artist himself this great-grandfather added the name Poussin to his own name of La Vallée. So Vallée Poussin came from a family with both artistic and scientific interests, but it was also a family with literary interests. From his boyhood he was encouraged by the mathematician Louis-Philippe Gilbert but at first Vallée Poussin thought he would become a Jesuit priest. He entered the Jesuit College at Mons but he found the teaching there unacceptable. He was particularly disappointed in the teaching of philosophy at the College, so he turned to a different topic although he still did not have mathematics as his main interest. He studied engineering and obtained his diploma in that subject. However soon after this he became absorbed by pure mathematics. He studied at the University of Louvain where he was taught by Gilbert who proved to be an inspiring teacher. Gilbert was an excellent mathematician and the author of a fine analysis textbook. Vallée Poussin also studied at the University of Paris and at the University of Berlin. In 1891 Vallée Poussin was appointed as an assistant of Gilbert's at the University of Louvain. However the collaboration was not to last for long since Gilbert died in 1892. Although only 26 years old at the time Vallée Poussin was elected to Gilbert's chair. Vallée Poussin's first mathematical research was on analysis, in particular concentrating on integrals and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vallee_Poussin.html (1 of 5) [2/16/2002 11:36:25 PM]

Vallee_Poussin

solutions of differential equations. One of his first papers in 1892 on differential equations was awarded a prize by the Belgium Academy. His best known work, however, appeared four years later in 1896 when he proved the prime number theorem. This states that (x), the number of primes x, tends to x/loge x as x tends to infinity. The prime number theorem had been conjectured in the 18th century, but in 1896 two mathematicians independently proved the result, namely Hadamard and Vallée Poussin. The first major contribution to proving the result was made by Chebyshev in 1848, then the proof was outlined by Riemann in 1851. The clue to two independent proofs being produced at the same time is that the necessary tools in complex analysis had not been developed until that time. In fact the solution of this major open problem was one of the major motivations for the development of complex analysis during the period from 1851 to 1896. In 1900, while on holiday in Norway, Vallée Poussin met a Belgium family. He married the talented daughter of this family and it was a very happy marriage. The result was that he had a home where he and his wife were happy and contented. He lived in Louvain from the time he was first appointed there except for a few periods abroad. During the First World War he was invited to Harvard in 1915 and then to Paris in 1916. Among a number of famous lectures he gave were those in Fribourg in 1918, Rome in 1923 and Houston in 1924. Other than the prime number theorem, Vallée Poussin's only contributions to prime numbers were contained in two papers on the Riemann zeta function which he published in 1916. The Riemann hypothesis, perhaps the most famous of all the still open questions of mathematics, is that all the complex zeros of the zeta function lie on the line 1/2 + i b. Vallée Poussin strengthened results proved by Hardy in 1914 which showed that an infinite number of the zeros were on that line. Vallée Poussin's results were of passing interest, however, for Hardy and Littlewood proved still stronger results in 1918. Vallée Poussin also worked on approximation to functions by algebraic and trigonometric polynomials from 1908 to 1918. Let us quote Vallée Poussin's own description of the problem approximation as given in a lecture which he gave in Houston in 1924:The most important of the problems which have been attacked in the study of approximation is that of the order of approximation. Let us define first what we mean by approximation. For example, let a continuous function f(x) be represented by means of a polynomial of degree n, and let Pn(x) be such a polynomial. The difference f - Pn is the error of the approximation, and is a function of x; its maximum value in the interval of representation is the approximation on. This positive number approaches 0 as 1/n approaches zero, if the polynomial Pn is well chosen. ... The problem of the order of approximation is the following: To determine the relation which exists between the order of approximation on, which f(x) may admit for a finite expression of order n, and the differential properties of the function. He then continued to put his own contribution to this problem into context, although one must say that it is phrased in a very modest way:I offered myself the beginnings of an answer to this very problem in 1908, while studying the approximation given by Landau's integral. I showed also that the function |x| admits an approximation of the order of 1/n by a polynomial of degree n, and I raised the question of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vallee_Poussin.html (2 of 5) [2/16/2002 11:36:25 PM]

Vallee_Poussin

deciding whether or not that was the order of the best possible approximation. This definite question had much more importance for the development of the subject than the few isolated results which I had obtained, because that question caused the writing of the two most important memoirs on the subject, one by D Jackson and the other by Sergi Bernstein. Vallée Poussin's most major work was Cours d'analyse . Burkill writes in [2]:It was [Jordan's Cours d'Analyse] which, as is recorded by Hardy and other mathematicians of his generation, opened their eyes to what analysis really was. If Jordan's is the most noble of the Cours d'Analyse and perhaps Goursat's (helped by its translation by Hedrick) the most widely read, it can hardly be doubted that Vallée Poussin's is the most elegant and lucid. Vallée Poussin's Cours d'analyse went through several editions, each containing new material. By 1899, several years before the publication of the first edition, much of the material already existed in the form of lecture notes. The first edition of Volume 1 appeared in 1903, and the first edition of Volume 2 in 1906. Volume I covered differentiation of functions of one or more variables, and integration of functions of a single variable. Volume 2 covered multiple integrals, differential equations, and differential geometry. The treatise was written in an interesting way, combining an introductory text with an advanced work for specialists. The way this was achieved was having two different type sizes. If a reader only read the larger type then it was a complete introduction to the subject for beginners or those interested in applications to engineering. The smaller type material was aimed at the pure mathematical specialist interested in the deeper subtleties. The work changed dramatically when a second edition appeared, Volume 1 in 1909 and Volume 2 in 1912. Most of the additional material appeared in small type and covered topics such as set theory, in particular the Schröder-Bernstein theorem, the Lebesgue integral, functions of bounded variation, the Jordan curve theorem, polynomial approximation, Parseval's theorem on trigonometric series, results of Fejér, etc. The third edition of Volume I again contained new material and was published in 1914. However World War II disrupted Vallée Poussin's work. The promised German translation failed to appear and the third edition of Volume 2 was burned by the German army when it overran Louvain. It would have discussed the Lebesgue integral, work which was never to be published in this form but a lot of it was incorporated into a later monograph. Unlike many similar books of its time Cours d'analyse contains no complex function theory. The fourth edition appeared in 1921 and 1922. It ended the larger/smaller print distinction and became a work aimed at beginners. The two volumes had reached their seventh edition by 1938 but it went through much fewer changes after the fourth edition. After 1925 Vallée Poussin turned to complex variable, potential theory and conformal representation. Further important texts published by him were his Borel tract on the Lebesgue integral (1916), approximation theory (1919), mechanics (1924), and potential theory (1937). In 1930 Vallée Poussin was revising his 1916 tract Lebesgue integrals: Set functions: Baire classes when Luzin's Lectures on analytic sets and their applications was published. The paper [5] contains three letters written by Vallée Poussin to Luzin dated 4 February 1933, 8 March 1933 and 21 March 1933. Vallée Poussin comments in these letters on the fact, which is of great interest to him, that Luzin used slightly different classifications of the same sets as he had studied. He gives high praise to Luzin's book.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vallee_Poussin.html (3 of 5) [2/16/2002 11:36:25 PM]

Vallee_Poussin

Publication of Vallée Poussin's work Le potential logarithmique was held up by World War II and only published in 1949. Vallée Poussin was elected to the Belgium Academy in 1909. More honours were to follow including election to the Madrid Academy of Sciences, the Naples Society of Science, the American Academy of Arts and Science, the Institute of France, the Accademia dei Lincei, the Paris Academy of Science, and the American National Academy of Sciences. There were celebrations in 1928 when Vallée Poussin had held the chair at the University of Louvain for 35 years and again celebrations in 1943 when he had been 50 years in the chair of mathematics at Louvain. In 1928, when he had held the chair at Louvain for 35 years, the King of Belgium conferred the title Baron on Vallée Poussin at the celebrations for this event. In 1961 he fractured his shoulder and since Vallée Poussin was in his mid 90s it failed to heal. His death followed a few months later.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Charles De la Vallée Poussin

Mathematicians born in the same country

Cross-references to History Topics

Prime numbers

Other references in MacTutor

1. Prime Number Theorem 2. Chronology: 1890 to 1900

Other Web sites

1. The Prime Pages (The Prime Number Theorem) 2. Steve Finch (Hadamard-de la Vallée Poussin or Meissel-Mertens constants) Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vallee_Poussin.html (4 of 5) [2/16/2002 11:36:25 PM]

Vallee_Poussin

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Vallee_Poussin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vallee_Poussin.html (5 of 5) [2/16/2002 11:36:25 PM]

Van_Ceulen

Ludolph van Ceulen Born: 28 Jan 1540 in Hildesheim, Germany Died: 31 Dec 1610 in Leiden, Netherlands

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ludolph Van Ceulen does not appear to have had a university education as his parents were not sufficiently wealthy to pay for one. He held a number of posts not only as a teacher of mathematics but also as a fencing teacher. He taught fencing and mathematics in Delft, then in 1594, opened a fencing school in Leiden. In 1600 he was appointed to the Engineering School in Leiden. For the last ten years of his life he taught arithmetic, surveying and fortification in the engineering school. Van Ceulen is famed for his calculation of to 35 places which he did using polygons with 262 sides. He spent most of his life doing this and it is fitting that the 35 places of are engraved on his tombstone. In Germany

was called the Ludolphine number for a long time.

Van Ceulen had several friends among the mathematicians of the time. In particular his friendships with Simon Stevin and Adriaan Van Roomen were important for van Ceulen's career. His most famous student, who studied under him at Leiden, was Willebrord Snell. Van Ceulen wrote a number of works including On the Circle. Snell translated two of Van Ceulen's books into Latin to make them more accessible to the world-wide mathematical community. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Van_Ceulen.html (1 of 2) [2/16/2002 11:36:27 PM]

Van_Ceulen

A Poster of Ludolph van Ceulen

Mathematicians born in the same country

Cross-references to History Topics

1. Pi through the ages 2. A chronology of pi

Other Web sites

1. The Galileo Project 2. Kelly MacKenzie

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Van_Ceulen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Van_Ceulen.html (2 of 2) [2/16/2002 11:36:27 PM]

Van_Vleck

Edward Burr Van Vleck Born: 7 June 1863 in Middletown, Connecticut, USA Died: 2 June 1943 in Madison, Wisconsin, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edward Van Vleck's father was a teacher of mathematics at the Wesleyan University where he taught H S White and F S Woods. His father was one of the two vice-presidents of the American Mathematical Society in 1904, the other being Bolza. Edward Van Vleck attended Middletown High School, then Wilbraham Academy before entering the Wesleyan University in 1880. Here he was taught mathematics by his father. At this stage his interests were in mathematics, physics and astronomy and after graduating with an A.B. in 1884 he spent a year as an assistant in the physics laboratory at the Wesleyan University. In 1885 Van Vleck became a graduate student at Johns Hopkins University where his interests still ranged through mathematics, physics and astronomy. He studied there under Craig, Newcomb and Story for two years before becoming convinced that mathematics was the topic for him and travelling to Germany to continue his studies. At the University of Göttingen, Van Vleck attended lectures by Klein, Burkhardt, Fricke, Schwarz, Voigt, Weber and others. His doctorate was awarded by Göttingen in 1893 for the dissertation entitled Zur Kettenbruchentwicklung Laméscher und ähnlicher Integrale written under Klein's supervision. After returning to the United States Van Vleck was appointed as an instructor at the University of Wisconsin. Van Vleck returned to the Wesleyan University in 1895 as an associate professor and in 1898 was promoted to professor, again moving to the University of Wisconsin in 1906 where he remained a professor until he retired in 1929.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Van_Vleck.html (1 of 2) [2/16/2002 11:36:30 PM]

Van_Vleck

Almost all Van Vleck's research papers were in the fields of function theory and differential equations. He was American Mathematical Society Colloquium lecturer in 1903 giving six lectures on divergent series and continued fractions. The American Mathematical Society was fortunate to have Van Vleck's support as well as that of his father. Van Vleck was an editor of the Transactions of the American Mathematical Society from 1905 to 1910, vice-president in 1909 and president from 1913 to 1914. His interests are described in [1] as:Travelling and collecting Japanese art... His collection of these prints, numbering thousands of items, is very remarkable, and one of the major private collections. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

A Quotation

Mathematicians born in the same country Honours awarded to Edward Van Vleck (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1913 - 1914

AMS Colloquium Lecturer

1903

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Van_Vleck.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Van_Vleck.html (2 of 2) [2/16/2002 11:36:30 PM]

Van_der_Waerden

Bartel Leendert van der Waerden Born: 2 Feb 1903 in Amsterdam, Netherlands Died: 12 Jan 1996 in Zurich, Switzerland

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

As a school pupil B L van der Waerden showed remarkable promise and he developed for himself the laws of trigonometry. He studied mathematics at the universities of Amsterdam and Göttingen from 1919 until 1925. The year 1924 he spent in Göttingen studying with Emmy Noether. His doctorate, supervised by Hendrik de Vries, was awarded by Amsterdam for a thesis on the foundations of algebraic geometry. In 1928 he received his habilitation from Göttingen. The year 1928 was a busy one for van der Waerden. He received a position at the University of Rostock but was appointed to a lectureship at Groningen in the same year. In 1931 he was appointed professor of mathematics at the University of Leipzig where he became a colleague of Heisenberg. Before and after World War II van der Waerden had problems as a foreigner from the Nazis. Although working in Germany he refused to give up his Dutch citizenship and his life was made difficult. After the War van der Waerden worked for Shell in Amsterdam in applied mathematics. In 1947 he visited the USA going to Johns Hopkins University. He returned in 1948 to a chair of mathematics at Amsterdam where he remained until 1951. In 1950 Karl Fueter died and van der Waerden was appointed to fill the vacant chair in Zurich in 1951. His impact on the department in Zurich was very great. As well as an almost unbelievable range of mathematical research interests, van der Waerden stimulated research in Zurich by supervising over 40 doctoral students during his years there. In fact van der Waerden was to remain in Zurich for the rest of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Van_der_Waerden.html (1 of 3) [2/16/2002 11:36:32 PM]

Van_der_Waerden

his life. Van der Waerden worked on algebraic geometry, abstract algebra, groups, topology, number theory, geometry, combinatorics, analysis, probability theory, mathematical statistics, quantum mechanics, the history of mathematics, the history of modern physics, the history of astronomy and the history of ancient science. In algebraic geometry van der Waerden defined precisely the notions of dimension of an algebraic variety, a concept intuitively defined before. His work in algebraic geometry uses the ideal theory in polynomial rings created by Artin, Hilbert and Emmy Noether. His work also makes considerable use of the algebraic theory of fields. Van der Waerden's most famous work is Algebra published in 1930. This two volume work reports on the algebra developed by Emmy Noether, Hilbert, Dedekind and Artin. In Galois theory he showed the asymptotic result that almost all integral algebraic equations have the full symmetric group as Galois group. He produced results in invariant theory, linear groups, Lie groups and generalised some of Emmy Noether's results on rings. In group theory he studied the Burnside groups B(3, r) with r generators and exponent 3. These are solutions of the Burnside problem. These groups were shown to be finite by Burnside. In 1933 van der Waerden found the exact order and structure of the groups B(3, r). He showed that the order of B(3, r) is 3N(r) where the exponent N(r) = r + r(r-1)/2 + r(r-1)(r-2)/6. Among his many historical books are Ontwakende wetenschap (1950) translated into English as Science Awakening (1954), Geometry and Algebra in Ancient Civilizations (1983) and A History of Algebra (1985). The history of mathematics was not a topic he just turned to late in life: his important paper Die Arithmetik der Pythagoreer appeared in 1947. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Cross-references to History Topics

Greek Astronomy

Other references in MacTutor

Chronology: 1930 to 1940

Other Web sites

1. AMS (An interview with van der Waerden) 2. Clark Kimberling

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Van_der_Waerden.html (2 of 3) [2/16/2002 11:36:32 PM]

Van_der_Waerden

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Van_der_Waerden.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Van_der_Waerden.html (3 of 3) [2/16/2002 11:36:32 PM]

Vandermonde

Alexandre-Théophile Vandermonde Born: 28 Feb 1735 in Paris, France Died: 1 Jan 1796 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alexandre-Theophile Vandermonde's father was a medical doctor who was originally from Landrices but had spent 12 years in the Orient. He had set up a medical practice in Paris and was working there as a doctor when his son Alexandre-Théophile was born. He did not encourage his son to follow a medical profession but rather encouraged him to take up a career in music. Certainly he was not interested in mathematics when he was young. Alexandre-Théophile was awarded his bachelier on 7 September 1755 and his licencie on 7 September 1757. His first love was music and his instrument was the violin. He pursued a music career and he only turned to mathematics when he was 35 years old. It was Fontaine des Bertins whose enthusiasm for mathematics rubbed off on Vandermonde. Perhaps surprisingly he was elected to the Académie des Sciences in 1771 with little evidence of his mathematical genius other than his first paper which, although he was not a member at the time, was read to the Academy in November 1770. However, he did make quite a remarkable contribution to mathematics in this paper and three further papers which he presented to the Academy between 1771 and 1772. These four papers represent his total mathematical output and we will discuss their content below together with the views of a number of historians of mathematics on his contribution. Vandermonde's election to the Académie des Sciences did motivate him to work hard for the Academy and to publish other works on science and music. In 1777 he published the results of experiments he had carried out with Bezout and the chemist Lavoisier on low temperatures, in particular investigating the effects of a very severe frost which had occurred in 1776. Ten years later he published two papers on manufacturing steel, this time joint work with Monge and Bertholet. The aim of this research was to improve the steel used for bayonets but experimenting with different mixtures of iron and carbon. That he work closely with Monge reflected the fact that the two were very close friends, in fact he so close that he was known as femme de Monge. In 1778 Vandermonde presented the first of a two part work on the theory of music to the Académie des Sciences. The second part was presented two years later. This work Système d'harmonie applicable à l'état actuel de la musique did not propose a mathematical theory of music as one might have expected from someone who was an expert in both fields. On the contrary the aim of the work was to put forward the idea that musicians should ignore all theory of music and rely solely on their trained ears when judging music. As one might expect this proved a controversial work with musicians being sharply http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vandermonde.html (1 of 4) [2/16/2002 11:36:33 PM]

Vandermonde

divided as to whether they agreed with Vandermonde or not. Despite the opposition of many musicians at first, the ideas put forward by Vandermonde gained favour over the years and by the beginning of the nineteenth century the Académie des Sciences had moved music from the mathematical area to the arts area. It is worth repeating that it is strange that a mathematician of the highest rank should have argued against music as a mathematical art, a position it had held since the days of ancient Greece. Positions which Vandermonde held include director of the Conservatoire des Arts et Métiers in 1782 and chief of the Bureau de l'Habillement des Armées in 1792. In the same year of 1792 he sat with Lagrange on a committee of the Académie des Sciences which had to examine the violon harmonique, a newly invented musical instrument. He was involved with the Ecole Normale, which was founded in October 1794, and was on the team designing a course in political economy. His friend Monge was also involved with the Ecole Normale as were Lagrange and Laplace. However the establishment only operated for six months after it opened in the Muséum d'Histoire Naturelle in January 1795 before being closed down. Like Monge, Vandermonde was a strong supporter of the Revolution which began with the storming of the Bastille on 14 July 1789. The politics of Revolution in France long before this event had been so exciting for Vandermonde that it diverted him from a possible longer mathematical and scientific career. However the truth of the matter is that he suffered from poor health all his life and, but for this, he might well have been able to be highly involved in politics yet continue with mathematical and scientific activities. Perhaps the name of Vandermonde is best known today for the Vandermonde determinant. Now it is certainly true that he made a major contribution to the theory of determinants yet nowhere in his four mathematical papers does this determinant appear. It is rather strange, therefore, that this determinant should be named after him and several authors have puzzled over the fact for some time. Lebesgue's conjecture in [3] (first published in 1940) that it resulted for someone misreading Vandermonde's notation, and therefore believing that this determinant was in his work, seems the most likely. Vandermonde's four mathematical papers, with their dates of publication by the Académie des Sciences, were Mémoire sur la résolution des équations (1771), Remarques sur des problèmes de situation (1771), Mémoire sur des irrationnelles de différens ordres avec une application au cercle (1772), and Mémoire sur l'élimination (1772). The first of these four papers presented a formula for the sum of the mth powers of the roots of an equation. It also presented a formula for the sum of the symmetric functions of the powers of such roots. Neither of these were new having appeared in Waring's work shortly before but, although he was aware of this Vandermonde claimed, rightly in my [EFR] opinion, that his approach was sufficiently different to make publication of these results for a second time worthwhile. The paper also shows that if n is a prime less than 10 the equation xn - 1 = 0 can be solved in radicals. Jones writes in [1]:... Vandermonde's real and unrecognised claim to fame was lodged in his first paper, in which he approached the general problem of the solubility of algebraic equations through a study of functions invariant under permutations of the roots of the equation. Kronecker claimed in 1888 that the study of modern algebra began with this first paper of Vandermonde. Cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea which eventually led to the study of group theory.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vandermonde.html (2 of 4) [2/16/2002 11:36:33 PM]

Vandermonde

In his second paper Vandermonde considered the problem of the knight's tour on the chess board. This paper is an early example of the study of topological ideas. Vandermonde considers the intertwining of the curves generated by the moving knight and his work in this area marks the beginning of ideas which would be extended first by Gauss and then by Maxwell in the context of electrical circuits. In his third paper Vandermonde studied combinatorial ideas. He defined the symbol [p]n = p(p-1)(p-2)(p-3)...(p-n+1) and [p]-n = 1/(p+1)(p+2)(p+3)...(p+n). He gave an identity for the expansion of [x + y]n and also proved that /2 = [1/2]1/2.[-1/2]-1/2. It is interesting to note that at this time no notation existed for n! yet with his notation Vandermonde had defined something more general. Clearly [n]n = n! The final of Vandermonde's four papers studied the theory of determinants. Muir [4] claims that because of this paper Vandermonde was:The only one fit to be viewed as the founder of the theory of determinants. The reason for this strong claim by Muir is that, although mathematicians such as Leibniz had studied determinants earlier than Vandermonde, all earlier work had simply used the determinant as a tool to solve linear equations. Vandermonde, however, thought of the determinant as a function and gave properties of the determinant function. He showed the effect of interchanging two rows and of interchanging two columns. From this he deduced that a determinant with two identical rows or two identical columns is zero. Finally he gave a remarkably clever notation for determinants which has not survived. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Mathematical games and recreations 2. Matrices and determinants

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vandermonde.html (3 of 4) [2/16/2002 11:36:33 PM]

Vandermonde

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Vandermonde.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vandermonde.html (4 of 4) [2/16/2002 11:36:33 PM]

Vandiver

Harry Schultz Vandiver Born: 21 Oct 1882 in Philadelphia, Pennsylvania, USA Died: 9 Jan 1973 in Austin, Texas, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Harry Vandiver developed an antagonism to public education and left school at an early age to work for his father's firm. In 1904 he collaborated with Birkhoff on a paper on the prime factors of an - bn. Birkhoff persuaded him to accept a post at Cornell University in 1919. He worked during the summer with Dickson at Chicago on the History of the Theory of Numbers. In 1924 he moved to the University of Texas. Vandiver was awarded the Cole Prize by the American Mathematical Society in 1931 for his papers on Fermat's last theorem published in the Transactions of the American Mathematical Society and in the Annals of Mathematics during the preceding five years. In particular special mention was made of the paper entitled On Fermat's last theorem published in the Transactions of the American Mathematical Society in 1929. In 1952 Vandiver used a computer to study Fermat's Last Theorem and was able to prove it for all primes less than 2000. It is this work on Fermat's Last Theorem for which he is best known. Vandiver never owned a house and lived with his wife in the Alamo Hotel where he had a large collection of classical recordings. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles)

A Quotation

Mathematicians born in the same country Cross-references to History Topics

Fermat's last theorem

Honours awarded to Harry S Vandiver (Click a link below for the full list of mathematicians honoured in this way) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vandiver.html (1 of 2) [2/16/2002 11:36:35 PM]

Vandiver

AMS Colloquium Lecturer

1935

AMS Cole Prize

Awarded 1931

Other Web sites

University of Texas

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vandiver.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vandiver.html (2 of 2) [2/16/2002 11:36:35 PM]

Varahamihira

Varahamihira Born: 505 in Kapitthaka, India Died: 587 in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Our knowledge of Varahamihira is very limited indeed. According to one of his works, he was educated in Kapitthaka. However, far from settling the question this only gives rise to discussions of possible interpretations of where this place was. Dhavale in [3] discusses this problem. We do not know whether he was born in Kapitthaka, wherever that may be, although we have given this as the most likely guess. We do know, however, that he worked at Ujjain which had been an important centre for mathematics since around 400 AD. The school of mathematics at Ujjain was increased in importance due to Varahamihira working there and it continued for a long period to be one of the two leading mathematical centres in India, in particular having Brahmagupta as its next major figure. The most famous work by Varahamihira is the Pancasiddhantika (The Five Astronomical Canons) dated 575 AD. This work is important in itself and also in giving us information about older Indian texts which are now lost. The work is a treatise on mathematical astronomy and it summarises five earlier astronomical treatises, namely the Surya, Romaka, Paulisa, Vasistha and Paitamaha siddhantas. Shukla states in [11]:The Pancasiddhantika of Varahamihira is one of the most important sources for the history of Hindu astronomy before the time of Aryabhata I I. One treatise which Varahamihira summarises was the Romaka-Siddhanta which itself was based on the epicycle theory of the motions of the Sun and the Moon given by the Greeks in the 1st century AD. The Romaka-Siddhanta was based on the tropical year of Hipparchus and on the Metonic cycle of 19 years. Other works which Varahamihira summarises are also based on the Greek epicycle theory of the motions of the heavenly bodies. He revised the calendar by updating these earlier works to take into account precession since they were written. The Pancasiddhantika also contains many examples of the use of a place-value number system. There is, however, quite a debate about interpreting data from Varahamihira's astronomical texts and from other similar works. Some believe that the astronomical theories are Babylonian in origin, while others argue that the Indians refined the Babylonian models by making observations of their own. Much needs to be done in this area to clarify some of these interesting theories. In [1] Ifrah notes that Varahamihira was one of the most famous astrologers in Indian history. His work Brihatsamhita (The Great Compilation) discusses topics such as [1]:... descriptions of heavenly bodies, their movements and conjunctions, meteorological

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Varahamihira.html (1 of 3) [2/16/2002 11:36:36 PM]

Varahamihira

phenomena, indications of the omens these movements, conjunctions and phenomena represent, what action to take and operations to accomplish, sign to look for in humans, animals, precious stones, etc. Varahamihira made some important mathematical discoveries. Among these are certain trigonometric formulas which translated into our present day notation correspond to sin x = cos( /2 - x), sin2x + cos2x = 1, and (1 - cos 2x)/2 = sin2x. Another important contribution to trigonometry was his sine tables where he improved those of Aryabhata I giving more accurate values. It should be emphasised that accuracy was very important for these Indian mathematicians since they were computing sine tables for applications to astronomy and astrology. This motivated much of the improved accuracy they achieved by developing new interpolation methods. The Jaina school of mathematics investigated rules for computing the number of ways in which r objects can be selected from n objects over the course of many hundreds of years. They gave rules to compute the binomial coefficients nCr which amount to nCr

= n(n-1)(n-2)...(n-r+1)/r!

However, Varahamihira attacked the problem of computing nCr in a rather different way. He wrote the numbers n in a column with n = 1 at the bottom. He then put the numbers r in rows with r = 1 at the left-hand side. Starting at the bottom left side of the array which corresponds to the values n = 1, r = 1, the values of nCr are found by summing two entries, namely the one directly below the (n, r) position and the one immediately to the left of it. Of course this table is none other than Pascal's triangle for finding the binomial coefficients despite being viewed from a different angle from the way we build it up today. Full details of this work by Varahamihira is given in [5]. Hayashi, in [6], examines Varahamihira's work on magic squares. In particular he examines a pandiagonal magic square of order four which occurs in Varahamihira's work. Article by: J J O'Connor and E F Robertson List of References (12 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 500 to 900

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Varahamihira.html (2 of 3) [2/16/2002 11:36:36 PM]

Varahamihira

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Varahamihira.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Varahamihira.html (3 of 3) [2/16/2002 11:36:36 PM]

Varignon

Pierre Varignon Born: 1654 in Caen, France Died: 23 Dec 1722 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Varignon was educated at the Jesuit College in Caen. He then studied at the University of Caen where he received his M.A. in 1682. The following year he became a priest in Caen. Led into mathematics by reading Euclid, he read Descartes' Géométrie and thereafter devoted himself to mathematical sciences. Varignon became professor of mathematics at the Collège Mazarin, Paris in 1688, then in 1704 in addition to the chair at Collège Mazarin, he became professor of mathematics at the Collège Royal. His lectures at the Collège Mazarin were published in Eléments de mathematiques (1731). Varignon's chief contributions were to graphical statics and mechanics. He published Projet d'une nouvelle méchanique in 1687. He was one of the first French scholars to recognise the value of the calculus and developed analytic dynamics by adapting Leibniz's calculus to the inertial mechanics of Newton's Principia. Among his other work was a publication in 1699 on applications the differential calculus to fluid flow and to water clocks. In 1702 he applied the calculus to clocks driven by a spring. Varignon was elected to the Académie Royal des Sciences in 1688, the Berlin Academy in 1713 and the Royal Society in 1718. Article by: J J O'Connor and E F Robertson List of References (11 books/articles) Mathematicians born in the same country Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Varignon.html (1 of 2) [2/16/2002 11:36:37 PM]

Varignon

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Varignon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Varignon.html (2 of 2) [2/16/2002 11:36:37 PM]

Vashchenko

Mikhail Egorovich Vashchenko-Zakharchenko Born: 12 Nov 1825 in Maliivka, Zolotonosha, Poltava, Ukraine Died: 27 Aug 1912 in Kiev, Ukraine

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vashchenko-Zakharchenko attended Kiev University, then went to France where he studied at the Sorbonne and then he spent academic year 1847-48 studying at the College de France. After returning from France, Vashchenko-Zakharchenko continued his studies at Kazan University where he obtained a further degree in 1854. After this he taught at Kiev Cadet School from 1855 to 1862, receiving his Master's Degree in 1862 for a dissertation on the operational method and its application to solving linear differential equations. After receiving his Master's Degree, Vashchenko-Zakharchenko was appointed to the University of Kiev in 1863. He remained there for the rest of his life, retiring from teaching in 1902. Vashchenko-Zakharchenko made contributions to several areas of mathematics. In particular he worked on the theory of linear differential equations, the theory of probability (see [1]) and non-euclidean geometry. He exerted great influence on a number of other mathematicians who joined him at Kiev, and his interests were taken up by others there, particularly Bukreev. With wide ranging interests, it was natural for Vashchenko-Zakharchenko to write on a variety of topics and his twelve textbooks indeed cover many different topics. He wrote an important work on the history of mathematics in 1883 in which he discussed the history of mathematics up to the 15th century. He wrote on other historical topics too, for example he wrote a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vashchenko.html (1 of 2) [2/16/2002 11:36:39 PM]

Vashchenko

history of the development of analytic geometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Vashchenko-Zakharchenko

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vashchenko.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vashchenko.html (2 of 2) [2/16/2002 11:36:39 PM]

Veblen

Oswald Veblen Born: 24 June 1880 in Decorah, Iowa, USA Died: 10 Aug 1960 in Brooklyn, Maine, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oswald Veblen made important contributions to projective and differential geometry, and topology. Veblen attended school in Iowa City before entering the University of Iowa in 1894 receiving his A.B. in 1898. After a year spent as a laboratory assistant, Veblen spent a year at Harvard University before going to the University of Chicago to undertake research. Archibald writes in [4]:He received the major part of his mathematical training at the University of Chicago from that inspiring trio Bolza, Maschke, and Eliakim Moore. Under their direction he laid the basis for the important work he was later to achieve in the fields of foundations of geometry, projective geometry, topology, differential invariants and spinors. His often quoted dissertation under Eliakim Moore, on a system of axioms of Euclidean geometry, followed the trend of development of Pasch (1882) and Peano (1889, 1894) rather than that of Hilbert (1899) and Pieri (1899) Veblen's doctoral dissertation was entitled A System of Axioms for Geometry and he was awarded his doctorate from the University of Chicago in 1903. He taught mathematics at Princeton University from 1905 to 1932. In the academic year 1928-29 he taught at Oxford as part of an exchange with G H Hardy. In 1932 he helped organise the Institute for Advanced Study in Princeton and he became a professor there in 1932. Veblen's first work on topology appeared just before he arrived in Princeton and Veblen went on to establish Princeton as one of the leading centres in the World for topology research.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Veblen.html (1 of 3) [2/16/2002 11:36:41 PM]

Veblen

His interest in the foundations of geometry led to his work on the axiom systems of projective geometry. Together with John Wesley Young he published Projective geometry (1910-18). The introduction to this work justifies the study of foundations:Even the limited space devoted in this volume to the foundations may seem a drawback from the pedagogical point of view to some mathematicians. To this we can only reply that, in our opinion, an adequate knowledge of geometry cannot be obtained without attention to its foundations. We believe, moreover, that the abstract treatment is particularly desirable in projective geometry, because it is through the latter that the other geometric disciplines are most readily coordinated. Since it is most natural to derive the geometrical disciplines associated with the names of Euclid, Descartes, Lobachevsky etc. from projective geometry than to derive projective geometry from one of them, it is natural to take the foundations of projective geometry as the foundations of all geometry. Veblen's Analysis Situs (1922) provided the first systematic coverage of the basic ideas of topology and contributed to the development of modern topology. Soon after Einstein's theory of general relativity appeared Veblen turned his attention to differential geometry. This work led to important applications in relativity theory, and much of his work also found application in atomic physics. His work The invariants of quadratic differential forms (1927) is a systematic treatment of Riemann geometry while his work, written jointly with his student Henry Whitehead, The foundations of differential geometry (1933) gives the first definition of a differentiable manifold. In Projective relativity theory (1933) he gave a new treatment of spinors, used to represent electron spin. Veblen was an active member of the American Mathematical Society, serving the Society as Vice-President in 1915 and President in 1923-24. He was the Colloquium Lecturer for the Society in 1916 when he gave a series of lectures on topology. He was honoured with memberships of other societies around the World. For example he was a member of the London Mathematical Society, serving on the council in 1928 when he was replacing Hardy at Oxford. Oxford further honoured him with an Honorary D.Sc. in 1929, while in the same year he was honoured by the University of Oslo on the occasion of the centenary celebrations for Abel. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) A Poster of Oswald Veblen Cross-references to History Topics

Mathematicians born in the same country 1. The four colour theorem 2. Mathematical games and recreations

Honours awarded to Oswald Veblen (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Veblen.html (2 of 3) [2/16/2002 11:36:41 PM]

Veblen

Fellow of the Royal Society of Edinburgh American Maths Society President

1923 - 1924

AMS Colloquium Lecturer

1916

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Veblen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Veblen.html (3 of 3) [2/16/2002 11:36:41 PM]

Vega

Georg Freiherr von Vega Born: 23 Mar 1754 in Zagorica, Ljubljana, Slovenia Died: 26 Sept 1802 in Vienna, Austria

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

The parents of Jurij Vega (Georg is the German version of his name)were poor farmers, his father dying when Jurij was 6 years old. He attended school in Ljubljana until the age of 19 when he became a navigational engineer. He entered military service in 1780 as professor of mathematics at the Artillery School in Vienna. He was a talented teacher and writer, with a great skill as a calculator. Vega wrote on artillery but he is best remembered for his tables of logarithms and trigonometric functions. His seven figure log tables were based on those of Vlacq and were famed for their accuracy. The calculations were done with the help of soldiers who were given a gold ducat for every mistake they found. His first book of logarithms appeared in 1783. The tables Thesaurus logarithmorm completus appeared in 1794 and the 90th edition appeared in 1924. Vega wrote a four volume textbook Vorlesungen über die Mathematik (1782, 1784, 1788, 1800). This book also contains trigonometric tables. Vega calculated to 140 places, a record which stood for over 50 years. This appears in a paper which he published in 1789. As a military man Vega was involved in several wars. In 1788 he fought against the Turks at a battle near Belgrade commanding gun positions. He also fought against the French at many battles in this time of revolution and wars in Europe. In September 1802 Jurij Vega was reported missing. A search was unsuccessful until his body was found in the Danube near Vienna. The official cause of death was an accident but many suspect that he was

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vega.html (1 of 2) [2/16/2002 11:36:43 PM]

Vega

murdered. The picture displayed is from a Slovenian 50 Tolar bank note issued in his honour. A Slovenian stamp has also been issued to honour him. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) A Poster of Jurij Vega

Mathematicians born in the same country

Cross-references to History Topics

1. Pi through the ages 2. A chronology of pi

Honours awarded to Jurij Vega (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Vega

Other Web sites

Ljubljana, Slovenia

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vega.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vega.html (2 of 2) [2/16/2002 11:36:43 PM]

Venn

John Venn Born: 4 Aug 1834 in Hull, England Died: 4 April 1923 in Cambridge, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Venn came from a Low Church Evangelical background and when he entered Gonville and Caius College Cambridge in 1853 he had so slight an acquaintance with books of any kind that he may be said to have begun there his knowledge of literature. He graduated in 1857, was elected a Fellow in that year and two years later was ordained a priest. For a year he was curate at Mortlake. In 1862 he returned to Cambridge University as a lecturer in Moral Science, studying and teaching logic and probability theory. He developed Boole's mathematical logic and is best known for his diagrammatic way of representing sets and their unions and intersections. Venn considered three discs R, S, and T as typical subsets of a set U. The intersections of these discs and their complements divide U into 8 nonoverlapping regions, the unions of which give 256 different Boolean combinations of the original sets R, S, T. Venn wrote Logic of Chance in 1866 which Keynes described as strikingly original and considerably influenced the development of the theory of statistics. Venn published Symbolic Logic in 1881 and The Principles of Empirical Logic in 1889. The second of these is rather less original but the first was described by Keynes as probably his most enduring work on logic. In 1883 Venn was elected a Fellow of the Royal Society. About this time his career changed direction. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Venn.html (1 of 2) [2/16/2002 11:36:45 PM]

Venn

He had already left the Church in 1870 but his interest now turned to history. He wrote a history of his college, publishing The Biographical History of Gonville and Caius College 1349-1897 in 1897. He then undertook the immense task of compiling a history of Cambridge University which, the first volume of which was published in 1922. He was assisted by his son in this task which was described by another historian in these terms: It is difficult for anyone who has not seen the work in its making to realise the immense amount of research involved in this great undertaking. Venn had other skills and interests too, including a rare skill in building machines. He used his skill to build a machine for bowling cricket balls which was so good that when the Australian Cricket team visited Cambridge in 1909, Venn's machine clean bowled one of its top stars four times. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1880 to 1890

Honours awarded to John Venn (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1883

Other Web sites

Frank Ruskey

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Venn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Venn.html (2 of 2) [2/16/2002 11:36:45 PM]

Verhulst

Pierre François Verhulst Born: 28 Oct 1804 in Brussels, Belgium Died: 15 Feb 1849 in Brussels, Belgium Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Verhulst was educated in Brussels, then in 1822 he entered the University of Ghent. He received his doctorate in 1825 after only three years study and returned to Brussels. There he worked on the theory of numbers, and, influenced by Quetelet, he became interested in social statistics. He had been intending to publish the complete works of Euler but he became more and more interested in social statistics. In 1829 Verhulst published a translation of John Herschel's Theory of light. However he became ill and decided to travel to Italy in the hope that his health would improve. In 1830 Verhulst arrived in Rome. However his visit there was not a quiet one. Quetelet wrote:Whilst on a trip to Rome he conceived the idea of carrying out reform in the Papal States and of persuading the Holy Father to give a constitution to his people. This plan did not meet with approval and Verhulst was ordered to leave Rome. He returned to Belgium. On 28 September 1835 Verhulst was appointed professor of mathematics at the Université Libre of Brussels. There he gave courses on astronomy, celestial mechanics, the differential and integral calculus, the theory of probability, geometry and trigonometry. In 1840 Verhulst moved to the military school, the Ecole Royale Militaire. He continued to be influenced by Quetelet although he was not always in agreement with Quetelet's ideas. Verhulst's research on the law of population growth is important. The assumed belief before Quetelet and Verhulst worked on population growth was that an increasing population followed a geometric progression. Quetelet believed that there are forces which tend to prevent this population growth and that they increase with the square of the rate at which the population grows. Verhulst showed in 1846 that forces which tend to prevent a population growth grow in proportion to the ratio of the excess population to the total population. The non-linear differential equation describing the growth of a biological population which he deduced and studied is now named after him. Based on his theory Verhulst predicted the upper limit of the Belgium population would be 9,400,000. In http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Verhulst.html (1 of 2) [2/16/2002 11:36:47 PM]

Verhulst

fact the population in 1994 was 10,118,000 and, but for the affect of immigration, his prediction looks good. In 1841 Verhulst was elected to the Belgium Academy and in 1848 he became its president. However, the bad health which he had suffered from earlier returned to make his life increasing difficult over the last years of his life. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Verhulst.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Verhulst.html (2 of 2) [2/16/2002 11:36:47 PM]

Vernier

Pierre Vernier Born: 19 Aug 1584 in Ornans, Franche-Comté, Spanish Habsburgs (now France) Died: 14 Sept 1638 in Ornans, Franche-Comté, Spanish Habsburgs (now France) Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Vernier was taught mathematics and science by his father who was a lawyer and engineer who held government office. His father introduced Pierre to the works of Clavius and Brahe. Being born in Franche-Comté (Free Country) meant that Vernier (and his father) were involved, not with the government of France but with that of Spain. Franche-Comté was a Habsburg possession controlled by the Spanish Habsburgs throughout Vernier's life. In fact the period from 1598 to 1635 was one of peace. Vernier became a government official holding various positions such as military engineer for the Hapsburgs and director general of the treasury in Dole and Besançon, the capital of Franche- Comté. Vernier also held various government posts with the government of Spain and became a Conseiller du Roi. He worked for much of the time as an engineer, working on the fortifications of various cities. In 1623 he was given the title of citizen from the city of Besançon for his work on the defences of the city. In fact the threat of war was never far away and during the last two years of Vernier's life Franche-Comté was frequently invaded by France. Like many other mathematicians and scientists of this period, Vernier worked on cartography and on surveying. He collaborated with his father in making a map of the Franche-Comté area. His interest in surveying led to develop instruments for surveying and this prompted the invention for which he is remembered by all scientists. His most famous publication is La Construction, l'usage, et les propriétés du quadrant nouveau de mathématiques (1631). In this book Vernier gives a table of sines and a method for deriving the angles of a triangle if its sides are known. He also describes his most famous invention, that of the vernier caliper, an instrument for accurately measuring length. It has two graduated scales, a main scale like a ruler and a second scale, the vernier, that slides parallel to the main scale and enables readings to be made to a fraction of a division on the main scale.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vernier.html (1 of 2) [2/16/2002 11:36:48 PM]

Vernier

Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Pierre Vernier (Click a link below for the full list of mathematicians honoured in this way) Paris street names

Rue Vernier (17th Arrondissement)

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vernier.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vernier.html (2 of 2) [2/16/2002 11:36:48 PM]

Veronese

Giuseppe Veronese Born: 7 May 1854 in Chioggia, Italy Died: 17 July 1917 in Padua, Italy

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giuseppe Veronese was born and brought up in Chioggia which was a small fishing village not far from Venice. His father was a house painter and his mother Elenora Duse was related to a famous actress. However, it was a poor family and they could not afford to finance Giuseppe through university. In 1872, at age eighteen, Veronese had give up his education to take a job in Vienna. Veronese was fortunate, however, and the following year he was able to begin his studies again. He found himself a patron in Count Nicol˜ Papadopoli who supported him financially so that he could go to Zurich Polytechnic in 1873. There he set out on a course of study which involved both engineering and mathematics. However, he began to correspond with Cremona, who was at the University of Rome, on mathematical topics. He started work on a paper on Pascal's hexagram but, following Cremona's advice, he moved to Rome to complete his undergraduate degree. In 1876 Veronese was appointed as assistant in analytical geometry on the strength of his paper on Pascal's hexagram which he had completed by this time. This is quite remarkable for one should remember that at this point Veronese was still studying the undergraduate course in Rome. He graduated in 1877 and continued to work for his doctorate in Rome. Veronese was in contact with Klein who was about to take up a chair of geometry at the University of Leipzig. It was arranged that Veronese would go to Leipzig in 1880 and to spend the year 1880-81 undertaking research under Klein. Bellavitis died in November 1880 and his chair of algebraic geometry in Padua became vacant. The chair was filled by holding a competition which Veronese won and he was appointed to the chair in 1881. He held this chair throughout his life. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Veronese.html (1 of 3) [2/16/2002 11:36:50 PM]

Veronese

Freguglia, in [5], describes Veronese's study of geometry in higher dimensions. In 1880 Veronese described an n-dimensional projective geometry, showing that simplifications could be obtained in passing to higher dimensions. He illustrated the fact that difficulties arose when a simple surface in high dimension was projected onto 3-space. This was a very original approach to higher-dimensional projective geometry that Veronese developed. He is certainly considered to be one of the founders of that topic for with him what others had considered as linear algebra viewed geometrically became geometry. This original approach was based on the supremacy of geometric intuitive techniques over the analytic and algebraic viewpoints. Veronese provided both logical and psychological motivations for his approach which greatly influenced the Italian school of geometry for many years. Veronese invented non-Archimedean geometries which he proposed around 1890. However Peano strongly criticised the notion due to the lack of rigour of Veronese's description and also for the fact that he did not justify his use of infinitesimal and infinite segments. The resulting argument was extremely useful to mathematics since it helped to clarify the notion of the continuum. Any fears that non-Archimedean systems would not be consistent were shown to unnecessary soon after this when Hilbert proved that indeed they were consistent. We should mention one or two further aspects of Veronese's life. He wrote a number of useful secondary school texts on mathematics and he also became involved in politics. He served as a member of the Parliament of Chioggia from 1897 to 1900, then later he served as a member of the Padua City Council, finally being a Senator from 1904 until his death. He had two particularly famous pupils. Castelnuovo, one of the greatest algebraic geometers of the Italian school, was his pupil in the mid 1880s and Levi-Civita was one of his pupils about ten years later. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Veronese.html (2 of 3) [2/16/2002 11:36:50 PM]

Veronese

JOC/EFR May 2000

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Veronese.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Veronese.html (3 of 3) [2/16/2002 11:36:50 PM]

Vessiot

Ernest Vessiot Born: 8 March 1865 in Marseilles, France Died: 17 Oct 1952 in La Bauche, Savoie, France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernest Vessiot's father was a school teacher, then later he was appointed inspector general of primary schools. Vessiot therefore came from an academic background. He attended the lycée at Marseilles, then sat the entrance examination for the Ecole Normale Supérieure in Paris. In the entrance examination Vessiot was placed second to Hadamard and thereafter he studied in the same class as Hadamard. After graduating from the Ecole Normale Supérieure, Vessiot accepted a teaching post at Lyon in 1887. In 1892 he submitted his doctoral dissertation on groups of linear transformations, in particular studying the action of these groups on the independent solutions of a differential equation. After the award of his doctorate, Vessiot taught in a number of places, Lille, Toulouse, Lyon and finally Paris in 1910. He was appointed to the prestigious post of Director of the Ecole Normale Supérieure in Paris and he continued to hold this post until he retired in 1935. In his role of director he supervised the construction of new physical laboratories at the Ecole Normale Supérieure. Vessiot applied continuous groups to the study of differential equations. He extended results of Drach (1902) and Cartan (1907) and also extended Fredholm integrals to partial differential equations. Vessiot was assigned to ballistics during World War I and made important discoveries in this area. He was honoured by election to the Académie des Sciences in 1943. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vessiot.html (1 of 2) [2/16/2002 11:36:52 PM]

Vessiot

Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vessiot.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vessiot.html (2 of 2) [2/16/2002 11:36:52 PM]

Viete

François Viète Born: 1540 in Fontenay-le-Comte, Poitou (now Vendée), France Died: 13 Dec 1603 in Paris, France

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

François Viète's father was Etienne Viète, a lawyer in Fontenay-le-Comte in western France about 50 km east of the coastal town of La Rochelle. François' mother was Marguerite Dupont. He attended school in Fontenay-le-Comte and then moved to Poitiers, about 80 km east of Fontenay-le-Comte, where he was educated at the University of Poitiers. Given the occupation of his father, it is not surprising that Viète studied law at university. After graduating with a law degree in 1560, Viète entered the legal profession but he only continued on this path for four years before deciding to change his career. In 1564 Viète took a position in the service of Antoinette d'Aubeterre. He was employed to supervise the education of Antoinette's daughter Catherine, who would later become Catherine of Parthenay (Parthenay is about half-way between Fontenay-le-Comte and Poitiers). Catherine's father died in 1566 and Antoinette d'Aubeterre moved with her daughter to La Rochelle. Viète moved to La Rochelle with his employer and her daughter. This was a period of great political and religious unrest in France. Charles IX had become king of France in 1560 and shortly after, in 1562, the French Wars of Religion began. It is a gross over-simplification to say that these wars were between Protestants and Roman Catholics but fighting between the various factions would continue on and off until almost the end of the century. In 1570 Viète left La Rochelle and moved to Paris. Although he was never employed as a professional scientist or mathematician, Viète was already working on topics in mathematics and astronomy and his first published mathematical work appeared in Paris in 1571. While Viète was in Paris, Charles IX authorised the massacre of the Huguenots, who were an increasingly powerful group of French Protestants, on 23 August 1572. This must have been an extremely difficult time for Viète for, although not active in the Protestant cause, he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Viete.html (1 of 6) [2/16/2002 11:36:54 PM]

Viete

was a Huguenot himself. Charles did not live very long after this event, the massacre apparently haunting him for the rest of his life. However, on 24 October 1573 Charles appointed Viète to the government of Brittany which was based at Rennes. Viète moved to Rennes to take up his position of counsellor there. He remained at Rennes until March 1580 when he returned to Paris. Charles IX had died on 30 May 1574 and, on Charles' death Henry III became king. Henry made concessions to the Protestant Huguenots in 1576 and the Roman Catholics formed the Holy League to look after their own interests by military actions. In this tense atmosphere Viète was appointed by Henry III as royal privy counsellor on 25 March 1580, and he was attached to the parliament in Paris. In 1584 the Holy League was strengthened when Henry III's brother died and the Protestant Henry of Navarre became heir to the throne. Fearing that the Protestants might gain control in France, the Holy League fought more vigorously for the Roman Catholic cause. The royal court contained factions with different political aims and in 1584 Viète's position as a known Huguenot became untenable and he was banished by his political enemies from the court. Leaving Paris, Viète went to Beauvoir-sur-Mer, on the coast about 130 km northwest of his home town of Fontenay-le-Comte. During the five years that he spent at Beauvoir-sur-Mer, Viète was able to devote himself entirely to his mathematical studies. In many ways Viète's enemies did mathematics a favour, for it was during this period without formal duties that Viète's most important mathematics was done. In 1587, Henry of Navarre, defeated the army of Henry III. A rising of the people of Paris, a Holy League stronghold, on 12 May 1588, caused the king to flee to Chartres. At this stage Henry III sent for Viète and, in April 1589, brought him back into his parliament which was now set up at Tours. Henry III was reconciled with Henry of Navarre (since it suited them to combine forces) and together they tried to retake Paris in 1589. Henry III was, however, assassinated by a Jacobin friar on 1 August of that year. Philip II of Spain, a champion of the Roman Catholic Counter-Reformation, supported the Holy League by sending money and troops to France. After the murder of Henry III, Philip claimed the throne of France for his daughter, Isabella Clara Eugenia. A letter to Philip dated 28 October 1589 written in code fell into the hands of Henry of Navarre who was to become the next king, Henry IV. Following the assassination of Henry III, Viète worked for Henry IV. He was now in a sounder position, as a Protestant supporter of a Protestant King. Viète was certainly well known for his mathematical abilities by this time and, as one of the Henry IV's most loyal supporters, it was natural for Henry to turn to Viète to decode messages being sent to his enemy Philip II of Spain. It took Viète some time to crack the complicated code. At first he was only able to decode parts of the message and forwarded parts to Henry IV, but eventually Viète sent him the fully decoded message on 15 March 1590. However [2]:... when Philip, assuming that the cipher could not be broken, discovered that the French were aware of his military plans, he complained to the Pope that black magic was being employed against his country. Although Viète was never a professional mathematician, he did lecture on mathematics. For instance in 1592 he lectured at Tours and discussed recent claims that the circle could be squared, an angle trisected, and the cube doubled using only ruler and compass. He showed in these lectures that the "proofs" which had been published earlier in the year were fallacious. In 1592 Henry IV did not control Paris, and he was still opposed by the Holy League in France who were http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Viete.html (2 of 6) [2/16/2002 11:36:54 PM]

Viete

supported by Spain. Henry converted back to Roman Catholicism in July 1593, perhaps for political rather than religious reasons. Viète followed the example of his king and also converted to Roman Catholicism. Henry's conversion was certainly effective for resistance against him lessened and he took Paris on 22 March 1594. Henry declared war on Philip II of Spain in January 1595 and continued to wipe out resistance by the League and its Spanish allies. During the period referred to in the previous paragraph, Viète had again come to the King's rescue by solving a mathematical problem. In 1593 Roomen had proposed a problem which involved solving an equation of degree 45. The ambassador from the Netherlands made comments to Henry IV on the poor quality of French mathematicians saying that no Frenchman could solve Roomen's problem. Henry put the problem to Viète who solved it by realising that there was an underlying trigonometric relation. As a result of this a friendship grew up between Viète and Roomen. Viète proposed the problem of drawing a circle to touch 3 given circles to Roomen (the Apollonian Problem) and Roomen solved it using hyperbolas, publishing the result in 1596. Viète himself, published his answer to Roomen's problem in 1595, stating in the introduction [1]:I, who do not profess to be a mathematician, but who, whenever there is leisure, delight in mathematical studies ... Viète continued to serve Henry IV in Paris until 1597 when he went back to his home town of Fontenay-le-Comte. Two years later he was back in Paris, again in the service of Henry IV, but he was dismissed by Henry on 14 December 1602. He died almost exactly one year later. Some of Viète's first work was directed towards the production of a major work on mathematical astronomy Ad harmonicon coeleste. It was a work which was never published but four manuscripts, one of them an autograph, have survived and were rediscovered by Libri. The contents of these manuscripts are described in [22] where it is stated that Viète was interested purely in the geometry of the planetary theories of both Ptolemy and Copernicus, and did not consider the question of whether the theories represented the actual physical reality. Perhaps rather surprisingly Viète came to the conclusion that Copernicus's theory was not valid geometrically. Although the Ad harmonicon coeleste was never published, Viète did begin publishing the Canon Mathematicus in 1571 which was intended as a mathematical introduction to the astronomy treatise. The Canon Mathematicus covers trigonometry; it contains trigonometric tables, it also gives the mathematics behind the construction of the tables, and it details how to solve both plane and spherical triangles. It is interesting that in the second part of the Canon Mathematicus Viète [1]:... wrote decimal fractions with the fractional part printed in smaller type than the integral and separated from the latter by a vertical line. Viète introduced the first systematic algebraic notation in his book In artem analyticam isagoge published at Tours in 1591. The title of the work may seem puzzling, for it means "Introduction to the analytic art" which hardly makes it sound like an algebra book. However, Viète did not find Arabic mathematics to his liking and based his work on the Italian mathematicians such as Cardan, and the work of ancient Greek mathematicians. One would have to say, however, that had Viète had a better understanding of Arabic mathematics he might have discovered that many of the ideas he produced were already known to earlier Arabic mathematicians. In his treatise In artem analyticam isagoge Viète demonstrated the value of symbols introducing letters http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Viete.html (3 of 6) [2/16/2002 11:36:54 PM]

Viete

to represent unknowns. He suggested using letters as symbols for quantities, both known and unknown. He used vowels for the unknowns and consonants for known quantities. The convention where letters near the beginning of the alphabet represent known quantities while letters near the end represent unknown quantities was introduced later by Descartes in La Gèometrie. This convention is used today, often without people realising that a convention is being used at all. (If I asked for a solution to ax = b nobody asks: "For which quantity do I solve the equation ?") Viète made many improvements in the theory of equations. However, if we are to be strictly accurate we should say that he did not solve equations as such but rather he solved problems of proportionals which he states quite explicitly is the same thing as solving equations. However, he was restricted by a condition of homogeneity of dimension. The problem is that if we ask for a solution of x3 + x = 1 then we ask for the solution to a problem which does not make sense geometrically. For x3 is a cube while x is a line and clearly it makes no sense to add a three dimensional object to a one dimensional object. Viète therefore looked for solutions of equations such as A3 + B2A = B2Z where, using his convention, A was unknown and B and Z were knowns. The dimensions here are "correct" each term being of dimension 3. Viète wrote in the In artem analyticam isagoge (see [7] or [3]):The first and permanent law of equalities or proportions which, because it is conceived from homogeneous quantities is called the law of homogeneous quantities, is this: homogeneous quantities must be compared with homogeneous quantities. He presented methods for solving equations of second, third and fourth degree. He knew the connection between the positive roots of equations and the coefficients of the different powers of the unknown quantity. Perhaps it is worth noting that the word "coefficient" is actually due to Viète. When Viète applied numerical methods to solve equations as he did in De numerosa potestatum he gave methods which were similar to those given by earlier Arabic mathematicians. For example his methods are compared with those of Sharaf al-Din al-Tusi in the paper [11] and [19]. In the first the author argues that although the methods appear to be similar at first sight, there are many important differences. He deduces that the work of Viète is not based on that of Sharaf al-Din al-Tusi. In [19], however, Rashed argues that the methods of Sharaf al-Din al-Tusi and of Viète are very close indeed. Viète also wrote books on trigonometry and geometry such as Supplementum geometriae (1593). He gave geometrical solutions to doubling a cube and trisecting an angle in this book. In 1593 Viète published a second book, which in many ways was motivated by his lecture course at Tours in the previous year (which we mentioned above), examining various problems such as doubling the cube, trisecting an angle and the construction the tangent at any point on an Archimedian spiral. Also, in this book, he calculated to 10 places using a polygon of 6.216 = 393216 sides. He also represented as an infinite product which, as far as is known, is the earliest infinite representation of . Finally we should mention that Viète is often called "the father of algebra". As the author of [9] argues this, on the one hand, is unfair on the many fine algebraists who preceded Viète. On the other hand it is unfair to Viète since his contributions were of much wider mathematical importance. It would also be interesting to know how much Viète's ideas were influenced by those of Harriot. In [3] a quotation from a book about Harriot written in 1900 by H Stevens is given:... it appears that Harriot's system of analytics or algebra was based on that of his friend

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Viete.html (4 of 6) [2/16/2002 11:36:54 PM]

Viete

and correspondent François Viète as Viète's was avowedly based on that of the ancients. ... Full credit was given by Harriot and his friends to the distinguished French mathematician. Although this seems to make Harriot's dependence on Viète clear, one would have to say that the two men give very similar approaches to solving equations algebraically, yet Harriot shows deeper understanding than does Viète. I [EFR] feel that one must allow the possibility that ideas flowed in both directions and that Viète's algebra must have benefited from his correspondence with Harriot.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles)

Some Quotations (3)

A Poster of François Viète

Mathematicians born in the same country

Some pages from publications

Extract from Works (printed 1609) showing the use of decimal fractions. A page from Isagoge (printed 1646)

Cross-references to History Topics

1. Thomas Harriot's manuscripts 2. The fundamental theorem of algebra 3. An overview of the history of mathematics 4. The trigonometric functions 5. Pi through the ages 6. A chronology of pi 7. Quadratic, cubic and quartic equations

Cross-references to Famous Curves

Archimedian spiral

Other references in MacTutor

Chronology: 1500 to 1600

Honours awarded to François Viète (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Vieta

Paris street names

Rue Viète (17th Arrondissement)

Other Web sites

1. The Catholic Encyclopedia 2. The Galileo Project 3. Karen H Parshall 4. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Viete.html (5 of 6) [2/16/2002 11:36:54 PM]

Viete

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR January 2000 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Viete.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Viete.html (6 of 6) [2/16/2002 11:36:54 PM]

Vijayanandi

Vijayanandi Born: about 940 in Benares (now Varanasi), India Died: about 1010 in India Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vijayanandi (or Vijayanandin) was the son of Jayananda. He was born into the Brahman caste which meant he was from the highest ranking caste of Hindu priests. He was an Indian mathematician and astronomer whose most famous work was the Karanatilaka. We should note that there was another astronomer named Vijayanandi who was mentioned by Varahamihira in one of his works. Since Varahamihira wrote around 550 and the Karanatilaka was written around 966, there must be two astronomers both named "Vijayanandi". The Karanatilaka has not survived in its original form but we know of the text through an Arabic translation by al-Biruni. It is a work in fourteen chapters covering the standard topics of Indian astronomy. It deals with the topics of: units of time measurement; mean and true longitudes of the sun and moon; the length of daylight; mean longitudes of the five planets; true longitudes of the five planets; the three problems of diurnal rotation; lunar eclipses, solar eclipses; the projection of eclipses; first visibility of the planets; conjunctions of the planets with each other and with fixed stars; the moon's crescent; and the patas of the moon and sun. The Indians had a cosmology which was based on long periods of time with astronomical events occurring a certain whole number of times within the cycles. This system led to much work on integer solutions of equations and their application to astronomy. In particular there was, according to Aryabhata I, a basic period of 4320000 years called a mahayuga and it was assumed that the sun, the moon, their apsis and node, and the planets reached perfect conjunctions after this period. Hence it was assumed that the periods were rational multiples of each other. All the planets and the node and apsis of the moon and sun had to have an integer number of revolutions in the mahayuga. Many Indian astronomers proposed different values for these integral numbers of revolutions. For the number of revolutions of the apsis and node of the moon per mahayuga, Aryabhata I proposed 488219 and 232226, respectively. However Vijayanandi changed these numbers to 488211 and 232234. The reasons for giving the new numbers is unclear. Like other Indian astronomers, Vijayanandi made contributions to trigonometry and it appears that his calculation of the periods was computed by using tables of sines and versed sines. It is significant that accuracy was need in trigonometric tables to give accurate astronomical theories and this motivated many of the Indian mathematicians to produce more accurate methods of approximating http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vijayanandi.html (1 of 2) [2/16/2002 11:36:56 PM]

Vijayanandi

entries in tables. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vijayanandi.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vijayanandi.html (2 of 2) [2/16/2002 11:36:56 PM]

Vinogradov

Ivan Matveevich Vinogradov Born: 14 Sept 1891 in Milolyub, Velikie Luki, Pskov province, Russia Died: 20 March 1983 in Moscow, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ivan Vinogradov studied at St Petersburg, beginning his studies in 1910. Two of his teachers there, A A Markov and Ya V Uspenskii, both had interests in probability and number theory and Vinogradov's interest in number theory stems from this period. He graduated with his first degree in 1914 and continued his studies, supervised by Uspenskii. His master's degree was completed in 1915 and he worked on quadratic residues. He generalised results of Voronoy on the Dirichlet divisor problem. Vinogradov taught at the State University of Perm from 1918 to 1920. The State University of Perm had been founded in 1916, was called Molotov University for a time, and is now the Gorky State University. He returned to St Petersburg to two posts, one at the polytechnic and the other at the university. He gave a course on number theory at the university which was to be the basis for his famous text on the subject. He was promoted to professor at the university in 1925, becoming head of the probability and number theory section. From around 1930 he became heavily involved with mathematics administration on a national level but his research work was amazingly unaffected by the heavy workload. He moved to Moscow to become the first director of the Steklov Institute in 1934, a post he held until his death. As an indication of his research activity during this period it is worth noting that he published around 12 papers in each of the years 1934 to 1938. The importance of trigonometric sums in the theory of numbers was first shown by Weyl in 1916. In the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vinogradov.html (1 of 3) [2/16/2002 11:36:58 PM]

Vinogradov

1920's the work of Hardy and Littlewood developed Weyl's methods to attack other problems in analytic number theory. However it was Vinogradov who, in a series of papers in the 1930's, brought the method to its full potential. His methods reached their height in Some theorems concerning the theory of prime numbers written in 1937 which provides a partial solution to the Goldbach conjecture. In it Vinogradov proved that every sufficiently large odd integer can be expressed as the sum of three odd primes. In [3] the authors write:He introduced and developed two fundamental methods, which could be briefly described as 'the bilinear form technique' and 'the mean value theorem'. They have enabled progress to be made on a whole range of problems. For example, in what is probably his most celebrated piece of work [Some theorems concerning the theory of prime numbers (1937)], he was able to combine the bilinear form technique with the Hardy-Littlewood method so as to reduce the Goldbach ternary problem to that of checking a finite number of cases. Recent research on the type of problems studied by Vinogradov shows that his methods are still the most powerful available to obtain yet further results. Vinogradov made many other contributions, for example to the theory of distribution of power residues, non-residues, indices and primitive roots. He often returned to the topic of his first research paper on the error term in an asymptotic formula discovered by Gauss. Vinogradov's influence outside the Soviet Union was soon apparent. Even in Landau's three volume work on number theory, published in 1927, prominence is given to Vinogradov's methods. However he seldom travelled outside the Soviet Union although he did visit St Andrews in 1958 as the leader of the Soviet delegation to the International Mathematical Union. He then went on to the International Congress at Edinburgh. He did welcome mathematicians who visited him in Moscow. One such visitor wrote:He was a marvellous and meticulous host. ... No one who has been at his home as a guest can forget his bountiful hospitality. An international conference was held in Moscow to mark his 80th birthday. Vinogradov gave a dinner for the participants at his own expense and personally addressed the invitation cards. The proceeding of the conference were published in 1973 with Vinogradov as editor-in-chief. Always a fit man, and proud of his physical fitness, he remained healthy and active into his early 90s. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (12 books/articles) A Poster of Ivan Vinogradov

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1930 to 1940

Honours awarded to Ivan Vinogradov (Click a link below for the full list of mathematicians honoured in this way) http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vinogradov.html (2 of 3) [2/16/2002 11:36:58 PM]

Vinogradov

Fellow of the Royal Society

Elected 1942

Lunar features

Crater Vinogradov

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vinogradov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vinogradov.html (3 of 3) [2/16/2002 11:36:58 PM]

Vitali

Giuseppe Vitali Born: 26 Aug 1875 in Ravenna, Italy Died: 29 Feb 1932 in Bologna, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Giuseppe Vitali graduated from the Scuola Normale Superiore in Pisa in 1899. He assisted Dini for 2 years from 1899 then left mathematics. His move away from mathematics was probably due to financial problems and he taught in schools. From 1904 until 1923 he taught at the Liceo C Colombo in Genoa where he became involved in politics becoming a Socialist councillor. However, when the Fascists came to power in 1922, they dissolved the Socialist Party. His political career at an end, Vitali returned to mathematics. First Vitali was appointed to a chair in Modena. This was as a result of him winning a competition for the chair of infinitesimal analysis in 1923. Then the following year Vitali was appointed to the chair of mathematics at Padua and finally, in 1930, to the chair of mathematics at the University of Bologna. His significant mathematical discoveries include a theorem on set-covering, the notion of an absolutely continuous functions and a criteria for the closure of a system of orthogonal functions. Since he worked very much on his own, his work involves some rediscovering of known results but also some remarkably original discoveries. From 1926 he developed a serious illness and he could no longer write. Nevertheless about half his research papers were written in the last four years of his life after the illness struck. In his last years he worked on a new absolute differential calculus and a geometry of Hilbert spaces. These topics were not followed up by later mathematicians. After his death his work Moderna teoria delle funzoni d variabile reale was completed and published in 1935. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vitali.html (1 of 2) [2/16/2002 11:36:59 PM]

Vitali

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vitali.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vitali.html (2 of 2) [2/16/2002 11:36:59 PM]

Viviani

Vincenzo Viviani Born: 5 April 1622 in Florence, Italy Died: 22 Sept 1703 in Florence, Italy

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vincenzo Viviani studied at a Jesuit school. His intelligence was quickly seen and a scholarship was provided by the Grand Duke Ferdinando de'Medici to purchase mathematical books for Viviani. Viviani became a pupil of Torricelli and a disciple of Galileo and worked on physics and geometry. He became a companion and pupil of Galileo from 1639 until Galileo died in 1642. When Torricelli died in 1647 Viviani was appointed to fill the lectureship at the Accademia del Disegno in Florence. The Grand Duke also appointed Viviani engineer with the Uffiziali dei Fiumi, a position he held for the rest of his life. In 1660, together with Borelli, Viviani measured the velocity of sound by timing the difference between the flash and the sound of a cannon. They obtained the value of 350 metres per second, which is considerably better than the previous value of 478 metres per second obtained by Gassendi (the currently accepted value is 331.29 metres per second at 0 C). His reputation as a mathematician was high throughout Europe. Louis XIV of France offered him a position at the Académie Royale in 1666, John II Casimir of Poland offered Viviani a post as his astronomer, also in 1666. The Grand Duke, not wishing to lose Viviani, appointed him as his mathematician. Viviani accepted this post and turned down the offers from Louis XIV and John II. Viviani determined the tangent to the cycloid but he was not the first to succeed in this. As he was an engineer all his life Viviani published on engineering. In particular he published Discorso intorno al difendersi da' riempimenti e dalle corrosione de' fiumi (1687). http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Viviani.html (1 of 2) [2/16/2002 11:37:01 PM]

Viviani

In 1692 Viviani proposed the following problem which aroused much interest. A hemisphere has 4 equal windows of such a size that the remaining surface can be exactly squared - how is this possible? On his death Viviani left an almost completed work on the resistance of solids which was completed and published by Grandi. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (17 books/articles) Mathematicians born in the same country Cross-references to History Topics

Longitude and the Académie Royale

Other references in MacTutor

Chronology: 1650 to 1675

Honours awarded to Vincenzo Viviani (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1696

Lunar features

Crater Viviani

Other Web sites

The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Viviani.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Viviani.html (2 of 2) [2/16/2002 11:37:01 PM]

Vlacq

Adriaan Vlacq Born: 1600 in Gouda, Netherlands Died: 1667 in The Hague, Netherlands Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adriaan Vlacq was a bookseller and publisher. In 1632 he settled in London and opened a bookshop. Tension between King Charles I and the House of Commons steadily increased during 1641 and after his unsuccessful attempt to arrest five members of Parliament on 4 January 1642 Charles left London on 10 January and King and Parliament prepared for war. Vlacq decided that London was not a good place to sell books with the approaching unrest so he left for Paris in 1642. A dedication in a book he published 10 years later suggests that he sympathised with the Royalist side in the Civil War. In Paris Vlacq again set up a book business. He remained there for six years before returning to The Hague where he again sold and published books. Vlacq published a table of Briggs logarithms from 1 to 100,000 to 10 decimal places in Arithmetica logarithmica in 1628. Briggs' own tables were only 1 to 20,000 and 90,000 to 100,000 so Vlacq added 70,000 values. Vlacq also constructed log trigonometric tables which he published in 1633 during his time in London. Vlacq also published many mathematical works by other authors including, perhaps surprisingly, by Briggs himself. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country Honours awarded to Adriaan Vlacq (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Vlacq

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vlacq.html (1 of 2) [2/16/2002 11:37:02 PM]

Vlacq

Other Web sites

1. The Galileo Project 2. Nijmegen, Netherlands (in Dutch, but with some pictures)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vlacq.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vlacq.html (2 of 2) [2/16/2002 11:37:02 PM]

Volterra

Vito Volterra Born: 3 May 1860 in Ancona, Papal States (now Italy) Died: 11 Oct 1940 in Rome, Italy

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Vito Volterra's interest in mathematics started at the age of 11 when he began to study Legendre's Geometry. At the age of 13 he began to study the Three Body Problem and made some progress by partitioning the time into small intervals over which he could consider the force constant. His family were extremely poor (his father had died when Vito was two years old) but after attending lectures at Florence he was able to proceed to Pisa in 1878. At Pisa he studied under Betti, graduating Doctor of Physics in 1882. His thesis on hydrodynamics included some results of Stokes, discovered later but independently by Volterra. He became Professor of Mechanics at Pisa in 1883 and, after Betti's death, he occupied the Chair of Mathematical Physics. After being appointed to the Chair of Mechanics at Turin he was appointed to the Chair of Mathematical Physics at Rome in 1900. Volterra conceived the idea of a theory of functions which depend on a continuous set of values of another function in 1883. Hadamard was later to introduce the word 'functional' which replaced Volterra's original terminology. In 1890 Volterra showed by means of his functional calculus that the theory of Hamilton and Jacobi for the integration of the differential equations of dynamics could be extended to other problems of mathematical physics. During the years 1892 to 1894 Volterra published papers on partial differential equations, particularly the equation of cylindrical waves.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Volterra.html (1 of 3) [2/16/2002 11:37:04 PM]

Volterra

His most famous work was done on integral equations. He began this study in 1884 and in 1896 he published papers on what is now called 'an integral equation of Volterra type'. He continued to study functional analysis applications to integral equations producing a large number of papers on composition and permutable functions. During the First World War Volterra joined the Air Force. He made many journeys to France and England to promote scientific collaboration. After the War he returned to the University of Rome and his interests moved to mathematical biology. He studied the Verhulst equation and the logistic curve. He also wrote on predator-prey equations. In 1922 Fascism siezed Italy and Volterra fought against it in the Italian Parliament. However by 1930 the Parliament was abolished and when Volterra refused to take an oath of allegience to the Fascist Government in 1931 he was forced to leave the University of Rome. From the following year he lived mostly abroad, mainly in Paris but also Spain and other countries. Volterra was offered an honorary degree by the University of St Andrews in 1938 but his doctor did not allow him to travel to Scotland to receive it. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (34 books/articles) A Poster of Vito Volterra

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1880 to 1890

Honours awarded to Vito Volterra (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1910

Lunar features

Crater Volterra

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Volterra.html (2 of 3) [2/16/2002 11:37:04 PM]

Volterra

JOC/EFR December 1996

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Volterra.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Volterra.html (3 of 3) [2/16/2002 11:37:04 PM]

Von_Dyck

Walther Franz Anton von Dyck Born: 6 Dec 1856 in Munich, Germany Died: 5 Nov 1934 in Munich, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Perhaps we should first comment that Walther von Dyck was not known by precisely that name in his youth, but rather by the name Walther Dyck. He was ennobled with the "von" only much later in his life. Walther's father was Hermann Dyck while his mother was Marie Royko. Hermann Dyck was a painter and he was the Director of the Kunstgewerbeschule in Berlin, that is the Berlin School of Industrial Art. Walther certainly inherited his father's artistic abilities but in addition he had a wide range of skills across the arts and sciences. He studied at a number of German universities, which was the standard pattern for students at that time. After beginning his studies at the University of Munich, he then spent time studying at the Universities of Berlin and Leipzig. He received his doctorate from Munich in 1879 for a thesis entitled Uber regulär verzweigte Riemannsche Flächen und die durch sie definierten Irrationalitäten. His thesis supervisor was Klein who had been appointed to a chair at the Technische Hochschule at Munich a few years before in 1875. In 1880 Klein left Munich to take up the chair of geometry at Leipzig. Dyck went to Leipzig to take up a position as Klein's assistant and there he submitted his habilitation thesis, receiving the university lecturing qualification in 1882. During this period Dyck made important contributions to group theory with the publication of two papers; Gruppentheoretische Studien in Mathematische Annalen the first in 1882 and the second in the following year. Klein would remain at Leipzig until 1886, but Dyck left two years before that when he was appointed

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Dyck.html (1 of 3) [2/16/2002 11:37:06 PM]

Von_Dyck

professor at newly established Munich Polytechnikum. In [3] Dyck's work at the Munich Polytechnikum is studied in detail. In particular Hashagen, the author of [3], examines the way von Dyck tried to solve the problems which arose in the training of engineers. Von Dyck stressed how important a basic knowledge in mathematics was to engineers and he worked hard to construct a consistent course to include varied topics. Dyck would remain at the Munich Polytechnikum for the rest of his career and gave outstanding service to it in many important ways. He was appointed Director of the Polytechnikum in 1900 and under his inspired leadership the institution rose to university status becoming the Technische Hochschule of Munich. He served as rector of the Technische Hochschule for two terms, the first from 1903 to 1906 and the second from 1919 to 1925. There was another important project in which von Dyck played an important role. This was the creation of the Deutsches Museum of Natural Science and Technology. The idea was first suggested by Oskar von Miller who was an electrical engineer who was instrumental in setting up the electric power industry in Germany. In 1903 Miller enlisted the help of von Dyck and of Carl von Linde who had been appointed extraordinary professor of machine design at the Munich Polytechnikum in the same year as von Dyck was appointed. They proposed that a museum be built in Munich which would both preserve technological artefacts and let visitors learn about the scientific principles through interactive displays. The Deutsches Museum was first of its kind and its ideas were soon copied by other science museums around the world. Not only was von Dyck one of the three to establish and develop the museum in its early stages, but he was also appointed as the second Director of the Museum in 1906. Von Dyck made important contributions to function theory, group theory (where a fundamental result on group presentations is named after him), topology (where he was influenced by the work of Riemann), and to potential theory. He made significant contributions to the Gauss-Bonnet theorem. Another important project which von Dyck undertook was one to publish the complete works of Kepler, including all Kepler's letters. He undertook this in his role as class secretary of the Bayerische Akademie der Wissenschaften in 1906. This project has extended well beyond von Dyck's lifetime with Volume 7 appearing in 1953, and Volume 8 in 1963. Von Dyck's character and contribution are summed up in [1] as follows:Linguistically gifted and a warm, kind-hearted man of wide-ranging and liberal interests, including art and music, Dyck was an outstanding scholar and organiser and an enthusiastic and inspiring teacher. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) A Poster of Walther von Dyck

Mathematicians born in the same country

Cross-references to History Topics

The development of group theory

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Dyck.html (2 of 3) [2/16/2002 11:37:06 PM]

Von_Dyck

Other Web sites

Munich, Germany (A biography -- in German)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Von_Dyck.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Dyck.html (3 of 3) [2/16/2002 11:37:06 PM]

Von_Neumann

John von Neumann Born: 28 Dec 1903 in Budapest, Hungary Died: 8 Feb 1957 in Washington D.C., USA

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John von Neumann was born János von Neumann. He was called Jancsi as a child, a diminutive form of János, then later he was called Johnny in the United States. His father, Max Neumann, was a top banker and he was brought up in a extended family setting in Budapest where as a child he learnt languages from the German and French governesses that were employed. Although the family were Jewish, Max Neumann did not observe the strict practices of that religion and the household seemed to mix Jewish and Christian traditions. It is also worth explaining how Max Neumann's son acquired the "von" to become János von Neumann. In 1913 Max Neumann purchased a title but did not change his name. His son, however, used the German form von Neumann where the "von" indicated the title. As a child von Neumann showed he had an incredible memory. Poundstone, in [7], writes:At the age of six, he was able to exchange jokes with his father in classical Greek. The Neumann family sometimes entertained guests with demonstrations of Johnny's ability to memorise phone books. A guest would select a page and column of the phone book at random. Young Johnny read the column over a few times, then handed the book back to the guest. He could answer any question put to him (who has number such and such?) or recite names, addresses, and numbers in order. In 1911 von Neumann entered the Lutheran Gymnasium. The school had a strong academic tradition which seemed to count for more than the religious affiliation both in the Neumann's eyes and in those of the school. His mathematics teacher quickly recognised von Neumann's genius and special tuition was put on for him. The school had another outstanding mathematician one year ahead of von Neumann, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Neumann.html (1 of 7) [2/16/2002 11:37:09 PM]

Von_Neumann

namely Eugene Wigner. World War I had relatively little effect on von Neumann's education but after the war ended Béla Kun controlled Hungary for five months in 1919 with a Communist government. The Neumann family fled to Austria as the affluent came under attack. However, after a month, they returned to face the problems of Budapest. When Kun's government failed, the fact that it had been largely composed of Jews meant that Jewish people were blamed. Such situations are devoid of logic and the fact that the Neumann's were opposed to Kun's government did not save them from persecution. In 1921 von Neumann completed his education at the Lutheran Gymnasium. His first mathematics paper written jointly with Fekete, the assistant at the University of Budapest who had been tutoring him, was published in 1922. However Max Neumann did not want his son to take up a subject that would not bring him wealth. Max Neumann asked Theodore von Kármán to speak to his son and persuade him to follow a career in business. Perhaps von Kármán was the wrong person to ask to undertake such a task but in the end all agreed on the compromise subject of chemistry for von Neumann's university studies. Hungary was not an easy country for those of Jewish descent for many reasons and there was a strict limit on the number of Jewish students who could enter the University of Budapest. Of course, even with a strict quota, von Neumann's record easily won him a place to study mathematics in 1921 but he did not attend lectures. Instead he also entered the University of Berlin in 1921 to study chemistry. Von Neumann studied chemistry at the University of Berlin until 1923 when he went to Zurich. He achieved outstanding results in the mathematics examinations at the University of Budapest despite not attending any courses. Von Neumann received his diploma in chemical engineering from the Technische Hochschule in Zürich in 1926. While in Zurich he continued his interest in mathematics, despite studying chemistry, and interacted with Weyl and Pólya who were both at Zurich. He even took over one of Weyl's courses when he was absent from Zurich for a time. Pólya said [17]:Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper. Von Neumann received his doctorate in mathematics from the University of Budapest, also in 1926, with a thesis on set theory. He published a definition of ordinal numbers when he was 20, the definition is the one used today. Von Neumann lectured at Berlin from 1926 to 1929 and at Hamburg from 1929 to 1930. However he also held a Rockefeller fellowship to enable him to undertake postdoctoral studies at the University of Göttingen. He studied under Hilbert at Göttingen during 1926-27. By this time von Neumann had achieved celebrity status [7]:By his mid-twenties, von Neumann's fame had spread worldwide in the mathematical community. At academic conferences, he would find himself pointed out as a young genius. Veblen invited von Neumann to Princeton to lecture on quantum theory in 1929. Replying to Veblen that he would come after attending to some personal matters, von Neumann went to Budapest where he married his fiancée Marietta Kovesi before setting out for the United States. In 1930 von Neumann became a visiting lecturer at Princeton University, being appointed professor there in 1931. Between 1930 and 1933 von Neumann taught at Princeton but this was not one of his strong points [7]:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Neumann.html (2 of 7) [2/16/2002 11:37:09 PM]

Von_Neumann

His fluid line of thought was difficult for those less gifted to follow. He was notorious for dashing out equations on a small portion of the available blackboard and erasing expressions before students could copy them. He became one of the original six mathematics professors (J W Alexander, A Einstein, M Morse, O Veblen, J von Neumann and H Weyl) in 1933 at the newly founded Institute for Advanced Study in Princeton, a position he kept for the remainder of his life. During the first years that he was in the United States, von Neumann continued to return to Europe during the summers. Until 1933 he still held academic posts in Germany but resigned these when the Nazis came to power. Unlike many others, von Neumann was not a political refugee but rather he went to the United States mainly because he thought that the prospect of academic positions there was better than in Germany. In 1933 von Neumann became co-editor of the Annals of Mathematics and, two years later, he became co-editor of Compositio Mathematica. He held both these editorships until his death. Von Neumann and Marietta had a daughter Marina in 1936 but their marriage ended in divorce in 1937. The following year he married Klára Dán, also from Budapest, whom he met on one of his European visits. After marrying, they sailed to the United States and made their home in Princeton. There von Neumann lived a rather unusual lifestyle for a top mathematician. He had always enjoyed parties [7]:Parties and nightlife held a special appeal for von Neumann. While teaching in Germany, von Neumann had been a denizen of the Cabaret-era Berlin nightlife circuit. Now married to Klára the parties continued [17]:The parties at the von Neumann's house were frequent, and famous, and long. Ulam summarises von Neumann's work in [34]. He writes:In his youthful work, he was concerned not only with mathematical logic and the axiomatics of set theory, but, simultaneously, with the substance of set theory itself, obtaining interesting results in measure theory and the theory of real variables. It was in this period also that he began his classical work on quantum theory, the mathematical foundation of the theory of measurement in quantum theory and the new statistical mechanics. His text Mathematische Grundlagen der Quantenmechanik (1932) built a solid framework for the new quantum mechanics. Van Hove writes in [35]:Quantum mechanics was very fortunate indeed to attract, in the very first years after its discovery in 1925, the interest of a mathematical genius of von Neumann's stature. As a result, the mathematical framework of the theory was developed and the formal aspects of its entirely novel rules of interpretation were analysed by one single man in two years (1927-1929). Self-adjoint algebras of bounded linear operators on a Hilbert space, closed in the weak operator topology, were introduced in 1929 by von Neumann in a paper in Mathematische Annalen. Kadison explains in [21]:His interest in ergodic theory, group representations and quantum mechanics contributed significantly to von Neumann's realisation that a theory of operator algebras was the next important stage in the development of this area of mathematics.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Neumann.html (3 of 7) [2/16/2002 11:37:09 PM]

Von_Neumann

Such operator algebras were called "rings of operators" by von Neumann and later they were called W*-algebras by some other mathematicians. J Dixmier, in 1957, called them "von Neumann algebras" in his monograph Algebras of operators in Hilbert space (von Neumann algebras). In the second half of the 1930's and the early 1940s von Neumann, working with his collaborator F J Murray, laid the foundations for the study of von Neumann algebras in a fundamental series of papers. Ulam explains [34] how von Neumann was led towards game theory:Von Neumann's awareness of results obtained by other mathematicians and the inherent possibilities which they offer is astonishing. Early in his work, a paper by Borel on the minimax property led him to develop ... ideas which culminated later in one of his most original creations, the theory of games. In game theory von Neumann proved the minimax theorem. He gradually expanded his work in game theory, and with co-author Oskar Morgenstern, he wrote the classic text Theory of Games and Economic Behaviour (1944). Ulam continues in [34]:An idea of Koopman on the possibilities of treating problems of classical mechanics by means of operators on a function space stimulated him to give the first mathematically rigorous proof of an ergodic theorem. Haar's construction of measure in groups provided the inspiration for his wonderful partial solution of Hilbert's fifth problem, in which he proved the possibility of introducing analytical parameters in compact groups. In 1938 the American Mathematical Society awarded the Bôcher Prize to John von Neumann for his memoir Almost periodic functions and groups. This was published in two parts in the Transactions of the American Mathematical Society, the first part in 1934 and the second part in the following year. Around this time von Neumann turned to applied mathematics [34]:In the middle 30's, Johnny was fascinated by the problem of hydrodynamical turbulence. It was then that he became aware of the mysteries underlying the subject of non-linear partial differential equations. His work, from the beginnings of the Second World War, concerns a study of the equations of hydrodynamics and the theory of shocks. The phenomena described by these non-linear equations are baffling analytically and defy even qualitative insight by present methods. Numerical work seemed to him the most promising way to obtain a feeling for the behaviour of such systems. This impelled him to study new possibilities of computation on electronic machines ... Von Neumann was one of the pioneers of computer science making significant contributions to the development of logical design. Shannon writes in [28]:Von Neumann spent a considerable part of the last few years of his life working in [automata theory]. It represented for him a synthesis of his early interest in logic and proof theory and his later work, during World War II and after, on large scale electronic computers. Involving a mixture of pure and applied mathematics as well as other sciences, automata theory was an ideal field for von Neumann's wide-ranging intellect. He brought to it many new insights and opened up at least two new directions of research. He advanced the theory of cellular automata, advocated the adoption of the bit as a measurement of computer memory, and solved problems in obtaining reliable answers from unreliable computer

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Neumann.html (4 of 7) [2/16/2002 11:37:09 PM]

Von_Neumann

components. During and after World War II, von Neumann served as a consultant to the armed forces. His valuable contributions included a proposal of the implosion method for bringing nuclear fuel to explosion and his participation in the development of the hydrogen bomb. From 1940 he was a member of the Scientific Advisory Committee at the Ballistic Research Laboratories at the Aberdeen Proving Ground in Maryland. He was a member of the Navy Bureau of Ordnance from 1941 to 1955, and a consultant to the Los Alamos Scientific Laboratory from 1943 to 1955. From 1950 to 1955 he was a member of the Armed Forces Special Weapons Project in Washington, D.C. In 1955 President Eisenhower appointed him to the Atomic Energy Commission, and in 1956 he received its Enrico Fermi Award, knowing that he was incurably ill with cancer. Eugene Wigner wrote of von Neumann's death [17]:When von Neumann realised he was incurably ill, his logic forced him to realise that he would cease to exist, and hence cease to have thoughts ... It was heartbreaking to watch the frustration of his mind, when all hope was gone, in its struggle with the fate which appeared to him unavoidable but unacceptable. In [4] von Neumann's death is described in these terms:... his mind, the amulet on which he had always been able to rely, was becoming less dependable. Then came complete psychological breakdown; panic, screams of uncontrollable terror every night. His friend Edward Teller said, "I think that von Neumann suffered more when his mind would no longer function, than I have ever seen any human being suffer." Von Neumann's sense of invulnerability, or simply the desire to live, was struggling with unalterable facts. He seemed to have a great fear of death until the last... No achievements and no amount of influence could save him now, as they always had in the past. Johnny von Neumann, who knew how to live so fully, did not know how to die. It would be almost impossible to give even an idea of the range of honours which were given to von Neumann. He was Colloquium Lecturer of the American Mathematical Society in 1937 and received the its Bôcher Prize as mentioned above. He held the Gibbs Lectureship of the American Mathematical Society in 1947 and was President of the Society in 1951-53. He was elected to many academies including the Academia Nacional de Ciencias Exactas (Lima, Peru), Academia Nazionale dei Lincei (Rome, Italy), American Academy of Arts and Sciences (USA), American Philosophical Society (USA), Instituto Lombardo di Scienze e Lettere (Milan, Italy), National Academy of Sciences (USA) and Royal Netherlands Academy of Sciences and Letters (Amsterdam, The Netherlands). Von Neumann received two Presidential Awards, the Medal for Merit in 1947 and the Medal for Freedom in 1956. Also in 1956 he received the Albert Einstein Commemorative Award and the Enrico Fermi Award mentioned above. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Neumann.html (5 of 7) [2/16/2002 11:37:09 PM]

Von_Neumann

List of References (38 books/articles)

Some Quotations (7)

A Poster of John von Neumann

Mathematicians born in the same country

Cross-references to History Topics

1. The beginnings of set theory 2. The quantum age begins 3. Memory, mental arithmetic and mathematics

Other references in MacTutor

1. Chronology: 1920 to 1930 2. Chronology: 1930 to 1940 3. Chronology: 1940 to 1950

Honours awarded to John von Neumann (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1951 - 1952

AMS Colloquium Lecturer

1937

AMS Gibbs Lecturer

1944

AMS Bôcher Prize

Awarded 1938

Lunar features

Crater Von Neumann

Other Web sites

1. Virginia Tech 2. Kalamazoo 3. Paul Walker (A history of Game Theory) 4. US News 5. Kevin Brown (Von Neumann's 5th postulate) 6. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Neumann.html (6 of 7) [2/16/2002 11:37:09 PM]

Von_Neumann

JOC/EFR June 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Von_Neumann.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Neumann.html (7 of 7) [2/16/2002 11:37:09 PM]

Von_Staudt

Karl Georg Christian von Staudt Born: 24 Jan 1798 in Imperial Free City of Rothenburg (now Rothenburg ob der Tauber, Germany) Died: 1 June 1867 in Erlangen, Bavaria (now Germany)

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karl Von Staudt attended Göttingen from 1818 to 1822. His early work was on determining the orbit of a comet and, based on this work, he received a doctorate from Erlangen in 1822 . Staudt was appointed professor of mathematics at the Polytechnic School at Nürnberg in 1827 and he was appointed to the University of Erlangen in 1835. Von Staudt showed how to construct a regular inscribed polygon of 17 sides using only compasses. He turned to projective geometry and Bernoulli numbers. An important work on projective geometry, Geometrie der Lage was published in 1847. It was the first work to completely free projective geometry from any metrical basis. Another of his publications on projective geometry was Beiträge zur Geometrie der Lage (1856-60). He also gave a nice geometric solution to quadratic equations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Staudt.html (1 of 2) [2/16/2002 11:37:11 PM]

Von_Staudt

Some pages from publications

The title page of Beitrage zur Geometrie der Lage (1856)

Other references in MacTutor

Chronology: 1840 to 1850

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Von_Staudt.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Von_Staudt.html (2 of 2) [2/16/2002 11:37:11 PM]

Voronoy

Georgy Fedoseevich Voronoy Born: 28 April 1868 in Zhuravka, Poltava guberniya, Russia (now Ukraine) Died: 20 Nov 1908 in Warsaw, Poland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Georgy Voronoy studied at the gymnasium in Priluki graduating in 1885. He then entered the University of St Petersburg, joining the Faculty of Physics and Mathematics. After graduating from St Petersburg in 1889, Voronoy decided to remain there and work for his teaching qualification. He was awarded a Master's Degree in 1894 for a dissertation on the algebraic integers associated with the roots of an irreducible cubic equation. Voronoy lectured at Warsaw University, being appointed professor of pure mathematics there. He wrote his doctoral thesis on algorithms for continued fractions which he submitted to the University of St Petersburg. In fact both Voronoy's master's thesis and his doctoral thesis were of such high quality that they were awarded the Bunyakovsky prize by the St Petersburg Academy of Sciences. Later Voronoy worked on the theory of numbers, in particular he worked on algebraic numbers and the geometry of numbers. He extended work by Zolotarev and his work was the starting point for Vinogradov's investigations. His methods were also used by Hardy and Littlewood. In 1904 Voronoy attended the Third International Congress of Mathematicians at Heidelberg. There he met Minkowski and they discovered that they were each working on similar topics. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Voronoy.html (1 of 2) [2/16/2002 11:37:13 PM]

Voronoy

Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Voronoy.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Voronoy.html (2 of 2) [2/16/2002 11:37:13 PM]

Vranceanu

Gheorghe Vranceanu Born: 30 June 1900 in Valea Hogii, Vaslui, Romania Died: 27 April 1979 in Bucharest, Romania

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gheorghe Vranceanu was born into a family of poor peasants. He attended primary school in his own village and, while at primary school his teacher, G Arnautescu, realised his great potential. Arnautescu helped Gheorghe to move to Vaslui High School in 1912, where Gheorghe discovered his love of mathematics. Vranceanu was awarded an Adamachi scholarship to study at Iasi University, which he entered in 1919, and there he was taught mathematics by Alexandru, Vera Myller, Simeon Sanielevici, Victor Valcovici and Simeon Stoilow, all famous Romanian mathematicians of that time. Vranceanu was very highly regarded by his professors and, on 15 February 1922, he graduated from the University of Iasi. However, while still a student in his third year, Vranceanu was appointed, on 1 December 1921, as an assistant to the mathematics seminar at the request of S Sanielevici. After a brilliant undergraduate career Vranceanu went first to Göttingen in 1923 where he studied under Hilbert, then he went to Rome to study for his doctorate in mathematics. In Rome Vranceanu studied under Levi-Civita, obtaining his doctorate on 5 November 1924 for a dissertation Sopra una teorema di Weierstrass e le sue applicazioni alla stabilita which gave a new proof of a theorem on the decomposition of analytical functions of more variables and also studied applications of the theorem to mechanics. The examining board consisted of 11 professors, headed by Volterra. His doctoral thesis, and all his earlier publications, concerned applications of analysis to mechanics. Vranceanu returned to Iasi and, in 1926, still developing ideas suggested by Levi-Civita, Vranceanu discovered the notion of a non-holonomic space. Today this concept is named after Vranceanu.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vranceanu.html (1 of 3) [2/16/2002 11:37:15 PM]

Vranceanu

Although only 26 years old when he made this remarkable discovery, it quickly turned him into a celebrity. He was appointed a lecturer at Iasi University and then, during 1927-1928, he was awarded a Rockefeller scholarship to study in France and in the United States. In Paris he worked with Elie Cartan and then he went to the United States where he studied at Harvard University and Princeton University. He met Birkhoff and Veblen and later they became good friends. When his scholarship came to an end he was offered a position as a professor but he preferred to return to Romania, taking up his post in Iasi. In 1929 Vranceanu moved to Cernauti University where he was appointed professor of analytical geometry, then still at Cernauti he was appointed professor of Differential and Integral Geometry in the following year. After 10 years of great mathematical activity at Cernauti University, he was asked to fill in the professorship at Bucharest University which had become vacant on the death of Gheorghe Titeica in 1939. In 1948 Vranceanu was appointed Head of Geometry and Topology at Bucharest University. He retired from his chair in 1970, continuing to take an active interest in mathematics at the university. During his time at Cernauti University Vranceanu became known as one of the leading geometers in the world. In 1928 at the International Congress of Mathematics in Bologna, the notion of a non-holonomic space which he had discovered was studied by Schouten and Cartan. Meanwhile Vranceanu made new discoveries in global geometry. At Bucharest University Vranceanu began to organise the mathematics library in a similar way to the one in Iasi. He formed his own group of young geometers and together they wrote teaching texts, as well as the 4 volumes of a differential geometry text, later translated in German and French. Besides his major contributions to science, Vranceanu took an active interest in politics. In 1944 he was one of the founders of a movement which tried to prevent Romania from fighting against Russia. Vranceanu organised the Mathematical Institute of the Romanian Academy, a very important step for theoretical and applied researches in his country. Until his death, he was an editor of the Mathematical Studies and Researches and the Revue Roumaine de Mathématique Pure et Appliquée. He tried to make known all the Romanian discoveries in mathematics to the international mathematical community. He also organised many scientific conferences, both inside and outside Romania, the last one being held in September 1978 in Craiova. He was much in demend as a lecturer, being invited to lecture at over 30 institutions world-wide, for example he lectured at universities in Paris, Rome, Princeton, Moscow, Peking, Berlin, London, Salamanca, Geneva and many others. During his career, Vranceanu published over 300 articles in journals throughout the world. They cover all the branches of modern geometry, from the classical theory of surfaces to the notion of non-holonomic spaces which he discovered, creating efficient methods and solving fundamental problems. Other topics he studied include the absolute differential calculus of congruences, analytical mechanics, partial differential equations of the second order, non-holonomic unitary theory, conformal connection spaces, metrics in spherical and projective spaces, Lie groups, global differential geometry, discrete groups of affine connection spaces, locally Euclidean connection spaces, Riemannian spaces of constant connection, differentiable varieties, embedding of lens spaces into Euclidean space, tangent vectors of spheres and exotic spheres, the equivalence method, non-linear connection spaces, and the geometry of mechanical systems. Vranceanu won many honours, both in his own country and elsewhere. He was elected to the Romanian Academy as a corresponding member in 1946, then as a full member in 1955. From 1964 he was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vranceanu.html (2 of 3) [2/16/2002 11:37:15 PM]

Vranceanu

President of the Mathematics Section of the Romanian Academy. He won a Government Award (1952) and other medals and awards for excellence. He was awarded honorary degrees from Bologna University (1967) and Iasi University (1970). He was also elected to the Peloritana dei Pericolanti University in Messina (1968) and the Royal Flamand Academy of Brussels (1970). He was elected a member of the Royal Society of Liège in 1972. Vranceanu served as a member of the International Committee of The International Union of Mathematicians for many years and, in that capacity, he was involved in publishing the complete works of Elie Cartan. In 1975 Vranceanu was elected Vice-president of the International Union of Mathematicians. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Vranceanu.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Vranceanu.html (3 of 3) [2/16/2002 11:37:15 PM]

Wald

Abraham Wald Born: 31 Oct 1902 in Kolozsvár, Hungary (now Cluj, Romania) Died: 13 Dec 1950 in Travancore, India

Click the picture above to see four larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Abraham Wald was born into a Jewish family in Hungary. It was a family of intellectuals but, being Jewish, they were forced to earn their living in trades well below their abilities. At this time both primary and secondary schools in Hungary required pupils to attend on Saturdays and the Wald family could not allow their son to attend school on the Jewish Sabbath. As a result Abraham did not attend either primary of secondary school and was educated at home by members of his family. This certainly did not put him at a disadvantage from an educational point of view for his family were very knowledgeable and competent teachers. After World War I much of the land that had been part of Hungary was given to neighbouring countries and at that time Cluj became part of Romania. Wald was allowed to attend the University of Cluj but it appears that this was not made easy for him because he was Jewish. However his outstanding abilities in mathematics led him to wish to continue to undertake mathematical research and in 1927 he entered the University of Vienna to study with Karl Menger. He worked under Menger's supervision on geometry and was awarded his doctorate in 1931. Vienna in the 1930s was no place for a young Jewish man to obtain an academic position, no matter how talented. In fact there were few academic positions open at all. The only way that Wald could support himself so as to be able to continue with his research was to take employment. This he did taking the position of mathematics tutor to Karl Schlesinger, a leading Austrian banker and economist. Between 1931 and 1937 Wald published 21 papers on geometry which Menger describes in [5] as:... deep, beautiful and of fundamental importance. However his work with Schlesinger did not only give him financial security and hence the opportunity to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wald.html (1 of 4) [2/16/2002 11:37:17 PM]

Wald

undertake research in geometry. It also gave him an interest in applying his mathematical skills to the problems in economics and econometrics which interested Schlesinger. During this period Wald published 10 papers on economics and econometrics, and he also published an important monograph in 1936 on seasonal movements in time series. The main aim was to give methods to eliminate this seasonal variation. In [8] Morgenstern writes that in this monograph Wald:... developed techniques superior to all others. An indication of the problems that Wald had in Vienna at this time because he was Jewish is indicated by the fate of Menger's seminar. Wald was a member of this seminar which included lectures by both members of the seminar and also invited guests. Speakers raised open problems, and reported on recent publications and research. Wald reported to the seminar on his work in econometrics, in particular he wrote a paper for the seminar on the existence of a solution to the competitive economic model. However, the seminar was forced to stop work in 1936 after it was criticised for its Jewish contributors, one of whom of course was Wald. If life was difficult for a Jew in Vienna in 1936, they would soon become much worse. In 1938 the Nazi forces invaded Austria. For a Jewish person like Wald conditions under the Nazis were at best extremely difficult and at worst very dangerous. The Cowles Commission invited him to the United States to do econometric research in the United States and he left Austria in the summer of 1938. The move to the United States almost certainly saved his life for all but one of the nine members of his family left behind died in the gas chambers of the Nazi concentration camp at Auschwitz. By September 1938 Wald was a Fellow of the Carnegie Corporation studying statistics at Columbia University in New York under Hotelling. Wald remained a Fellow of the Carnegie Corporation until 1941 but by that time he had already begun lecturing at Columbia University which he began in academic year 1939-40. He was appointed to the Faculty of Columbia University in 1941 and he remained on the staff there until his death. In addition to his teaching and research at Columbia, he undertook war work after the United States entered World War II, working on military projects with the Statistics Research Group at Columbia. He used his statistical expertise to develop a method to estimate aircraft vulnerability. As we mentioned above, in Vienna Wald worked on pure mathematics, mostly geometry, and on econometrics. His first pure mathematical work was on metric spaces, an extension of Steinitz's work to infinite dimensional vector spaces, and some beautiful results on differential geometry. Wald's most important work, however, was in statistics. In [14] Wolfowitz, who was first his student, then his colleague and collaborator, described a paper Wald published in the Annals of Mathematical Statistics in 1939 as:... probably Wald's most important single paper. In this 1939 paper Wald [13]:... points out that the two major problems of statistical theory at that time, testing hypotheses and estimation, can both be regarded as simple special cases of a more general problem known nowadays as a "statistical decision problem". ... He defines loss functions, risk functions, a priori distributions, Bayes decision rules, admissible decision rules, and minimax decision rules, and proves that a minimax decision rule has a constant risk under certain regularity conditions.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wald.html (2 of 4) [2/16/2002 11:37:17 PM]

Wald

Wald also developed generalizations of the problem of gambler's ruin which play an important role in statistical sequential analysis. He invented the topic of sequential analysis in response to the demand for more efficient methods of industrial quality control during World War II. The idea here is a simple one yet Wald was the first to build it into a statistical theory. It is better to analyse data produced sequentially rather than collect all the data and then analyse it. In this approach one does not choose a fixed sample size but can end the sampling at any time if the results justify it. Wald was the first to solve the general problem of sequential tests of statistical hypotheses. The optimum property of the sequential probability ratio test was conjectured by Wald in 1943 and, in a joint paper with Wolfowitz in 1948, he proved this property. This and related work was very much aimed at practical applications and his theorems on the distribution of the required number of observations, and on the probabilities associated with errors, found immediate applications. His main results on sequential analysis and the theory of decision functions, another topic which was founded by him, were gathered together in his monograph Sequential Analysis (1947). One of Wald's continuing interests from his time working with Karl Schlesinger was economics. He proved important results, perhaps the most significant being the existence of a solution to the competitive economic model which, as we noted above was written for Menger's seminar. His other work in this area related to [5]:... seasonal corrections to time series, approximate formulas for economic index numbers, indifference surfaces, the existence and uniqueness of solutions of extended forms of the Walrasian system of equations of production, the Cournot duopoly problem, and finally, in his much used work written with Mann (1943), stochastic difference equations. Wolfowitz writes in [1] that:One of his great contributions to statistics was to bring to it mathematical precision in the formulation of problems and mathematical rigour in argument. These qualities, which were often lacking when he began his statistical career in 1938, have transformed the subject although not necessarily to the satisfaction of everyone. It was not only in research that Wald had a remarkable influence on statistics, but although he only taught for about ten years, he also had a marked influence as a teacher. The same qualities of precision and rigour he showed in research were brought to his teaching but this did not mean that his lectures were complicated. On the contrary his lectures were renowned for their clarity and [13]:He was a master at deriving complicated results in amazingly simple ways. The notes which his students took during his lectures in Columbia were circulated and because of their outstanding clarity they reached students studying statistics at many different universities in the United States. After Wald emigrated to the United States he met Lucille Lang and the two were married. In 1950 Wald received an invitation from the Indian government to lecture on statistics in that country. He went to India with his wife and tragically they were both killed in a plane crash. Freeman, who attended Wald's lectures at Columbia, writes in [5] about Wald's character:Wald was a quite and gentle man, deeply immersed in his work. He was fairly aloof from small talk, and he had few hobbies. But he was not indifferent to recognition...

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wald.html (3 of 4) [2/16/2002 11:37:17 PM]

Wald

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (14 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wald.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wald.html (4 of 4) [2/16/2002 11:37:17 PM]

Walker_Arthur

Arthur Geoffrey Walker Born: 17 July 1909 in Watford, Hertfordshire, England Died: 31 March 2001 in Sussex, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Geoffrey Walker attended Watford Grammar School and from there he entered Balliol College Oxford. He received an M.A. from Oxford and then moved to Edinburgh to undertake research. After completing his Ph.D. at Edinburgh, Walker was appointed as a lecturer in mathematics at Imperial College in London. This was a post for the academic year 1935-36 and after completing this temporary appointment he received his first permanent post as a lecturer in mathematics at Liverpool University. He was appointed in 1936 and was to remain in Liverpool until 1947 when he was offered the chair of mathematics at the University of Sheffield. In 1952, after five years in Sheffield, Walker was to return to Liverpool University, this time as Professor of Pure Mathematics. He held this post until he retired in 1974. Walker worked on geometry, in particular differential geometry, relativity and cosmology. His papers include ones on relativistic mechanics, completely symmetric spaces, completely harmonic spaces and Riemannian manifolds. He wrote an article Note on locally symmetric vector fields in a Riemannian space, published in 1976, in memory of Evan Tom Davies. This is concerned with the restrictions imposed on a Riemannian n-space by the existence of a locally symmetric vector field and it continues work begun by Walker in a paper on possible orientation of galaxies published early in his career in 1940. In 1962 Walker published Harmonic Spaces, a joint work with H S Ruse and T J Willmore. In 1975 he published Introduction to geometrical cosmology a survey which arose out of a course that Walker gave at the University of Arizona. The lectures consider the red-shift, the number of galaxies, and the distance http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Walker_Arthur.html (1 of 2) [2/16/2002 11:37:19 PM]

Walker_Arthur

between galaxies. Walker writes in the introduction:This is an account of a course of twelve lectures given at the University of Arizona on the geometry of cosmology. It is entirely concerned with what might be called the classical theory, leading up to and discussing the standard model with the Robertson-Walker metric; it contains no new results though some of the methods may not have appeared in print. Walker was elected both a fellow of the Royal Society of Edinburgh and a fellow of the Royal Society of London. The Royal Society of Edinburgh honoured Walker by awarding him their Keith Medal in 1950. The election to the Royal Society of London took place in 1955 and he served on the Council of the Royal Society in 1962-63. He was also a strong supporter of the London Mathematical Society and was awarded the Junior Berwick Prize of that Society in 1947. He served the Society by being the 50th President in 1963-65. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

Mathematicians born in the same country Honours awarded to Geoffrey Walker (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1955

Fellow of the Royal Society of Edinburgh London Maths Society President

1963 -1965

LMS Berwick Prize winner

1947

Other Web sites

The Times (London) (Obituary)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Walker_Arthur.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Walker_Arthur.html (2 of 2) [2/16/2002 11:37:19 PM]

Walker_John

John James Walker Born: 2 Oct 1825 in Kennington, Surrey, England Died: 15 Feb 1900 in Hampstead, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John James Walker's father, John Walker, was the headmaster of various schools. In fact Walker attended London High School, the school at which his father was then headmaster until his father moved to become headmaster of Plymouth New Grammar School. Walker completed his school education at Plymouth New Grammar School before entering Trinity College Dublin as an undergraduate. Although his father's family were originally from Yorkshire, several generations had lived in Ireland. His mother's family were English but Walker followed his father's family tradition in attending Trinity College Dublin. Walker's grandfather had been a fellow of Trinity and published classical texts as well as elementary mathematics and logic texts. Things were not entirely straightforward for Walker at Dublin for his family had ceased to be conformists so he was debarred from holding a scholarship. To add to his problems his father died young and Walker was left having little in the way of funds to see out his education. However he excelled at Trinity College, winning gold medals for both mathematics and physics as well as winning other distinctions. After Walker graduated he became tutor to the Guinness family, a wealthy Irish family who had made their money from brewing. Arthur Guinness bought a small brewery in Dublin which began brewing a dark beer with a rich head in 1799; this became the national beer of Ireland. Walker became a private tutor to the family in 1853, two years before the death of the founder Arthur Guinness, and he continued as a tutor until 1862. Walker returned to England and, from 1865, he worked in London as a lecturer in applied mathematics http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Walker_John.html (1 of 3) [2/16/2002 11:37:21 PM]

Walker_John

and physics at University College. In the same year 1865 he joined the newly formed London Mathematical Society and he was the president of the Society in 1888-90. Walker was honoured with election as a fellow of the Royal Society in 1883. However his post at University College ended in 1888 and after that he devoted himself to mathematical research without holding any further jobs. The range of Walker's mathematical research was quite impressive. He wrote some articles on theoretical mechanics but his more elaborate papers were on advanced algebra and geometry. Walker was a strong advocate of Hamilton's quaternions and strongly believed that they had not been given as wide a use as they merited. He applied quaternions to a variety of problems, mostly of an elementary nature. The three most important papers that Walker wrote were on the analysis of plane curves and curved lines. The papers were closely connected and all appeared in the Proceedings of the London Mathematical Society. He wrote further articles on cubic curves and in this area he wrote the memoir On the diameters of cubic curves which was published in the Transactions of the Royal Society in 1889. Walker's character is described in [1] in the following terms:Mr Walker was of a reserved temperament, marked by somewhat precise courtesy of manner which seemed to belong to a bygone generation. His real kindness was shown by genial estimates of character and liberal appreciation of the labours of others engaged in kindred studies. It is an interesting thought what someone writing in 1901 about 'courtesy of manner which seemed to belong to a bygone generation' might think of Walker now, almost 100 years later. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to John James Walker (Click a link below for the full list of mathematicians honoured in this way) London Maths Society President

1888 - 1890

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Walker_John.html (2 of 3) [2/16/2002 11:37:21 PM]

Walker_John

JOC/EFR April 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Walker_John.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Walker_John.html (3 of 3) [2/16/2002 11:37:21 PM]

Wall

Charles Terence Clegg Wall Born: 14 Dec 1936 in Bristol, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Terry Wall's father Charles Wall was a schoolteacher. Terry Wall attended Marlborough College. He then entered Trinity College, Cambridge from where he was awarded a B.A. and, in 1960, a Ph.D. Wall had already been awarded a fellowship at Trinity College in 1959 before the award of his doctorate and he held this fellowship until 1964. He did not remain in Cambridge for all of this period, for instance he was awarded a Harkness Fellowship which allowed him to spend the academic year 1960-61 at the Institute for Advanced Study at Princeton. In addition to his fellowship, Wall was appointed a College Lecturer at Cambridge when he returned from the United States in 1961. In 1964 Wall moved from Cambridge to Oxford where he was appointed Reader in Mathematics and a fellow of St Catherine's College. After a year he was appointed to the chair of Pure Mathematics at the University of Liverpool, taking up the professorship in Liverpool in 1965. During 1967 was a Royal Society Leverhulme visiting professor in Mexico. A SERC senior research fellowship from 1983 to 1988 enabled him to concentrate on research over this period. Wall's research is mostly in the area of geometric topology and related algebra. In particular he has made substantial contributions to the study of singularities, especially isolated singularities, of differentiable maps and algebraic varieties. He has written a number of highly influential books including Surgery on compact manifolds (1970) and A geometric introduction to topology (1972). This latter work is an introduction to algebraic topology for a reader without background in general topology. The book builds up to a proof of the Alexander duality theorem in the plane; a result which generalises the Jordan curve theorem. In 1995 Wall published The geometry of topological stability written jointly with A A du Plessis. His http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wall.html (1 of 3) [2/16/2002 11:37:23 PM]

Wall

current research continues the studies of applications of techniques developed in this book which is having spin-offs to results about hypersurfaces in projective space. To date he has published over 160 research articles on mathematical topics. Wall's work has led to him receiving many awards. He was elected a fellow of the Royal Society in 1969 and the Royal Society has honoured him further with the award of its Sylvester Medal in 1988. He was a member of the Council of the Royal Society from 1974 to 1976. He has served the London Mathematical Society in many ways over a large number of years being on the Council from 1972 to 1980 and then for a second spell from 1992 to 1996. He served as the 59th President of the London Mathematical Society from 1978-80. The Society has made him a number of awards to mark his fine mathematical achievements, including the award of their Junior Berwick Prize, their Whitehead Prize in 1976 and their Pólya Prize in 1988. The Whitehead Prize and the Pólya Prize were awarded for his work on surgery on manifolds and L-theory. In 1990 Wall was elected a foreign member of the Royal Danish Academy. Outside mathematics Wall has many interests including home wine making and an interest in politics which has seen him as treasurer of the Wirral Area SDP from 1985 to 1988 and then becoming a member of the Wirral West Liberal Democrat Party. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Terry Wall (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1969

Royal Society Sylvester Medal

Awarded 1988

London Maths Society President

1978 - 1980

LMS Berwick Prize winner

1965

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wall.html (2 of 3) [2/16/2002 11:37:23 PM]

Wall

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wall.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wall.html (3 of 3) [2/16/2002 11:37:23 PM]

Wallace

William Wallace Born: 23 Sept 1768 in Dysart, Scotland Died: 28 April 1843 in Edinburgh, Scotland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Wallace was self-taught in mathematics earning his living working for a bookbinder and tutoring mathematics. He became a mathematics teacher at Perth Academy in 1794. Playfair advised him to apply for the post of professor at the Royal Military College at Great Marlow where he was a colleague of Ivory. Then, in 1819, he was appointed professor of mathematics at Edinburgh University. Wallace's work was on geometry and Simson's line (which is definitely not due to Simson!) appears first in a paper of Wallace in 1799. One of Wallace's theorems, if 4 lines intersect each other to form 4 triangles (omit one line in turn) then the circumcircles of the triangles have a point in common, was generalised to 2n lines by Clifford. He published two books, A New Book of Interest containing Aliquot Tables (aliquot = fractional) and Geometrical Theorems and Analytical Formulae. Wallace also invented the pantograph, an instrument for duplicating a geometric shape at a reduced or enlarged scale. In addition to mathematical articles, he wrote articles on astronomy which he published in the Transactions of the Royal Astronomical Society. In [4] it is said that:He took an active interest in the erection of the Observatory on the Carlton Hill and the monument to Napier. As a Professor, Wallace was regarded as an able teacher, he was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wallace.html (1 of 2) [2/16/2002 11:37:25 PM]

Wallace

popular alike with pupils and colleagues. In recognition of his services to learning and to the University, he was made an honorary Doctor of Laws. Wallace retired from his chair at Edinburgh in 1838 due to ill health. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) A Poster of William Wallace

Mathematicians born in the same country

Other references in MacTutor

The Simson line configuration.

Honours awarded to William Wallace (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wallace.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wallace.html (2 of 2) [2/16/2002 11:37:25 PM]

Wallis

John Wallis Born: 23 Nov 1616 in Ashford, Kent, England Died: 28 Oct 1703 in Oxford, England

Click the picture above to see five larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Wallis went to school in Ashford, then moved to Tenterden where he first showed his great potential as a scholar. In 1630 he went to Felsted where he became proficient in Latin, Greek and Hebrew. From there he went to Emmanual College Cambridge where he first became interested in mathematics. Since nobody at Cambridge at this time could direct his mathematical studies, his main topic of study became divinity. He was ordained in 1640. Wallis was skilled in cryptography and decoded Royalist messages for the Parliamentarians during the Civil War. It is suggested that he was appointed to the Savilian Chair of geometry at Oxford in 1649 because of this. Certainly the holder of the chair, Peter Turner, was dismissed for his Royalist views. Wallis held the chair for over 50 years until his death and, even if the reason for his appointment is true, he most certainly deserved to hold the chair. Wallis was part of a group interested in natural and experimental science who started to meet in London. This group was to became the Royal Society, so Wallis is a founder member of the Royal Society and one of its first Fellows. Wallis contributed substantially to the origins of calculus and was the most influential English mathematician before Newton. He studied the works of Kepler, Cavalieri, Roberval, Torricelli and Descartes. Then Wallis introduced ideas of the calculus going beyond that of these authors. In Arithmetica infinitorum (1656) Wallis evaluated the integral of (1 - x2)n from 0 to 1 for integral values of n, building on Cavalieri's method of indivisibles. He devised a method of interpolation in an attempt to

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wallis.html (1 of 3) [2/16/2002 11:37:27 PM]

Wallis

compute the integral of (1 - x2)1/2 from 0 to 1. Using Kepler's concept of continuity he discovered methods to evaluate integrals which were later used by Newton in his work on the binomial theorem. Newton says About the beginning of my mathematical studies, as soon as the works of our celebrated countryman, Dr Wallis, fell into my hands, by considering the Series, by the Intercalation of which, he exhibits the Area of the Circle and the Hyperbola.... In Arithmetica infinitorum Wallis also established the formula /2 = (2.2.4.4.6.6.8.8.10..)/(1.3.3.5.5.7.7.9.9...) In his Tract on Conic Sections (1656) Wallis described the curves that are obtained as cross sections by cutting a cone with a plane as properties of algebraic coordinates without the embranglings of the cone. He followed methods in the style of Descartes' analytical treatment. Wallis was also an important early historian of mathematics and in his Treatise on Algebra he has a wealth of historical material. However the most important feature of this work, which appeared in 1685, is that it brought to mathematicians the work of Harriot in a clear exposition. In Treatise on Algebra Wallis accepts negative roots and complex roots. He shows that a3 - 7a = 6 has exactly three roots and that they are all real. He also criticises Descartes' Rule of Signs stating, quite correctly, that the rule which determines the number of positive and the number of negative roots by inspection, is only valid if all the roots of the equation are real. Wallis introduced our present symbol

for infinity.

He also restored some ancient Greek texts such as Ptolemy's Harmonics, Aristarchus's On the magnitudes and distances of the sun and moon and Archimedes' Sand-reckoner. His non-mathematical works include many religious works, a book on etymology and grammar Grammatica linguae Anglicanae (Oxford, 1653) and a logic book Institutio logicae (Oxford, 1687). Wallis had a bitter dispute with Hobbes, who although a fine scholar, was far from Wallis's class as a mathematician. In 1655 Hobbes claimed to have discovered a method to calculate the area of a circle by integration. Wallis's book with his methods was in press at the time and he refuted Hobbes's claims. Hobbes replied to the ... insolent, injurious, clownish language of Wallis with the pamphlet Six lessons to the Professors of Mathematics at the Institute of Sir Henry Savile. Wallis replied with the pamphlet Due Correction for Mr Hobbes, or School Discipline for not saying his Lessons Aright to which Hobbes wrote the pamphlet The Marks of the Absurd Geometry, Rural Language etc. of Doctor Wallis. The dispute continued for over 20 years, becoming extended to include Boyle, and ending only with Hobbes's death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (32 books/articles)

A Quotation

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wallis.html (2 of 3) [2/16/2002 11:37:27 PM]

Wallis

A Poster of John Wallis

Mathematicians born in the same country

Some pages from publications

Extract from Arithmetica infinitorum (1656) Page from De postulato quinto: et definitione quinta (1656)

Cross-references to History Topics

1. Non-Euclidean geometry 2. Elliptic functions. 3. Memory, mental arithmetic and mathematics 4. English attack on the Longitude Problem 5. The trigonometric functions 6. Pi through the ages

Cross-references to Famous Curves

1. Cissoid of Diocles 2. Neile's semi-cubical parabola

Other references in MacTutor

1. Chronology: 1650 to 1675 2. Chronology: 1675 to 1700

Honours awarded to John Wallis (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1663

Savilian Professor of Geometry

1649

Other Web sites

1. The Galileo Project 2. Rouse Ball 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wallis.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wallis.html (3 of 3) [2/16/2002 11:37:27 PM]

Wang

Hsien Chung Wang Born: 18 April 1918 in Peking (now Beijing), China Died: 25 June 1978 in New York, USA Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hsien Chung Wang studied at Nankai High School in Tientsin. He began his university studies at Tsing Hua University in Peking in 1936. His original intention was to take a degree in physics so he began his studies taking courses which would lead to this degree. On 7 July 1937 Japanese and Chinese troops clashed near Peking. In late July further fighting broke out and the Japanese quickly captured Peking and Tientsin. Tsing Hua University was moved to Southwest China where it was amalgamated with Nankai and Peking universities. Wang had to journey to the new site of his university and begin his studies again. Perhaps the political events had a positive effect as far as mathematics was concerned since Wang changed his studies to mathematics when he took them up again at the re-established university. Wang graduated in 1941 and began to study under S S Chern. He was awarded a master's degree in 1944 and began teaching in a school. However, after one year, he was awarded a British Council Scholarship to continue his studies in England. After a while at Sheffield he studied under Newman at Manchester and received a Ph.D. in 1948. On his return to China, Wang took up a research post at the Chinese National Academy of Sciences. However political events were again to play a major part in Wang's career. Between early November 1948 and early January 1949 the Communists and Nationalists fought for control. The National Government re-established itself on Taiwan where it had withdrawn early in 1949. The Chinese National Academy of Sciences was set up on Taiwan and Wang followed the Academy there. From 1949 Wang lived in the United States. This was not an easy time to obtain a mathematics post in the United States and Wang, although he had an impressive reputation as a mathematician by this time, could only manage a succession of temporary posts. First he taught at Louisiana State, then for two years at Baton Rouge before he spent his first year at Princeton in 1951-52. Again he held temporary posts, this time for two years at Alabama Polytechnic, then 1954-55 at Princeton again, 1955-57 at the University of Washington in Seattle followed by a time at Columbia in New York. Wang married during his time in Seattle. The year 1957 saw Wang receive an offer of a permanent post for the first time. This was at Northwestern University where he remained, having further spells at Princeton during this time, until 1966 when he was appointed to Cornell.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wang.html (1 of 2) [2/16/2002 11:37:29 PM]

Wang

Wang worked on algebraic topology and discovered the 'Wang sequence', an exact sequence involving homology groups associated with fibre bundles over spheres. These discoveries were made while he worked with Newman in Manchester. Wang also solved, at that time, an important open problem in determining the closed subgroups of maximal rank in a compact Lie group. Wang's most important work was on discrete subgroups of Lie groups, a topic on which he continued to work. He published Two-point homogeneous spaces in 1952 which dealt with a homogeneous space of a compact Lie group. In 1960 he studied transformation groups of n-spheres and wrote the highly original paper Compact transformation groups of Sn with an (n-1)-dimensional orbit. The latter part of Wang's life is described in [1] as follows:Wang's last paper was published in 1973, after which his research was much curtailed because of anxiety for his wife, who had developed cancer. His teaching and other mathematical and administrative activities continued unabated, however, and he played an important role in the department at Cornell. He was very much liked there, as everywhere, for his modesty, generosity, kindness and curtsey. He was a fine teacher and lecturer. ... He enjoyed excellent health until he was suddenly stricken with leukaemia in June 1978. He succumbed within weeks, to be survived for only a few months by his wife. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wang.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wang.html (2 of 2) [2/16/2002 11:37:29 PM]

Wangerin

Albert Wangerin Born: 18 Nov 1844 in Greiffenberg, Pomerania, Germany Died: 25 Oct 1933 in Halle, Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Albert Wangerin entered the University of Halle in 1862. From Halle he moved to the University of Königsberg where he studied under Franz Neumann being awarded his doctorate from Königsberg in 1866. It is clear, both from Wangerin's subsequent career, and from his writings, that he was greatly influenced by Franz Neumann. Wangerin would undertake research for the rest of his life on topics suggested by Franz Neumann and Wangerin [1]:... later wrote a book (1907) and a highly appreciative article on his former teacher. Although Jacobi had died over ten years before Wangerin began his studies at Königsberg, his influence was still strongly felt and it would not be unreasonable to say that Wangerin, through his teachers at Königsberg, was strongly influenced by Jacobi's style of mathematics. After he was awarded his doctorate, Wangerin was appointed as a teacher of mathematics in a Gymnasium in Posen. Wangerin moved from Posen to Berlin where again he taught mathematics in a Gymnasium until 1876. In the spring of 1876 he was appointed as an extraordinary professor at the University of Berlin. Wangerin remained in Berlin until 1882 when he was offered the position of ordinary professor at the University of Halle. Back in the university in which he had studied as an undergraduate, Wangerin held his professorship there until he retired in 1919. He continued to live and do mathematics in Halle after his retirement until his death at age almost 89. Wangerin's research was on potential theory, spherical functions and differential geometry. He wrote an important two volume treatise on potential theory and spherical functions. Despite great expertise in applications to mathematical physics, research was not the most important of Wangerin's contributions to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wangerin.html (1 of 2) [2/16/2002 11:37:31 PM]

Wangerin

the development of mathematics. Wangerin's main contribution was his writing of textbooks, writing for encyclopaedias and his historical writings. As examples of Wangerin's historical writing, in addition to the article on his teacher Franz Neumann which we mentioned above, we should point in particular to the article he wrote on Eduard Heine in 1928 as well as to his input to editing the works of Gauss, Euler, Lambert, and Lagrange. His two major encyclopaedia articles were both written for Encyklopädie der mathematischen Wissenschaften, the first in 1904 being on functions such as the Lamé and Bessel functions, while his second written in 1907 was for the physics volume of the encyclopaedia and was on optics. He also played a major role in the reviewing of mathematical papers. As a coeditor of Fortschritte der Mathematik from 1869 to 1921 he had a major influence in the policy of what during that period was the only reviewing journal for mathematics. His influence as a teacher was also strongly felt [1]:While at Berlin he directed his lectures to a fairly broad audience, and even at Halle he continued to be greatly interested in the training of high school teachers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wangerin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wangerin.html (2 of 2) [2/16/2002 11:37:31 PM]

Wantzel

Pierre Laurent Wantzel Born: 5 June 1814 in Paris, France Died: 21 May 1848 in Paris, France Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Pierre Wantzel's father served in the army for seven years after the birth of Pierre, then became professor of applied mathematics at the Ecole speciale du Commerce. Pierre Wantzel attended primary school in Ecouen, near Paris, where the family lived. Even at a very young age he showed great aptitude for mathematics, and Saint-Venant relates in [4] that ... he showed, with his great memory, a marvellous aptitude for mathematics, a subject about which he read with extreme interest. He soon surpassed even his master, who sent for the young Wantzel, at age nine, when he encountered a difficult surveying problem. In 1826, while still only 12 years old, Wantzel entered the Ecole des Arts et Métiers de Châlons. Here he was extremely fortunate in having Etienne Bobillier as a mathematics teacher. However France at this time was filled with political unrest and revolts, one of which caused the school to be reorganised in 1827. Unhappy with the less academic nature of the school in 1828, Wantzel entered the Collège Charlemagne after being coached in Latin and Greek by a M Lievyns (whose daughter he was later to marry). By 1829, at the remarkably young age of 15, he edited a second edition of Reynaud's Treatise on arithmetic giving a proof of a method for finding square roots which was widely used but previously unproved. In [2] de Lapparent describes his successes at the Collège Charlemagne:In 1831, the first prize of French dissertation from the Collège Charlemagne was awarded to him, and better yet, first prize in Latin dissertation, acquired in open contest, attested with splendour to the universality of Wantzel's aptitude. He was placed first in 1832 in the entrance examination to the Ecole Polytechnique and also first for the science section of the Ecole Normale. This had never previously been achieved and, as related in [3]:... he threw himself into mathematics, philosophy, history, music, and into controversy, exhibiting everywhere equal superiority of mind. He entered the engineering school of Ponts et Chaussées in 1834 and was sent to the Ardennes in 1835, then to Berry in 1836. However Saint-Venant in [4] says that Wantzel:... said merrily to his friends that he would be but a mediocre engineer. He preferred the teaching of mathematics... In order to further his career in mathematics he asked for leave of absence. He became a lecturer in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wantzel.html (1 of 3) [2/16/2002 11:37:32 PM]

Wantzel

analysis at the Ecole Polytechnique in 1838 but, in addition, he was made an engineer in 1840 and from 1841 became professor of applied mechanics at the Ecole des Ponts et Chaussées. Wantzel was not one to take life easy and he took on additional duties taking charge of the entrance examinations at the Ecole Polytechnique in 1843 and in addition taught various mathematics and physics courses at various schools around Paris, including at the Collège Charlemagne. Wantzel is famed for his work on solving equations by radicals. In 1837 Wantzel published proofs of what are some of the most famous mathematical problems of all time in a paper in Liouville's Journal on ... the means of ascertaining whether a geometric problem can be solved with ruler and compasses. Gauss had stated that the problems of duplicating a cube and trisecting an angle could not be solved with ruler and compasses but he gave no proofs. In this 1837 paper Wantzel was the first to prove these results. Improved proofs were later given by Charles Sturm but he did not publish them. In 1845 Wantzel, continuing his researches into equations, gave a new proof of the impossibility of solving all algebraic equations by radicals. Wantzel writes in the introduction:Although [Abel's] proof is finally correct, it is presented in a form too complicated and so vague that it is not generally accepted. Many years previous, Ruffini, an Italian mathematician, had treated the same question in a manner much vaguer still and with insufficient developments, although he had returned to this subject many times. In meditating on the researches of these two mathematicians, and with the aid of principles we established in an earlier paper, we have arrived at a form of proof which appears so strict as to remove all doubt on this important part of the theory of equations. In fact Wantzel published over 20 works which are listed in [4]. Three of these works are written jointly with Saint-Venant and concern the flow of air when there is a large pressure difference. De Lapparent in [2] sums up his other work as follows:We owe to him a note on the curvature of elastic rods, several works on the flow of air ... finally, in 1848, an important posthumous note on the rectilinear diameters of curves. It was he who first gave the integration of differential equations of the elastic curve. According to Saint-Venant in [4] his death was the result of overwork. Saint-Venant wrote:... one could reproach him for having been too rebellious against those counselling prudence. He usually worked during the evening, not going to bed until late in the night, then reading, and got but a few hours of agitated sleep, alternatively abusing coffee and opium, taking his meals, until his marriage, at odd and irregular hours. Wantzel certainly published some important results, although it must be understood that his proofs of the impossibliity of solving the classical ruler and compass problems were built on the work of others. Saint-Venant, in [4], ponders the question of why Wantzel with one of the most impressive early achievements of any mathematician, should have failed to achieve even more innovative results despite his early death. Saint-Venant writes:... I believe that this is mostly due to the irregular manner in which he worked, to the excessive number of occupations in which he was engaged, to the continual movement and feverishness of his thoughts, and even to the abuse of his own facilities. Wantzel improvised

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wantzel.html (2 of 3) [2/16/2002 11:37:32 PM]

Wantzel

more than he elaborated, he probably did not give himself the leisure nor the calm necessary to linger long on the same subject. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. Doubling the cube 2. Trisecting an angle

Other references in MacTutor

Chronology: 1830 to 1840

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wantzel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wantzel.html (3 of 3) [2/16/2002 11:37:32 PM]

Waring

Edward Waring Born: 1736 in Old Heath (near Shrewsbury), Shropshire, England Died: 15 Aug 1798 in Pontesbury, Shropshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edward Waring's father, John Waring, was a farmer. Several generations of his family lived at Mytton in Shropshire. John Waring married Elizabeth and their son Edward was educated at Shrewsbury school. Waring entered Magdalene College, Cambridge on 24 March 1753. He won a Milbridge scholarship and he was admitted as a sizar, meaning that he paid a reduced fee but essentially worked as a servant to make good the fee reduction. He immediately impressed his teachers with his mathematical ability and he graduated B.A. in 1757 as senior wrangler. On 24 April 1754 Waring was elected a fellow of Magdalene College. Waring's most famous work was Meditationes Algebraicae which he worked on during the next few years. He submitted the first chapter of this work to the Royal Society but following this nothing happened for two years. When Waring was nominated for the Lucasian Chair of Mathematics at Cambridge in 1759, the work was distributed as Miscellanea Analytica to prove he was qualified for the post despite his youth. William Powell of St John's College Cambridge had his own ideas about who should fill the Lucasian Chair of Mathematics and attempted to prevent Waring being appointed. He put out a pamphlet entitled Observations which criticised Waring and doubted his mathematical abilities. Waring responded to this criticism on 25 January 1760 with the pamphlet A reply to the observations. Now Powell, still following his own agenda, was not going to give up that easily and responded immediately with Defence of the observations. John Wilson now wrote A letter to support Waring and this was sufficient to see him confirmed as Lucasian professor on 28 January 1760 at the age of 23. When Miscellanea Analytica was published as a complete work in 1762, Waring chose to call it a second http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Waring.html (1 of 4) [2/16/2002 11:37:34 PM]

Waring

edition. Rather strangely, despite it being a second edition, it was given a new title Meditationes Algebraicae. We shall comment further below on this important work, covering topics in the theory of equations, number theory and geometry. On 2 June 1763 Waring was elected a Fellow of the Royal Society. Now in 1764 Lalande published Life of Condorcet. In this work it was claimed that there were no first-class analysts in England. Waring responded quickly to this comment by writing a letter to Nevil Maskelyne, the Astronomer Royal. In the letter Waring pointed out that D'Alembert, Euler and Lagrange had all praised his 1762 work. Waring wrote that in the book he had given:... somewhare between three and four hundred new propositions of one kind or another, considerably more than have been given by any other English writer. However, Waring knew that his work had not been widely read for he added that he:... never could hear of any reader in England, out of Cambridge, who took pains to read and understand it ... The reason that so few had read the book was partly because the subject matter was difficult, partly because Waring was a poor communicator, and partly because he did not have a good algebraic notation. It would be reasonable to compare Waring with Ruffini who, about 150 years later, suffered the same fate with his work in algebra for much the same reasons. One would not expect the Lucasian professor of mathematics to take a medical degree but that is exactly what Waring did, graduating with his M.D. in 1767. For a short time he practised medicine in various London hospitals, then Addenbroke hospital in Cambridge and finally at a hospital in St Ives, Huntingtonshire. However, he gave up practising medicine by 1770 [7]:... he was very short-sighted and very shy in manner, so that he quickly abandoned his profession. One might ask how Waring could practise medicine and hold the Lucasian Chair at the same time. Well, he never lectured as part of his duties. Some claim that it was because his ideas were so profound that they could not be communicated in lectures, but if truth be told it is more likely that the reason was because he was a poor communicator with handwriting which was almost impossible to read. [I will agree that such problems have not stopped others lecturing!] In 1776 Waring married Mary Oswell. They lived for a while in Shrewsbury but the town was not to Mary's liking and the couple moved to Waring's estate at Plealey in Pontesbury. Waring's Miscellane analytica... of 1762 formed the basis, as we have noted, of further books. Proprietates algebraicarum curvarum, covering geometry, was published in 1772. A further work Miscellanea Analytica appeared in 1776 with a new expanded edition in 1785. Meditationes Algebraicae, covering the theory of equations and number theory, appeared in 1770 with an expanded version in 1782. In Meditationes Algebraicae Waring proves that all rational symmetric functions of the roots of an equation can be expressed as rational functions of the coefficients. He derived a method for expressing symmetric polynomials and he investigated the cyclotomic equation xn - 1 = 0. This work makes Waring one of the earliest contributers to Galois theory. In particular, discussing Problem 22 of Chapter 3, Weeks writes [3]:The most significant aspect of Waring's treatment of this example is the symmetric relation http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Waring.html (2 of 4) [2/16/2002 11:37:34 PM]

Waring

between the roots of the quartic equation and its resolvent cubic. This is, in essence, the first result in the theory of symmetric functions (beyond the basic building blocks which appeared in Chapter 1), a theory whose systematic development was not to appear until the 19th century (Lagrange, Gauss, and others) and was ultimately followed by the theory of permutation groups (Galois, Jordan, ...). Chapter 4 of Meditationes Algebraicae contains results such as: k equations in k unknowns can be reduced to one equation with one unknown. His result that the product of the degrees of the original equations is the degree of the single reduced equation is known as the Generalised Theorem of Bezout. The rest of the book deals with number theory, a topic in which Waring made some interesting advances. He stated that any even integer can be written as the sum of two primes and every odd integer is either a prime or the sum of three primes. This result, now known as the Goldbach conjecture, is one of the most famous unsolved problems of mathematics. Although Goldbach proposed his question in a letter to Euler long before Waring published Meditationes Algebraicae it is still worth noting that Waring's version was the first to be published. Waring also stated, without giving a proof, what is now known as 'Waring's theorem':Every integer is equal to the sum of not more than 9 cubes. Also every integer is the sum of not more than 19 fourth powers, and so on .... In 1909 Hilbert proved that given any integer n there is an integer m (depending on n) such that every integer is a sum of m nth powers. It is reasonable to assume that Waring had this type of result in mind when he stated 'Waring's theorem'. Hilbert's proof led to major new theorems in number theory. Waring also wrote on algebraic curves, classifying quartic curves into 12 main divisions with 84551 subdivisions. Several descriptions of Waring given by authors from his own period are not too flattering. One writes that he was:... one of the strongest compounds of vanity and modesty which the human character exhibits. The former, however, is his predominant feature. Another says that he is:... one of the greatest analysts that England has produced ... [ near the end of his life being] sunk into a deep religious melancholy approaching to insanity. This last statement may partly explain the strange fact that although Waring was elected a Fellow of the Royal Society in 1763 and had the great distinction of being awarded its Copley Medal in 1784, he resigned from the Society in 1795 claiming poverty. He was awarded other honours, however, such as election to the Royal Society of Göttingen and the Royal Society of Bologna. Perhaps the most accurate assessment of Waring was made by Thomas Thomson:Waring was one of the profoundest mathematicians of the eighteenth century; but the inelegance and obscurity of his writings prevented him from obtaining that reputation to which he was entitled. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Waring.html (3 of 4) [2/16/2002 11:37:34 PM]

Waring

Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) A Poster of Edward Waring

Mathematicians born in the same country

Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Honours awarded to Edward Waring (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1763

Royal Society Copley Medal

Awarded 1784

Lucasian Professor of Mathematics

1760

Other Web sites

1. Bob Bruen 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Waring.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Waring.html (4 of 4) [2/16/2002 11:37:34 PM]

Watson

George Neville Watson Born: 31 Jan 1886 in Westward Ho!, Devon, England Died: 2 Feb 1965 in Leamington Spa, Warwickshire, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

G N Watson was educated at Cambridge where he was a pupil of Whittaker and graduated in 1907, becoming a Fellow of Trinity in 1910. From 1918 to 1951 he was professor at Birmingham. Watson worked on a wide variety of topics, all within the area of complex variable theory, such as difference equations, differential equations, number theory and special functions. He is best known as a joint author with Whittaker of A Course of Modern Analysis published in 1915. The first edition of the book has only Whittaker as an author. Watson undertook a major work on Ramanujan's notebooks, extending his results and supplying proofs. He enjoyed numerical calculations and spent many happy hours doing numerical work on his calculating machine. He was elected to the Royal Society of London in 1919. In 1946 he received the Sylvester Medal of the Royal Society:... in recognition of his distinguished contributions to pure mathematics in the field of mathematical analysis and in particular for his work on asymptotic expansion and on general transforms. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Watson.html (1 of 2) [2/16/2002 11:37:36 PM]

Watson

List of References (4 books/articles) Mathematicians born in the same country Honours awarded to G N Watson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1919

Royal Society Sylvester Medal

Awarded 1946

Fellow of the Royal Society of Edinburgh London Maths Society President

1933 - 1935

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Watson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Watson.html (2 of 2) [2/16/2002 11:37:36 PM]

Watson_Henry

Henry William Watson Born: 25 Feb 1827 in Marylebone, London, England Died: 11 Jan 1903 in Berkswell (near Coventry), England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henry Watson was educated at King's College, London, then in 1846 he went to Trinity College, Cambridge. He graduated in 1850 as second wrangler and Smith's prizeman. In 1851 Watson became a fellow of Trinity College and from 1851 until 1853 he was a tutor in mathematics at Cambridge. In 1854 he became mathematics master at the City of London School, then in 1857 he lectured at King's College, London. In fact Watson has been ordained a deacon in 1856 and he had taken priest's orders two years later. From 1857 until 1865 he was mathematics master at Harrow School. He left in 1865, however, to become Rector of Berkswell with Barston near Coventry where he remained until he retired. Watson wrote a number of mathematics books during the time he was rector of Berkswell. He wrote The elements of plane and solid geometry (1871), Treatise on the kinetic theory of gases (1876). After this book appeared Watson corresponded with Maxwell and the results of this correspondence is contained in the second edition of the book which appeared in 1893. After publishing on dynamics in 1879, Watson wrote a two volume work The mathematical theory of electricity and magnetism. The first volume Electrostatics appeared in 1885, the second Magnetism and electrodynamics appeared in 1889. In addition to these books he wrote on Lagrange's method and Monge's method for solving partial differential equations and, jointly with Galton, he wrote On the probability of extinction of families. This paper was written after a correspondence between the authors in 1873 and the paper contains a version of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Watson_Henry.html (1 of 2) [2/16/2002 11:37:38 PM]

Watson_Henry

the 'Criticality Theorem' which is the foundation of the modern theory of branching processes. Kendall describes in [4] how the work started:Galton ... gave to the problem [of extinction of families] a precise mathematical formulation, and communicated it in 1873 to ... the Educational Times. ... The only solution received did not please Galton ... He therefore persuaded Watson to take up the matter. ... Galton had consulted Watson a year or so earlier about a device for obtaining useful work from the energy of waves... Watson was elected a fellow of the Royal Society in 1881 and given an honorary D.Sc. by Cambridge in 1883. Among his interests was mountaineering and he was an early member of the Alpine Club. He was also a founder member of the Birmingham Philosophical Society and served as its president in 1880-81. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Honours awarded to Henry Watson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1881

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Watson_Henry.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Watson_Henry.html (2 of 2) [2/16/2002 11:37:38 PM]

Wazewski

Tadeusz Wazewski Born: 24 Sept 1896 in Galicia, Poland Died: 5 Sept 1972 Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Tadeusz Wazewski was born at a difficult time in Polish history. Poland had been partitioned in 1772 and the south, called Galicia, was under Austrian control. Russia controlled much of the country and in the years prior to Wazewski's birth there had been strong moves by Russia to make "Vistula Land", as it was called, be dominated by Russian culture. In a policy implemented between 1869 and 1874, all secondary schooling was in Russian. Warsaw only had a Russian language university after the University of Warsaw became a Russian university in 1869. Galicia, although under Austrian control, retained Polish culture and only there could Poles receive a Polish education. Wazewski attended secondary school in Tarnow, today in the south-east of Poland, but at that time part of Austrian controlled Galicia. On graduating from secondary school his intention was to study physics and he entered the Jagiellonian University in Kraków to study that subject. However, he was strongly influenced by Zaremba and changed his course from physics to mathematics. Under Zaremba, Wazewski became interested in set theory and topology and decided to study in Paris for his doctorate. This was a typical route for Polish mathematicians of this period, forced to study abroad (as Poland was partitioned) many chose the same route as Wazewski studying in Polish universities in Austrian controlled Galicia and completing their education in France. Wazewski studied in Paris between 1921 and 1923 continuing his interest in topology acquired during his studies at Kraków under Zaremba. His doctoral dissertation, on topological results relating to dendrites, was examined in 1923 by the powerful examining committee consisting of Borel, Denjoy and Montel. Having secured his doctorate, Wazewski returned to the Jagiellonian University in Kraków where he was he was appointed a docent in 1927. At about this time his interests shifted away from set theory and topology and he became interested in analysis. In 1933 he was appointed extraordinary professor at the Jagiellonian University. Kraków was taken by the German army at the beginning of World War II. A German governor was installed in Wawel Castle, and executions members of the teaching staff of the Jagiellonian University took place. Some 55,000 Jews from the city were sent to the Auschwitz-Birkenau concentration camp. Wazewski was sent to the Sachsenhausen- Oranienburg concentration camp north-west of Berlin where he survived for two years. Some were not so lucky, for example A Hoborski was a mathematics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wazewski.html (1 of 3) [2/16/2002 11:37:40 PM]

Wazewski

professor at the Jagiellonian University who died in the Sachsenhausen- Oranienburg concentration camp in February 1940. Wazewski returned to Kraków and taught in the underground university there despite the severe risk to his life. In 1945 Kraków was liberated in a surprise attack made by Soviet forces. Wazewski was appointed a full professor at the Jagiellonian University and put all his efforts into restoring the educational system which had been destroyed. Wazewski made important contributions to the theory of ordinary differential equations, partial differential equations, control theory and the theory of analytic spaces. The contribution for which he is most famous was made after his appointment as professor at the Jagiellonian University after the end of World War II. Kuratowski explains in [1] how his idea:... was to bring him fame and lead to the development of a new school of differential equations. ... he succeeded in applying with amazing effect the topological notion of retract (introduced by K Borsuk) to the study of the solutions of differential equations. Wazewski was invited to explain his ideas in a plenary address at the International Congress of Mathematicians in Amsterdam in 1954 and [1]:Lefschetz considered his method of retracts one of the most important achievements in the theory of differential equations since the war. We mentioned above that Wazewski worked in control theory. His interest in that topic began around 1960 and he published a series of important papers on the topic through the 1960s. His work on the time optimal control problem, to which he took a topologically oriented approach, is described in [4]. By this time he was heading his own school of mathematics which was highly successful because of the [1]:... broad scope of his problems, his skill in putting forward deeply motivated questions and his great teaching talent. Kuratowski also comments in [1] about Wazewski's personality describing him as:... gifted with immense qualities of character: his proverbial modesty, kindness, and the great care he showed his students meant that, besides general respect for an excellent scholar, everybody who had the good fortune to know him was charmed by his extraordinary personality. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wazewski.html (2 of 3) [2/16/2002 11:37:40 PM]

Wazewski

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wazewski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wazewski.html (3 of 3) [2/16/2002 11:37:40 PM]

Weatherburn

Charles Ernest Weatherburn Born: 18 June 1884 in Australia Died: 1974 in Australia

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Charles Weatherburn studied under H S Carslaw at the University of Sydney graduating with an M.A. in 1906. He then came to England, after the award of a scholarship, and studied at Trinity College Cambridge where he attended lectures by Whitehead, Whittaker and Hardy. He sat the Mathematical Tripos examinations in 1908, the same year as Brodetsky, and was awarded a First Class degree. Returning to Australia, Weatherburn was appointed to Ormond College of the University of Melbourne. Pitman, writing in [1], describes how Weatherburn taught him as a student at the University of Melbourne (1916,1917,1920):There were very few honours students, and I was the only Ormond student doing honours mathematics in my year. I went to his room once a week, and sat near his desk while he talked and wrote notes for me. Always he wrote on the back of foolscap paper, the front of which was filled with an early draft of a section of one of his books. He took me through the topics in his two books on vector analysis, and perhaps also some differential geometry... He was neat and clear and interesting, and for me it was a very easy and efficient way of mastering vector analysis. Certainly vector analysis was not universally accepted at this time and Weatherburn fought the battle for its acceptance against opposition from people such as Harold Jeffreys. Gibbs and Heaviside had been early exponents of the vector calculus while its chief opponents had been Tait. When Weatherburn published the first of his two volumes on vector analysis in 1921 he wrote in the introduction:The work of Gibbs and Heaviside drew forth denunciations from Professor Tait, who considered any departure from quaternionic usage in the treatment of vectors to be an enormity. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weatherburn.html (1 of 2) [2/16/2002 11:37:42 PM]

Weatherburn

Weatherburn left Sydney in 1923 to take up the chair of mathematics in Canterbury College, University of New Zealand. At about this time his research interests changed from vector analysis to differential geometry. He wrote two major volumes Differential geometry of three dimensions (1927, 1930) as well as nearly 30 papers on this topic. Hodge, reviewing the second volume, wrote:Much of the volume is devoted to subjects to which the author has himself contributed in the last few years, particularly in the theory of families of curves and surfaces, and of small deformations. Other topics are however included, with the result that the two volumes together give an account of most of the principal branches of classical differential geometry. An elementary account of Levi-Civita's theory of parallel displacements is given. In 1929 Weatherburn returned to Australia taking up the chair of mathematics at the University of Western Australia (founded 1911), becoming the first holder of this chair. He held this post until he retired in 1950 but his excellent sequence of research papers stopped in 1939. He published An Introduction to Riemannian Geometry and the Tensor Calculus in 1938 and it was reissued in 1966. After 1939 his only publication was a textbook on statistics which he published in 1946. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Charles Weatherburn

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Weatherburn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weatherburn.html (2 of 2) [2/16/2002 11:37:42 PM]

Weber

Wilhelm Eduard Weber Born: 24 Oct 1804 in Wittenberg, Saxony (now Germany) Died: 23 June 1891 in Göttingen, Germany

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Weber entered the University of Halle in 1822 and wrote his doctoral dissertation in 1826. After that he taught at Halle. In 1831 Weber was appointed to the chair of physics at Göttingen and there followed 6 years of close friendship and collaboration with Gauss. Weber developed sensitive magnetometers and other magnetic instruments during this time. When Victoria became Queen of Britain in 1837 her uncle became ruler of Hanover and revoked the liberal constitution. Weber was one of 7 professors at Göttingen to sign a protest and all were dismissed. He remained at Göttingen without a position until 1843 when he became professor of physics at Leipzig. In 1848 he returned to his old position in Göttingen and, in 1855, he and Dirichlet became temporary directors of the astronomical observatory there. His work on the ratio between the electrodynamic and electrostatic units of charge in 1855 proved extremely important and was crucial to Maxwell in his electromagnetic theory of light. Weber found the ratio was 3.1074 notice of the fact that this was close to the speed of light.

108 m/sec but failed to take any

Weber's later years at Göttingen were devoted to work in electrodynamics and the electrical structure of matter. He was described by Thomas Hirst in the following way: He speaks and stutters on unceasingly, one has nothing to do but listen. Sometimes he laughs for no earthly reason, and one feels sorry at being not able to join him.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weber.html (1 of 2) [2/16/2002 11:37:43 PM]

Weber

Article by: J J O'Connor and E F Robertson List of References (11 books/articles) A Poster of Wilhelm Weber

Mathematicians born in the same country

Cross-references to History Topics

A comment from Thomas Hirst's diary

Honours awarded to Wilhelm Weber (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1850

Royal Society Copley Medal

Awarded 1859

Lunar features

Crater Weber

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Weber.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weber.html (2 of 2) [2/16/2002 11:37:43 PM]

Weber_Heinrich

Heinrich Martin Weber Born: 5 May 1842 in Heidelberg, Germany Died: 17 May 1913 in Strasbourg, Germany (now France)

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Heinrich Weber was born in Heidelberg, the son of G Weber who was an historian. He entered the University of Heidelberg in 1860 but, as was the common practice of German students at this time, he spent part of his time studying at a different university. For Weber it was the University of Leipzig that he moved to in the middle of his studies, spending a year there before returning to Heidelberg to complete his education. He was awarded a doctorate from the University of Heidelberg in 1863. In order to become a university teacher, Weber needed to write a further thesis, his habilitation thesis. He went to Königsberg where he studied under Franz Neumann and Friedrich Richelot, who had been a student of Jacobi. Although Jacobi had died over ten years before Weber began his studies at Königsberg, his influence was still strongly felt and it would not be unreasonable to say that Weber, through his teachers at Königsberg, was strongly influenced by Jacobi's style of mathematics. There were other students at Königsberg at this time who would become important in the development of mathematics, in particular Wangerin studied for his doctorate around the same time as Weber worked for his habilitation. In 1866 Weber's habilitation thesis was accepted and he became a privatdozent at Heidelberg in that year. Three years later he was appointed as extraordinary professor at Heidelberg. Over the next twenty-five years, Weber taught at a number of different institutions. He taught in Zurich at the Eidgenössische Polytechnikum, at the University of Königsberg, and at the Technische Hochschule on Charlottenburg. His final post was in Strasbourg where he was appointed in 1895. The city of Strasbourg was German at this time (and called Strassburg) and it had been since it was

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weber_Heinrich.html (1 of 3) [2/16/2002 11:37:45 PM]

Weber_Heinrich

annexed by Germany during the Franco-German War of 1870-71. Weber remained in Strasbourg for the rest of his life and during this time it remained a German city; only at the end of the World War I in 1918 did the city revert to France Weber's main work was in algebra, number theory, analysis and applications of analysis to mathematical physics. This seems a contradiction in terms, for we have now almost said that Weber's main work spans the whole spectrum of mathematics. In fact this is not far from the truth for Weber work was characterised by its breadth across a wide range of topics. To a certain extent this breadth can be attributed to the various influences on Weber from colleagues around him. The applications to mathematical physics certainly grew from working with Franz Neumann in Königsberg. But in Königsberg there was also the Jacobi influence, particularly coming through one of his other teacher Friedrich Richelot, which saw Weber doing important work on algebraic functions. Perhaps Weber is best known for his outstanding text Lehrbuch der Algebra published in 1895 and it is his work in algebra and number theory for which he is best known. If he was influenced by his colleagues to work in different areas of mathematics then it is a very fair question to ask where the influence came from which prompted Weber to work on algebra and number theory. The answer must be Dedekind. He wrote an important paper with Dedekind in 1882 in which they examined algebraic functions from an algebraic rather than analytic point of view. Weber's Lehrbuch der Algebra is an outstanding work but, although he tried hard to connect the various algebraic theories, even fundamental concepts such as a field and a group are only seen as tools and not properly developed as theories in their own right. It was, however, a remarkable book which was effective over many years as a teaching tool. In fact [1]:Weber was an enthusiastic and inspiring teacher who took great interest in educational questions. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

The development of group theory

Other references in MacTutor

Chronology: 1890 to 1900

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weber_Heinrich.html (2 of 3) [2/16/2002 11:37:45 PM]

Weber_Heinrich

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Weber_Heinrich.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weber_Heinrich.html (3 of 3) [2/16/2002 11:37:45 PM]

Wedderburn

Joseph Henry Maclagen Wedderburn Born: 2 Feb 1882 in Forfar, Angus, Scotland Died: 9 Oct 1948 in Princeton, New Jersey, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Wedderburn's father, Alexander Stormonth Maclagen Wedderburn, was a medical doctor. Alexander Wedderburn came from a family of Ministers of the Church with his father being the Parish Minister of Kinfauns and his grandfather (Joseph's great-grandfather) being Parish Minister of Blair Atholl. Joseph's mother was Anne Ogilvie and she came from a family of lawyers; Anne's father had been a lawyer in Dundee. Anne and Alexander Wedderburn had a large family, Joseph being one of fourteen children, eight boys and six girls. In fact Joseph was the tenth child of the family. Joseph was brought up in Forfar, north of Dundee, and he attended Forfar Academy from the age of five until he was thirteen. He then went to George Watson's College, an independent school in Edinburgh, for three years. In 1898 he completed his school education and won a scholarship to study at the University of Edinburgh. He entered Edinburgh University in 1898, at the age of sixteen and a half. It was a time when Wedderburn made remarkable progress with his mathematics and in addition during 1902-03 he worked as an assistant in the Physical Laboratory of the University. He began mathematical research while still an undergraduate and his first paper, On the isoclinal lines of a differential equation of the first order was published in the Proceedings of The Royal Society of Edinburgh in 1903. Two other papers which he published in the same year in publications of the Royal Society of Edinburgh were on the scalar functions of a vector and on an application of quaternions to differential equations. He obtained an M.A. degree with First Class Honours in mathematics from the University of Edinburgh in 1903. Wedderburn then pursued postgraduate studies in Germany spending session 1903-1904 at the University of Leipzig and then the summer semester of 1904 at the University of Berlin. Already Wedderburn's http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wedderburn.html (1 of 4) [2/16/2002 11:37:47 PM]

Wedderburn

mathematical interests were in algebra and his German trip allowed him to interact with Frobenius and Schur. He was awarded a Carnegie Scholarship to study in the United States and he spent 1904-1905 at the University of Chicago where he did joint work with Veblen. Chicago was, of course, an excellent place to continue his deepening interest in algebra for, in addition to Veblen, Eliakim Moore and L E Dickson were there at this time. The determination of finite division algebras was a very natural problem in the light of the work being undertaken in Chicago, and as soon as he arrived at Chicago, Wedderburn started to work on it, in close contact with Dickson. Returning to Scotland in 1905, Wedderburn worked for four years at the University of Edinburgh as an assistant to George Chrystal. The depth of Wedderburn's contribution to algebra during these years in Edinburgh was remarkable. In 1905 he showed that a non-commutative finite field could not exist. In the paper he published in that year he gave three proofs of this theorem which were all based on a clever use of the interplay between the additive group of a finite division algebra A, and the multiplicative group A* = A-{0}. In [9] Parshall discusses this theorem. She notes that the first of the three proofs has a gap in it which was not noticed at the time. This is in fact significant since Dickson also found a proof of this result but, since Wedderburn had already found his first "proof" (which Dickson believed to be correct), Dickson acknowledged Wedderburn's priority in a paper he wrote on the topic. Dickson noted in the paper that it is only after having seen his proof that Wedderburn constructed his second and third proofs. Parshall's work here shows that really Dickson should be credited with having found the first correct proof. This theorem gave, as a corollary, the complete structure of all finite projective geometries. These geometries consisted of a set of "points", a set of "lines" and an "incidence relation" between points and lines, subject only to the conditions that two distinct points are on a single line, two distinct lines have a single common point and a line contains at least three points. Wedderburn and Veblen showed that in all these geometries Pascal's theorem is a consequence of Desargues' theorem. They published the paper Non-Desarguesian and non-Pascalian geometries in the Transactions of the American Mathematical Society in 1907 in which they constructed finite projective geometries which are neither "Desarguesian" nor "Pascalian" (this is Hilbert's terminology). In 1907 Wedderburn published what is perhaps his most famous paper on the classification of semisimple algebras. In this paper On hypercomplex numbers which appeared in the Proceedings of the London Mathematical Society, he showed that every semisimple algebra is a direct sum of simple algebras and that a simple algebra was a matrix algebra over a division ring. From 1906 to 1908 he served as editor of the Proceedings of the Edinburgh Mathematical Society. In 1909 Wedderburn returned to the United States being appointed a Preceptor in Mathematics at Princeton where he joined Veblen. We should say a word about the Preceptors at Princeton. They were the idea of Woodrow Wilson (who was to become the 28th President of the United States in 1913). Woodrow Wilson had been Professor of Political Science at Princeton and, in 1902, he was appointed President of Princeton. He set out to change the nature of Princeton by making it a leading research active university. To do this, Wilson said:... required a large scale infusion of new blood, of scholars who would assume an intimate personal relation with small groups of undergraduates and impart to them something of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wedderburn.html (2 of 4) [2/16/2002 11:37:47 PM]

Wedderburn

their own enthusiasm for things of the mind. Fifty Preceptors were to be appointed to achieve this in the whole university and Henry Fine, Dean of Mathematics, was put in charge of finding young mathematicians to fill the mathematics posts. Between 1905 and 1909 Eisenhart, Veblen, Bliss, George Birkhoff, and Wedderburn were appointed. The next five years were especially happy ones for Wedderburn and his fellow Preceptors described him during this time:They recall his passion for play as well as for work, his desire for companionship and association with men. He loved the out-of-doors, found deep satisfaction in the wilderness, in the woods, canoeing along rivers and streams in the company of thoughtful men. As in his scientific work, he brought to the construction of the camp-site, the erection of the tent, the paddling of the canoe up- and down-stream, the qualities of a complete perfectionist. In the wilds of Northern Canada, with congenial men, he found complete happiness. ... His taste in literature ran to books of travel and he accumulated a large library of travel. However the five happy years came to an end with the outbreak of the First World War. Immediately Wedderburn volunteered for the British Army but, being an exceptionally modest man, he enlisted only in the role of private. Records show that he was the first person at Princeton to volunteer for war service and that he had the longest war service of anyone on the staff. He served in France between January 1918 and March 1919, making use of his scientific skills. In France, as a Captain in the 4th Field Survey Battalion, he devised sound-ranging equipment to pinpoint the positions of enemy guns. On his return to Princeton he took up his post as Preceptor in Mathematics but he was soon promoted to Assistant Professor in 1920, obtaining permanent tenure as Associate Professor in 1921. He served as Editor of the Annals of Mathematics from 1912 to 1928. From about the end of this period Wedderburn seemed to suffer a mild nervous breakdown and became an increasingly solitary figure. It looks as if from this time on he suffered from depression. Certainly he stopped seeing his friends and although he seemed to recognise that his problems came from loneliness, rather than seek to be with people he deliberately cut himself off. Some of his friends made a strenuous effort to penetrate the barrier he was erecting and found that underneath was still the friendly, deep thinking, brilliant mathematician. By 1945 Princeton gave him early retirement in his own best interests. From this time on his isolation became almost total. Although we have given 9 October 1948 as the date of his death, in fact he probably died a few days earlier than this. The people who looked after the house and grounds in Princeton were he lived found him on that day but the subsequent medical examination revealed that he had died of a heart attack several days earlier. Parshall writes in [8]:According to officials at the bank which settled Wedderburn's estate ..., the papers remaining at his death were subsequently destroyed, thereby limiting historical study of Wedderburn's life and work almost exclusively to published sources. Wedderburn made important advances in the theory of rings, algebras and matrix theory. His best mathematical work was done before his war service and we have referred to some of it above. In total he published around 40 works mostly on rings and matrices. His famous book is Lectures on Matrices (1934). This work was described by Jacobson, who was a student of Wedderburn's, in [10]. Jacobson writes:That this was the result of a number of years of painstaking labour is evidenced by the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wedderburn.html (3 of 4) [2/16/2002 11:37:47 PM]

Wedderburn

Bibliography of 661 items (in the revised printing) covering the period 1853 to 1936. The work is, however, not a compilation of the literature, but a synthesis that is Wedderburn's own. It contains a number of original contributions to the subject. Though he did not follow the abstract point of view that had just become dominant, neither did he commit the error made by others of treating matrix theory as an art of juggling elements in an array. The important ideas of linear transformations, vector spaces, bilinear forms, though not set off, as is common in most modern treatments, do appear in Wedderburn's book. also, as in his best work, one finds here some neat and suggestive algebraic devices that make the book a very valuable reference book ... Among the honours which Wedderburn received were the MacDougall-Brisbane Gold Medal and Prize from the Royal Society of Edinburgh in 1921, and election to the Royal Society of London in 1933. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) A Poster of Joseph Wedderburn

Mathematicians born in the same country

Honours awarded to Joseph Wedderburn (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1933

Fellow of the Royal Society of Edinburgh Honorary Fellow of the Edinburgh Maths Society

Elected 1946

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wedderburn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wedderburn.html (4 of 4) [2/16/2002 11:37:47 PM]

Weierstrass

Karl Theodor Wilhelm Weierstrass Born: 31 Oct 1815 in Ostenfelde, Westphalia (now Germany) Died: 19 Feb 1897 in Berlin, Germany

Click the picture above to see six larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Karl Weierstrass's father, Wilhelm Weierstrass, was secretary to the mayor of Ostenfelde at the time of Karl's birth. Wilhelm Weierstrass was a well educated man who had a broad knowledge of the arts and of the sciences. He certainly was well capable of attaining higher positions than he did, and this attitude may have been one of the reasons that Karl Weierstrass's early career was in posts well below his outstanding ability. Weierstrass's mother was Theodora Vonderforst and Karl was the eldest of Theodora and Wilhelm's four children, none of whom married. Wilhelm Weierstrass became a tax inspector when Karl was eight years old. This job involved him in only spending short periods in any one place so Karl frequently moved from school to school as the family moved around Prussia. In 1827 Karl's mother Theodora died and one year later his father Wilhelm remarried. By 1829 Wilhelm Weierstrass had become an assistant at the main tax office in Paderborn, and Karl entered the Catholic Gymnasium there. Weierstrass excelled at the Gymnasium despite having to take on a part-time job as a bookkeeper to help out the family finances. While at the Gymnasium Weierstrass certainly reached a level of mathematical competence far beyond what would have been expected. He regularly read Crelle's Journal and gave mathematical tuition to one of his brothers. However Weierstrass's father wished him to study finance and so, after graduating from the Gymnasium in 1834, he entered the University of Bonn with a course planned out for him which included the study of law, finance and economics. With the career in the Prussian administration that was planned for him by his father, this was indeed a well designed course. However, Weierstrass suffered from the conflict of either obeying his father's wishes or studying the subject he loved, namely mathematics. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html (1 of 7) [2/16/2002 11:37:50 PM]

Weierstrass

The result of the conflict which went on inside Weierstrass was that he did not attend either the mathematics lectures or the lectures of his planned course. He reacted to the conflict inside him by pretending that he did not care about his studies, and he spent four years of intensive fencing and drinking. As Biermann writes in [1]:... the conflict between duty and inclination led to physical and mental strain. He tried, in vain, to overcome his problems by participating in carefree student life ... He did study mathematics on his own, however, reading Laplace's Méchanique céleste and then a work by Jacobi on elliptic functions. He came to understand the necessary methods in elliptic function theory by studying transcripts of lectures by Gudermann. In a letter to Lie, written nearly 50 years later, he explained how he came to make the definite decision to study mathematics despite his father's wishes around this time (see [1]):... when I became aware of [a letter from Abel to Legendre] in Crelle's Journal during my student years, [it] was of the utmost importance. The immediate derivation of the form of the representation of the function given by Abel ..., from the differential equation defining this function, was the first mathematical task I set myself; and its fortunate solution made me determined to devote myself wholly to mathematics; I made this decision in my seventh semester ... Now Weierstrass had made a decision to become a mathematician but he was still supposed to be on a course studying public finance and administration. After his decision, he spent one further semester at the University of Bonn, his eighth semester ending in 1838, and having failed to study the subjects he was enrolled for he simply left the University without taking the examinations. Weierstrass's father was desperately upset by his son giving up his studies. He was persuaded by a family friend, the president of the law courts at Paderborn, to allow Karl to study at the Theological and Philosophical Academy of Münster so that he could take the necessary examinations to become a secondary school teacher. On 22 May 1839 Weierstrass enrolled at the Academy in Münster. Gudermann lectured in Münster and was the reason that Weierstrass was so keen to study there. Weierstrass attended Gudermann's lectures on elliptic functions, some of the first lectures on this topic to be given, and Gudermann strongly encouraged Weierstrass in his mathematical studies. Leaving Münster in the autumn of 1839, Weierstrass studied for the teacher's examination which he registered for in March 1840. By this time, however, Weierstrass's father had moved jobs yet again, becoming director of a salt works in January 1840, and the family was now living in Westernkotten near Lippstadt on the Lippe River, west of Paderborn. At Weierstrass's request he was given a question on the paper he received in May 1840 on the representation of elliptic functions and he presented his own important research as an answer. Gudermann assessed the paper and rated Weierstrass's contribution:... of equal rank with the discoverers who were crowned with glory. When, in later life, Weierstrass learnt of Gudermann's comments he said that he would have published his results had he known. Weierstrass also commented on how generous Gudermann had been in his praise particularly since he had been highly critical of Gudermann's methods. By April 1841 Weierstrass had taken the necessary oral examinations and he began one year probation as a teacher at the Gymnasium in Münster. Although he did not publish any mathematics at this time, he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html (2 of 7) [2/16/2002 11:37:50 PM]

Weierstrass

wrote three short papers in 1841 and 1842 which are described in [3]:The concepts on which Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Münster from 1841 through 1842, while still under the influence of Gudermann. The transformation of his conception of an analytic function from a differentiable function to a function expansible into a convergent power series was made during this early period of Weierstrass's mathematical activity. Weierstrass began his career as a qualified teacher of mathematics at the Pro-Gymnasium in Deutsche Krone in 1842 where he remained until he moved to the Collegium Hoseanum in Braunsberg in 1848. As a teacher of mathematics he was required to teach other topics too, and Weierstrass taught physics, botany, geography, history, German, calligraphy and even gymnastics. In later life Weierstrass described the "unending dreariness and boredom" of these miserable years in which [1]:... he had neither a colleague for mathematical discussions nor access to a mathematical library, and that the exchange of scientific letters was a luxury that he could not afford. From around 1850 Weierstrass began to suffer from attacks of dizziness which were very severe and which ended after about an hour in violent sickness. Frequent attacks over a period of about 12 years made it difficult for him to work and it is thought that these problems may well have been caused by the mental conflicts he had suffered as a student, together with the stress of applying himself to mathematics in every free minute of his time while undertaking the demanding teaching job. It is not surprising that when Weierstrass published papers on abelian functions in the Braunsberg school prospectus they went unnoticed by mathematicians. However, in 1854 he published Zur Theorie der Abelschen Functionen in Crelle's Journal and this was certainly noticed. This paper did not give the full theory of inversion of hyperelliptic integrals that Weierstrass had developed but rather gave a preliminary description of his methods involving representing abelian functions as constantly converging power series. With this paper Weierstrass burst from obscurity. The University of Königsberg conferred an honorary doctor's degree on him on 31 March 1854. In 1855 Weierstrass applied for the chair at the University of Breslau left vacant when Kummer moved to Berlin. Kummer, however, tried to influence things so that Weierstrass would go to Berlin, not Breslau, so Weierstrass was not appointed. A letter from Dirichlet to the Prussian Minister of Culture written in 1855 strongly supported Weierstrass being given a university appointment. Details are given in [10]. After being promoted to senior lecturer at Braunsberg, Weierstrass obtained a year's leave of absence to devote himself to advanced mathematical study. He had already decided, however, that he would never return to school teaching. Weierstrass published a full version of his theory of inversion of hyperelliptic integrals in his next paper Theorie der Abelschen Functionen in Crelle's Journal in 1856. There was a move from a number of universities to offer him a chair. While universities in Austria were discussing the prospect, an offer of a chair came from the Industry Institute in Berlin (later the Technische Hochschule). Although he would have prefered to go to the University of Berlin, Weierstrass certainly did not want to return to the Collegium Hoseanum in Braunsberg so he accepted the offer from the Institute on 14 June 1856. Offers continued to be made to Weierstrass so that when he attended a conference in Vienna in http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html (3 of 7) [2/16/2002 11:37:50 PM]

Weierstrass

September 1856 he was offered a chair at any Austrian university of his choice. Before he had decided what to do about this offer, the University of Berlin offered him a professorship in October. This was the job he had long wanted and he accepted quickly, although having accepted the offer from the Industry Institute earlier in the year he was not able to formally occupy the University of Berlin chair for some years. Weierstrass's successful lectures in mathematics attracted students from all over the world. The topics of his lectures included:- the application of Fourier series and integrals to mathematical physics (1856/57), an introduction to the theory of analytic functions (where he set out results he had obtained in 1841 but never published), the theory of elliptic functions (his main research topic), and applications to problems in geometry and mechanics. In his lectures of 1859/60 Weierstrass gave Introduction to analysis where he tackled the foundations of the subject for the first time. In 1860/61 he lectured on the Integral calculus. In 1861 his emphasis on rigour led him to discover a function that, although continuous, had no derivative at any point. Analysts who depended heavily upon intuition for their discoveries were rather dismayed at this counter-intuitive function. We described above the health problems that Weierstrass suffered from 1850 onwards. Although he had achieved the positions that he had dreamed of, his health gave out in December 1861 when he collapsed completely. It took him about a year to recover sufficiently to lecture again and he was never to regain his health completely. From this time on he lectured sitting down with a student wrote on the blackboard for him. The attacks that he had suffered from 1850 stopped and were replaced by chest problems. In his 1863/64 course on The general theory of analytic functions Weierstrass began to formulate his theory of the real numbers. In his 1863 lectures he proved that the complex numbers are the only commutative algebraic extension of the real numbers. Gauss had promised a proof of this in 1831 but had failed to give one. Weierstrass's lectures developed into a four-semester course which he continued to give until 1890. The four courses were 1. Introduction to the theory of analytic functions, 2. Elliptic functions, 3. Abelian functions, 4. Calculus of variations or applications of elliptic functions. Through the years the courses developed and a number of versions have been published such as the notes by Killing made in 1868 and those by Hurwitz from 1878. Weierstrass's approach still dominates teaching analysis today and this is clearly seen from the contents and style of these lectures, particularly the Introduction course. Its contents were: numbers, the function concept with Weierstrass's power series approach, continuity and differentiability, analytic continuation, points of singularity, analytic functions of several variables, in particular Weierstrass's "preparation theorem", and contour integrals. At Berlin, Weierstrass had two colleagues Kummer and Kronecker and together the three gave Berlin a reputation as the leading university at which to study mathematics. Kronecker was a close friend of Weierstrass's for many years but in 1877 Kronecker's opposition to Cantor's work cause a rift between the two men. This became so bad that at one stage, in 1885, Weierstrass decided to leave Berlin and go to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html (4 of 7) [2/16/2002 11:37:50 PM]

Weierstrass

Switzerland. However, he changed his mind and remained in Berlin. A large number of students benefited from Weierstrass's teaching. We name a few who are mentioned elsewhere in our archive: Bachmann, Bolza, Cantor, Engel, Frobenius, Gegenbauer, Hensel, Hölder, Hurwitz, Killing, Klein, Kneser, Königsberger, Lerch, Lie, Lueroth, Mertens, Minkowski, Mittag-Leffler, Netto, Schottky, Schwarz and Stolz. One student in particular, however, deserves special mention. In 1870 Sofia Kovalevskaya came to Berlin and Weierstrass taught her privately since she was not allowed admission to the university. Clearly she was a very special student as far as Weierstrass was concerned for he wrote to her that he:... dreamed and been enraptured of so many riddles that remain for us to solve, on finite and infinite spaces, on the stability of the world system, and on all the other major problems of the mathematics and the physics of the future. ... you have been close ...throughout my entire life ... and never have I found anyone who could bring me such understanding of the highest aims of science and such joyful accord with my intentions and basic principles as you. It was through Weierstrass's efforts that Kovalevskaya received an honorary doctorate from Göttingen, and he also used his influence to help her obtain the post in Stockholm in 1883. Weierstrass and Kovalevskaya corresponded for 20 years between 1871 to 1890. More than 160 letters were exchanged (see [5], [7] etc.), but Weierstrass burnt Kovalevskaya's letters after her death. The standards of rigour that Weierstrass set, defining, for example, irrational numbers as limits of convergent series, strongly affected the future of mathematics. He also studied entire functions, the notion of uniform convergence and functions defined by infinite products. His effort are summed up in [2] as follows:Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations. He also advanced the theory of bilinear and quadratic forms. Weierstrass published little [1]:... because his critical sense invariably compelled him to base any analysis on a firm foundation, starting from a fresh approach and continually revising and expanding. However, he did edit the complete works of Steiner and those of Jacobi. He decided to supervise the publication of his own complete works, in his case this would involve a great deal of unpublished material from his lecture courses and Weierstrass realised that without his help this would be a difficult task. The first two volumes appeared in 1894 and 1895, being the only ones to appear before his death in 1897. His last years were difficult [1]:During his last three years he was confined to a wheelchair, immobile and dependent. He died of pneumonia. The remaining volumes of his Complete Works appeared slowly; volume 3 in 1903, volume 4 in 1902, volumes 5 and 6 in 1915, and volume 7 in 1927. The seven volumes were reprinted in 1967. More work continues to be published today, particularly versions of his lecture courses taken from the notes made by those who attended the lectures.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html (5 of 7) [2/16/2002 11:37:50 PM]

Weierstrass

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (40 books/articles)

A Quotation

A Poster of Karl Weierstrass

Mathematicians born in the same country

Cross-references to History Topics

1. The beginnings of set theory 2. Topology enters mathematics 3. The fundamental theorem of algebra 4. An overview of the history of mathematics 5. Matrices and determinants

Other references in MacTutor

1. Chronology: 1850 to 1860 2. Chronology: 1860 to 1870 3. Chronology: 1880 to 1890

Honours awarded to Karl Weierstrass (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1881

Royal Society Copley Medal

Awarded 1895

Lunar features

Crater Weierstrass

Other Web sites

1. Interactive Real Analysis 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1998

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html (6 of 7) [2/16/2002 11:37:50 PM]

Weierstrass

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Weierstrass.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weierstrass.html (7 of 7) [2/16/2002 11:37:50 PM]

Weil

André Weil Born: 6 May 1906 in Paris, France Died: 6 Aug 1998 in Princeton, New Jersey, USA

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

André Weil was born in Paris, the son of Jewish parents. He studied at universities in Paris, Rome and Göttingen, receiving his D.Sc. from the University of Paris in 1928. He then taught at different universities for example the Aligarh Muslim University in India from 1930 to 1932, and the University of Strasbourg, France from 1933 until the outbreak of World War II. The war was a disaster for Weil who was a conscientious objector and so wished to avoid military service. He fled to Finland as soon as war was declared in an attempt to avoid becoming forced into the army, but it was not a simple matter to escape from the war in Europe at this time. He was sent from Finland back to France where he was put in prison. Weil was certainly in great danger at this time, partly because he was Jewish, partly because he had a sister, Simone Weil who was a mystic philosopher and a leading figure in the French Resistance. The dangers of his predicament made Weil decide that being in the army was a better bet and he was able to argue successfully for his release on the condition that indeed he did join the army. Having used the army as a reason to get out of prison, Weil had no intention of serving any longer than he possibly could. As soon as the chance to escape to the United States came, he took it at once. In the United States he went to Pennsylvania where he taught from 1941 at Haverford College and at Swarthmore College. In 1945 he accepted a position in Sao Paulo University, Brazil where he remained until 1947. In 1947 Weil returned to the United States and he was appointed to the faculty of the University of Chicago, a position he continued to hold until 1958. From 1958 he worked at the Institute for Advanced Study at Princeton University. He retired in 1976, becoming Professor Emeritus at that time.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weil.html (1 of 3) [2/16/2002 11:37:52 PM]

Weil

Weil's research was in number theory, algebraic geometry and group theory. His work is summarised in [3]:Beginning in the 1940s, Weil started the rapid advance of algebraic geometry and number theory by laying the foundations for abstract algebraic geometry and the modern theory of abelian varieties. His work on algebraic curves has influenced a wide variety of areas, including some outside mathematics, such as elementary particle physics and string theory. In fact Weil's work in this area was basic to work by mathematicians such as Yau who was awarded a Fields Medal in 1982 for work in three dimensional algebraic geometry which has major applications to quantum field theory. Yau is not the only mathematician who received a Fields Medal for work which continued that begun by Weil. In 1978 Deligne was awarded a Fields Medal for solving the Weil Conjectures. Again we quote [3] for a description of Weil's fundamental contribution:One of Weil's major achievements was his proof of the Riemann hypothesis for the congruence zeta functions of algebraic function fields. In 1949 he raised certain conjectures about the congruence zeta function of algebraic varieties over finite fields. These Weil conjectures, as they came to be called, grew out of his deep insight into the topology of algebraic varieties and provided guiding principles for subsequent developments in the field. Weil's work on bringing together number theory and algebraic geometry was highly fruitful. The foundations of many topics studied in depth today were laid by Weil in this work, such as the foundations of the theory of modular forms, automorphic functions and automorphic representations. However, Weil's work was of major importance in a number of other new mathematical topics. He contributed substantially to topology, differential geometry and complex analytic geometry. It was not just to these areas that he contributed but, even more importantly, his work brought out fundamental relationships between the areas when he studied harmonic analysis on topological groups and characteristic classes. Also bringing these areas together was his work on the geometric theory of the theta function and Kähler geometry. Together with Dieudonné and others, Weil wrote under the name Nicolas Bourbaki, a project they began in the 1930s, in which they attempted to give a unified description of mathematics. The purpose was to reverse a trend which they disliked, namely that of a lack of rigour in mathematics. The influence of Bourbaki has been great over many years but it is now less important since it has basically succeeded in its aim of promoting rigour and abstraction. Weil's most famous books include Foundations of Algebraic Geometry (1946) and Elliptic Functions According to Eisenstein and Kronecker (1976). Weil received many honours for his outstanding mathematics. Among these has been honorary membership of the London Mathematical Society in 1959 and election to a Fellowship of the Royal Society of London in 1966. In addition he has been elected to the Academy of Sciences in Paris and to the National Academy of Sciences in the United States. Weil was an invited speaker at the International Congress of Mathematicians in 1950 at Harvard and

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weil.html (2 of 3) [2/16/2002 11:37:52 PM]

Weil

again at the following International Congress in 1954. In 1979 Weil was awarded the Wolf Prize and, in the following year, the American Mathematical Society awarded him their Steele Prize. In 1994 he received the Kyoto Prize from the Inamori Foundation of Japan:... for outstanding achievement and creativity. The citation for the Kyoto Prize reads:The results achieved and problems raised by André Weil through his deep understanding of and sharp insight into mathematical sciences in general will continue to have immeasurable influence on the development of mathematical sciences, and to contribute greatly to the development of science, as well as the deepening and uplifting of the human spirit. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

Some Quotations (4)

Mathematicians born in the same country Cross-references to History Topics

Fermat's last theorem

Other references in MacTutor

Chronology: 1940 to 1950

Honours awarded to André Weil (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1966

Other Web sites

1. AMS (Special issue of the AMS Notices) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Weil.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weil.html (3 of 3) [2/16/2002 11:37:52 PM]

Weingarten

Julius Weingarten Born: 2 March 1836 in Berlin, Germany Died: 16 June 1910 in Freiburg im Breisgau, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Julius Weingarten was born in Germany, but his family were Polish and had emigrated to Germany. He certainly did not come from an academic family for his father was a weaver, and the family were not well off which would have a serious effect on the whole of Weingarten's career. Weingarten attended the Municipal Trade School in Berlin. He completed his studies there in 1852 and, in the same year, he entered the University of Berlin to embark on a course of study which involved mainly mathematics and physics. At the University of Berlin Weingarten attended lectures on potential theory given by Dirichlet. These lectures were particularly inspiring and, although this would not be Weingarten's main area of research, he continued to work, from time to time, on problems related to this topic throughout his career. He also studied chemistry at the Berlin Gewerbeinstitut (the Institute for Crafts) during these years. Coming from a poor family Weingarten did not have the financial support to allow him to complete his doctorate at Berlin without earning his living so, in 1858, he began teaching at a school in Berlin. Despite having to work as a teacher at various schools while he undertook research, his work on the theory of surfaces progressed remarkably well. In fact the work was of such quality that Weingarten received a prize for work on the lines of curvature of a surface in 1857. In 1864 he received a doctorate from the University of Halle for the same work which had won him the prize from the University of Berlin, but he had been far from idle over the years for he had published other important work on the theory of surfaces. The theory of surfaces was the most important topic in differential geometry and [1]:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weingarten.html (1 of 3) [2/16/2002 11:37:54 PM]

Weingarten

... one of its main problems was that of stating all the surfaces isometric to a given surface. The only class of such surfaces known before Weingarten consisted of the developable surfaces isometric to the plane. In 1863 Weingarten was able to make a major step forward in the topic when he gave a class of surfaces isometric to a given surface of revolution. Surfaces of constant mean curvature or constant Gaussian curvature are now called the Weingarten surfaces. Now having produced work of outstanding quality, while one must remember he was teaching in schools, it would be reasonable to expect that Weingarten would find a good academic post. However, this was not easy at that time except for those who had the necessary funds to allow them the luxury of starting an academic career with little income. Weingarten had to take the option which would provide him with an income so he accepted a rather unsatisfactory position at the Bauakademie in Berlin. Weingarten was promoted to professor at the Bauakademie in 1871 but left the rather unsatisfactory post to take on what was another rather unsatisfactory position at the Technische Hochschule in Berlin. By 1902, at the age of 66, his health began to fail and for that reason he moved to Friburg im Breisgau where he was appointed as an honorary professor. He taught there until 1908 in what was in many ways the most satisfactory of his teaching positions. Weingarten's work on the infinitesimal deformation of surfaces, undertaken around 1886, was praised by Darboux who included it in his four volume treatise on the theory of surfaces. In fact Darboux said that Weingarten's work was worthy of Gauss, a compliment indeed. The interest which Darboux showed in his work, encouraged Weingarten to push his results further and he wrote a long paper which won the Grand Prix of the Académie des Sciences in Paris in 1894. The work was published in Acta mathematica in 1897 and was another major step forward in solving the problems on which Weingarten had worked all his life. In this work he reduced the problem of finding all surfaces isometric to a given surface to the problem of determining all solutions to a partial differential equation of the Monge-Ampère type. Darboux was not the only leading mathematician in Weingarten's time who was also interested in the theory of surfaces. Another was Bianchi and a major correspondence grew up between Weingarten and Bianchi. In fact in [2], which is a 304 page book containing all Bianchi's correspondence, the most extensive correspondence of all is the one with Weingarten. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weingarten.html (2 of 3) [2/16/2002 11:37:54 PM]

Weingarten

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Weingarten.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weingarten.html (3 of 3) [2/16/2002 11:37:54 PM]

Weinstein

Alexander Weinstein Born: 21 Jan 1897 in Saratova, Russia Died: 6 Nov 1979 in Washington DC, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alexander Weinstein studied at Astrakhan, at this stage planning to study astronomy. After graduating he studied at the University of Würzburg and the University of Göttingen during 1913/14. He moved to Zurich and he continued his interest in astronomy carrying out observations. He studied under Weyl and was awarded a doctorate in 1921. During 1922 Weinstein worked as an assistant at the University of Leipzig. Weyl recommended Weinstein for a Rockefeller Fellowship and after this was awarded Weinstein spent 1926 and 1927 in Rome working with Levi-Civita. He returned to Zurich as Weyl's assistant, then in 1928 he was appointed to Hamburg. From Hamburg Weinstein moved to Breslau and, by 1933, he was being sought by Einstein as a collaborator in Berlin. However 1933 was the year that the Nazis came to power and Weinstein, being of Jewish background, could not remain in Germany. He therefore had to give up the chance of working with Einstein and instead he went to the Collège de France in Paris where he worked with Hadamard. He was awarded the degree of Docteur ès Sciences Mathématique by Paris in 1937. By 1940 World War II caught up with Weinstein in Paris and he left for the United States. There he taught at a number of different places such as the Free French University in New York, the Carnegie Institute of Technology and the University of Maryland. He also worked in Canada at the University of Toronto for a while. He was also a member of Birkhoff's research group at Harvard doing war work. For 18 years he was principle investigator at the Institute for Fluid Dynamics and Applied Mathematics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weinstein.html (1 of 2) [2/16/2002 11:37:56 PM]

Weinstein

at the University of Maryland. Weinstein's research covered a wide range of topics. He is famed for solving a variety of boundary value problems. For example he solved Helmholtz's problem for jets, giving the first uniqueness and existence theorems for free jets in a series of papers from 1923 to 1929. He examined boundary problems in an infinite strip, giving hydrodynamic and electromagnetic applications. Weinstein's method was developed to give accurate bounds for eigenvalues of plates and membranes. In examining singular partial differential equations he introduced a new branch of potential theory and applied the results to many different situations including flow about a wedge, flow around lenses and flow around spindles. After retiring in 1967, Weinstein continued research at the American University, then, from 1968 to 1972 he worked at Georgetown University. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) A Poster of Alexander Weinstein

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Weinstein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weinstein.html (2 of 2) [2/16/2002 11:37:56 PM]

Weisbach

Julius Lugwig Weisbach Born: 10 Aug 1806 in Mittelschmiedeberg (near Annaberg), Germany Died: 24 Feb 1871 in Freiberg, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Julius Weisbach's father was Christian Gottlieb Weisbach and his mother was Christina Rebekka Stephan. Julius's father was the foreman at a mine, so the family were not well off, particularly since there were nine children. Of the children, Julius was the second youngest. Weisbach was educated at the lyceum in Annaberg near his home town, then at the Bergschule in Freiberg. Having completed the courses at these schools by 1822, Weisbach wanted to continue his education. However, his parents could not afford to finance him so he borrowed money to enable him to attend the Bergakademie in Freiberg. In 1827 Weisbach was advised by Friedrich Mohs to study at Göttingen. Mohs was the famous German mineralogist who, in 1812, devised a scale, now known as the Mohs scale, to determine the hardness of a mineral by observing whether its surface is scratched by various substances of known hardness. Weisbach followed Mohs' advice and spent two years in Göttingen before moving to the University of Vienna. The reason for this move was that Mohs had gone to Vienna. In Vienna, Weisbach studied mathematics, physics and mechanics. During 1830 Weisbach travelled on foot for six months through Hungary, the Tyrol, Bavaria, and Bohemia. From 1831 he taught mathematics at the Freiberg Gymnasium and, starting in 1832, he also taught mathematics at the Bergakademie in Freiberg. In this same year of 1832, Weisbach married Marie Winkler and they would have a son who was to became the Professor of Mineralogy at the Bergakademie in Freiberg. In 1836 Weisbach was promoted to professor, and his expertise was now such that in addition to http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weisbach.html (1 of 2) [2/16/2002 11:37:58 PM]

Weisbach

mathematics he was professor of mine machinery and of surveying. He had, at this stage only one publication, but this would soon change. From around 1839 he became interested in hydraulics and some of his most important work would be done on this topic. This interest in hydraulics seems to have been as a result of Weisbach visiting the Paris Industrial Exposition in 1839. His interests were always wide and this is reflected in the range of courses that Weisbach was teaching around this time: descriptive geometry, crystallography, optics, mechanics and machine design. In 1855 Zürich Polytechnikum opened, and in the year prior to that they made a serious attempt to appoint quality staff. For example they appointed Clausius as professor of physics, but Weisbach was not tempted by the offer that was made to him and preferred to remain in Freiberg. In 1855 Weisbach was back in Paris, this time visiting the World Exposition which was held there. We have indicated the range of Weisbach's interests and this can be seen from the topics of the fourteen books and 59 papers he wrote on mechanics, hydraulics, surveying, and mathematics. It is in hydraulics that his work was most influential, with his books on the topic continuing to be of importance well into the 20th century. Among the honours Weisbach received for his contributions to science were honorary degrees from Leipzig, and election to membership of various learned societies such as the St Petersburg Academy of Sciences, the Royal Swedish Academy of Sciences and the Accademia dei Lincei. Article by: J J O'Connor and E F Robertson List of References (3 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Weisbach.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weisbach.html (2 of 2) [2/16/2002 11:37:58 PM]

Weldon

Walter Frank Raphael Weldon Born: 15 March 1860 in Highgate, London, England Died: 13 April 1906 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Raphael Weldon was tutored by a local clergyman then, in 1873, he went to boarding school at Caversham. After three years at the school, followed by several months of private study he entered University College London. There he studied a wide range of subjects and was taught mathematics by Henrici. Weldon was most impressed with Henrici's lectures and felt that he was the first naturally gifted teacher he studied under. In 1877 he went to King's College, his aim being to enter the medical profession. He moved to St John's College in Cambridge in 1878 and his interests turned to zoology. After graduating he went to Naples working at the Zoological Station. He undertook research June to January, teaching at Cambridge for two terms each year. His teaching was described in these glowing terms:Seldom is it given to a man to teach as Weldon taught. He lectured almost as one inspired. His extreme earnestness was only equalled by his lucidity. He awoke enthusiasm even in the dullest, and had the divine gift of compelling interest. Why is this zoologist in the History of Mathematics Archive? His work soon began to involve statistical analysis. He was led in this direction by work of Galton and he carried the statistical analysis that Galton and Quetelet had applied to humans to other zoological species. His work involved organic correlation coefficients and is important in the beginnings of biometry. Realising that his mathematical skills were somewhat less than he wished, Weldon read widely the leading works by the French mathematicians on the calculus of probability. He was appointed professor at University College London in 1891. By 1893 he was serving on a Royal http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weldon.html (1 of 2) [2/16/2002 11:38:00 PM]

Weldon

Society Committee along with Galton and Pearson 'For the Purpose of conducting Statistical Enquiry into the Variability of Organisms'. Weldon proposed a journal for biometrics in a letter dated 16 November 1899. He wrote:The contention 'that numbers mean nothing and do not exist in Nature' is a very serious thing, which will have to be fought. ... Do you think it would be too hopelessly expensive to start a journal of some kind? ... The journal Biometrika was named within weeks and Pearson and Weldon became joint editors. Weldon was appointed to a chair in Oxford in 1900 and he held this post until his death in 1906 from pneumonia. It is likely that he illness became serious due to overwork and Weldon's refusal to stop work when he became ill. In [1] Pearson gives this rather touching description:He was by nature a poet, and these give the best to science, for they give ideas. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Raphael Weldon

Mathematicians born in the same country

Honours awarded to Raphael Weldon (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1890

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Weldon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weldon.html (2 of 2) [2/16/2002 11:38:00 PM]

Werner

Johann Werner Born: 14 Feb 1468 in Nuremberg, Germany Died: May 1522 in Nuremberg, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johann Werner's main work was on astronomy, mathematics and geography. In astronomy he followed Regiomontanus and was a skilled maker of instruments. A skilled observer, he recorded a comet on 1 June 1500 and kept a record of observations until 24 June 1500. Werner's most famous work on astronomy and geography is In Hoc Opere Haec Cotinentur Moua Translatio Primi Libri Geographicae Cl'Ptolomaei written in 1514. This book contains a description of an instrument with an angular scale on a staff from which degrees could be read off. It also gives a method to determine longitude based on eclipses of the Moon. In mathematics Werner worked on spherical trigonometry and conic sections. He was the first to use the formula 2sin(a)sin(b) = cos(a - b) - cos(a + b) as an aid to calculation. This was used by Rheticus and Brahe and others up to the invention of logarithms. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

Longitude and the Académie Royale

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Werner.html (1 of 2) [2/16/2002 11:38:02 PM]

Werner

Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Werner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Werner.html (2 of 2) [2/16/2002 11:38:02 PM]

Wessel

Caspar Wessel Born: 8 June 1745 in Vestby (near Dröbak), Norway Died: 25 March 1818 in Copenhagen, Denmark Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Caspar Wessel's father, Jonas Wessel, and his grandfather were both ministers in the church. His mother, Helene Marie Schumacher and Jonas had a large family consisting of fourteen children; Caspar was the sixth of the fourteen. Caspar, together with his two elder brothers Johan Herman Wessel and Ole Christopher Wessel, attended Christiania Cathedral School in the city of Christiania from 1757 until 1763 (Christiania was later renamed Kristiania, then Oslo in 1925). Caspar could not attend university in Norway for, at that time, there were no Norwegian universities. Since Norway was united with Denmark, the natural place for Norwegians to go for a university education was Denmark, and Caspar's brothers Johan Herman and Ole Christopher were already at the University of Copenhagen, having entered in 1761. Caspar spent one year, from 1763 to 1764, studying at the University of Copenhagen but, as one might imagine, the large family was putting quite a strain on his parent's finances. As we shall see below, this led to both Caspar and his brother Ole Christopher obtaining positions as surveyors with the Royal Danish Academy. Ole Christopher and Caspar were studying at the University of Copenhagen for a law degree. Ole Christopher started to work as a surveyor to help pay his way through university while Caspar was still at school. Obtaining his law degree in 1770, Ole Christopher went on to reach a very elevated position in the legal profession in Norway. Johan Herman (three years older than Caspar) became a poet and is the only one of the Wessel children to have their own entry in Encyclopaedia Britannica (although Caspar is mentioned in three of the mathematics articles). Johan Herman is described as a:... writer and wit, known for his epigrams and light verse and for a famous parody of neoclassical tragedy. The Royal Danish Academy began an ambitious project to undertake a topographical survey of Denmark and also to use triangulation to determine geographical coordinates. The project was led by Thomas Bugge, a professional surveyor, and Christen Hee, the professor of mathematics in Copenhagen. Ole Christopher was employed as a surveyor on this project from 1762, and, when he needed an assistant in 1764, his brother Caspar joined the project to help him. Caspar, like his brother Ole Christopher, continued to study for his law degree which he eventually achieved after fifteen years. However by that time he was so involved with surveying that he remaining in that profession the rest of his working life. Throughout his life Wessel suffered financial hardship. Certainly he earned to little as an assistant that he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wessel.html (1 of 4) [2/16/2002 11:38:03 PM]

Wessel

requested he might be allowed to draw maps, as well as surveying, so that his income might be sufficient to allow him to survive. This request was granted and he was given quite a lot of increased responsibility drawing maps based on the data which was being gathered from the triangulation survey. The maps he drew marked [4]:... the locations of towns, churches, castles, mills, and woods, the courses of roads and streams, and the positions of coastlines and islands. Work which had originally been intended to provide Wessel with the financial support to complete his university course had become such a major undertaking that there was not enough time left for him to study. Fearing that he would never complete his degree with his work for the Academy running at this high level, yet knowing that he could not survive financially without this income, he requested a sabbatical year on full pay to complete his degree course. Christen Hee, the professor of mathematics, strongly supported Wessel's application writing (see for example [4]):None of the surveyors has been more useful to us than [Wessel] has, during the summers he has been surveying and in the winter time he has been working as a designator, which in the fourteen years he has stayed with the surveying has ruined his health and been an obstacle to his studies in such a way that if he once again has to interrupt his studies he is lost and will never pick them up again. Last winter when he half unwillingly, half willingly had to draw the general map of Zealand he was once more distracted in his studies, and then I promised him never again to disturb his circles. Wessel's sabbatical was granted and he was indeed able to complete his law degree. However, after the sabbatical year he returned to his tasks of surveying and map construction. Such tasks required demanding mathematical skills and Wessel was an innovator in finding new methods and techniques. When he compiled reports on his work, Wessel appended short articles explaining the theoretical ideas behind the methods he was employing. In May 1782 Wessel was released from his work with the Royal Danish Academy so that he could conduct a trigonometrical survey of the duchy of Oldenburg. Oldenburg had come under Danish control in 1667 but in 1773, not long before Wessel was asked to survey it, Oldenburg had been exchanged by Christian VII of Denmark for Holstein-Gottorp. Bugge, who headed the survey work at the Royal Danish Academy wrote a letter recommending Wessel for the post (see for example [4]):He possesses a lot of theoretical knowledge of algebra, trigonometry and mathematical geometry, and as far as the last point is concerned, he has come up with some new and beautiful solutions to the most difficult problems in geographical surveying. Wessel worked on the survey of Oldenburg until the summer of 1785 when he returned to his work with the Royal Danish Academy. He had been developing more and more sophisticated mathematical methods of surveying and these he explained fully in a report he wrote in 1787. This report already contains Wessel's brilliant mathematical innovation, namely the geometric interpretation of complex numbers. By 1796 Wessel had completed the triangulation of Denmark and used the data to produce the first really accurate map of the country. In the same year he wrote his one and only mathematical paper and presented it to a meeting of the Royal Danish Academy on 10 March 1797. Only in the year before had the Academy relaxed its rule that all papers must be written by members of the Academy, and Wessel's paper was the first to be accepted which was not authored by a member. Wessel's fame as a mathematician rests solely on this paper, which was published in 1799, giving for the

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wessel.html (2 of 4) [2/16/2002 11:38:03 PM]

Wessel

first time a geometrical interpretation of complex numbers. Today we call this geometric interpretation the Argand diagram but Wessel's work came first. It was rediscovered by Argand in 1806 and again by Gauss in 1831. (It is worth noting that Gauss redid another part of Wessel's work, for he retriangulated Oldenburg in around 1824.) Of course it is not unreasonable to call the geometrical interpretation of complex numbers the Argand diagram since it was Argand's work which was influential. It was so named before the world of mathematics learnt of Wessel's prior publication. In fact Wessel's paper was not noticed by the mathematical community until 1895 when Juel draw attention to it and, in the same year, Sophus Lie republished Wessel's paper. A French translation, made by Zeuthen, was published in 1897 but an English translation of this most remarkable work was not published until 1999 (exactly 200 years after it was first published). Johannes Nikolaus Tetens was a professor of mathematics and philosophy at the University of Copenhagen near the end of the 19th century, and it was because of his encouragement that Wessel presented his paper to the Academy. In fact Wessel, not being a member of the Academy, was not present when his paper was read. It was Tetens who presented the paper to the Academy on 10 March 1797. One can only suppose that, despite Tetens encouraging Wessel, he could not have realised its importance for otherwise he certainly could have translated it from Danish to German and thus ensured for Wessel the world-wide fame as a mathematician which he has not achieved until very recent times. We have called Wessel's work remarkable, and indeed although the credit has gone to Argand, many historians of mathematics feel that Wessel's contribution was [1]:... superior to and more modern in spirit to Argand's. In the [1] article the approaches by Argand and Wessel are compared and contrasted. Of course Wessel was a surveyor and his paper was motivated by his surveying and cartography work:Wessel's development proceeded rather directly from geometric problems, through geometric-intuitive reasoning, to an algebraic formula. Argand began with algebraic quantities and sought a geometric representation for them. ... Wessel's initial formulation was remarkably clear, direct, concise and modern. It is regrettable that it was not appreciated for nearly a century and hence did not have the influence it merited. However more is claimed for Wessel's single mathematical paper than the first geometric interpretation of complex numbers. In [3] Crowe credits Wessel with being the first person to add vectors. Again this shows the depth of Wessel's thinking but again, as the paper was unnoticed it had no influence on mathematical development despite appearing in the Mémoires of the Royal Danish Academy which by any standard was a major source of publications. In many ways Wessel was a remarkable person, and here we are not only referring to his mathematical brilliance. Despite his poverty, he refused to accept payment for maps which he had been commissioned to make. He did accept the award of a silver medal from the Royal Danish Academy for his work on maps but medals did not make him well off. He resigned from his work with the Royal Danish Academy in 1805 when he was 60 years old. He did not give up drawing maps, however, and he still helped the Royal Danish Academy when necessary. One of the maps he drew after he retired was a map of Schleswig-Holstein which had been requested by Napoleon Bonaparte.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wessel.html (3 of 4) [2/16/2002 11:38:03 PM]

Wessel

In 1815 Wessel was made a Knight of the Order of Dannebrog:... in recognition of his exceptional contribution to surveying. Article by: J J O'Connor and E F Robertson List of References (8 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1780 to 1800

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wessel.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wessel.html (4 of 4) [2/16/2002 11:38:03 PM]

West

John West Born: 10 April 1756 in Logie (near St Andrews), Scotland Died: 17 Oct 1817 in Morant Bay, Jamaica Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John West was educated at St Andrews and was assistant to Nicholas Vilant from 1775 to 1784, during which time he taught John Leslie and James Ivory. For financial reasons, he went to Jamaica in 1784, despite having just published his Elements of Mathematics and A new system of shorthand. Apart from a few visits to England, he spent the remainder of his life in Jamaica becoming an Anglican priest and was for 28 years rector of St Thomas in the East, Morant Bay. Two manuscript treatises were sent, after his death, to John Leslie, but these were not published until 1838. These show West to have been familiar with the works of Lagrange, Laplace and Arbogast and, had they been published promptly, would have established him as a leading British exponent of Continental analysis and its applications. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/West.html (1 of 2) [2/16/2002 11:38:04 PM]

West

JOC/EFR January 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/West.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/West.html (2 of 2) [2/16/2002 11:38:04 PM]

Weyl

Hermann Klaus Hugo Weyl Born: 9 Nov 1885 in Elmshorn (near Hamburg), Germany Died: 8 Dec 1955 in Zürich, Switzerland

Click the picture above to see eight larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hermann Weyl (known as Peter to his close friends) was educated at the universities of Munich and Göttingen. His doctorate was from Göttingen where his supervisor was Hilbert. After submitting a doctoral dissertation Singuläre Integralgleichungen mit besonder Berücksichtigung des Fourierschen Integraltheorems he was awarded the degree in 1908. It was at Göttingen that he held his first teaching post. From 1913 to 1930 he held the chair of mathematics at Zürich Technische Hochschule, from 1930 to 1933 he held the chair of mathematics at Göttingen and from 1933 until he retired in 1952 he worked at the Institute for Advanced Study at Princeton. He attempted to incorporate electromagnetism into the geometric formalism of general relativity. Weyl published Die Idee der Riemannschen Fläche (1913) which united analysis, geometry and topology. He produced the first guage theory in which the Maxwell electromagnetic field and the gravitational field appear as geometrical properties of space-time. From 1923-38 he evolved the concept of continuous groups using matrix representations. With his application of group theory to quantum mechanics he set up the modern subject. He also made contributions on the uniform distribution of numbers modulo 1 which are fundamental in analytic number theory. More recently attempts to incorporate electromagnetism into general relativity have been made by John Wheeler, Kaluza and others. These theories, like Weyl's, lack the connection with quantum phenomena

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weyl.html (1 of 2) [2/16/2002 11:38:07 PM]

Weyl

that is so important for interactions other than gravitation. Weyl's own comment, although half a joke, sums up his personality. My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (25 books/articles)

Some Quotations (14)

A Poster of Hermann Weyl

Mathematicians born in the same country

Cross-references to History Topics

General relativity

Other references in MacTutor

Chronology: 1910 to 1920

Honours awarded to Hermann Weyl (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1936

AMS Gibbs Lecturer

1948

Lunar features

Crater Weyl

Other Web sites

1. Clark Kimberling 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Weyl.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weyl.html (2 of 2) [2/16/2002 11:38:07 PM]

Weyr

Emil Weyr Born: 21 July 1848 in Austria Died: 25 Jan 1894

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Emil Weyr taught in Prague and Vienna after studying at Vienna and in Italy. He, together with his brother Eduard Weyr, were the main members of the Austrian geometric school. They were interested in descriptive geometry, then in projective geometry and their interests turned towards algebraic and synthetic methods in geometry. Emil Weyr led the geometry school in Vienna throughout the 1880's up until his death. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Emil Weyr

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weyr.html (1 of 2) [2/16/2002 11:38:09 PM]

Weyr

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Weyr.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Weyr.html (2 of 2) [2/16/2002 11:38:09 PM]

Wheeler

Anna Johnson Pell Wheeler Born: 5 May 1883 in Calliope (now Hawarden), Iowa, USA Died: 26 March 1966 in Bryn Mawr, Pennsylvania, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Anna Johnson was the daughter of Swedish immigrants who came to the United States about 10 years before Anna was born. When she was nine years old the family moved to Akron, Iowa where Anna attended private school. In 1899 she entered the University of South Dakota where she showed great promise in mathematics. The professor of mathematics, Alexander Pell, recognised her talents and helped persuade Anna Johnson that she should follow a career in mathematics. She received an A.B. degree in 1903. After winning a scholarship to study for her master's degrees at the University of Iowa, she was awarded the degree for a thesis The extension of Galois theory to linear differential equations in 1904. A second master's degree from Radcliffe was awarded in 1905 and she remained there to study under Bôcher and Osgood. Anna Johnson was awarded the Alice Freeman Palmer Fellowship from Wellesley College to study for a year at Göttingen University. There she attended lectures by Hilbert, Klein, Minkowski, Herglotz and Schwarzschild. She worked for her doctorate at Göttingen. While there Alexander Pell, her former mathematics professor came to Göttingen so that they could marry. After returning to the United States, where her husband was by now Dean of Engineering, she taught courses in the theory of functions and differential equations. In 1908 Anna Pell returned to Göttingen where she completed the work for her doctorate but, after a disagreement with Hilbert, she returned to Chicago, where her husband was now on the university staff, without the degree being awarded.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wheeler.html (1 of 3) [2/16/2002 11:38:11 PM]

Wheeler

At Chicago she became a student of Eliakim Moore and received her Ph.D. in 1909, her thesis Biorthogonal Systems of Functions with Applications to the Theory of Integral Equations being the one written originally at Göttingen. From 1911 Anna Pell taught at Mount Holyoke College and then at Bryn Mawr from 1918. Anna Pell's husband, who was 25 years order than she was, died in 1920. In 1924 Anna Pell became head of mathematics when Scott retired, becoming a full professor in 1925. After a short second marriage to Arthur Wheeler, during which time they lived at Princeton and she taught only part-time, her second husband died in 1932. After this Anna Wheeler returned to full time work at Bryn Mawr where Emmy Noether joined her in 1933. However Emmy Noether died in 1935. The period from 1920 until 1935 certainly must have been one with much unhappiness for Anna Wheeler since during those years her father, mother, two husbands and close friend and colleague Emmy Noether died. Anna Wheeler remained at Bryn Mawr until her retirement in 1948. The direction of Anna Wheeler's work was much influenced by Hilbert. Under his guidance she worked on integral equations studying infinite dimensional linear spaces. This work was done in the days when functional analysis was in its infancy and much of her work has lessened in importance as it became part of the more general theory. Perhaps the most important honour she received was becoming the first woman to give the Colloquium Lectures at the American Mathematical Society meetings in 1927. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Anna J P Wheeler (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1927

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wheeler.html (2 of 3) [2/16/2002 11:38:11 PM]

Wheeler

JOC/EFR January 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wheeler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wheeler.html (3 of 3) [2/16/2002 11:38:11 PM]

Whiston

William Whiston Born: 9 Dec 1667 in Norton, Leicestershire, England Died: 22 Aug 1752 in Lyndon, Rutland, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Whiston was taught by his father until he was 17 when he went to school in Tamworth. In 1686, after two years of school, he entered Clare College, Cambridge. He was very poor since his father died shortly before he went to university. William attended Newton's lectures while at Cambridge and he showed great promise in mathematics. He obtained his B.A. in 1690, then obtained a fellowship in 1691, two years later he was awarded an M.A. and was ordained in the same year. He was encouraged by David Gregory at this time to study Newton's Principia. He returned to Cambridge, intending to take mathematics pupils, but ill health made him give up teaching. From 1694 to 1698 he was chaplain to the bishop of Norwich. During this period he wrote A New Theory of the Earth (1696), in which he claimed that the biblical stories of the creation, flood etc. could be explained scientifically as descriptions of events with historical bases. For example he claimed that the biblical flood was due to a comet hitting the Earth, an interesting theory given the current interest in theories of this type. In 1698 he obtained a vicarage in Suffolk at Lowestoft-with-Kissingland. He was appointed assistant to Newton at Cambridge from 1701 and published an edition of Euclid for student use at this time. He fell out with Newton over Bible chronology. His cosmology involved direct intervention by God. In 1703 he succeeded Newton as Lucasian professor. He lectured on mathematics and natural philosophy and, after Roger Cotes was appointed to the Plumian professorship in 1706, with strong

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whiston.html (1 of 2) [2/16/2002 11:38:12 PM]

Whiston

recommendations from Whiston, they undertook joint research. Whiston came to believe that the doctrine of the Trinity was incorrect. His views were not popular and he was deprived of his chair in 1710. He went to London where a court was set up by the lord chancellor but, after the death of Queen Anne, proceedings against him were dropped. He was now poor and lived off the income of a small farm near Newmarket. He lectured in the coffee-houses of London, being one of the first to lecture making science experiments during the lectures. He tried hard to solve the problem of the longitude; there was a lot of money in that but he never succeeded. He left the Church of England in 1747 and joined the Baptists. His publications were mostly on religious topics with a small number of mathematical ones. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) A Poster of William Whiston

Mathematicians born in the same country

Cross-references to History Topics

English attack on the Longitude Problem

Honours awarded to William Whiston (Click a link below for the full list of mathematicians honoured in this way) Lucasian Professor of Mathematics

1702

Other Web sites

1. The Galileo Project 2. Bob Bruen 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Whiston.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whiston.html (2 of 2) [2/16/2002 11:38:12 PM]

White

Henry Seely White Born: 20 May 1861 in Cazenovia, New York, USA Died: 20 May 1943 in Poughkeepsie, New York, USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henry White attended Cazenovia Seminary where his father was a teacher of elementary mathematics and surveying. From there White entered the Wesleyan University, in Middletown, Connecticut, which was founded in 1831 by a group of Methodists. At the Wesleyan University White was taught mathematics by Van Vleck's father. After graduating with an A.B. in 1882, White was appointed an assistant in the astronomical observatory at the Wesleyan University then he moved to Centenary Collegiate Institute, Hackettstown, New Jersey where he taught mathematics and chemistry for a year before returning to the Wesleyan University to become a tutor. In 1887 he decided to study for his doctorate and travelled to Germany to study at the University of Göttingen for his doctorate between 1887 and 1890 under Klein's supervision, the doctorate being awarded for the thesis Abelsche Integrale auf singularitätenfreien, einfach überdeckten, vollständigen Schnittkurven eines beliebig ausgedehnten Raumes in 1891. On his return to the United States, White was appointed to Clark University, then professor of pure mathematics at Northwestern University before being appointed, in 1905, as professor at Vassar College in Poughkeepsie, New York. Vassar College was then a college for women, founded to provide women with a quality of education previously only available to men. Grace Hopper was one who benefitted from the quality education provided there. White was an excellent research mathematician. He worked on invariant theory, the geometry of curves and surfaces, algebraic curves and twisted curves. In [1] Archibald describes in detail a theorem proved by White in 1915:-

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/White.html (1 of 2) [2/16/2002 11:38:14 PM]

White

If seven points on a twisted cubic be joined, two and two, by twenty-one lines, then any seven planes that contain these 21 lines will osculate a second cubic curve. This theorem is more strictly fundamental than von Staudt's ... [which] can be deduced from White's. Despite White's impressive mathematical contributions, he may be most important for his work for the American Mathematical Society where he instigated the Colloquium Lectures in 1896. He was a Colloquium Lecturer himself in 1903 when he lectured on Linear systems of curves on algebraic surfaces. White was president of the American Mathematical Society from 1907 to 1908. In [1] Archibald notes his love of music and the fact that White's wife was the composer Mary Gleason. White is described as:Wise, kind, the soul of courtesy. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Henry White

Mathematicians born in the same country

Honours awarded to Henry White (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1907 - 1908

AMS Colloquium Lecturer

1903

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/White.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/White.html (2 of 2) [2/16/2002 11:38:14 PM]

Whitehead

Alfred North Whitehead Born: 15 Feb 1861 in Ramsgate, Isle of Thanet, Kent, England Died: 30 Dec 1947 in Cambridge, Massachusetts, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alfred Whitehead's father Alfred Whitehead was an Anglican clergyman from Ramsgate. He is said to have been an upright man with countless friends and Alfred North Whitehead's son North Whitehead wrote of his grandfather:He never asked a favour of anyone and never shirked what he considered to be a duty, but it cannot be said that he spent more time absorbing the lessons of the New Testament than was necessitated by his calling. Alfred Whitehead married Maria Sarah Buckmaster, who came from London, on the 20th of December 1851. She is described as (see [6]);... an unimaginative, small minded woman with some wit but no sense of humour. Alfred and Maria Whitehead had four children with Alfred North Whitehead as the youngest of the family. He had two brother who were seven and eight years older than he was, and a sister who was two years older. Whitehead was always treated by his parents as the baby of the family and, rather surprisingly, they considered him a sickly and frail child when it appears that this was not the case. Whitehead was not sent to primary school because his parents thought that he was too delicate so he was taught at home by his father until he was 14. Other than the usual childhood illnesses he was, despite his parents views, a healthy child. He received much affection from his father and brothers (but sadly little from his mother) and he seems to have had a childhood which was not unhappy, even though he was on his own a great deal and must have been somewhat lonely. Whitehead's father taught him Latin from the age of ten and Greek from the age of twelve. His ability in these subjects could certainly be classed as competent but it was certainly not outstanding; there was no http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitehead.html (1 of 5) [2/16/2002 11:38:17 PM]

Whitehead

sign of the genius that he showed later in life. He did learn a little mathematics from his father but quite how he developed an interest in the subject is a mystery. In September 1875 he left his father's vicarage and entered Sherborne Independent School. His oldest brother became a teacher at the school in 1876 when Whitehead was entering his second year of study. The course he followed at Sherborne was a fairly standard one for the time. There was little choice of subjects and all the boys studied as their major subjects Latin, Greek and English, with the minor subjects of mathematics, physical sciences, history, geography and modern languages receiving less attention. Whitehead showed a special gift for mathematics and was allowed to devote extra time to that subject in his final school year, dropping composition and reading of Latin poetry to make way for the extra mathematics. In 1879 Whitehead took the entrance examinations for Trinity College, Cambridge, and he won a scholarship. Following this he spent his final year at Sherborne as Head Boy and Captain of Games, before he entered university in October 1880. As the holder of a scholarship, Whitehead lived in College. He attended only mathematics lectures and was taught by J W L Glaisher, H M Taylor, and W D Niven. He also attended lectures by Stokes and Cayley while his coach was the famous E J Routh. Among his close friends at Cambridge was D'Arcy Thompson. Whitehead won a second scholarship, a College Foundation, and so by the time he entered his second year of study he was quite well off. He took the Mathematical Tripos examinations in 1883 and was placed Fourth Wrangler; the Senior Wrangler that year was G B Mathews. In the following year he was also placed in the First Class of Part III of the Tripos. He presented a dissertation on Maxwell's theory of electricity and magnetism in the competition for a Fellowship in 1884. Thomson and Forsyth were appointed to examine Whitehead and, much to his surprise, he won one of the five scholarships available that year. After winning the Fellowship, Whitehead was appointed to an assistant lectureship. He taught mostly applied mathematics but, surprisingly, he published no papers during the first five years of his tenure of the Fellowship. It is not known if he worked on mathematical research over this period. Certainly he was very much of a loner and did not talk much with the other mathematicians. In the twelve years following taking up the teaching position at Cambridge he published only two papers, both in 1889 on the motion of viscous fluids. The topic almost certainly interested him because he had attended lectures by Stokes on the topic. Despite his poor publication record, Whitehead was promoted to a Lectureship at Cambridge in 1888. He took up additional teaching duties by accepting a teaching position at Girton College. All the signs at this time would point to him having decided that his strength was in teaching and not in publishing. A rather remarkable change came, however, when he married Evelyn Wade in London on 16 December 1890 [6]:Whereas he was quiet and restrained, she was active and outgoing. He had become interested in pure mathematics and he started work on Treatise on Universal Algebra in January 1891, just weeks after his marriage. The work would take him seven years to complete, not finally being published until 1898. Other changes in his life took place around the time of his marriage. We have already indicated that

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitehead.html (2 of 5) [2/16/2002 11:38:17 PM]

Whitehead

Whitehead's father was an Anglican vicar and, of course, Whitehead was brought up as an Anglican. However around 1889-90 he began to move towards the Roman Catholic Church. He debated with himself for seven years whether to remain an Anglican or join the Roman Catholic Church. In the end he chose neither and became an agnostic around the mid 1890s. He himself stated that the biggest factor in his becoming an agnostic was the rapid developments in science; particularly his view that Newton's physics was false. It may seem surprising to many that the correctness of Newton's physics could be a major factor in deciding anyone's religious views. However one has to understand the complex person that Whitehead was, and in particular the interest which he was developing in philosophy and metaphysics. We should return to the story of the Treatise on Universal Algebra which Whitehead worked on for much of the 1890s. Perhaps the first comment we should make is that the work is not on the modern topic of universal algebra for the term 'universal algebra' had quite a different meaning to Whitehead. In the Preface to the treatise he writes that his aim is:... to present a thorough investigation of the various systems of symbolic reasoning allied to ordinary algebra ... . The chief examples of such systems are Hamilton's Quaternions, Grassmann's Calculus of Extension, and Boole's Symbolic Algebra. Also in the Preface Whitehead also gives his views on the nature on mathematics and the philosophy of mathematics:Mathematics in its widest signification is the development of all types of formal, necessary, deductive reasoning. The reasoning is formal in the sense that the meaning of propositions forms no part of the investigation. The sole concern of mathematics is the inference of proposition from proposition. ... The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every providence of thought, or external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus. Although Whitehead became very productive after his marriage, he never considered himself a creator of new areas of mathematics, but rather as a developer of ideas introduced by others. This does not mean that his contribution should be considered any less important because of this but certainly Cambridge seems to have undervalued his contribution. In 1894 Whitehead became an examiner for the Mathematical Tripos. In 1903 he was promoted to Senior Lecturer, a position which had only just been established at Cambridge. Whitehead is perhaps best known for his collaboration with Bertrand Russell. We shall give details of this collaboration below, but first we shall complete the details of Whitehead's career. He remained at Cambridge until 1910 but, in some sense, having not made the grade in mathematics and, having little prospects of a mathematics chair at Cambridge, he moved to the University of London. This explanation of his move is almost certainly basically correct and this indeed was the motivation behind Whitehead's thinking; on the face of it, however, rather different and dramatic events ended his association with Cambridge. In 1910 Andrew Forsyth, who had been a close friend of Whitehead's since his student days, had a love affair with Marion Amelia Boys, the wife of C V Boys, and the scandal forced him to resign his chair at Cambridge. Whitehead did everything he could to ensure that Forsyth kept his Fellowship. The decision http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitehead.html (3 of 5) [2/16/2002 11:38:17 PM]

Whitehead

as to whether he could keep the Fellowship was taken by the Council of Trinity and Whitehead, as a member of that Council, argued strongly that Forsyth should be allowed to remain a Fellow of Trinity. Whitehead was outvoted on the Council, however, and shortly after this he resigned his Senior Lectureship and his Fellowship. The Council then voted that Whitehead had served as a Lecturer for over 25 years (the maximum period) so must leave his post. Whitehead's appointment as Senior Lecturer still had three years to run but he did not stay to argue his case. He moved to London in the summer of 1910 with no job to go to. In 1914, after four years without a proper position, he became Professor of Applied Mathematics at the Imperial College of Science and Technology in London. He accepted a chair in philosophy at Harvard University in 1924, and he taught at Harvard until his retirement in 1937. Bertrand Russell entered Cambridge in 1890 and Whitehead, as examiner for the entrance examinations, spotted Russell's brilliance in his papers. Whitehead argued that Russell should be awarded a more prestigious scholarship than his marks would have merited and indeed this was awarded. When Russell was in his second year as an undergraduate he was taught by Whitehead. Their collaboration on Principia Mathematica appears to have begun near the end of 1900, although both men failed to remember the exact time their collaboration began when interviewed late in their lives. At the time they began collaborating Whitehead was working on his article Memoir on the algebra of symbolic logic while Russell was close to finishing the first draft of his Principles of mathematics. Whitehead was planning a second volume of Treatise on Universal Algebra but both their plans were somewhat disrupted in 1901 when Russell discovered his famous set theory paradox. After the initial worry over the paradox they joined forces on volume 2 of Russell's work so by 1903 Whitehead was working simultaneously on two different second volumes. Realising that this was not the optimal course for him he abandoned the second volume of his own work to concentrate on his collaboration with Russell. Their joint work attempted to construct the foundations of mathematics on a rigorous logical basis and it was carried out with Russell as the philosopher on the project and Whitehead the mathematician. Working with Russell did not occupy Whitehead completely for he continued to produce work of his own. In 1906 he published The axioms of projective geometry and, in the following year, The axioms of descriptive geometry. The first volume of Principia Mathematica was published in 1910, the second in 1912, and the third in 1913. He also wrote the popular mathematics book An introduction to mathematics which was published in 1911, between volumes 1 and 2 of the Principia. As the Principia Mathematica neared completion, Whitehead turned his attention to the philosophy of science. This interest arose out of the attempt to explain the relation of formal mathematical theories in physics to their basis in experience and was sparked by the revolution brought on by Einstein's general theory of relativity. In The Principle of Relativity (1922), Whitehead presented an alternative to Einstein's views. Science and the Modern World (1925), a series of lectures given in the United States, served as an introduction to his later metaphysics. Whitehead's most important book, Process and Reality (1929), took this theory to a level of even greater generality. Article by: J J O'Connor and E F Robertson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitehead.html (4 of 5) [2/16/2002 11:38:17 PM]

Whitehead

Click on this link to see a list of the Glossary entries for this page List of References (24 books/articles)

Some Quotations (41)

A Poster of Alfred Whitehead

Mathematicians born in the same country

Other references in MacTutor

1. Chronology: 1910 to 1920 2. Chronology: 1920 to 1930

Honours awarded to Alfred Whitehead (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1903

Royal Society Sylvester Medal

Awarded 1925

LMS Berwick Prize winner

1948

Other Web sites

1. Stanford Encyclopedia of Philosophy 2. Stanford Encyclopedia of Philosophy (Principia Mathematica) 3. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Whitehead.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitehead.html (5 of 5) [2/16/2002 11:38:17 PM]

Whitehead_Henry

John Henry Constantine Whitehead Born: 11 Nov 1904 in Madras, India Died: 8 May 1960 in Princeton, New Jersey, USA

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Henry Whitehead's father was The Right Rev Henry Whitehead, Bishop of Madras in India. His mother, Isobel Duncan, was the daughter of the Rector of Calne, Wiltshire, so Henry came from a family deeply involved with the Church. However, Henry Whitehead's family also had strong academic connections; in particular there was a strong tradition of mathematical excellence. His mother had studied mathematics at Oxford University, being one of the early women undergraduates, while the famous mathematician and philosopher, A N Whitehead, was his uncle. Although Henry was born in India he lived in England from the age of about eighteen months. When he was that age his parents brought him back from India and left him in the care of his maternal grandmother who lived in Oxford. His parents then returned to India and Henry saw little of them while he was growing up. It would not be until his father retired when Henry was sixteen years old that they returned to England. Henry's childhood in Oxford was a quiet one since [6]:... it was a very peaceful place and he would recall going for drives with his grandmother in her carriage and seeing the horsedrawn buses in town. Henry did quite well at primary school, both academically, socially and in sport. He was [6]:... of above average intelligence, good at games, prone to be careless in his work, but with a great capacity for enjoying life. If he had worked harder he might have won a scholarship to Eton... Despite the lack of a scholarship to Eton, Whitehead was successful in the Entrance Examination and began a happy period at Eton where he specialised in mathematics, yet never showed himself as a mathematical genius. One reason why this outstanding mathematician only appeared "good" at school was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitehead_Henry.html (1 of 4) [2/16/2002 11:38:19 PM]

Whitehead_Henry

a whole range of other interests which occupied him [6]:His exuberance, gaiety and intelligence made him many friends and his irrepressible high spirits and disregard for authority sometimes strained the patience of his tolerant and long suffering housemaster. His personal popularity got him elected to Pop, and his athletic prowess won him a place in the cricket second eleven, his fives colours and a silver cup at boxing. Another reason why he failed to shine academically may have been due to an inner sadness at being separated from his parents. Whatever the reasons it made his desired progress harder than it might otherwise have been. Whitehead wanted to go to Balliol College, Oxford, to study mathematics but his mathematics teacher at Eton did not think that he stood a chance of winning a scholarship. He wrote [6]:In pure geometry he has not been over diligent ... he would have been more successful in mathematics if he had been less so at cricket. The mathematics teacher was wrong, however, and in March 1923 Whitehead did win a scholarship to study at Balliol College. At Balliol Whitehead was tutored by J W Nicholson, who had been a student of Whitehead's uncle A N Whitehead. However, Nicholson's health was poor and Whitehead was tutored frequently by H Newboult at Merton College. As Eton had done, however, Oxford also provided a whole range of distractions to Whitehead. He played many sports including cricket, squash, tennis, and boxing. His interest in cricket brought him into contact with G H Hardy, so the sporting interests had some academic benefits. At Oxford Whitehead developed another passion, namely playing poker. He claimed that his mother taught him how to play the game when he was a young child recovering from an illness. Whitehead played poker for quite large sums of money while at Oxford although his friends did not always pay what they owed him. At Oxford Whitehead showed himself to be better than the "good" at mathematics which he had displayed at Eton. Despite his success, and the award of a First Class degree, he did not consider himself sufficiently talented for an academic career so, in 1927, he joined the firm of stockbrokers Buckmaster and Moore. By this time his parents had returned from India and were living in the village of Sulham in Berkshire and Whitehead lived there and travelled to his job in the city of London every day. It took not much over a year of work at the stockbrokers to convince Whitehead that the City was not the life for him so, in 1928, he returned to the University of Oxford. While at Oxford Whitehead met Veblen, who was on leave from Princeton. He attended a seminar which Veblen gave on differential geometry and it must have been a very fine talk for it persuaded Whitehead that he would undertake research in that topic. Veblen supported Whitehead's application for a Commonwealth Fellowship to enabled him to study for a Ph.D. at Princeton. Whitehead arrived at Princeton in the summer of 1929 to begin his research. He worked mainly on differential geometry although towards the end of his three years there he became interested in topology. He was awarded his doctorate from Princeton in 1932 for a dissertation entitled The Representation of Projective Spaces. Whitehead's joint work with his doctoral supervisor Veblen led to The Foundations of Differential Geometry (1932), now considered a classic. It contains the first proper definition of a differentiable manifold. As we mentioned Whitehead's interests turned more towards topology near the end of his three years in Princeton when he collaborated with Lefschetz in proving that all analytic manifolds can be triangulated.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitehead_Henry.html (2 of 4) [2/16/2002 11:38:19 PM]

Whitehead_Henry

In this area he is best remembered for his work on homotopy equivalence. The three years at Princeton were happy ones for Whitehead and he [6]:... had throughout his life a really deep affection for [Princeton] and its inhabitants, ranging from the Dean of the Graduate College to the barman at "Andy's". Whitehead returned to Oxford after being awarded his doctorate and he was elected to a Fellowship at Balliol College in 1933. In the following year Whitehead married Barbara Smyth (one of the authors of [6]) who was a concert pianist. They lived at first in St Giles, Oxford, but moved to North Oxford after their first son was born. Their home there [6]:... became a meeting place for mathematicians, where there was generally a mug of beer or a cup of tea and always a warm welcome, and a pencil and a block of paper each for host and guest to write their thoughts on. Many ideas were exchanged and many informal seminars took place in his study. Soon after his return to England, Whitehead wrote another major work on differential geometry On the Covering of a Complete Space by the Geodesics Through a Point (1935). Whitehead also studied Stiefel manifolds and set up a school of topology at Oxford. However, events would soon cause a break in Whitehead's career. The Nazi moves against Jewish mathematicians gave Whitehead great distress, and he actively helped many to escape to safety. In particular he helped Eilenberg and Dehn, while Schrödinger came to live in his home after escaping from Austria. Whitehead left Oxford in 1940 to undertake war work in London, spending [6]:... the night of the worse blitz on London sitting in his friend's wine cellar placidly working at mathematics. He afterwards congratulated himself on his high standard of morality as not one bottle was [opened]. After working at the Board of Trade, at the Admiralty, and finally at the Foreign Office, during the War, Whitehead returned to his home in North Oxford when World War II ended. In 1947 he was appointed to the Waynflete Chair of Pure Mathematics at Oxford. At that time Whitehead moved from Balliol College to Magdalen College. Whitehead's father died in 1947 and his mother died six years later in 1953. She had owned a small farm and when Whitehead inherited the cattle he and his wife decided to buy a farm north of Oxford. The farm was run mainly by Whitehead's wife but he took a keen interest in the farm when the couple lived until Whitehead's death. His death, while on a visit to Princeton, was totally unexpected [6]:In May 1960, without any pervious warning symptoms or illness, he died of a heart attack in Princeton, where his mathematical life had begun. Whitehead's personality is clearly described in [6]:He was able to reach across the barriers of age, class and nationality to talk on equal terms with anyone who shared his passion for mathematics. The long series of collaborative papers written between 1950 and 1960 reflects his eagerness to share his ideas and to interest himself in the results of others, which remained undiminished to the end of his life. It was in long mathematical conversations, in which ever detail had to be hammered out till he had it quite correct and secure that he most delighted and it is by these conversations, gay and informal, in which he contrived to make everyone his own equal, that he will be best remembered by those who knew him.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitehead_Henry.html (3 of 4) [2/16/2002 11:38:19 PM]

Whitehead_Henry

His career produced mathematical results which would have a major influence on the directions of mathematical research. However, as the authors of [6] relate, his influence extended beyond this:His influence on the development of mathematics during his active lifetime can be partly measured by the innumerable references, implicit and explicit, in current mathematical literature on algebraic and geometric topology; but it could not have been so great without the generosity and enthusiasm which he poured into every mathematical enterprise and which inspired such deep affection in all who knew him well. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

A Quotation

A Poster of Henry Whitehead

Mathematicians born in the same country

Honours awarded to Henry Whitehead (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1944

London Maths Society President

1953 - 1955

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR August 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Whitehead_Henry.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitehead_Henry.html (4 of 4) [2/16/2002 11:38:19 PM]

Whitney

Hassler Whitney Born: 23 March 1907 in New York, USA Died: 10 May 1989 in Mount Dents Blanches, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hassler Whitney attended Yale University where he received his first degree in 1928, then continued to undertake mathematical research at the University of Harvard from where his doctorate was awarded in 1932. His doctorate was awarded for a dissertation The Coloring of Graphs written under Birkhoff's supervision. He continued to work at Harvard, being an instructor in mathematics from 1930 until 1935, although the years 1931-33 were spent as a National Research Council Research Fellow. From 1935 he was promoted to assistant professor, then from 1940 associate professor. Harvard made him a full professor in 1946 and he held this professorship until he accepted an offer from the Institute for Advanced Study at Princeton of a chair in 1952. Whitney's main work was in topology, particularly in the theory of manifolds. Continuing work started by Veblen and Henry Whitehead, Whitney produced fundamental work in differential topology in 1935. Whitney also wrote on graph theory, in particular the colouring of graphs and chromatic polynomials. Other work on algebraic varieties and integration theory was important. Outside mathematical research Whitney contributed in many ways to his subject. He was chairman of the National Science Foundation mathematics panel from 1953 until 1956. He was editor of the American Journal of Mathematics from 1944 to 1949, then editor of Mathematical Reviews from 1949 until 1954. Ulam writing about Whitney said:He was friendly, but rather taciturn - psychologically of a type one encounters in this http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitney.html (1 of 2) [2/16/2002 11:38:21 PM]

Whitney

country more frequently than in central Europe - with wry humour, shyness but self-assurance, a probity which shines through, and a certain genius for persistent and deep follow-through in mathematics. Princeton was to remain Whitney's base from 1952 until he retired in 1977. The year before he retired he was awarded the National Medal of Science. Then in 1983 he received the Wolf Prize and, two years later, the Steele Prize. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Hassler Whitney (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1946

Other Web sites

Whitney-Mikhlin Extension Constants

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Whitney.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whitney.html (2 of 2) [2/16/2002 11:38:21 PM]

Whittaker

Edmund Taylor Whittaker Born: 24 Oct 1873 in Southport, Lancashire, England Died: 24 March 1956 in Edinburgh, Scotland

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edmund Whittaker was educated at Manchester Grammar School and from there he went up to Trinity College, Cambridge in 1892. At Cambridge he was a pupil of G H Darwin and A R Forsyth. His interests at this time were on the applied side of mathematics which is certainly illustrated by the fact that, in 1894, he was awarded the Sheepshanks Exhibition in Astronomy. Whittaker graduated as Second Wrangler in the examination of 1895. He was beaten into second place by Bromwich. He became first Smith's prizeman in 1896 for a work on pure mathematics, namely on uniform functions. In 1896 Whittaker became a Fellow of Trinity College and began to lecture in Cambridge. His first pupils at Cambridge included G H Hardy and J H Jeans. Whittaker made revolutionary changes to the topics taught at Cambridge. He taught a course based on his famous book A Course of Modern Analysis (1902). This work is important in the study of functions of a complex variable. It also studies special functions and their related differential equations. Other courses Whittaker taught at Cambridge included Astronomy, Geometrical Optics, and Electricity and Magnetism. Whittaker's interest in astronomy is illustrated by the courses he taught, but he also joined the Royal Astronomical Society serving as its secretary from 1901 to 1906. He became Astronomer Royal of Ireland in 1906 and moved to Dunsink Observatory where Hamilton had worked. He was at the same time Professor of Astronomy at the University of Dublin. In 1912 Whittaker took up Chrystal's chair in Edinburgh and remained in Edinburgh for the rest of his career. In fact he was due to retire in 1943 but due to World War II he agreed to carry on for a further three years. His eldest daughter married Copson. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whittaker.html (1 of 3) [2/16/2002 11:38:23 PM]

Whittaker

Whittaker is best known work is in analysis, in particular numerical analysis, but he also worked on celestial mechanics and the history of applied mathematics and physics. He wrote papers on algebraic functions and automorphic functions. He found expressions for the Bessel functions as integral involving Legendre functions. He studied these special functions as arising from the solution of differential equations derived from the hypergeometric equation. His results in partial differential equations (described as most sensational by Watson) included a general solution of the Laplace equation in three dimensions in a particular form and the solution of the wave equation. Soon after he arrived in Edinburgh, Whittaker set up the Edinburgh Mathematical Laboratory to give a practical side to his interest in numerical analysis. His many lecture courses on this topic were collected into a book which he published in 1924 The Calculus of Observations: a treatise on numerical mathematics. On the applied side of mathematics he was interested in relativity theory for many years publishing at least five articles on the topic. He also worked on electromagnetic theory, and it was through this topic that his interest in relativity arose. One of his most important historical studies was A History of the Theories of Aether and Electricity, from the Age of Descartes to the Close of the Nineteenth Century (1910). In 1953 he produced a revised version including the work of the first 1900 to 1925. In [8] McCrea describes Whittaker's research lectures which he gave twice a week throughout the whole academic year while he was professor in Edinburgh:Either he discussed his own current work or he gave his own development of topics of current interest in mathematics. One marvels at the mathematical power that enabled him always, year after year, to have material for these lectures - he never repeated the same ones - just as though he had nothing else to think about, when actually he was inundated with other duties. As to Whittaker's character McCrea writes in [8]:He grasped new ideas with unbelievable rapidity and he had an infallible memory for everything he read. ... He was the most unselfish of men with a delicate sense of what would give help or pleasure to others. Always he seemed to have his vast number of friends at he tip of his mind so that he never missed an opportunity to do or say something on behalf of any one of them. He had a quick wit and an ever-present sense of humour and liked telling harmlessly mischievous stories about people he had known. Whittaker received many honours. He was a member of the London Mathematical Society, being President in 1928-29. He won the De Morgan Medal of the Society in 1935. He was elected a Fellow of the Royal Society in 1905 being awarded the Society's Sylvester Medal in 1931 and the Copley Medal in 1954:... for his distinguished contributions to both pure and applied mathematics and to theoretical physics. He was knighted in 1945. He was a Fellow of the Royal Society of Edinburgh and served the Society as President for most of the years of World War II. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whittaker.html (2 of 3) [2/16/2002 11:38:23 PM]

Whittaker

Mention has been made of Whittaker's pupils Hardy and Jeans. These were not the only famous mathematicians which Whitttaker taught at Cambridge. His pupils included Bateman, Eddington, Littlewood, Turnbull and Watson. Whittaker's son, John Whittaker, also became a mathematician and his daughter married Edward Copson. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) A Poster of Edmund Whittaker

Mathematicians born in the same country

Honours awarded to Edmund Whittaker (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1905

Royal Society Copley Medal

Awarded 1954

Royal Society Sylvester Medal

Awarded 1931

Fellow of the Royal Society of Edinburgh London Maths Society President

1928 - 1929

LMS De Morgan Medal

Awarded 1935

Honorary Fellow of the Edinburgh Maths Society

Elected 1937

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Whittaker.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whittaker.html (3 of 3) [2/16/2002 11:38:23 PM]

Whittaker_John

John Macnaughten Whittaker Born: 7 March 1905 in Cambridge, England Died: 29 Jan 1984 in Sheffield, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Whittaker's father was Edmund Whittaker and his sister was married to Copson. His parents moved to Dublin when Jack (the name by which he was known) was one year old. His parents moved to Edinburgh in Scotland in 1912 and Jack attended school, first Watson's in Edinburgh, then he attended school in St Andrews until he won a scholarship to Fettes Academy. Jack attended Fettes from 1918 until 1920 during which time he was introduced to calculus. Then at the age of 15 he entered Edinburgh University where he became friends with Hodge who was in the same class. Jack entered Trinity College, Cambridge in 1923 and there he was taught by Littlewood, Milne, R H Fowler and S Pollard. His fellow students included Coxeter and Todd. After two years as an assistant in Edinburgh and four years as a lecturer at Pembroke College, Cambridge Whittaker accepted the chair of pure mathematics at the University of Liverpool. World War II saw Whittaker doing war service, spending a year in Cairo were he made contacts with mathematicians which were to forge strong links between Cairo and Liverpool which lasted many years. Whittaker returned to his chair at Liverpool in 1945 and became involved in administrative duties. By 1953 he had decided that if he was to be an administrator he would be a professional one and he became Vice-Chancellor of Sheffield University. He successfully led Sheffield until his retirement in 1965. Whittaker's first four papers in the late 1920s were on quantum theory. He then did some of his most important work on complex analysis. He extended results of his father E T Whittaker on the cardinal function. He also made important contributions to Nevanlinna theory with results on meromorphic http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whittaker_John.html (1 of 2) [2/16/2002 11:38:25 PM]

Whittaker_John

functions. He was interested over many years in expanding functions in a series of polynomials and Whittaker's constant is named after him. His interests and character are described in [1] as follows:... he collected 18th and 19th century water colours and other pictures and drawings. ... In his youth Jack had been active in hockey, rugger and tennis ... He remained a keen gardener .. He neither smoked nor drank but later in life he came increasingly to appreciate the fine food his wife provided, especially sweet things. He was a good after-dinner speaker, always without notes. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to John Whittaker (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1949

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Whittaker_John.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whittaker_John.html (2 of 2) [2/16/2002 11:38:25 PM]

Whyburn

Gordon Thomas Whyburn Born: 7 Jan 1904 in Lewisville, Texas, USA Died: 8 Sept 1969 in Charlottesville, Virginia, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Gordon Whyburn's parents were Thomas Whyburn and Eugenia Elizabeth Whyburn. His school education was in his home town of Lewisville and then, after graduating from school, he entered the University of Texas. He did not embark on a mathematics course at university, however, for his first love was chemistry and it was this topic which he studied for his first degree and he was awarded his A.B. in Chemistry in 1925. Robert Moore had been appointed as associate professor at the University of Texas in 1920 and it was Moore who taught Whyburn calculus early in his university studies. Moore quickly saw the mathematical potential in Whyburn, and Whyburn was soon attending further mathematics courses given by Moore who encouraged him greatly towards the study of mathematics. Even before he obtained his first degree in chemistry, Whyburn was undertaking research in mathematics with Moore. Of course with Moore having a deep interest in topology, that was the direction that Whyburn took and it was to become the topic of his research throughout his life. Whyburn had other connections with mathematics at the University of Texas in addition to his work with Moore. His elder brother, William Marvin Whyburn (1901-1972), was also at the University of Texas at the same time as Gordon Whyburn, and unlike Gordon he was studying mathematics. William Whyburn went on to become Chairman of the Mathematics Department at the University of California, Los Angeles and then at the University of North Carolina. His research was mostly on second order ordinary differential equations, see [3] for details. Another mathematician at Texas at this time was Lucille Smith, who was also from Lewisville, and Whyburn married Lucille in 1925. Lucille Whyburn has, since her husband's death, written four fascinating papers (see for example [4] and [5]) relating to mathematicians and events from her life with Gordon Whyburn. Despite having these connections with mathematics and despite Moore strongly encouraging him to move from chemistry to mathematics, Whyburn continued with his chemistry studies being awarded his Master's Degree in 1926. Only at that point did he see that continuing to study chemistry was foolish when he had completed high quality research in mathematics. Whyburn presented his first paper, which was on cyclic elements for locally connected plane continua, at the Western Christmas Meeting of the American Mathematical Society in Chicago on 31 December 1926

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whyburn.html (1 of 3) [2/16/2002 11:38:27 PM]

Whyburn

(see [5] for more details). The road to a doctorate in mathematics was now easy, and indeed he was awarded his Ph.D. in mathematics in 1927. Following this he was appointed as adjunct professor of mathematics at the University of Texas, holding this post from 1927 until 1929. In 1929 Whyburn, together with his wife, was able to go to Europe, financed by a Guggenheim Fellowship, where he spent academic year 1929-30. He spent most time in Vienna working with Hahn, but also visited Warsaw and made important links with Kuratowski and Sierpinski. In [3] letters between Whyburn and Robert Moore are discussed. These concern Moore's arcwise connectivity theorem which he had first proved for connected open sets in 1916 but had still not published by 1930. Back in the United States, Whyburn was appointed associate professor of mathematics at Johns Hopkins University, then in 1933 he was approached by the University of Virginia who asked him to accept an appointment as professor and chairman of the Department of Mathematics. He accepted the appointment in 1934 and planned [2]:... to get together a few young and congenial mathematicians of topflight accomplishments. their fields should be different but overlapping so that the students would not be confronted with choosing between absolutely unrelated areas. The plan worked beautifully. E H McShane joined the department in 1935, and G H Hedlund in 1939. The three of them ran a program of charm and high standards. Whyburn's first research contributions were on cyclic elements and the structure of continua. This work aimed at examining a locally connected plane continuum and the regions of the plane created by it. The theory was based on cyclic elements, that is a region C such that any two points of C are contained in a simple closed curve of C. Around the time Whyburn went to the University of Virginia he began working on homology theory and examined different notions of convergence in the space of all subsets of a compact metric space. From around 1936 Whyburn looked at open maps and their applications to complex function theory. In 1942 he published his famous text Analytic Topology in which he says [1]:Analytic topology is meant to cover those phases of topology which are being developed advantageously by methods in which continuous transformations play the essential role. A reviewer of the book [1] wrote:Were the volume a mere collection of the theory developed in diverse papers in recent years, it would be worthwhile as a source book for present and future workers in transformations. However, it is much more than that. Some of the results are new and much of the treatment is new, some of the proofs acquiring an elegance and polish they sadly lacked in the original papers. The book was the result of further developing material which Whyburn had presented as American Mathematical Society Colloquium lecturer in 1940. Later major texts by Whyburn were Topological analysis (1958) and Dynamic topology which was jointly authored by Edwin Duda and was published 10 years after Whyburn's death. The term "dynamic topology", writes Whyburn [6]:... refers to the body of results of a topological nature in which the function concept plays a central role.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whyburn.html (2 of 3) [2/16/2002 11:38:27 PM]

Whyburn

Although Whyburn spent his career from 1934 at Virginia, he did make frequent visits for summer teaching. He taught at Stanford, the University of California, UCLA and the University of Colorado. In addition he took study leave in 1952-53 at Stanford, and 1956-57 spent in England and Switzerland. He retired from his role as chairman at Virginia after he suffered a heart attack in 1966. He regained his health but died three years later of a heart attack Whyburn's character is summed up in [1]:Whyburn was a very private man. He was quiet and shy, and remarkably gentle with students and family. But in moments of administrative crisis, he could be extremely tough when he had to be. A man of brilliance, with a remarkable speed in research, he nevertheless believed deeply in continuity and patience, and that it was the total record of accomplishment of a lifetime that mattered most. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Whyburn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Whyburn.html (3 of 3) [2/16/2002 11:38:27 PM]

Widman

Johannes Widman Born: 1462 in Eger, Bohemia (now Cheb, Czech Republic) Died: 1498 in Leipzig, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Johannes Widman attended the University of Leipzig graduating in 1482. His Master's Degree was awarded in 1485 and he then taught at the University of Leipzig on the fundamentals of arithmetic and algebra. Widman's 1486 algebra lecture is the first given in Germany. He used Cossist notation as was usual at that time. He considered computation with irrational numbers and polynomials to be part of algebra. Widman is best remembered for an early arithmetic book in German in 1489 which contains the first appearance of + and - signs. It is an early example of a printed arithmetic book and it is better than those before it in having more examples and a wider range of examples. His book was reprinted until 1526 when it was superseded by one by Adam Ries and others. There are no direct references to Widman after 1489 but it is believed that he was still working on mathematical topics in 1498. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1300 to 1500

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Widman.html (1 of 2) [2/16/2002 11:38:28 PM]

Widman

Glossary index

Quotations index

Mathematicians of the day JOC/EFR December 1996

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Widman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Widman.html (2 of 2) [2/16/2002 11:38:28 PM]

Wielandt

Helmut Wielandt Born: 19 Dec 1910 in Niedereggenen, Lörrach, Germany Died: 14 Feb 2001

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Helmut Wielandt entered the University of Berlin in 1929 and there he studied mathematics, physics and philosophy. There he was greatly influenced by Schmidt and Schur. In his speech accepting membership of the Heidelberg Academy in 1960 he said:It is to one of Schur's seminars that I owe the stimulus to work with permutation groups, my first research area. At that time the theory had nearly died out. It had developed last century, but at about the turn of the century had been so completely superseded by the more generally applicable theory of abstract groups that by 1930 even important results were practically forgotten - to my mind unjustly. It was on the topic of permutation groups that Wielandt wrote his doctoral dissertation and he was awarded a doctorate in 1935. From 1934 until 1938 he worked on the editorial staff of Jahrbuch über die Fortschritte der Mathematik in Berlin. From 1938 he was an assistant at Tübingen where he submitted his habilitation thesis in 1939. Wielandt described the contents of his habilitation thesis in the following way:The work on permutation groups led me inevitably to involvement with the structure theory of finite groups. In the twenties this theory had fallen into neglect ... But Philip Hall's fundamental papers had already revitalised it. Where Hall had started from arithmetical questions and product decompositions, my own work was triggered by a question of Robert Remak of a quite different type: is the group generated by two subgroups that occur in composition series always of the same kind? In my Habilitationsschrift I expanded the discovery that this question can be answered affirmatively to a detailed study of the normal http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wielandt.html (1 of 3) [2/16/2002 11:38:30 PM]

Wielandt

structure of finite groups. Although Wielandt was formally on the staff at Tübingen until 1946, during World War II he was on leave for military service. He underwent basic training in 1939 and artillery training in 1940. Then from 1941 he was engaged in research on meteorology, cryptology and aerodynamics. Then in 1942 he was attached to the Kaiser Wilhelm Institute and the Aerodynamics Research Institute at Göttingen. Describing this work Wielandt said:... I had to work on vibration problems. I am indebted to that time for valuable discoveries: on the one hand the applicability of abstract tools to the solution of concrete problems, on the other hand, the - for a pure mathematician - unexpected difficulty and unaccustomed responsibility of numerical evaluation. It was a matter of estimating eigenvalues of non-self-adjoint differential equations and matrices. At the end of World War II Wielandt was appointed associate professor at the University of Mainz. He remained there until 1951 when he was appointed Ordinary Professor at the University of Tübingen. He remained at Tübingen until he retired in 1977 but during this time he spent two periods at the University of Wisconsin, Madison, one in 1963 and the second from 1965 to 1967. He also held a number of visiting positions in the United States, at the University of Warwick, England and at the University of Brazil. For 20 years beginning in 1952 Wielandt was managing editor of Mathematische Zeitschrift. Wielandt's research work continued on finite groups and on permutation groups. One of the areas which his work took him into was infinite permutation groups. He contributed greatly to linear algebra and matrix theory. Among his contributions was a shorter, elegant proof of the Perron-Frobenius Theorem. Wielandt believed that the axiomatic method had produced major advances:By turning increasingly towards the abstract, revolutionary unification has been achieved in mathematics. It is as if some areas of mathematics which earlier could hardly be reached on foot are now connected by motorways. However he was convinced, from his student days onwards, that the axiomatic method had limitations and carried his research forward on with this belief:... I could not share the general opinion that this would henceforth be the only rewarding direction for research. It seemed to me that, like all great deductive systems, it was threatened by the danger that the problems which it could not properly accommodate would be dismissed as uninteresting, whereas on the contrary, these ought to provide a stimulus to broaden the foundation. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Helmut Wielandt

Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wielandt.html (2 of 3) [2/16/2002 11:38:30 PM]

Wielandt

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wielandt.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wielandt.html (3 of 3) [2/16/2002 11:38:30 PM]

Wien

Wilhelm Carl Werner Otto Fritz Franz Wien Born: 13 Jan 1864 in Gaffken, East Prussia (now Poland) Died: 30 Aug 1928 in Munich, Germany

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Wien worked at the Physikalisch- Technische Reichsanstalt in Berlin- Charlottenburg where he was a colleague of Planck. Wien was appointed professor of physics at Giessen in 1899 and professor of physics at Munich in 1920. In 1893 Wien stated his displacement law of blackbody radiation spectra at different temperatures. His method is described in [2]:It was [Wien's] idea to use as a good approximation for the ideal blackbody an oven with a small hole. Any radiation that enters the small hole is scattered and reflected from the inner walls of the oven so often that nearly all incoming radiation is absorbed and the chance of some of it finding its way out of the hole again can be made exceedingly small. The radiation coming out of this hole is then very close to the equilibrium blackbody electromagnetic radiation corresponding to the oven temperature. In 1896 Wien derived a distribution law of radiation. Planck, who was a colleague of Wien's when he was carrying out this work, later, in 1900, based quantum theory on the fact that Wien's law, while valid at high frequencies, broke down completely at low frequencies. While studying streams of ionized gas Wien, in 1898, identified a positive particle equal in mass to the hydrogen atom. Wien, with this work, laid the foundation of mass spectroscopy. J J Thomson refined Wien's apparatus and conducted further experiments in 1913 then, after work by E Rutherford in 1919, Wien's particle was accepted and named the proton.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wien.html (1 of 3) [2/16/2002 11:38:32 PM]

Wien

Wien received the 1911 Nobel Prize for his work on heat radiation. In [4] a letter from Einstein to Wien is described in which he asks Wien to conduct an experimental proof of the principle of equivalence which Einstein had proposed from purely theoretical considerations in 1907:In 1912 [Einstein] turned by letter to W Wien with the request to measure the difference between the periods of oscillation of pendulums made of uranium and lead, as well as the proportionality of inertial and gravitational masses of a uranium and a lead weight, respectively, namely with a torsion balance. The letter testifies that Einstein was not aware of the Eötvös experiment when he formulated the principle of equivalence ... Wien also made important contributions to the study of cathode rays, X-rays and canal rays. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. The quantum age begins 2. Special relativity

Honours awarded to Wilhelm Wien (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1911

Other Web sites

1. Nobel prizes site (A biography of Wien and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wien.html (2 of 3) [2/16/2002 11:38:32 PM]

Wien

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wien.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wien.html (3 of 3) [2/16/2002 11:38:32 PM]

Wiener_Christian

Ludwig Christian Wiener Born: 7 Dec 1826 in Darmstadt, Germany Died: 31 July 1896 in Karlsruhe, Germany Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Christian Wiener attended school in Darmstadt, then studied engineering and architecture at the University of Giessen from 1843 to 1847. In 1848 he became a teacher of physics, mechanics, hydraulics and descriptive geometry at the Technische Hochschule in Darmstadt. (It was called the Höhere Gewerbeschule at that time.) Wiener obtained a doctorate in 1850 and taught at the University of Giessen. After working at the Technical University of Karlsruhe for a year he returned to Giessen in 1851. In 1852 he was appointed to the chair of descriptive geometry at the Technical University of Karlsruhe. Wiener's chief work is a 2 volume book on geometry Lehrbuch der darstellenden Geometrie which supplements Chasles's work and contains important historical information. Wiener extended work on descriptive geometry to physics and calculated the amount of solar radiation received at different latitudes during the varying lengths of days in the course of the uear. This work was important in atmospheric studies and climatic studies. At Clebsch's suggestion Wiener constructed plaster of Paris models of mathematical surfaces which were exhibited in London, Munich and Chicago. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wiener_Christian.html (1 of 2) [2/16/2002 11:38:33 PM]

Wiener_Christian

Glossary index

Quotations index

Mathematicians of the day JOC/EFR February 1997

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wiener_Christian.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wiener_Christian.html (2 of 2) [2/16/2002 11:38:33 PM]

Wiener_Norbert

Norbert Wiener Born: 26 Nov 1894 in Columbia, Missouri, USA Died: 18 March 1964 in Stockholm, Sweden

Click the picture above to see seven larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Norbert Wiener received his Ph.D. from Harvard at the age of 18 with a dissertation on mathematical logic. From Harvard to went to Cambridge, England to study under Russell, then he went to Göttingen to study under Hilbert. He was influenced by both Hilbert and Russell but also, perhaps to an even greater degree, by Hardy. After various occupations (journalist, university teacher, engineer, writer) in which he was very unhappy, he began a long association with MIT in 1919. His work on generalised harmonic analysis and Tauberian theorems won the Bôcher Prize in 1933 when he received the prize from the American Mathematical Society for his memoir Tauberian theorems published in Annals of Mathematics in the previous year. Wiener had an extraordinarily wide range of interests and contributed to many areas including cybernetics (a term he coined), stochastic processes, quantum theory and during World War II he worked on gunfire control. His wide dealings with other scientists led him to say One of the chief duties of the mathematician in acting as an adviser to scientists is to discourage them from expecting too much from mathematics. Some of Wiener's most important publications include The Fourier Integral, and Certain of Its Applications (1933), Cybernetics: or, Control and Communication in the Animal and the Machine (1948), Nonlinear Problems in Random Theory (1958) and God and Golem, Inc.: A Comment on Certain Points Where Cybernetics Impinges on Religion (1964).

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wiener_Norbert.html (1 of 2) [2/16/2002 11:38:35 PM]

Wiener_Norbert

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles)

Some Quotations (9)

Mathematicians born in the same country Other references in MacTutor

Chronology: 1940 to 1950

Honours awarded to Norbert Wiener (Click a link below for the full list of mathematicians honoured in this way) Lunar features

Crater Wiener

AMS Colloquium Lecturer

1934

AMS Gibbs Lecturer

1949

AMS Bôcher Prize

Awarded 1933

Other Web sites

1. AMS (An electronic version of the AMS article) 2. AMS 3. San Francisco (Extracts from Wiener's article on Cybernetics) 4. Some stories (Norbert Wiener as the prototype of the absent-minded professor) 5. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wiener_Norbert.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wiener_Norbert.html (2 of 2) [2/16/2002 11:38:35 PM]

Wigner

Eugene Paul Wigner Born: 17 Nov 1902 in Budapest, Hungary Died: 1 Jan 1995 in Princeton, New Jersey, USA

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

The Hungarian version of Eugene Paul Wigner's name was Jenó Pál Wigner. His father, Antal Wigner, was the director of a leather-tanning factory while his mother, Erzsébet Wigner, looked after the family of three children. Both Antal and Erzsébet were from a Jewish background but they did not practice Judaism. Paul was born in Pest, the eastmost of the two towns which, together with Buda, formed the Hungarian capital of Budapest. He was the middle of his parents three children, having both an older and younger sister. From the time he was five years old Wigner was given private tuition at home. When he was ten years old he entered an elementary school but about a year after he began his studies at the school he was told that he had tuberculosis. The cure was to be found in sending him to a sanatorium in Breitenstein in Austria and he spent six weeks there before being told that the diagnosis had been wrong and that he had never had tuberculosis. However, one advantage of his six weeks was that he began to think about mathematical problems [13]:I had to lie on a deck chair for days on end, and I worked terribly hard on constructing a triangle if the three altitudes are given. In 1915 Wigner entered the Lutheran High School in Budapest. Here he met John von Neumann who was in the class below him. However he wrote [8]:I never felt I knew von Neumann well at Gymnasium. Perhaps no one did; he always kept a bit apart. The school provided a solid education for Wigner in mathematics, literature, classics and religion. It did provide science teaching, but there was less emphasis on this than on other subjects. He was still at the http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wigner.html (1 of 6) [2/16/2002 11:38:38 PM]

Wigner

Gymnasium when the communists took control in Hungary in March 1919 and the whole Wigner family fled the country. They lived in Austria until the communists were overthrown in November 1919 when they returned to Budapest and Wigner completed has school education. When he was in his late teens the whole Wigner family became converts to Lutheranism but it did not mean a great deal to Wigner who in later life described himself as "only mildly religious". In 1920 Wigner left school being one of the top students in his class. Already he knew that mathematics and physics were the topics for him but he realised that von Neumann [8]:... was a much better mathematician than I was and a better scientist. But I knew more physics. Now Wigner wanted to be a physicist but his father expected him to join the family business and he believed that a degree in chemical engineering would be useful to his son in the family's leather-tanning factory. Wigner followed his father's wishes and took his first degree in chemical engineering spending one year at the Technical Institute in Budapest, then moving to the Technische Hochschule in Berlin. He said [13]:I went to practically no classes ... but worked extremely hard in the laboratory. I loved inorganic chemistry. Despite working for a degree in chemical engineering, Wigner studied mathematics and physics in his own time. He attended colloquia at the University of Berlin with Einstein, Planck, von Laue, and Nernst. Wigner obtained the degree of Dr. Ing. in 1925 from the Technische Hochschule in Berlin with a thesis Bildung und Zerfall von Molekulen supervised by Michael Polyani, who was a fellow countryman also from Budapest. Wigner's thesis contains the first theory of the rates of association and dissociation of molecules. Wigner and Polyani published a joint paper on this work in 1925. Having completed his doctorate, Wigner returned to Budapest to join his father's tannery firm as planned. However, things did not go too well [13]:I did not get along very well in the tannery. ... I did not feel at home there. ... I did not feel that this was my life. ... [In 1926] I received a letter from a crystallographer at the Kaiser Wilhelm Institute [who] wanted an assistant ... to find out why the atoms occupy positions in the crystal lattices which correspond to symmetry axes. ... He also told me that this had to do with group theory and that I should read a book on group theory and then work it out and tell him. Wigner's father supported him taking the post in Berlin. There he read Heisenberg's papers but in developing his own ideas he realised that the mathematics presented problems. He submitted a paper on the spectrum of atoms with 3 electrons to Zeitschrift der Physik on 12 November 1926 extending Heisenberg's results for 2 electrons. The paper ends with Wigner writing that his methods would be prohibitively complicated for atoms with more than three electrons. However, he asked von Neumann for advice on the mathematical difficulties and was told to read about the theory of group characters in Schur's papers. Wigner, because of his interest in crystals, had already read Heinrich Weber's Lehrbuch der Algebra and, already having an expertise in matrices from Weber's text, he found Schur's papers easy to understand. He also studied the representation theory of the symmetric group due to Frobenius and Burnside. The theory, as von Neumann suggested, was exactly what he needed to develop a theory of the spectrum of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wigner.html (2 of 6) [2/16/2002 11:38:38 PM]

Wigner

atoms with n electrons. He then began the work for which he is famous, namely applying group theory to quantum mechanics. His paper on the case of n electrons was submitted to the Zeitschrift der Physik on 26 November 1926. Wigner was invited to Gottingen in 1927 to become Hilbert's assistant. Hilbert, already interested in quantum mechanics, felt that he needed a physicist as an assistant to complement his own expertise. This was an important time for Wigner who produced papers of great depth and significance, introducing in his paper On the conservation laws of quantum mechanics (1927) the new concept of parity. However his collaboration with Hilbert was less successful for they only met five times during the year [8]:I found him painfully withdrawn. ... His enormous fatigue was plain. Wigner returned to Berlin after the year in Göttingen where he lectured on quantum mechanics, worked on writing his famous text Group theory and its application to the quantum mechanics of atomic spectra and continued his research. In fact Wigner's book on the applications of group theory to quantum mechanics was not the first to appear, since Weyl published his a little before Wigner. However, as Mackey writes in [3]:Weyl's ideas differed from those of Wigner in that he wanted to apply group representations to get a better understanding of the foundations of quantum mechanics in general and not so much to gain insight into particular problems. An offer to spend a term in Princeton saw him travel to the United States at the end of 1930. From 1930 to 1933 Wigner spent part of the year at Princeton, part at Berlin. His Berlin post vanished under the Nazi rules passed in 1933 and from then, except for the years 1936 - 1938 in Wisconsin, Wigner spent the rest of his career at Princeton. In 1934 his younger sister Margit (always known as Manci) joined her brother in Princeton. There she met Dirac, who was a visitor, and the two married in January 1937. There is slight confusion about the reason that Wigner left Princeton in 1936. In [8] he said:In 1936 came a shock ... Princeton dismissed me ... they never explained why ... I could not help feeing angry. Pais points out in [16] however, that this statement by Wigner is not strictly accurate and he was not dismissed. Rather it appears that he was not receiving the promotion in Princeton which he felt that he deserved and so took leave of absence to accept a position of acting professor in Wisconsin. While in Wisconsin, Wigner became a U.S. citizen. Also while at the University of Wisconsin at Madison he met and married Amelia Frank. She was a physics student there but the happiness was soon repaced by much pain for she fell ill with cancer and died in 1937 less than a year after the marriage. While in Wisconsin Wigner showed the role of the special unitary group SU(4) in considering nuclear forces and he constructed a class of irreducible unitary representations of the Lorentz group. Kim writes in [12]:Wigner's 1939 paper on representations of the inhomogeneous Lorentz group [Ann. of Math. (2) 40 (1939), 149-204] is one of the most fundamental papers in physics. He was appointed to the Thomas D Jones Chair of Mathematical Physics at Princeton in 1938. He brought his parents to the United States in 1939. First they lived in Princeton, then they moved to a more country place in New York State. They were never happy in the United States and for that matter Wigner never really felt at home. Near the end of his life he wrote:After 60 years in the United States I am still more Hungarian than American. ... much of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wigner.html (3 of 6) [2/16/2002 11:38:38 PM]

Wigner

American culture escapes me. He met Mary Annette Wheeler, a physics teacher from Vassar College, in 1940 and they were married on 4 June 1941. They had two children, David Wigner who taught mathematics at the University of California in Berkeley, and Martha who worked on the transportation system in the Chicago area. Wigner worked on the Manhattan Project at the University of Chicago during World War II, from 1942 to 1945. His training as an engineer proved valuable background for his war work on nuclear fission. Wigner received the Nobel Prize for Physics in 1963. The presentation Speech by I Waller put Wigner's contributions into their context:In order to be able to calculate the motion of the nucleons it was ... necessary to know also the forces which act between them. A very important step in the investigation of these forces was taken by Wigner in 1933 when he found, deducing from some experiments, that the force between two nucleons is very weak except when their distance apart is very small but that the force is then a million times stronger than the electric forces between the electrons in the outer part of the atoms. Wigner discovered later other important properties of the nuclear forces. ... It was ... fundamentally important that Wigner could show that most essential properties of the nuclei follow from generally valid symmetries of the laws of motion. Earlier Wigner had performed pioneering work by studying such symmetries in the laws of motion for the electrons and had made important discoveries by investigating e.g. those symmetries which express the fact that the laws mentioned make no difference between left and right and that backward in time according to them is equivalent to forward in time. These investigations were extended by Wigner to the atomic nuclei at the end of the 1930's and he explored then also the newly discovered symmetry property of the force between two nucleons to be the same whether either of the nucleons is a proton or a neutron. This work by Wigner and his other investigations of the symmetry principles in physics are important far beyond nuclear physics proper. His methods and results have become an indispensable guide for the interpretation of the rich and complicated picture which has emerged from recent years' experimental research on elementary particles. They were also an important preliminary for the deeper penetration into and the partial revision of the earlier concepts concerning the right-left symmetry ... Wigner has made many other important contributions to nuclear physics. He has given a general theory of nuclear reactions and has made decisive contributions to the practical use of nuclear energy. He has, often in collaboration with younger scientists, broken new paths in many other domains of physics. R L Ingraham summarised some of the many contributions made by Wigner. These include his:... epoch-making work on how symmetry is implemented in quantum mechanics, the determination of all the irreducible unitary representations of the Poincaré group, and his work with Bargmann on realizing those irreducible unitary representations as the Hilbert spaces of solutions of relativistic wave equations, ... discrete symmetries and superselection rules in quantum mechanics, symmetry implications for atomic and molecular spectra, natural line-width theory, contrast of microscopic and macroscopic physics and of general relativity and quantum mechanics, explanation of why symmetry yields more information for quantum than for classical mechanics, philosophical questions such as what nature laws should be, limits on causality, and whether quantum mechanics could in principle explain life. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wigner.html (4 of 6) [2/16/2002 11:38:38 PM]

Wigner

His important works include Nuclear Structure (1958) with L Eisenbud, The Physical Theory of Neutron Chain Reactors (1958) with A Weinberg, Dispersion Relations and Their Connection with Causality (1964), and Symmetries and Reflections (1967). Wigner received many honours for his outstanding work. He was awarded the United States Medal for Merit in 1946, the Enrico Fermi Prize in 1958, and the Atoms for Peace Award in 1960, the Medal of the Franklin Society, the Max Planck Medal of the German Physical Society, the George Washington Award of the American-Hungarian Studies Foundation (1964), the Semmelweiss Medal of the American-Hungarian Medical Association (1965), and the National Medal of Science (1969). The list of universities which awarded him an honorary degree is extensive, University of Wisconsin, Washington University, Case Institute, University of Alberta, University of Chicago, Colby College, University of Pennsylvania, Yeshiva University, Thiel College, Notre Dame University, Technische Universität Berlin, Swarthmore College, Université de Louvain, Université de Liège, University of Illinois, Catholic University, and The Rockefeller University. He was elected a Fellow of the Royal Society of London in 1970 and other memberships of learned societies included the National Academy of Science, the American Academy of Arts and Sciences, the Royal Netherlands Academy of Sciences and Letters, the American Association for the Advancement of Science, the Austrian Academy of Sciences, and the Gesellschaft der Wissenschaften of Gottingen. A J Coleman writes of the:... course by Wigner on advanced quantum mechanics which I had the good fortune to attend at Princeton in 1940. I recall a person with razor-sharp mind and of a kind and gentle spirit. Many other references to Wigner's personality leave the feeling that, despite the extensive interviews such as [8] and [13], he is still someone who is slightly mysterious. As Pais writes in [16]:He was a very strange man and one of the giants of twentieth-century physics. Perhaps we should end with Wigner's own words:The promise of future science is to furnish a unifying goal to mankind rather than merely the means to an easy life, to provide some of what the human soul needs in addition to bread alone. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (20 books/articles)

Some Quotations (4)

Mathematicians born in the same country Honours awarded to Wigner (Click a link below for the full list of mathematicians honoured in this way) Nobel Prize

Awarded 1963

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wigner.html (5 of 6) [2/16/2002 11:38:38 PM]

Wigner

Fellow of the Royal Society

Elected 1970

AMS Gibbs Lecturer

1968

Other Web sites

1. Nobel prizes site (A biography of Wigner and his Nobel prize presentation speech) 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wigner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wigner.html (6 of 6) [2/16/2002 11:38:38 PM]

Wilczynski

Ernest Julius Wilczynski Born: 13 Nov 1876 in Hamburg, Germany Died: 14 Sept 1932 in Denver, Colorado, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernest Wilczynski attended school in Hamburg for two years. Then his family emigrated to the United States and settled in Chicago. Wilczynski attended schools in Chicago, but rather than take his university education in the United States he returned to Germany to study at the university of Berlin. Wilczynski was taught at Berlin by some outstanding mathematicians including Fuchs, Hensel, Planck, Pringsheim, Schlesinger and Schwarz. He received his doctorate from the University of Berlin in 1897 and returned to the United States. He did not find a university post and decided to take a temporary post in the Office of the Nautical Almanac in Washington D.C. However, in 1898, he was offered a position as instructor at the University of California. In 1902 Wilczynski was promoted to assistant professor at Berkeley, then in 1906 to associate professor. During the years from 1903 to 1905 Wilczynski worked in Europe as associate of the Carnegie Institution. During his time in Verona, Italy he met his future wife and they were married in 1906. Wilczynski moved to the University of Illinois in 1907, then in 1910 he was appointed to Chicago filling a vacancy caused by the death of Maschke in 1908. Promotion to a full professor came at Chicago in 1914 but by 1919 his health began to fail. He continued to teach and carry out his other duties under increasing difficulties. Lane writes in [3]:... in the midst of a lecture [in 1923] he finally realised that he could go no further and, with a simple statement to that effect, walked from his classroom never to return, leaving his students amazed by the classic self-restraint with which he accepted his tragic fate. It is characteristic of the man, however, that during the long illness which followed he never lost his interest in geometry and never gave up hope and the belief that he would some day be able to return to his academic duties. Wilczynski began his research career as a mathematical astronomer. This interest lasted until he was appointed to Berkeley. By that time he had published over a dozen papers in astronomy, but his interests moved towards differential equations which arose in his study of the dynamics of astronomical objects. From there his interests became pure mathematical interests in differential equations. However, Wilczynski's main work was in projective differential geometry and ruler surfaces. He extended Halphen's work, devised new methods and extended the theory of curves to surfaces. Lane [3] writes:It has often been stated that Wilczynski was the founder, or inventor, of projective http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilczynski.html (1 of 3) [2/16/2002 11:38:39 PM]

Wilczynski

differential geometry. This is not quite precise, for Halphen in the latter part of the nineteenth century was the first ever consciously to undertake and carry to fruition a systematic projective differential investigation. He was primarily interested in curves in the plane and in ordinary space. But Wilczynski was the first ever to appreciate, demonstrate and exploit the utility of completely integrable systems of linear homogeneous differential equations for projective differential geometry. Wilczynski received many honours for his outstanding research contributions. He was Colloquium lecturer for the American Mathematical Society in New Haven in 1906 together with Max Mason and Eliakim Moore. In addition he served as vice-president of the American Mathematical Society and as a council member of the Association of America. Among the prizes which he won was one awarded by the Royal Belgium Academy of Sciences in 1909. He was elected to the National Academy of Sciences in 1919, the year illness began to affect him. It is not only as a research mathematician that Wilczynski excelled. He was a talented teacher bringing clarity through his carefully prepared lectures and extremely talented in mathematical exposition whether it was in English or German. He was almost equally at home with French or Italian. He is described by Lane [3] as:... gentle, his manner mild. He was unselfish and habitually thoughtful to others. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country Honours awarded to Ernest Wilczynski (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1906

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilczynski.html (2 of 3) [2/16/2002 11:38:39 PM]

Wilczynski

JOC/EFR November 1997

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wilczynski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilczynski.html (3 of 3) [2/16/2002 11:38:39 PM]

Wiles

Andrew John Wiles Born: 11 April 1953 in Cambridge, England

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Andrew Wiles's interest in Fermat's Last Theorem began at a young age. He said:... I was a ten year old and one day I happened to be looking in my local public library and I found a book on maths and it told a bit about the history of this problem and I, a ten year old, could understand it. From that moment I tried to solve it myself, it was such a challenge, such a beautiful problem, this problem was Fermat's Last Theorem. In 1971, Wiles entered Merton College, Oxford, graduating with a B.A. in 1974. He then entered Clare College, Cambridge to study for his doctorate. His Ph.D. supervisor at Cambridge was John Coates who said:I have been very fortunate to have had Andrew as a student. Even as a research student he was a wonderful person to work with, he had very deep ideas then and it was always clear he was a mathematician who would do great things. Wiles did not work on Fermat's Last Theorem for his doctorate. He said:... the problem with working on Fermat is that you could spend years getting nothing so when I went to Cambridge my advisor John Coates was working on Iwasawa theory of elliptic curves and I started working with him... From 1977 until 1980 Wiles was a Junior Research fellow at Clare College, Cambridge and also a Benjamin Peirce Assistant Professor at Harvard University. In 1980 he was awarded his doctorate, then spent a while at the Sonderforschungsbereich Theoretische Mathematik in Bonn. He returned to the United States near the end of 1981 to take up a post at the Institute for Advanced Study in Princeton. He was appointed a professor at Princeton the following year and, also during 1982, he spent a while as a visiting professor in Paris. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wiles.html (1 of 4) [2/16/2002 11:38:41 PM]

Wiles

Wiles was awarded a Guggenheim Fellowship which enabled him to visit the Institut des Hautes Etudes Scientifique in Paris and also the Ecole Normale Supérieure in Paris during 1985-86. In [1] the important events which changed the direction of Wiles's research after this period are described:... about ten years ago, G Frey suggested and K Ribet proved (building on ideas of B Mazur and J-P Serre) that Fermat's Last Theorem follows from the Shimura-Taniyama conjecture that every elliptic curve defined over the rational numbers is modular. Precisely, if an + bn = cn is a counterexample to Fermat's Last Theorem, then the elliptic curve y2 = x(x-an)(x+bn) cannot be modular, thus violating the Shimura-Taniyama conjecture. This result set the stage for Wiles's work. In fact Wiles abandoned all his other research when he heard what had been proved and, for seven years, he concentrated solely on attempting to prove the Shimura-Taniyama conjecture, knowing that a proof of Fermat's Last Theorem then followed. Wiles said:... after a few years I realised that talking to people casually about Fermat was impossible because it generated too much interest and you cannot focus yourself for years unless you have this kind of undivided concentration which too many spectators would destroy... In fact married life was a rather restricted affair for Wiles who said:... my wife has only known me while I have been working on Fermat. I told her a few days after I got married. I decided that I really only had time for my problem and my family and while I was concentrating very hard then I found with young children that it was the best possible way to relax. When you're talking to young children they're simply not interested in Fermat... In 1988 Wiles went to Oxford University where he spent two years as a Royal Society Research Professor. While in Oxford he was elected, in 1989, a Fellow of the Royal Society. In [1] the course of his research is described:Using Mazur's deformation theory of Galois representations, recent results on Serre's conjecture on the modularity of Galois representations, and deep arithmetical properties of Hecke algebras, Wiles (with one key step due jointly to Wiles and R Taylor) succeeded in proving that all semistable elliptic curves defined over the rational numbers are modular. Although less than the full Shimura-Taniyama conjecture, this result does imply that the elliptic curve given above is modular, thereby proving Fermat's Last Theorem. In fact the path to the proof was not as smooth as suggested by this description. In 1993 Wiles told two other mathematicians that he was close to a proof of Fermat's Last Theorem. He filled what he thought were the remaining few gaps and gave a series of lectures at the Isaac Newton Institute in Cambridge ending on 23 June 1993. At the end of the final lecture he announced he had a proof of Fermat's Last Theorem. When the results were written up for publication, however, a subtle error was discovered. Wiles said:... the first seven years I had worked on this problem I loved every minute of it however hard it had been. There had been setbacks, things which had seemed insurmountable but it was a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wiles.html (2 of 4) [2/16/2002 11:38:41 PM]

Wiles

kind of private and very personal battle I was engaged in and then after there was a problem with it, doing mathematics in that kind of rather overexposed way is certainly not my style, I certainly have no wish to repeat it... Wiles worked hard for about a year, helped in particular by R Taylor referred to above, and by 19 September 1994, having almost given up, he decided to have one last try:... suddenly, totally unexpectedly, I had this incredible revelation. It was the most important moment of my working life. Nothing I ever do again ... it was so indescribably beautiful, it was so simple and so elegant, and I just stared in disbelief for twenty minutes, then during the day I walked round the department. I'd keep coming back to my desk to see it was still there - it was still there. In 1994 Wiles was appointed Eugene Higgins Professor of Mathematics at Princeton. His paper which proves Fermat's Last Theorem is Modular elliptic curves and Fermat's Last Theorem which appeared in the Annals of Mathematics in 1995. From 1995 Wiles began to receive many honours for this outstanding piece of work. He was awarded the Schock Prize in Mathematics from the Royal Swedish Academy of Sciences and the Prix Fermat from the Université Paul Sabatier. In 1996 he received further awards included the Wolf Prize and was elected as a foreign member to the National Academy of Sciences of the United States, receiving its mathematics prize. Wiles said:... there's no other problem that will mean the same to me. I had this very rare privilege of being able to pursue in my adult life what had been my childhood dream. I know its a rare privilege but I know if one can do this it's more rewarding than anything one can imagine. In [1] his worked is summed up:Wiles's work is highly original, a technical tour de force, and a monument to individual perseverance. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles)

A Quotation

A Poster of Andrew Wiles

Mathematicians born in the same country

Cross-references to History Topics

Fermat's last theorem

Other references in MacTutor

Chronology: 1990 to 2000

Honours awarded to Andrew Wiles (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1996

Fellow of the Royal Society

1989

Royal Society Royal Medal winner

1996

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wiles.html (3 of 4) [2/16/2002 11:38:41 PM]

Wiles

AMS Cole Prize winner

1997

Other Web sites

1. AMS (An article about the proof of Fermat's last theorem 2. AMS (The citation for the Cole prize) 3. PBS 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wiles.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wiles.html (4 of 4) [2/16/2002 11:38:41 PM]

Wilkins

John Wilkins Born: 1614 in Fawsley (4 km S of Daventry), Northamptonshire, England Died: 16 Nov 1672 in Chester, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Wilkins was described in a 1703 book in the following way He was taught his Latin and Greek by Edward Sylvester, a noted Grecian, who kept a Private School in the Parish of All Saints in Oxford: His Proficiency was such, that at Thirteen Years of Age he entered a Student in New-Inn, in Easter-Term 1627. He made no long stay there, but was removed to Magdalen Hall, ... and there he took his Degree in Arts in October 1631. He afterwards entered into Orders. He tutored in Oxford, then was appointed Vicar at Fawsley. He held a number of chaplaincies, then in 1648 he was appointed Warden of Wadham College, Oxford. In 1656 Wilkins married Robina, a sister of Oliver Cromwell, obtaining special dispensation since the statutes of Wadham College prohibited the Warden from marrying. This did his career no harm at all! In 1659 he became Master of Trinity College Cambridge. Deprived of his post when Charles II was restored to the throne, he became Bishop of Chester in 1668. He was a founder of the Royal Society. In fact while at Wadham he gathered round him a group of worthy persons inquisitive into natural philosophy and other parts of human learning. This group became the Royal Society in 1660. In 1638 Wilkins published his first book which shows his interest in astronomy. He believed that the Moon is a habitable planet and predicts that one day space travel to the Moon will be possible.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilkins.html (1 of 3) [2/16/2002 11:38:43 PM]

Wilkins

Wilkins worked on codes and ciphers publishing his work in 1641. Wilkins also wrote on mechanical devices, publishing Mathematical Magick, or the wonders that may be performed by mechanical geometry in 1648. It is an account of the fundamental principles of machines. One device he described was the smokejack, first sketched by Leonardo da Vinci. This machine consisted of horizontal sails mounted on a vertical shaft and driven by the hot air rising up a chimney. A gearing system was used so that the smokejack could turn a roasting spit. Wilkins would work on machines with Hooke for many years and it is reported that persons together were not to be found else where in Europ, for parts and ingenuity. Wilkins' last work, which had to be rewritten since the original was destroyed in the Great Fire of London in 1666, is on a universal language. It is described in [1] in the following way The First Part describes the origin of languages and alphabets... . The Second Part is an exhaustive classification of notions in all spheres of thought, P M Roget clearly based his Thesaurus on this work of Wilkins. The Third Part relates to grammar, syntax, orthography, vowels and consonants. The Fourth Part provides the symbols of the proposed new writing and language, and gives examples and instructions for its use. The difficulty in using Wilkins' new language is nicely seen by one of his contemporaries who wrote A Doctor counted very able Designs that all mankind converse shall, Spite o' the confusion was at Babell, By a character called universall. How long that character will be learning, That truly passeth my discerning. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) A Poster of John Wilkins

Mathematicians born in the same country

Cross-references to History Topics

English attack on the Longitude Problem

Other references in MacTutor

Chronology: 1625 to 1650

Honours awarded to John Wilkins (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society Other Web sites

Elected 1663 1. R Anderson 2. The Galileo Project

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilkins.html (2 of 3) [2/16/2002 11:38:43 PM]

Wilkins

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wilkins.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilkins.html (3 of 3) [2/16/2002 11:38:43 PM]

Wilkinson

James Hardy Wilkinson Born: 27 Sept 1919 in Strood, Kent, England Died: 5 Oct 1986 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jim Wilkinson won a Foundation Scholarship to Sir Joseph Williamson's Mathematical School, Rochester at the age of 11. At the age of 16 he won a scholarship to Trinity College, Cambridge. At Cambridge he was taught by Hardy, Littlewood and Besicovitch. His final examination results were sent to him by Besicovitch who wrote:Easily at the top of the First Class. Heartiest Congratulations. In 1940 Wilkinson began war work which involved mathematical and numerical work on ballistics. He also worked on the thermodynamics of explosions but asked for a transfer. He became Turing's assistant at the National Physical Laboratory in London in 1946. At the N.P.L. he worked on the ACE computer project. His work at this time was described as follows:Turing provided the blueprint, but could not get on with others. Wilkie [the name by which Wilkinson was known at that time] was able to cooperate as well as anyone with Turing, and had the tact and wisdom necessary to get things done. He grasped all the technical details very quickly, shared the lectures on ACE with Turing and wrote the definitive description of the design. Wilkinson continued work becoming more involved in writing many high quality papers on numerical analysis, particularly numerical linear algebra. In numerical linear algebra he developed backward error analysis methods. He worked on numerical methods for solving systems of linear equations and eigenvalue problems. As well as the large numbers of papers on his theoretical work on numerical analysis, Wilkinson http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilkinson.html (1 of 2) [2/16/2002 11:38:45 PM]

Wilkinson

developed computer software, working on the production of libraries of numerical routines. The NAG (Numerical Algorithms Group) began work in 1970 and much of the linear algebra routines were due to Wilkinson. He received many awards for his outstanding work. He was elected a Fellow of the Royal Society in 1969. He received the A M Turing award from the Association of Computing Machinery and the J von Neumann award from the Society for Industrial and Applied Mathematics both in 1970. He is described in [3] as having extrovert characteristics but being a reserved and private man:Discussion with him was not always easy because the competitive instinct led him to introduce topics about which he had read and remembered a great deal. He was described by a colleague as:always optimistic and jovial, very fair and impartial in his evaluations, liked virtually everybody, had a nearly overpowering enthusiasm for many things outside mathematics. .. he was always competitive. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Honours awarded to Jim Wilkinson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1969

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wilkinson.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilkinson.html (2 of 2) [2/16/2002 11:38:45 PM]

Wilks

Samuel Stanley Wilks Born: 17 June 1906 in Little Elm, Texas, USA Died: 7 March 1964 in Princeton, New Jersey, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Samuel Wilks attended school in Denton, then studied architecture at North Texas State Teachers College. He received a B.A. in architecture in 1926. However his eyesight was not too good, and he feared that this would be a handicap if he pursured architecture as a profession so he decided on a career in mathematics. During session 1926-27 Wilks taught at a school in Austin, Texas and at the same time he began to study mathematics at the University of Texas. Here he was taught set theory and other courses in advanced mathematics by Robert Moore and he took courses in probability and statistics with E L Dodd. Wilks received an M.A. in mathematics in 1928 and during this time, in fact from 1927 until 1929, he was an instructor in mathematics. Wilks was awarded a fellowship to the University of Iowa where he studied for his doctorate. Here H L Rietz, who supervised his doctorate, introduced him to Gosset's theory of small samples and R A Fisher's statistical methods. After receiving his doctorate in 1931, on small sample theory of 'matched' groups in educational psychology, he continued research at Columbia University in session 1931-32. In 1932 Wilks went to England where he spent a period in Karl Pearson's department in University College, London. In 1933 he went to Cambridge where he worked with John Wishart, who had been a research assistant to both Pearson and Fisher. He was appointed instructor of mathematics at Princeton in 1933. He was to remain there for the rest of his career, being promoted to professor of mathematical statistics in 1944.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilks.html (1 of 2) [2/16/2002 11:38:47 PM]

Wilks

Wilks's work was all on mathematical statistics. His early papers on multivariate analysis were his most important, one of most influential being Certain generalizations in the analysis of variance. He constructed multivariate generalisations of the correlation ratio and the coefficient of multiple correlation and studied random samples from a normal multivariate population. Three papers in 1931-33 concerned deriving the sample distributions of estimates of the parameters of a bivariate normal distribution when some of the individuals gave observations on both variables, some others on only one. In 1935 he investigated multinomial distributions. He advanced the work of Neyman on the theory of confidence-interval estimation. In 1941 Wilks developed his theory of 'tolerance limits'. Wilks was a founder member of the Institute of Mathematical Statistics (1935). He was editor of the Annals of Mathematical Statistics from 1938 until 1949. Wilks served the U.S. government in many roles. Among many other similar tasks, he worked for the U.S. Department of Agriculture and was a member of the National Defense Committee. In 1947 he was awarded the Presidential Certificate of Merit for his contributions to antisubmarine warfare and the solution of convoy problems. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wilks.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilks.html (2 of 2) [2/16/2002 11:38:47 PM]

Wilson_Alexander

Alexander Wilson Born: 1714 in St Andrews, Scotland Died: 18 Oct 1786 in Edinburgh, Scotland Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alexander Wilson was educated at the University of St Andrews. After this he was an apprentice to a surgeon in St Andrews, then he moved to London. In 1739 he returned to St Andrews and, in 1742, he set up a type foundry. Two years later he moved to Glasgow where, in 1760, he was appointed to the chair of astronomy, a post he held until 1784. Wilson made many observations of sunspots using a geometric argument to show that they were depressions in the Sun. A similar theory had been proposed by La Hire and by Cassini. Wilson also published Thoughts on General Gravitation (1770), in which he attempted to answer Newton's question What hinders the fixed stars from falling upon one another. Wilson's answer, that the entire universe rotates about a centre, is of course incorrect. He was awarded an honorary degree by the University of St Andrews in 1763 and was a founding member of the Royal Society of Edinburgh. Article by: J J O'Connor and E F Robertson List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Alexander Wilson (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society of Edinburgh Lunar features

Crater Wilson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilson_Alexander.html (1 of 2) [2/16/2002 11:38:48 PM]

Wilson_Alexander

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wilson_Alexander.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilson_Alexander.html (2 of 2) [2/16/2002 11:38:48 PM]

Wilson_Edwin

Edwin Bidwell Wilson Born: 25 April 1879 in Hartford, Connecticut, USA Died: 28 Dec 1964 in Brookline, Massachusetts, USA Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edwin Wilson's father, Edwin Horace Wilson, was a teacher in a secondary school. His mother was Jane Amelia Bidwell. Wilson attended Harvard University, graduating with a A.B. in 1899. He then decided to work for his doctorate at Yale and there he was a student of Gibbs. Wilson graduated from Yale with a Ph.D. in 1901 and, in the same year, a textbook which he had written on vector analysis was published. Vector analysis (1901) was based on Gibbs' lectures and [1]:This beautiful work, published when Wilson was only twenty-two years old, had a profound and lasting influence on the notation for and the use of vector analysis. Wilson was appointed an Instructor at Yale in 1900 and, after being awarded his doctorate, Wilson went to Paris where he studied at the Ecole Polytechnique, the Sorbonne and the College de France during 1902-3. On returning to the United States he continued to teach mathematics as an Instructor at Yale. He became interested in the foundations of geometry, particularly in projective and differential geometry and he published a paper The so-called foundations of geometry in 1903 which criticised Hilbert's approach to geometry. In 1906 Wilson was appointed as an assistant professor at Yale, then in 1907 he was appointed associate professor at the Massachusetts Institute of Technology. In 1911 he was promoted to full professor. Wilson had been inspired by Gibbs to work on mathematical physics and he began to write papers on mechanics and the theory of relativity. In 1912 Wilson published the first American advanced calculus text [1]:... a comprehensive text on advanced calculus that was the first really modern book of its kind in the United States. Holding a post of professor of physics at the Massachusetts Institute of Technology, he was appointed as Head of the Department of Physics there in 1917. World War I had seen another move in Wilson's research interests for he had undertaken war work which involved aerodynamics and this led him to study the effects of gusts of wind on a plane. In 1920 he published his third major text Aeronautics and gathered round him a group of students working on this topic. Wilson had already worked in a number of quite distinct areas and his work on aeronautics did not become the major topic for the rest of his career. Not long after the publication of his important text on

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilson_Edwin.html (1 of 3) [2/16/2002 11:38:50 PM]

Wilson_Edwin

Aeronautics his interests moved again, this time towards probability and statistics. He did not study statistics for its own, however, but he was interested in applying statistics both to astronomy and to biology. He was the first to study confidence intervals, later rediscovered by Neyman. In 1922 Wilson left the Massachusetts Institute of Technology to become Professor of Vital Statistics at the Harvard School of Public Health. He continued to hold this post until he retired in 1945, when he became professor emeritus. After he retired, Wilson spent a year in Glasgow, Scotland when he was Stevenson lecture on Citizenship. From 1948 he was a consultant to the Office of Naval Research in Boston. Gridgeman and MacLane in [1] sum up Edwin Wilson's contributions as follows:Wilson exhibited a constructively critical mind, quick to expose flaws and errors. Each of his books was an effective and timely exposition of a major subject, and his best papers made lasting impressions. He contributed to many disciplines other than his specialities, including epidemiology, socially, and economics. His greatest originality may have been reached in his papers on statistics - which, interestingly, was a subject he did not explore deeply until middle age. Wilson received many honours. He was a member of the National Academy of Sciences and he served as vice-president during 1949-53. He was elected a Fellow of the Royal Statistical Society of London, and he was a member of the American Statistical Association, serving as president in 1929. He was also a member of the American Academy of Arts and Sciences, again serving as president during 1927-31. Finally we record his membership of the American Philosophical Society. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country Honours awarded to Edwin Wilson (Click a link below for the full list of mathematicians honoured in this way) AMS Gibbs Lecturer

1930

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilson_Edwin.html (2 of 3) [2/16/2002 11:38:50 PM]

Wilson_Edwin

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wilson_Edwin.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilson_Edwin.html (3 of 3) [2/16/2002 11:38:50 PM]

Wilson_John

John Wilson Born: 6 Aug 1741 in Applethwaite, Westmoreland, England Died: 18 Oct 1793 in Kendal, Westmoreland, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Wilson attended school in Kendal. From there he entered Peterhouse, Cambridge and he was the Senior Wrangler in 1761. This means that he was the best of all the First Class students to graduate after taking the Mathematical Tripos. Before he took his final examinations Wilson had already gained a strong reputation and he had also attracted considerable attention by defending Waring, who was the Lucasian Professor of Mathematics, from strong attacks which had been made on him as a result of his text Miscellanea analytica. In 1764 Wilson was elected a Fellow of Peterhouse and he taught mathematics at Cambridge with great skill, quickly gaining an outstanding reputation for himself. However, he was not to continue in the world of university teaching, for in 1766 he began a legal career. It was a highly successful career, too. He became a member of the Court of Common Pleas in 1876 which had been set up to make judgements in cases of civil disputes between individuals. At the time that Wilson served on this body it was one of three courts which dealt with common-law business. He is best known for Wilson's theorem which states that ... if p is prime then 1 + (p - 1)! is divisible by p This result was first published by Waring, without proof, and attributed to Wilson. Leibniz appears to have known the result. It was first proved by Lagrange in 1773 who showed that the converse is true, namely ... if n divides 1 + (n - 1)! then n is prime. Almost certainly Wilson's theorem was a guess made by him, based on the evidence of made cases, which neither he nor Waring knew how to prove. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilson_John.html (1 of 2) [2/16/2002 11:38:51 PM]

Wilson_John

Mathematicians born in the same country Cross-references to History Topics

Arabic mathematics : forgotten brilliance?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wilson_John.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wilson_John.html (2 of 2) [2/16/2002 11:38:51 PM]

Winkler

Wilhelm Winkler Born: 29 June 1884 in Prague, Bohemia (Austro-Hungarian Empire, now Czech Republic) Died: 3 Sept 1984 in Vienna, Austria

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Winkler's father, Julius Winkler, was a music teacher in Prague and his mother was Anne Winkler. He came from a large German speaking family of eight children, Wilhelm being the fourth child. The size of the family naturally stretched Julius and Anne's finances so Wilhelm had to work to augment the income from the time he was 13 years old. He attended the Kleinseitner Gymnasium and then entered the German language Karl Friedrich University in Prague to study law. It was a time of financial hardship and difficulty for Winkler. After graduating, Winkler practiced law for a short time before volunteering to serve for a year in the Austrian army. After a year in the army he returned to civilian life obtaining a post in the Bohemian State Statistical Bureau in 1909. He now began to study statistics and mathematics seriously, attending statistics seminars at the university and advanced mathematics courses at the Technische Hochschule of Prague. This was a period in which [3]:... his interest in application of methods of mathematical statistics in social and economic matters continued to grow. However Winkler was not happy with the status of statistics in central Europe. He wrote in his memoirs (see for example [5]):I soon found out that the German statistical literature did not offer too many ideas. New life came into statistics from England and Russia. With the outbreak of World War I in 1914, Winkler enlisted in the Austrian army. He was awarded two medals for bravery before being wounded in November 1915. He was taken to a hospital in Prague where http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Winkler.html (1 of 3) [2/16/2002 11:38:53 PM]

Winkler

he took six months to regain his health. During this time he met one of his former teachers who was in the Ministry of War in Vienna. He invited Winkler to join the scientific committee which the Ministry had set up on war economy and, in June 1916 Winkler began work with the committee in Vienna. When the war ended in 1918 Winkler became Secretary of State for Military Affairs, but his progressive ideas meant that this was an unpopular appointment as far as certain traditionally minded colleagues were concerned. In this role Winkler attended the Versailles Peace Conference in 1919 as a member of the Austrian delegation. From 1921 Winkler began what is effectively two careers. He had been appointed to the Austrian Central Statistics Office in 1920 and from 1921 he also taught at the University of Vienna at a Privatdozent. Winkler also was by now a married man, having married Clara Deutch in 1918. Both Winkler's careers progressed in parallel. He was promoted to Head of the Division of Population Statistics in 1925 and at the university he became an extraordinary professor in 1929. As the authors of [3] write:Winkler's career is a remarkable conjunction of activity in two main spheres - the practical needs of government and the intellectual austerity of scientific research. Winkler's work in statistics achieved international recognition for him at this time. However things were to change rather dramatically. The Munich Agreement of 1938 saw large parts of the Czechoslovak republic surrendered to Germany. German troops along with Hitler himself entered Austria on 12 March 1938, and a Nazi government had been set up there. Political pressure was put on Winkler' who was forced to resign from both his government post and his university professorship. There followed an extremely difficult period for Winkler through the years of World War II. Despite the hardship he and his family suffered, Winkler worked on a book Basic Course in Demography which was eventually published in 1956. It was only after the war ended in 1945 that Winkler was reinstated to his university post when he was appointed to a chair in the University of Vienna. He retired from his chair in 1949 but continued to serve in various capacities such as Dean of the Faculty of Law until 1955. Schmetterer writes in [5]:Winkler published 20 books and about 200 papers covering a wide field of theoretical and applied statistics. He was widely recognised as a statistician of high reputation and received many Austrian and foreign distinctions and honours. Among these honours was election to the International Statistical Institute in 1927, being made an honorary member in 1965, and also serving as President of the Institute. He was also elected to the Austrian Academy of Sciences (1952) and the Royal Statistical Society (1961). The universities of Munich and Vienna awarded him honorary degrees. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Winkler.html (2 of 3) [2/16/2002 11:38:53 PM]

Winkler

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Winkler.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Winkler.html (3 of 3) [2/16/2002 11:38:53 PM]

Wintner

Aurel Freidrich Wintner Born: 8 April 1903 in Budapest, Hungary Died: 15 Jan 1958 in Baltimore, Maryland, USA Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Aurel Wintner's father, Eduard Wintner, was a businessman who had lived in Rotterdam and has emigrated to Budapest while his mother, Charlotte Eugenie Hirshfeld, was from Vienna. Aurel attended school in Budapest, completing his school education there in 1920. Wintner's mathematical ability was recognised by one of his school teachers and, realising that he was capable of progressing much faster than the other pupils, arrangements were made to enable Wintner to use the mathematics library at the University of Budapest. This was a major influence in allowing him to discover his love of the subject, for at school it was not clear that he would continue to study science. The reason was that Wintner was extremely musical and showed great talent for the violin. As he came towards the end of his school career in 1920 he realised that he could not pursue both music and science. Many might have decided to aim for a scientific career and still keep the violin as a major hobby - in fact many of the mathematicians in this archive played the violin to a very high standard. Not so Wintner, for once he had made the decision for a scientific career he never played the violin again. Although being allowed to use the mathematics library at the University of Budapest was a major influence in directing his interests in that direction, there was another influence which pushed him towards astronomy. This came from his uncle, S Oppenheim, who was professor of mathematics at the University of Vienna. As a schoolboy Wintner had spent holidays in Vienna and had used his uncle's astronomy library. After leaving school Wintner entered the University of Budapest in 1920. However, although he came from a fairly well-off family, this was a period of hyper-inflation and it became increasingly difficult for him to continue with his education. In 1924 he withdrew from university by this time he had reached a level where he was undertaking research. During the years 1924 to 1927 he published about 20 papers on astronomy and mathematics. The quality of the work that he was producing made a marked impression on the world of mathematics and Leon Lichtenstein, who taught at the University of Leipzig began to correspond with him on mathematical topics. Lichtenstein began to encourage Wintner to consider studying for his doctorate. Wintner entered the University of Leipzig to study for his doctorate in 1927. During the following two years he served as assistant to Lichtenstein as editor of Mathematische Zeitschrift and also Jahrbuch über die Fortschritte der Mathematik. Wintner always felt a great debt of gratitude towards Lichtenstein for supporting his studies and throughout his life Wintner kept a picture of Lichtenstein on the wall in his http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wintner.html (1 of 4) [2/16/2002 11:38:55 PM]

Wintner

office. George W Hill had published an account of his lunar theory in 1878. Hill's methods used infinite matrices and series expansions which he assumed were convergent but gave no proof. Wintner wrote a number of papers putting Hill's theory on a rigorous mathematical foundation. He published the first proofs of the basic facts about Hilbert spaces in 1929, the year in which he was awarded his doctorate by the University of Leipzig. In fact at this time the development of Hilbert spaces had become particularly important for the study of quantum theory since this mathematics underlay the theory. This should have made Wintner's work of particularly topical interest but unfortunately for him mathematicians were developing their theories in terms of operators, following the approach of von Neumann. This should not have meant that Wintner's approach via infinite matrices was not of great value but it had the effect that his contributions were not appreciated as they should have been. This in turn led to Wintner feeling bitter at his lack of recognition for his contribution. After completing his doctorate in 1929 Wintner won an International Board Fellowship to enable him to study abroad. He made two important visits during 1929-30, one to Rome where he worked with Levi-Civita, and the other to the observatory in Copenhagen where he worked with E Strömgren. While studying at Leipzig Wintner had been taught by Hölder. This became more than just a relationship between teacher and pupil for in 1930 Wintner married Irmgard Hölder, Otto Hölder's daughter. In the same year he accepted a post at Johns Hopkins University in the United States where he continued to be employed until his death. He spent the year 1937-38 at the Institute for Advanced Study at Princeton, also visiting Harvard during that period to work with G D Birkhoff. Wintner was awarded a Guggenheim Fellowship in 1941 which enabled him to visit Cambridge, Massachusetts. He had planned to write a book with Norbert Wiener during this time but Wiener was involved with war work which meant their plans could not be carried through as they had intended. In 1946 Wintner was appointed to a full professorship at Johns Hopkins University, a position which Wintner should have been offered many years before. In 1944 he became editor of the American Journal of Mathematics having already been associate editor from 1936. Clearly his editorial duties as Lichtenstein's assistant in 1927-29 stood him in good stead, and editorial duties were clearly something which gave Wintner great satisfaction. He played an important role in developing the editorial policy of the American Journal of Mathematics and, working with André Weil, he made major changes to the editorial policy from 1953 until his death, which resulted in significantly increasing the standard of the journal. Wintner published on analysis, number theory, differential equations and probability (with several joint papers with Norbert Wiener). These interests appear to be unconnected but this is in fact far from true. Although it is true that Wintner studied certain areas of mathematics for their own sake, he was led to these areas through his work in celestial mechanics. Along with Poincaré and George Birkhoff, he placed celestial mechanics on a more sound mathematical basis. These innovators were more concerned with the underlying theory, less concerned with quantitatively accurate prediction of celestial body motion. A study of certain astronomical equations led Wintner to consider almost periodic functions. Hartman, Wintner's student and then colleague, writes in [3] that an interest in perturbations of planets and other related work:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wintner.html (2 of 4) [2/16/2002 11:38:55 PM]

Wintner

... led Wintner to various interests: first, his interest in almost periodic functions as such; second, analytic number theory and summability; third, the asymptotic distributions of almost periodic functions; and finally, the theory of distribution functions as such. In this connection, Wintner used to like to point out the debt of analytic number theory to dynamics, noting that in a certain sense the oldest Tauberian theorems date back to the dynamical work of Sundman and Hadamard. Wintner published 437 papers during his career and 9 monographs. Of these monographs six were published after he emigrated to the United States. These were Lectures on asymptotic distributions and infinite convolutions (1938), Analytical foundations of celestial mechanics (1941), Eratosthenian averages (1943), Theory of measure in arithmetical semigroups (1944), The Fourier transforms of probability distributions (1947), and An arithmetical approach to ordinary Fourier series (1945). Of these monographs, those of 1943, 1944 and 1945 were published privately. Wintner [3]:... took rather a dim view of American publishing houses which would not publish specialised monographs with limited sales. He felt that he proved his point by not losing money in this capacity as entrepreneur. As to Wintner's character Sternberg writes in [1]:Wintner, a man of high moral principles, opposed direct government support of scholarly research, for fear of interference. He not only accepted considerable financial hardship by personally refusing such support but also was willing to forgo fruitful scientific collaboration in order to maintain his ideals. Hartman, in [3], writes of:... the intensity and the great energy which [Wintner] brought to every task, mathematical and non-mathematical. ... in the last few years of his life he seemed to prefer working closely with one or two students to lecturing, but earlier in his career he gave inspiring lecture courses in which he transmitted his love and enthusiasm for mathematics. As to his personal relations with students, he [3]:... sought out the promising young students and offered them friendship, help and suggestions. On the other hand, he could be very abrupt in dropping these students if he decided that they did not work as hard as he thought they should. His interests outside mathematics included hiking in the mountains but, in a similar way to never playing the violin past the age of 17, he strongly believed that one should not have non-essential interests. As a result he had many friends among mathematicians, but few others. He died suddenly from a heart attack while still at the height of his mathematical productivity. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wintner.html (3 of 4) [2/16/2002 11:38:55 PM]

Wintner

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Wintner.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wintner.html (4 of 4) [2/16/2002 11:38:55 PM]

Wirtinger

Wilhelm Wirtinger Born: 15 July 1865 in Ybbs a.d. Donau, Austria Died: 15 Jan 1945 in Ybbs a.d. Donau, Austria

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Wilhelm Wirtinger was born in Ybbs, a town on the Danube about half way between Vienna and Linz. Wirtinger studied at the University of Vienna, and it was from that University that he received his doctorate in 1887 and his habilitation in 1890. However, the largest influence on the direction of his research came from Klein before he was awarded his habilitation, when he studied at the University of Berlin and at the University of Göttingen. In 1895 Wirtinger was promoted to a chair at Vienna then, later the same year, he accepted a chair at the University of Innsbruck. Wirtinger returned to a chair at the University of Vienna in 1905. In 1896 Wirtinger published a work of major importance on the general theta function. In this work Wirtinger combined ideas from Riemann's function theory with ideas from Klein to prove results of great significance. This paper, which he had developed from work begun at Göttingen, brought Wirtinger's name to the fore as a leading mathematician. Caratheodory writes in [1]:However, Wirtinger was not a specialist who only worked on one problem and did not have a sense for the essentials of science. In his lectures he always stressed the historical context and had a remarkable interest in the philosophical basis of mathematics. He was economical with his publications, but every single paper - even if only a few pages long does not only contain surprising thought of exceptional beauty but also proof that he could combine his perfect geometrical insight with his rare skill of mastering the mathematical

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wirtinger.html (1 of 3) [2/16/2002 11:38:57 PM]

Wirtinger

symbolism. Wirtinger's range of mathematics was quite exceptional. Not only did he write beautiful papers on function theory, he also wrote on geometry, algebra, number theory, plane geometry and the theory of invariants. He also wrote several important papers on Lie's translation manifolds and their application to abelian integrals. This list would make one believe that Wirtinger's range within pure mathematics was very wide but his interests went well beyond pure mathematics. He published results on Einstein's theory of relativity and other areas of mathematical physics. He also worked in statistics and wrote a work on rainbows. Others taking their summer vacations on the Achensee would have relaxed and taken the chance to have a break from academic work. Not so Wirtinger, who sat and watched the waves on the lake and thought about the mathematical theory which lie behind them. After his holiday he wrote up his results to provide an improved theory of capillary waves. When Reidemeister was appointed as associate professor of geometry at the University of Vienna in 1923 he became a colleague of Wirtinger. At that time Wirtinger was interested in knot theory and he showed Reidemeister how to compute the fundamental group of a knot from its projection. Wirtinger's method was first published in work of Artin in 1925. Wirtinger certainly did not lessen his mathematical activity as he grew older. At the age of 71 he wrote the first of a series ground-breaking papers on higher dimensional spaces. Among the mathematicians who Wirtinger taught while he held the chair at Vienna are Schreier, Gödel, Radon and Taussky-Todd. Wirtinger received many honours for his achievements. In 1907 the Royal Society of London awarded him their Sylvester Medal. He was the third recipient of the medal which had been previously awarded to Poincaré and Cantor, so indeed this ranked him among the very greatest mathematicians of his day. Among other honours was his election to the Munich Academy in 1931. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) Mathematicians born in the same country Honours awarded to Wilhelm Wirtinger (Click a link below for the full list of mathematicians honoured in this way) Royal Society Sylvester Medal

Awarded 1907

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wirtinger.html (2 of 3) [2/16/2002 11:38:57 PM]

Wirtinger

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wirtinger.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wirtinger.html (3 of 3) [2/16/2002 11:38:57 PM]

Wishart

John Wishart Born: 28 Nov 1898 in Montrose, Scotland Died: 14 July 1956 in Acapulco, Mexico

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

John Wishart's family moved from Montrose to Perth in Scotland when John was two years old. He attended Perth Academy and then, in 1916, entered the University of Edinburgh. There he was taught mathematics by E T Whittaker. World War I meant that Wishart's university career was disrupted. He spent two years from 1917 to 1919 in the Black Watch regiment and served in France in 1918. He completed his university course in 1922, graduating with a First Class degree in mathematics and physics. He had taken a teacher training course at Moray House as part of his degree and, after graduating, he moved to Leeds accepting a post as mathematics teacher at West Leeds High School. In 1924, after a recommendation from Whittaker, Wishart was offered a post in University College London as assistant to Pearson. Pearson had a project for Wishart to work on and, given that Whittaker had set up his mathematical laboratory in Edinburgh, it was clear why Whittaker's advice on a possible assistant had been sought. Pearson had published his Tables of the Incomplete Gamma Function in 1922 and now he was looking for computational help in his next 'tables' project Tables of the Incomplete Beta Function. Wishart learned a great deal of statistics during his three years with Pearson. He attended Pearson's lectures and learnt how to go about statistical research. After a few months as a Mathematical Demonstrator at Imperial College, Wishart accepted an offer from R A Fisher to be his statistical assistant at Rothamsted.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wishart.html (1 of 3) [2/16/2002 11:38:59 PM]

Wishart

When Yule left his Cambridge lectureship in Statistics in 1931 there was a reorganisation of statistics teaching at Cambridge. A Readership in Statistics was created in the Faculty of Agriculture to teach courses in that Faculty and courses in Mathematics. A separate lectureship in Economic Statistics was also created. Wishart was appointed to the Readership in the Faculty of Agriculture. A laboratory was set up by Wishart at Cambridge for his postgraduate students. Cochran was one of these postgraduates and he described his time studying with Wishart:In those days he believed in his students keeping office hours. When he assigned me a desk in the Laboratory, he told me that he expected me to be sitting at the desk most of the day when not in class. He instructed me to do three hours computing a day on a table of the one-percent level of z to seven decimal places ... Having anticipated a free and easy life as a graduate student, punctuated of course by periods of esoteric thinking when the spirit moved me, I did not much like either the office hours or the computing, but I don't think they did me any harm. There were two aspects to Wishart's teaching at Cambridge since he taught both mathematics students and agriculture students. This suited him well since he had both a flair for mathematical statistics and a flair for very practical applications of experimental design. The arrangement did not suit other academics at Cambridge, however, and Wishart had to fight many academic battles. The problem that Wishart's position caused at Cambridge was that he was too high powered a statistician for those in Agriculture but the mathematicians were also unhappy to send their students to the Faculty of Agriculture for statistics courses, and they would have much preferred to have statistics completely within Mathematics. If World War II had not come along it is unclear how the problems would have resolved themselves. As it was, Wishart worked in army Intelligence from 1940-42 and then on statistical work for the Admiralty from 1942 to 1946. The problems he had been having at Cambridge before the War made him think long and hard about whether to return, but his love of teaching, more than anything else, took him back. At Cambridge more statisticians were taken on within Mathematics and a Statistical Laboratory was set up within the Mathematics Faculty in 1949. Wishart became Head of the Statistical Laboratory in 1953. In [2] Wishart's international connections are summed up:There are probably few statisticians who have had more friends scattered across the world that had John Wishart. Many of these friendships were made in the course of his work as a teacher of statistical method to practical agriculturists overseas. ... A pioneering visit to Nanking University in 1934 had been followed after the war by visits to Spain in 1947, to the United States in 1949, to India in 1954 and then, in his last year, to Mexico, where he was taking a leading part in the work of the Training Centre in Experimental Design arranged by the United Nations Food and Agricultural Organisation. Some of Wishart's most important publications were in the 1928-32 period before he became so involved with teaching at Cambridge. In 1928 he derived the generalised product-moment distribution which is now named the Wishart distribution. This distribution is described in [1] as:... fundamental to multivariate statistical analysis ... and is fully described on pages 154 to 163 of [2]. As well as further papers on the Wishart distribution,

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wishart.html (2 of 3) [2/16/2002 11:38:59 PM]

Wishart

he also studied properties of the distribution of the multiple correlation coefficient which Fisher had considered earlier. In addition he wrote many papers on agricultural applications of statistics such as fertiliser trials, sugar beet experiments, crop experimentation and pig nutrition. Wishart was also much involved with the work of the Royal Statistical Society. He was one of the Fellows who formed the organising committee of the Agriculture research section in 1933. In 1945 he became chairman of the Royal Statistical Society's Research Section. He also sat on two committees of the Royal Statistical Society on the Teaching of Statistics: the reports of these committees appearing in the Journal of the Royal Statistical Society in 1948 and 1955. One other service that Wishart performed for statistics was his editorial work for Biometrika. He served as Assistant editor from 1937 and Associate Editor from 1948. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wishart.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wishart.html (3 of 3) [2/16/2002 11:38:59 PM]

Witt

Ernst Witt Born: 26 June 1911 in Alsen (German Baltic island) (now Als, Denmark) Died: 3 July 1991

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernst Witt was born on the island of Alsen. Alsen together with the rest of North Schleswig became part of Germany in 1864. The island and was returned to Denmark by plebiscite in 1920, nine years after Witt's birth there, and is now known as Als. The island is separated from the Sundeved peninsula of southern Jutland by a narrow piece of water called Als Sound. Shortly after his birth Witt was taken to China where he was brought up. He returned to Europe when he was nine years old. Witt studied at the universities of Freiburg and Göttingen. His doctorate was obtained from Göttingen where his doctoral studies were supervised by Emmy Noether. The topic proposed by Emmy Noether was on a topic assiciated to the Riemann-Roch theorem and this was indeed the topic on which his dissertation was written. At Göttingen Witt joined Helmut Hasse's seminar on congruence function fields and p-adic numbers. Oswald Teichmüller and Ludwig Schmid were also members of the seminar, and Schmid collaborated with Witt on ideas which would lead to the Witt vector calculus. Emil Artin was not a Jew but his wife was a Jew so when the "New Official's Law" was passed by the Nazis in 1937 affecting those who were related to Jews by marriage he was forced from his post at the University of Hamburg. Artin left Germany for the United States. Witt was appointed to Artin's post at Hamburg and he remained there until he retired in 1979. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Witt.html (1 of 2) [2/16/2002 11:39:01 PM]

Witt

Witt's work was mainly concerned with quadratic forms and various related fields such as algebraic function fields, Witt vectors, Lie rings and Mathieu groups. He is best known for his introduction of Witt vectors which appeared in his paper in 1936 in J. Reine Angew. Math. The original construction of Witt vectors is given in the articles [2] and [3] ([2] is a German translation of [3]). The paper [4] is written by I Kersten who was one of Witt's pupils at Hamburg. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Witt.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Witt.html (2 of 2) [2/16/2002 11:39:01 PM]

Witten

Edward Witten Born: 26 Aug 1951 in Baltimore, Maryland, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Edward Witten studied at Brandis University and received his B.A. in 1971. From there he went to Princeton receiving his M.A. in 1974 and his Ph.D. in 1976. After completing his doctorate, Witten went to Harvard where he was postdoctoral fellow during session 1976-77 and then a Junior Fellow from 1977 to 1980. In September 1980 Witten was appointed professor of Physics at Princeton. He was awarded a MacArthur Fellowship in 1982 and remained as professor of Physics at Princeton until 1987 when he was appointed as a Professor in the School of Natural Sciences at the Institute for Advanced Study. Basically Witten is a mathematical physicist and he has a wealth of important publications which are properly in physics. However, as Atiyah writes in [3]:Although he is definitely a physicist (as his list of publications clearly shows) his command of mathematics is rivalled by few mathematicians, and his ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by his brilliant application of physical insight leading to new and deep mathematical theorems. Speaking at the American Mathematical Society Centennial Symposium in 1988, Witten explained the relation between geometry and theoretical physics:It used to be that when one thought of geometry in physics, one thought chiefly of classical physics - and in particular general relativity - rather than quantum physics. ... Of course, quantum physics had from the beginning a marked influence in many areas of mathematics functional analysis and representation theory, to mention just two. ... Several important

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Witten.html (1 of 3) [2/16/2002 11:39:03 PM]

Witten

influences have brought about a change in this situation. One of the principal influences was the recognition - clearly established by the middle 1970s - of the central role of nonabelian gauge theory in elementary particle physics. The other main influence came from the emerging study of supersymmetry and string theory. In his study of these areas of theoretical physics, Witten has achieved a level of mathematics which has led him to be awarded the highest honour that a mathematician can receive, namely a Fields Medal. He received the medal at the International Congress of Mathematicians which was held in Kyoto, Japan in 1990. The Proceedings of the Congress contains two articles describing Witten's mathematical work which led to the award. The main tribute is the article [3] by Atiyah, but Atiyah could not be in Kyoto to deliver the address so the address at the Congress was delivered by Faddeev [5] who quotes freely from Atiyah [3]. The first major contribution which led to Witten's Fields Medal was his simpler proof of the positive mass conjecture which had led to a Fields Medal for Yau in 1982. Gawedzki and Soulé describe this work by Witten, which appeared in 1981, in [9]:The proof ... employed in a subtle way the idea of supersymmetry. This became the centrepiece of many of Witten's subsequent works... One of Witten's subsequent works was a paper which Atiyah singles out for special mention in [3], namely Supersymmetry and Morse theory which appeared in the Journal of differential geometry in 1984. Atiyah writes that this paper is:... obligatory reading for geometers interested in understanding modern quantum field theory. It also contains a brilliant proof of the classic Morse inequalities, relating critical points to homology. ... Witten explains that "supersymmetric quantum mechanics" is just Hodge-de Rham theory. The real aim of the paper is however to prepare the ground for supersymmetric quantum field theory as the Hodge-de Rham theory of infinite dimensional manifolds. It is a measure of Witten's mastery of the field that he has been able to make intelligent and skilful use of this difficult point of view in much of his subsequent work. Since this highly influential paper, the ideas in it have become of central importance in the study of differential geometry. Further new ideas of fundamental importance were introduced by Witten and described in [9]:Witten subsequently gave a string interpretation of the elliptic genus and provided arguments for its rigidity ... Another piece of new mathematics stemmed from Witten's papers on global gravitational anomalies. ... In recent years, Witten focused his attention on topological quantum field theories. These correspond to Lagrangians ... formally giving manifold invariants. Witten described these in terms of the invariants of Donaldson and Floer (extending the earlier ideas of Atiyah) and generalised the Jones knot polynomial ... The authors of [9] sum up Witten's contributions to mathematics:Although mostly not in the form of completed proofs, Witten's ideas have triggered major mathematical developments by the force of their vision and their conceptual clarity, his main discoveries soon becoming theorems. His Fields Medal at the 1990 International Congress of Mathematicians acknowledged the growing impact of his work on contemporary mathematics. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Witten.html (2 of 3) [2/16/2002 11:39:03 PM]

Witten

Atiyah, in [3], expresses the same ideas in the following way:... he has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics. Of course physical insight does not always lead to immediately rigorous mathematical proofs but it frequently leads one in the right direction, and technically correct proofs can then hopefully be found. This is the case with Witten's work. So far the insight has never let him down and rigorous proofs, of the standard we mathematicians rightly expect, have always been forthcoming. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1980 to 1990

Honours awarded to Edward Witten (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1990

AMS Colloquium Lecturer

1987

AMS Gibbs Lecturer

1998

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Witten.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Witten.html (3 of 3) [2/16/2002 11:39:03 PM]

Wittgenstein

Ludwig Josef Johann Wittgenstein Born: 26 April 1889 in Vienna, Austria Died: 29 April 1951 in Cambridge, Cambridgeshire, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ludwig Wittgenstein's parents were both very musical and Ludwig was brought up in a home which was always filled with music. Ludwig's parents had eight children who were all highly gifted both artistically and intellectually. The family were wealthy industrialists having made a fortune in the steel industry and they were able to provide the best possible education for their children. Ludwig was the youngest of the children and was educated at home until he was fourteen years of age. In 1903 Wittgenstein began three years of schooling in Austria at a school which specialised in mathematics and natural science. This emphasis led Wittgenstein to decide to study engineering and, in 1906, he went to Berlin where began studies in mechanical engineering. Intending to study for his doctorate in engineering, Wittgenstein went to England in 1908 and registered as a research student in an engineering laboratory of the University of Manchester. His first project involved the study of the behaviour of kites in the upper atmosphere of the earth. He moved from this to further study of aeronautical research, this time examining the design of a propeller with a small jet engine on the end of each blade. At this stage Wittgenstein was much more practically minded than one might suppose, given his later highly theoretical work, and he not only studied the theoretical design of the propeller but he actually built and tested it. The tests of the propeller were successful but, needing to understand more mathematics for his research, he began a study which soon involved him in the foundations of mathematics. Russell had published his Principles of Mathematics in 1903 and Wittgenstein turned to this work as he sought a better understanding of foundations of his subject. He became so interested in Russell's work that he decided

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wittgenstein.html (1 of 5) [2/16/2002 11:39:05 PM]

Wittgenstein

that he wanted to learn more. Wittgenstein left Manchester in 1911 to study mathematical logic with Russell in Cambridge. Russell was not one to be easily impressed by a student, but he was certainly very impressed by Wittgenstein. Russell wrote that teaching Wittgenstein was:... one of the most exciting intellectual adventures [of my life]. ... [Wittgenstein had] fire and penetration and intellectual purity to a quite extraordinary degree. ... [He] soon knew all that I had to teach. By 1912 Russell had become convinced that Wittgenstein possessed a genius which should be directed towards mathematical philosophy. He therefore persuaded Wittgenstein to give up any ideas that he still had to resume his applied mathematical work on aeronautics. The first paper that Wittgenstein presented was to the Cambridge Philosophical Society in 1912. Entitled What is philosophy it [11]:... shows that from the very beginning Wittgenstein recognised the importance of understanding the nature of philosophical problems and of reflecting on the appropriate methods for approaching them. During this period at Cambridge Wittgenstein continued to work on the foundations of mathematics and also on mathematical logic. However Wittgenstein found Cambridge less than an ideal place to work since he felt that the academics there were merely trying to be clever in their discussions while their ideas lacked depth. He went to Skjolden in Norway and this was an extremely fruitful period during which lived in isolation working on his ideas on logic and language that would form the basis of his great work the Tractatus Logico-Philosophicus. When World War I broke out in 1914 Wittgenstein immediately travelled from Skjolden to Vienna to join the Austrian army. He served first in a ship then in an artillery workshop. In 1916 he was sent as a member of a howitzer regiment to the Russian front where he gained many distinctions for bravery. In 1918 he was sent to north Italy in an artillery regiment and he was there at the end of the war, becoming a prisoner of the Italians. During these four years of active service Wittgenstein had written his great work in logic, the Tractatus, and the manuscript was found in his rucksack when he was taken prisoner. He was allowed to send the manuscript to Russell while he was held in a prison camp in Italy. Released in 1919 he gave away the family fortune he had inherited and, in 1920, trained as a primary school teacher in Austria. He was trained in the methods of the Austrian School Reform Movement, which believed that the main aim of a teacher was to arouse a child's curiosity and to help the child develop as an independent thinker. The Movement rejected the method of teaching which encouraged children to simply learn to repeat facts. But although Wittgenstein was a strong believer in these principles and tried with great enthusiasm to provide the children that he taught with the best possible education, there were factors working against his success. Perhaps the biggest difficulty that Wittgenstein faced was that giving away the family fortune did nothing to enable someone with his highly privileged background to fit into the culture of the children of farmers who he taught. During this period Wittgenstein was desperately unhappy and came close to committing suicide on a number of occasions. The thought that he was appreciated by the children kept him at his task, but he found difficulties in keeping relations between himself and the other teacher on a friendly basis. Eventually, feeling largely that he had failed as a primary school teacher, Wittgenstein gave up in 1925. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wittgenstein.html (2 of 5) [2/16/2002 11:39:05 PM]

Wittgenstein

He still did not feel that he wanted to return to an academic life so he worked at a number of different jobs. First he worked as a gardener's assistant in a monastery near Vienna. Then he worked as an architect for two years occupied in the design and construction of a mansion house for his sister, Margaret Stoneborough, near Vienna. Although Wittgenstein had not wished to return to academic life during this period he was not completely isolated from the study of mathematical logic, the foundations of mathematics, and philosophy. He met with Ramsey, who was making a special study of the Tractatus and had travelled from Cambridge to Austria on several occasions to have discussions with Wittgenstein, and also with philosophers from the Vienna Circle. In 1929 Wittgenstein returned to Cambridge where he submitted the Tractatus as his doctoral thesis. In the Preface to Philosophical Investigations written sixteen years after he returned to Cambridge, Wittgenstein wrote:... since beginning to occupy myself with philosophy again, sixteen years ago, I have been forced to recognise grave mistakes in what I wrote in that first book. I was helped to realise these mistakes - to a degree which I myself am hardly able to estimate - by the criticism which my ideas encountered from Frank Ramsey, with whom I discussed them in innumerable conversations during the last two years of his life. However, it was not until 1953, two years after Wittgenstein's death, that this second great work Philosophical Investigations was published. In this work Wittgenstein studied [11]:... the philosophy of language and philosophical psychology. ... the form of the book is quite unique. ... we first get a part of 693 distinct, numbered remarks, varying in length from one line to several paragraphs, and a second part of fourteen sections, half a page to thirty-six pages long ... instead of presenting arguments and clearly stated conclusions, these remarks reflect on a wide range of topics without ever producing a clear final statement on any of them. Wittgenstein was appointed a lecturer at Cambridge and he was made a fellow of Trinity College. In the following years Wittgenstein lectured there on logic, language, and the philosophy of mathematics. He was appointed to the chair in philosophy at Cambridge in 1939. Malcolm, a student of Wittgenstein, writes in [9] about Wittgenstein's lectures which he attended in 1939:His lectures were given without preparation and without notes. He told me once that he tried to lecture from notes but was disgusted with the result; the thoughts that came out were 'stale', or, as he put it to another friend, the words looked like 'corpses' when he began to read them. In the methods that he came to use, his only preparation for the lecture, as he told me, was to spend a few minutes before the class met, recollecting the course that the inquiry had taken at the previous meeting. At the beginning of the lecture he would give a brief summary of this and then he would start from there, trying to advance the investigation with fresh thoughts. ... what occurred in these class meetings was largely new research. G H von Wright was a pupil of Wittgenstein at Cambridge. He writes [9]:Wittgenstein thought that his influence as a teacher was, on the whole, harmful to the development of independent minds in his disciples. I am afraid that he was right. And I believe that I can partly understand why it should be so. Because of the depth and originality of his thinking, it is very difficult to understand Wittgenstein's ideas and even http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wittgenstein.html (3 of 5) [2/16/2002 11:39:05 PM]

Wittgenstein

more difficult t incorporate them into one's own thinking. At the same time the magic of his personality and style was most inviting and persuasive. to learn from Wittgenstein without coming to adopt his forms of expression and catchwords and even to imitate his tone of voice, his mien and gestures was almost impossible. There is a suggestion here that Wittgenstein would never have fitted in as the leader of a large group of students and researchers. Although he did have students who went on produce important work, yet remain true to his way of thinking, Wittgenstein always seemed an isolated figure. He seemed to understand the reasons for this when he wrote:Am I the only one who cannot found a school or can a philosopher never do this? I cannot found a school because I do not really want to be imitated. Not at any rate by those who publish articles in philosophy journals. Wittgenstein remained at Cambridge until he resigned in 1947 except for the period of World War II during which he worked as a hospital porter in Guy's Hospital in London. He also spent time working as a laboratory assistant in the Royal Victoria Infirmary before returning to his duties at Cambridge in 1944. After three years back at Cambridge he retired and moved to an isolated cottage on the west coast of Ireland. His health deteriorated and in 1949 cancer was diagnosed. Wittgenstein did not seem unhappy at the diagnosis since he said that he did not wish to live any longer. He continued to work on his ideas until a few days before his death, the power and depth of his intellect being undiminished by illness. McGinn, in [11], gives a fair estimate of Wittgenstein:The power and originality of his thought show a unique philosophical mind and many would be happy to call him a genius. Wittgenstein was never happy with his own writings and as a result only the one major work, the Tractatus, was published during his life. A wealth of material from his lectures and notes has subsequently been published. That his ideas are found difficult is something that he was well aware of and he felt that in some way he did not fit into the world in which he lived. Let us end with a quote from his own writing about why ideas are found difficult:Why is philosophy so complicated? It ought to be entirely simple. Philosophy unties the knots in our thinking that we have, in a senseless way, put there. To do this it must make movements that are just as complicated as these knots. Although the result of philosophy is simple, its method cannot be if it is to succeed. The complexity of philosophy is not a complexity of its subject matter, but of our knotted understanding. Article by: J J O'Connor and E F Robertson List of References (16 books/articles)

Some Quotations (21)

A Poster of Ludwig Wittgenstein

Mathematicians born in the same country

Other Web sites

1. Internet Encyclopedia of Philosophy 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wittgenstein.html (4 of 5) [2/16/2002 11:39:05 PM]

Wittgenstein

Previous History Topics Time lines Glossary index

Next

Main index

Societies, honours, etc.

Famous curves

(Alphabetically)

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wittgenstein.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wittgenstein.html (5 of 5) [2/16/2002 11:39:05 PM]

Wolf

Johann Rudolf Wolf Born: 7 July 1816 in Fällanden (near Zurich), Switzerland Died: 6 Dec 1893 in Zurich, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Rudolf Wolf attended the University of Zurich where he studied under Raabe. He also studied at Vienna and Berlin where he attended lectures by Encke, Dirichlet, Poggendorf, Steiner and Crelle. In 1838 he visited Gauss then the following year he became a teacher of mathematics and physics at the University of Bern. He became professor of astronomy there in 1844. Wolf became director of the Bern Observatory in 1847. In 1855 he accepted a chair of astronomy at both the University of Zürich and the Federal Institute of Technology in Zürich. An observatory was opened at Zürich in 1864, largely due to Wolf's efforts. Wolf wrote on prime number theory and geometry, then later on probability and statistics - a long paper discussed Buffon's needle experiment. He estimated by Monte Carlo methods. Wolf's main contribution, however, was his work on the 11 year sunspot cycle. He extended earlier work by A H Schwabe and he was the codiscoverer of the connection of the cycle with geomagnetic activity on Earth. In 1848 he devised a system now known as Wolf's sunspot numbers. This system is still in use for studying solar activity by counting sunspots and sunspot groups. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wolf.html (1 of 2) [2/16/2002 11:39:07 PM]

Wolf

Mathematicians born in the same country Other Web sites

1. High Altitude Observatory 2. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wolf.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wolf.html (2 of 2) [2/16/2002 11:39:07 PM]

Wolfowitz

Jacob Wolfowitz Born: 19 March 1910 in Warsaw, Russian Poland Died: 16 July 1981 in Tampa, Florida, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jacob Wolfowitz's father emigrated to the USA in 1914 and Jacob joined him in 1920 when he was ten years old. Jacob attended High School in New York and, having graduated, entered the College of the City of New York. While Wolfowitz was in the middle of his undergraduate course the Great Depression began. The Great Depression began in 1929 and by 1932 one quarter of the workers in the United States were unemployed. When Wolfowitz graduated in 1931 there was little prospects of good employment so he the next ten years teaching mathematics in a number of different high schools while he worked towards his doctorate. In 1934 Wolfowitz married Lillian Dundes; they had one daughter, born in 1941 and a son Paul, born in 1943. Wolfowitz met Wald in 1938 and they began a collaboration which lasted until Wald's death [2]:They were the closest of friends, and Wolfowitz regarded Wald as his teacher as well as his co-worker. Their work together produced some of the most important and striking results in theoretical statistics. In fact the first paper which Wolfowitz wrote was a joint one with Wald. Wolfowitz's earliest interest was nonparametric inference and the joint paper we just mentioned presents ways of calculating confidence intervals which are not necessarily of fixed width, on a distribution function F based on the empiric independent identically distributed observations on F. It is in a paper by Wolfowitz in 1942 that the word 'nonparametric' appears for the first time. Wolfowitz obtained his doctorate from New York University in 1942 and that year joined the Statistical http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wolfowitz.html (1 of 3) [2/16/2002 11:39:09 PM]

Wolfowitz

Research Group at Columbia University. This research group was working on problems related to war work and one of the statistical methods it was working on was sequential analysis. The type of problem that this statistical method applies to is when the number of observations of a variable is not determined before the experiment begins, but rather the number of observations is determined by the observations themselves. Wald and Wolfowitz were both attached to the Statistical Research Group at Columbia and they led the research project to develop a theory for sequential analysis. Wolfowitz produced work on sequential estimators of a Bernoulli parameter, and results on the efficiency of certain sequential estimators. Again he collaborated with Wald on work in this area, and one particular result should be mentioned, namely their proof of the optimal character of the sequential probability ratio test for testing between two hypotheses. This result is described in [2] as:... one of the strikingly beautiful results of theoretical statistics. At the end of the war Wolfowitz left the Columbia research group and took up a position as associate professor at University of North Carolina. After spending the year 1945-46 there, he returned to Columbia University. He remained at Columbia until after the death of Wald, then he was appointed professor of mathematics at Cornell in 1951. While on the Faculty at Cornell he was visiting professor at the University of California at Los Angeles in 1952, at the University of Illinois in 1953, Technion in Israel in 1957. In 1967 he was visiting professor at both Technion and the University of Paris, and he spent a period at the University of Heidelberg in 1969. He left Cornell and joined the University of Illinois at Urbana in 1970 , retiring in 1978 when he then went to the University of South Florida at Tampa. In 1979 he was Shannon Lecturer of the Institute of Electrical and Electronic Engineers. As someone who collaborated with others frequently on research, it is worth hearing the opinions of collaborators who [2]:... attest to the stimulating experience of doing joint research with him. In research discussions he is energetic, probing, critical, humorous, and very inventive. We have mentioned Wolfowitz's work on nonparametric inference and his work on sequential analysis. He also studied asymptotic statistical theory, that is the theory of how statistical processes behave in the limit as the sample size gets larger and larger. The properties of consistency and efficiency are important here, the first ensuring convergence and the second relating to the rate of convergence. Wolfowitz looked at many aspects of the maximum likelihood method. Information theory, which had been started by Shannon, was another area to which Wolfowitz made important contributions. His book Coding Theorems of Information Theory (3rd ed. 1978) is a classic in the subject. It is [2]:... the only book which concentrates on statistical and probabilistic aspects of noisy channel communication theory. It is also a handy introductory text because of its brief and simple formulations of problems and estimates. Yet it is comprehensive and at the limits of present research. The completely revised third edition is indispensable for specialists, as the other two editions were before. It contains the core of the ideas of Wolfowitz's papers and of research influenced by him, which already means that the main stream of present research in this theory is covered. We should also mention what a fine teacher Wolfowitz was [2]:His lectures reflect his own insistence on understanding the essential features of a proof. "Lets see what makes things tick", his class hear, and his students and audiences at http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wolfowitz.html (2 of 3) [2/16/2002 11:39:09 PM]

Wolfowitz

scientific meetings have the privilege of receiving a lively and lucid exposition that enables them to appreciate the crucial ideas of a subject much more than does the customary formal lecture or line by line proof. ... His students ... always found generosity, patience, and the deep personal concern along with helpful criticism. Wolfowitz received many honours for his outstanding contributions to statistics. He was elected to the National Academy of Sciences, and the American Academy of Arts and Sciences. He was elected a Fellow of the Econometric Society, the International Statistics Institute, and the Institute of Mathematical Statistics. He was both Rietz Lecturer and Wald Lecturer for this latter Institute. Technion, in Israel, awarded him an honorary degree in 1975. Finally, a comment on his personality and interests outside mathematics and statistics:He is a voracious reader, and his knowledge of, and intense interest in, all facets of the state of the world, make him an interesting person with whom to discuss almost anything. Article by: J J O'Connor and E F Robertson List of References (6 books/articles) A Poster of Jacob Wolfowitz

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wolfowitz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wolfowitz.html (3 of 3) [2/16/2002 11:39:09 PM]

Wolstenholme

Joseph Wolstenholme Born: 30 Sept 1829 in Eccles (near Manchester), England Died: 18 Nov 1891 Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Joseph Wolstenholme's father was a Methodist minister. Joseph studied at Wesley College in Sheffield, then entered St John's College, Cambridge on 1 July 1846 and, four years later, he graduated as Third Wrangler (he was ranked third in the list of first class degrees given in mathematics that year). The second wrangler that year was Henry Watson. Wolstenholme was awarded a fellowship to St John's College in March 1852. He accepted a fellowship from Christ's College, Cambridge in November of the same year. It appears that men from Lancashire were given preference at this time in the competition for fellowships at Christ's College. The fact that he had been at John's College and not Christ's College did result in a protest being made against his election but it was soon dropped. On four occasions during the years that he held the fellowship at Christ's College Wolstenholme was an examiner for the mathematical tripos (1854, 1856, 1863, 1870). One of the conditions of fellowships was that the holder had to be unmarried, so when Wolstenholme wished to marry a Swiss girl on 27 July 1869, he had to resign his fellowship. However, Wolstenholme continued to take pupils at Cambridge for two years after resigning his fellowship. In 1871 Wolstenholme was appointed professor of mathematics at the Royal Indian Engineering College at Cooper's Hill, near London. He held this post until he retired in 1889. Forsyth is quoted in [1] (and a shorter quote in [2]) regarding Wolstenholme's contributions to mathematics. Gow notes in [2] that Forsyth campaigned strongly against the syllabus and the style of examinations at Cambridge so is likely to rate Wolstenholme's contribution to the traditional Cambridge system somewhat less than others:Wolstenholme was the author of a number of mathematical papers... . They were usually concerned with questions of analytical geometry, and they were marked they were marked by a peculiar analytical skill and ingenuity. His greatest contribution towards mathematics was his volume of mathematical problems. Forsyth notes ([1] and [2]):... Wolstenholme's problems have proved a help and a stimulus to many students. A collection of some three thousand problems naturally varies widely in value, but many of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wolstenholme.html (1 of 3) [2/16/2002 11:39:11 PM]

Wolstenholme

them contain important results, which in other places or at other times would not infrequently have been embodied in original papers. As they stand, they form a curious and almost unique monument of ability and industry, active within a restricted range of investigation. Gow in [2] shows up a rather different side of Wolstenholme's character. Leslie Stephen, the father of the author Virginia Woolf, studied mathematics at Cambridge and held a fellowship at Cambridge from 1854 till 1864. During that period he became friends with Wolstenholme. Several years after Wolstenholme's death, Stephen wrote down details of his own life for his children and in these he refers to Wolstenholme (see [2]):I think especially of poor old Wolstenholme, called 'the woolly' by you irreverent children, a man whom I had first known as a brilliant mathematician at Cambridge, whose Bohemian tastes and heterodox opinions had made a Cambridge career inadvisable, who tried to become a hermit at Wastdale. He had emerged, married an uncongenial and rather vulgar Swiss girl, and obtained a professorship at Cooper's Hill. His four sons were badly brought up: he was despondent and dissatisfied and consoled himself with mathematics and opium. I liked him or rather was very fond of him, partly from old association and partly because feeble and faulty as he was, he was thoroughly amiable and clung to my friendship pathetically. His friends were few and his home life wretched. ... [We] had him stay every summer with us in the country. There at least he could be without his wife. There is another side to this friendship between Wolstenholme and Stephen which is the main purpose of the article [2]. Stephen's daughter, Virginia Woolf, was a young girl when Wolstenholme shared the family holidays. She later incorporated Wolstenholme into one of her most famous books To the Lighthouse. In this book Mr Augustus Carmichael is based on Wolstenholme. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (2 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wolstenholme.html (2 of 3) [2/16/2002 11:39:11 PM]

Wolstenholme

JOC/EFR June 1998

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wolstenholme.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wolstenholme.html (3 of 3) [2/16/2002 11:39:11 PM]

Woodhouse

Robert Woodhouse Born: 28 April 1773 in Norwich, England Died: 28 Dec 1827 in Cambridge, England Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Woodhouse attended school in North Walsham. In 1790 he was admitted to Caius College, Cambridge where he became a fellow from 1798 to 1823. Woodhouse was Lucasian professor of mathematics from 1820 to 1822, then Plumian professor of astronomy and experimental philosophy from 1822 to 1827. He was made a Fellow of the Royal Society in 1802. Woodhouse was interested in the theoretical foundations of the calculus, the importance of notation, the nature of imaginary numbers and other similar topics. He wrote an important work Principles of Analytic Calculation in 1803 defending Lagrange's calculus method based on series expansions. This support of Continental methods was aimed at his fellow professors at Cambridge but it had little effect. Peacock, however, considered his work of major importance. Woodhouse, therefore, failed to have much influence as a reformer in mathematical studies at Cambridge but he wrote several widely used elementary texts which, during his lifetime, brought him more fame. Woodhouse's other works include History of the Calculus of Variations (1810), Treatise on Astronomy (1812) and a work on gravitation published in 1818. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Honours awarded to Robert Woodhouse (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1802

Lucasian Professor of Mathematics

1820

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Woodhouse.html (1 of 2) [2/16/2002 11:39:12 PM]

Woodhouse

Other Web sites

Bob Bruen

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Woodhouse.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Woodhouse.html (2 of 2) [2/16/2002 11:39:12 PM]

Woodward

Robert Simpson Woodward Born: 21 July 1849 in Rochester, Michigan, USA Died: 29 June 1924 in Washington, D.C., USA

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Robert Woodward attended school in Rochester before entering the University of Michigan in 1868 to study engineering. After graduating he spent 10 years (1872-1882) working for the U.S. Lake Survey where his work consisted of triangulation work on the great lakes. At this time he became interested in geology, in particular being interested in the shape of the Earth, the tides, the atmosphere and in astronomical studies. In 1882 he was appointed to a post as assistant astronomer, then two years later, he was appointed as astronomer, geographer to the U.S. Geological Survey. From 1890 to 1893 he worked for the U.S. Coast and Geodetic Survey and developed triangulation methods of surveying which were less expensive and more accurate than those employed up to that time. In 1893 Woodward was appointed professor of mechanics at Columbia University. From 1899 until 1904 he was professor of mechanics and mathematical physics at Columbia. He spent 12 years at Columbia during which [1]:... he was remarkably successful both as teacher and as administrator. He had a most attractive, genial, and lovable personality, and his advice was being so constantly sought by students and members of the faculty, that he found it very difficult to pursue mathematical work to which he had looked forward. During his time in New York, Woodward was closely associated with the American Mathematical Society. He was vice-president of the Society from 1897 to 1898 and president from 1899 to 1900. In 1904 Woodward left New York to take up the post of president of the newly formed Carnegie Institute

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Woodward.html (1 of 2) [2/16/2002 11:39:14 PM]

Woodward

of Washington which had been set up with ten million dollars gifted by Andrew Carnegie to promote study and research. Archibald [1] writes:At this critical period, his mature judgement and experience, his clarity of vision, common sense, enthusiasm, and geniality, led ... to the establishment on a firm foundation... Article by: J J O'Connor and E F Robertson A Reference (One book/article)

A Quotation

Mathematicians born in the same country Honours awarded to Robert Woodward (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1899 - 1900

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR October 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Woodward.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Woodward.html (2 of 2) [2/16/2002 11:39:14 PM]

Wren

Sir Christopher Wren Born: 20 Oct 1632 in East Knoyle, Wiltshire, England Died: 25 Feb 1723 in London, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Christopher Wren attended Westminster School, London. He entered Wadham College, Oxford in 1649 and received an M.A. from Oxford in 1653. In 1657 he became professor of astronomy at Gresham College, London. Wren became Savile Professor of Astronomy at Oxford in 1661 and held this post until 1673. His interests proved to be architecture and geometry as well as astronomy. In 1663 he designed the chapel at Pembroke College, Cambridge; in 1664, the Sheldonian Theatre, Oxford; and in 1665, buildings for Trinity College, Oxford. His greatest opportunity in architecture came with the rebuilding that followed the London fire of 1666. He replanned the entire city and supervised the rebuilding of 51 churches. His most famous design was that of Saint Paul's Cathedral. Although he is known today almost exclusively for his architectural achievement Wren was a very famous mathematician in his own day. Newton, never one to give excessive praise to others, states in the Principia that he ranks Wren together with Wallis and Huygens as the leading geometers of the day. Wren's fame in mathematics resulted from results he obtained in 1658. He found the length of an arc of the cycloid using an exhaustion proof based on disections to reduce the problem to summing segments of chords of a circle which are in geometric progression. He was the first to resolve Kepler's Problem on cutting a semicircle in a given ratio by a line through a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wren.html (1 of 3) [2/16/2002 11:39:16 PM]

Wren

given point on its diameter. This problem had arisen in Kepler's work on elliptical orbits. Wren independently proved Kepler's third law and formulated the inverse-square law of gravitational attraction. Another topic to which Wren contributed was optics. He published a description of a machine to create perspective drawings and he discussed the grinding of conical lenses and mirrors. Out of this work came another of Wren's important results, namely that the hyperboloid of revolution is a ruled surface. These results were published in 1669. Other work on the logarithmic spiral, which had been rectified by Wallis in the late 1650s, led Wren to note that it was possible to consider an area preserving transformation which would transform a cone into a solid logarithmic spiral which, he remarked, resembled snail shapes and sea shell shapes. D'Arcy Thompson was to examine such ideas 250 years later. Wren was the leader of a scientific discussion group at Gresham College London that, in 1660, initiated formal weekly meetings. In 1662 this body received its Royal Charter from Charles II and 'The Royal Society of London for the Promotion of Natural Knowledge' was formed. Wren was president of the Royal Society from 1680 to 1682. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (11 books/articles)

A Quotation

A Poster of Christopher Wren

Mathematicians born in the same country

Cross-references to History Topics

1. Orbits and gravitation 2. English attack on the Longitude Problem

Cross-references to Famous Curves

1. Cycloid 2. Logarithmic Spiral

Other references in MacTutor

Chronology: 1650 to 1675

Honours awarded to Christopher Wren (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1663

Lunar features

Crater Wren on Mercury

Savilian Professor of Astronomy

1661

Other Web sites

1. The Galileo Project 2. Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wren.html (2 of 3) [2/16/2002 11:39:16 PM]

Wren

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wren.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wren.html (3 of 3) [2/16/2002 11:39:16 PM]

Wronski

Josef Hoëné de Wronski Born: 23 Aug 1778 in Wolsztyn, Poland Died: 8 Aug 1853 in Neuilly (near Paris), France

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hoëné Wronski was born Josef Hoëné but he adopted the name Wronski around 1810 just after he married. He had moved to France and become a French citizen in 1800 and then, in 1810 he moved to Paris. His first memoir on the foundations of mathematics was published there in 1810 but, after it received less than good reviews from Lacroix and Lagrange, Wronski broke off relations with the Institute in Paris. Among other things he did was design caterpillar vehicles to compete with the railways. However they were never manufactured. His main work involved applying philosophy to mathematics, the philosophy taking precedence over rigorous mathematical proofs. He criticised Lagrange's use of infinite series and introduced his own ideas for series expansions of a function. The coefficients in this series are determinants now known as Wronskians (so named by Muir in 1882). In 1812 he published a work claiming to show that every equation had an algebraic solution, contradicting Ruffini's results which were already published. Wronski's work here, although of course wrong, nevertheless still has important applications. Wronski spent the years 1819 to 1822 in London. He came to England to try to obtain an award from the Board of Longitude but his instruments were detained by the Customs as he entered the country. He found himself in severe financial difficulties but, after his instruments had been returned to him, he was able to address the Board of Longitude. His address On the Longitude only contained generalities and did http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wronski.html (1 of 2) [2/16/2002 11:39:18 PM]

Wronski

not impress. His book Introduction to a course in mathematics was published in London in 1821. For many years Wronski's work was dismissed as rubbish. However a closer examination of the work in more recent times shows that, although some is wrong and he has an incredibly high opinion of himself and his ideas, there is also some mathematical insights of great depth and brilliance hidden within the papers. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Hoëné Wronski

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Wronski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Wronski.html (2 of 2) [2/16/2002 11:39:18 PM]

Xenocrates

Xenocrates of Chalcedon Born: 396 BC in Chalcedon (now Kadiköy, near Istanbul), Bithynia (now Turkey) Died: 314 BC in Athens, Greece Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Xenocrates of Chalcedon was a student of Plato who entered the Academy in Athens in about 376 BC. In about 367 BC Xenocrates accompanied Plato on his journey to Syracuse following the death of Dionysius I. Xenocrates left Athens with Aristotle after Plato's death in 347 BC when they were both invited to Assos. Xenocrates remained for around five years in Assos. Plato's nephew Speusippus had become head of the Academy on Plato's death, but in 340 BC he sent for Xenocrates to return to Athens to prepare to become his successor. Despite Xenocrates having been chosen to head the Academy by Speusippus, an election took place to find a successor to Speusippus after his death. It was a close battle between Xenocrates, Menedemus of Pyrrha and Heraclides Ponticus but Xenocrates triumphed by just a few votes. Although Xenocrates had been many years in Athens he had refused to become a citizen of that state since he did not approve of its close relations with Macedonia. In this respect he contrasted strongly with his predecessor Speusippus who had strongly supported the political ties between Athens and Macedonia. It is clear that the Academy at this time was far from what many picture it as, namely an institution where scholars sat thinking, isolated from the world around them. On the contrary, the Academy was highly involved in the politics of the day and different political views strove for supremacy. In 322 BC Xenocrates found himself in a directly political post when he headed a team negotiating a political settlement with Macedonia. To say 'political settlement' is perhaps rather wide of the mark since effectively they had to negotiate terms for the surrender of Athens. The fact that Xenocrates was not an Athenian citizen became a sore point with the Macedonians and he was deemed to be illegitimate as an ambassador for Athens. Xenocrates remained head of the Academy in Athens for the rest of his life. A hard working man, Xenocrates is described as [1]:... good-natured, gentle and considerate but ... he lacked the graciousness of his teacher Plato. Xenocrates wrote on philosophy and mathematics. Diogenes Laertius gives the titles of two mathematics books by Xenocrates, namely On numbers and The theory of numbers. All his books are lost and it would http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Xenocrates.html (1 of 3) [2/16/2002 11:39:20 PM]

Xenocrates

appear that there only ever existed a single copy of each in his own hand. In many ways Xenocrates was not a particularly original thinker. Certainly he saw it his duty as the head of the Academy to promote the views of Plato as exactly as he possibly could. As Dorrie writes in [1]:Xenocrates' lifework consisted of producing a kind of codification - and thus of necessity, a transformation - of Plato's philosophy. But it immediately became apparent that others, especially Aristotle, understood Plato in a wholly different way with respect to certain key questions. Xenocrates believed that matter is composed of indivisible units, so he may be regarded as an early believer in the atomic theory. He agreed with Pythagoras regarding the importance of numbers in philosophy and attributed to Pythagoras an atomic view of acoustics where sound, perceived as a single entity, consists of discrete sounds. Xenocrates believed in human beings having threefold existence, mind, body and soul. It is not clear whether he was the instigator of this belief. He also believed that people die twice, once on Earth, then for a second time on the Moon when the mind separates from the soul and travels to the Sun. Plutarch writes about an attempt by Xenocrates to calculate the total number of syllables which could be made from the letters of the alphabet. The result which Xenocrates obtained was, according to Plutarch, 1,002,000,000,000. If true this probably represents the first attempt at solving a combinatorial problem involving permutations. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) Mathematicians born in the same country Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Xenocrates.html (2 of 3) [2/16/2002 11:39:20 PM]

Xenocrates

JOC/EFR April 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Xenocrates.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Xenocrates.html (3 of 3) [2/16/2002 11:39:20 PM]

Yang

Yang Hui Born: about 1238 in China Died: about 1298 in China Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yang Hui was a minor Chinese official who wrote two books, dated 1261 and 1275, which use decimal fractions (in the modern form) and give the first account of the Pascal triangle. The 1275 work is called Cheng Chu Tong Bian Ben Mo which means Alpha and omega of variations on multiplication and division. One of the more remarkable aspects of this work is the document on mathematics education Xi Suan Gang Mu (A syllabus of mathematics) which prefaced the first chapter. Man Keung Siu, reviewing [10], writes that the syllabus:... is an important and unusual extant document in mathematics education in ancient China. Not only does it specify the content and the time-table of a comprehensive study program in mathematics, it also explains the rationale behind the design of such a curriculum. It emphasizes a systematic and coherent program that is based on real understanding rather than on rote learning. This program is a marked improvement on the traditional way of learning mathematics by which a student is assigned certain classical texts, to be studied one followed by the other, each for a period of one to two years! Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1100 to 1300

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yang.html (1 of 2) [2/16/2002 11:39:21 PM]

Search Form

Yang

Glossary index

Quotations index

Mathematicians of the day JOC/EFR January 1997

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Yang.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yang.html (2 of 2) [2/16/2002 11:39:21 PM]

Yates

Frank Yates Born: 12 May 1902 in Manchester, England Died: 17 June 1994 in Harpenden, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Frank Yates was educated at Wadham House, a private school where the mathematics master was an excellent mathematician and teacher who influenced Frank into this direction. He obtained a scholarship to Clifton College in 1916. Four years later he was awarded a scholarship to study at St John's College Cambridge. He graduated with First Class Honours in 1924 after doing very well at university but never looking like the outstanding scholar that he was to become. After two years teaching mathematics he decided to leave teaching and joined the Gold Coast Survey as mathematical advisor. Because of ill health he decided to try for a post in England and, after applying to R A Fisher, he was appointed assistant statistician at Rothamsted Experimental Station in 1931. When Fisher was appointed to a chair in University College London in 1933, Yates became Head of Statistics at Rothamsted. He held this post until he retired in 1968. Yates worked on experimental design, often collaborating with Fisher. During World War II he studied food supplies and application of fertilisers to improve crops. He applied his experimental design techniques to a wide range of problems such as control of pests. After 1945 he was to continue to apply his statistical techniques to problems of human nutrition. Yates was appointed to the United Nations Commission on Statistical Sampling and published Sampling Methods for Censuses and Surveys in 1949. He was an enthusiastic user of computers writing:... to be a good theoretical statistician one must also compute, and must therefore have the best computing aids.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yates.html (1 of 3) [2/16/2002 11:39:23 PM]

Yates

He was one of the people who were influential in establishing the British Computer Society, and he was president of the Society in 1960-61. Yates was an extremely good Departmental Head. In the address at his memorial service this was his style in this role was talked about:Frank Yates's method of managing his department was a remarkable one, in that it was totally invisible. There were almost no rules, apart from that which insisted that no scientific paper left the department without being read, and usually greatly improved, by him. After he retired, he became Senior Research Fellow at Imperial College, London. There he did some lecturing for the first time in his career, without having a great deal of success. In [2] his lecturing is described as follows:He was not an ideal lecturer, for he lacked concern for comprehensive formal presentation and preferred to talk about general ideas. The Royal Society of London awarded him their Royal Medal in 1966 in:... recognition of his profound and far-reaching contributions to the statistical methods of experimental biology. Article by: J J O'Connor and E F Robertson List of References (7 books/articles) Mathematicians born in the same country Honours awarded to Frank Yates (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1948

Royal Society Royal Medal

Awarded 1966

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 1997 The URL of this page is:

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yates.html (2 of 3) [2/16/2002 11:39:23 PM]

Yates

http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Yates.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yates.html (3 of 3) [2/16/2002 11:39:23 PM]

Yativrsabha

Yativrsabha Born: about 500 in India Died: about 570 in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Yativrsabha (or Jadivasaha) was a Jaina mathematician who studied under Arya Manksu and Nagahastin. We know nothing of Yativrsabha's dates except for a reference which he makes to the end of the Gupta dynasty which he says was after 231 years of ruling. Now this ended in 551 so we must assume that 551 AD is a date which occured during Yativrsabha's lifetime. This fits with the only other information regarding his dates which are that his work is referenced by Jinabhadra Ksamasramana in 609 and that Yativrsabha himself refers to a work written by Sarvanandin in 458. Yativrsabha's work Tiloyapannatti gives various units for measuring distances and time and also describes the system of infinite time measures. It is a work which describes Jaina cosmology and gives a description of the universe which is of historical importance in understanding Jaina science and mathematics. The Jaina belief was in an infinite world, both infinite in space and in time. This led the Jainas to devise ways of measuring larger and larger distances and longer and longer intervals of time. It led them to consider different measures of infinity, and in this respect the Jaina mathematicians would appear to be the only ones before the time when Cantor developed the theory of infinite cardinals to envisage different magnitudes of infinity. Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

1. An overview of Indian mathematics 2. Jaina mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

Search Form

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yativrsabha.html (1 of 2) [2/16/2002 11:39:24 PM]

Yativrsabha

Glossary index

Quotations index

Mathematicians of the day JOC/EFR November 2000

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Yativrsabha.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yativrsabha.html (2 of 2) [2/16/2002 11:39:24 PM]

Yau

Shing-Tung Yau Born: 4 April 1949 in Kwuntung, China

Click the picture above to see two larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Shing-Tung Yau studied for his doctorate at the University of California at Berkeley under Chern's supervision. He received his Ph.D. in 1971 and, during session 1971-72, Yau was a member of the Institute for Advanced Study at Princeton. Yau was appointed assistant professor at the State University of New York at Stony Brook in 1972. In 1974 he was appointed an associate professor at Stanford University. He was promoted to full professor at Stanford before returning to the Institute for Advanced Study at Princeton in 1979. In 1980 he was made a professor at the Institute for Advanced Study at Princeton, a position he held until 1984 when he moved to a chair at the University of California at San Diego. In 1988 he was appointed professor at Harvard University. Yau was awarded a Fields Medal in 1982 for his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampère equations. In fact the 1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982. They were not presented until the International Congress in Warsaw which could not be held in 1982 as scheduled and was delayed until the following year. Nirenberg described Yau's work at the International Congress in Warsaw in 1983. Writing in [5] after the Fields Medal awards were announced in 1982, Nirenberg wrote:S-T Yau has done extremely deep and powerful work in differential geometry and partial differential equations. He is an analyst's geometer (or geometer's analyst) with enormous technical power and insight. He has cracked problems on which progess has been stopped for years. Nirenberg describes briefly the areas of Yau's work. On the Calabi conjecture, which was made in 1954, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yau.html (1 of 3) [2/16/2002 11:39:26 PM]

Yau

he writes that this:... comes from algebraic geometry and involves proving the existence of a Kähler metric, on a compact Kähler manifold, having a prescribed volume form. The analytic problem is that of proving the existence of a solution of a highly nonlinear (complex Monge-Ampère ) differential equation. Yau's solution is classical in spirit, via a priori estimates. His derivation of the estimates is a tour de force and the applications in algebraic geometry are beautiful. Yau solved the Calabi conjecture in 1976. Another conjecture solved by Yau was the positive mass conjecture, which comes from Riemannian geometry. Yau, in joint work, constructed minimal surfaces, studied their stability and made a deep analysis of how they behave in space-time. His work here has applications to the formation of black holes. The Plateau problem was studied by Plateau, Weierstrass, Riemann and Schwarz but it was finally solved by Douglas and Radó. However, there were still questions relating to whether Douglas's solution, which was known to be a smooth immersed surface, is actually embedded. Yau, working with W H Meeks solved this problem in 1980. In 1981 Yau was awarded The Oswald Veblen Prize in Geometry:...for his work in nonlinear partial differential equations, his contributions to the topology of differentiable manifolds, and for his work on the complex Monge-Ampère equation on compact complex manifolds. In joint work of Yau with Karen Uhlenbeck On the existence of Hermitian Yang-Mills connections in stable bundles (1986), they solved higher dimensional versions of the Hitchin-Kobayashi conjecture. Their work extended that of Donaldson on this topic in 1985. The Crafoord Prize of the Royal Swedish Academy of Sciences was awarded to Yau in 1994:... for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems. G Tian [6] sums up Yau's work to date which led to his being awarded the Crafoord Prize:As a result of Yau's work over the past twenty years, the role and understanding of basic partial differential equations in geometry has changed and expanded enormously within the field of mathematics. His work has had, and will continue to have, a great impact on areas of mathematics and physics as diverse as topology, algebraic geometry, representation theory, and general relativity as well as differential geometry and partial differential equations. Yau was elected to the National Academy of Sciences in 1993. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yau.html (2 of 3) [2/16/2002 11:39:26 PM]

Yau

Mathematicians born in the same country Other references in MacTutor

Chronology: 1980 to 1990

Honours awarded to Shing-Tung Yau (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1982

AMS Colloquium Lecturer

1986

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Yau.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yau.html (3 of 3) [2/16/2002 11:39:26 PM]

Yavanesvara

Yavanesvara Born: about 120 in Western India Died: about 180 in India Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Indian astrology was originally known as Jyotisha, which means "science of the stars". Until around the first century AD no real distinction was made between astrology and astronomy and in fact most astronomical theories were propounded to support the theory that the positions of the heavenly bodies directly influenced human events. The Indian methods of computing horoscopes all date back to the translation of a Greek astrology text into Sanskrit prose by Yavanesvara in 149 AD. Yavanesvara (or Yavanaraja) literally means "Lord of the Greeks" and it was a name given to many officials in western India during the period 130 AD - 390 AD. During this period the Ksatrapas ruled Gujarat (or Madhya Pradesh) and these "Lord of the Greeks" officials acted for the Greek merchants living in the area. The particular "Lord of the Greeks" official Yavanesvara who we are interested in here worked under Rudradaman. Now Rudradaman became ruler of the Ksatrapas in around 130 AD and it was during the period of his rule that Yavanesvara worked as an official and made his translation. We know of Rudradaman because information is recorded in a lengthy Sanskrit inscription at Junagadh written around 150 AD. The Greek astrology text in question was written in Alexandria some time round about 120 BC. Yavanesvara did far more than just translate the Greek text for such a translation would have had little relevance to the Indians. He therefore not only translated the language but he translated the context too. Instead of the Greek gods who appear in the original, Yavanesvara used Hindu images. Again he worked the Indian caste system into the work and made the work one which would fit well with the Indian thought. The work was written with the aim of letting Indians became astrologers so it had to present astronomy in a form in which it could be used for astrology. In order to do this Yavanesvara put into his work an explanation of the Greek version of the Babylonian theory of the motions of the planets. All this he wrote in Sanskrit prose but sadly the original has not survived. We do have, however, a version written in Sanskrit verse 120 years after Yavanesvara's work appeared. Yavanesvara had an important influence on the whole of astrology in India for centuries after he made his popular translation. Although the influence was more than on astrology, as the science of astronomy split from astrology, the influence of Yavanesvara's work reached into astronomy too.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yavanesvara.html (1 of 2) [2/16/2002 11:39:28 PM]

Yavanesvara

Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country Cross-references to History Topics

An overview of Indian mathematics

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR November 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Yavanesvara.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yavanesvara.html (2 of 2) [2/16/2002 11:39:28 PM]

Yoccoz

Jean-Christophe Yoccoz Born: 29 May 1957 in France

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Jean-Christophe Yoccoz was placed first in the entrance examination for the Ecole Normale Supérieure and also for the Ecole Polytechnique in 1975. He received the Agrégation de Mathematique in 1977 in joint first position. Yoccoz did his military service in Brazil in 1981-1983. He was a student of M Herman and submitted his doctoral thesis in 1985. As a student of Herman, a leading word expert on dynamical systems, it was not surprising that Yoccoz would himself work on dynamical systems. However, it was remarkable how quickly Yoccoz was to establish himself as the most brilliant researcher in this area. In his thesis, Yoccoz improved theorems of his supervisor Herman by giving simpler proofs but also obtaining the same results under weaker hypotheses. Given his outstanding work it was clear that he would quickly be offered appointments and indeed he was appointed as professor at the University of Paris-Sud (Orsay). He became a member of the Institut Universitaire de France and a member of the Unité Recherche Associé "Topology and Dynamics" of the Centre National de la Recherche Scientifique at Orsay. At the International Congress of Mathematicians in Zurich in 1994, Yoccoz received his greatest honour for this work on dynamical systems when he was awarded a Fields Medal. In his own address to the Congress, he began by stating the aim of his subject:Broadly speaking, the goal of the theory of dynamical systems is, as it should be, to understand most of the dynamics of most systems. The theory of dynamical systems really began with Poincaré who was studying the stability of the solar system. This is typical of what the theory of a dynamical system tries to do: it describes how a system evolves over time given a rule which, for any particular state of the system, describes the following state. The solar system example shows exactly what one wants to know - given any configuration of the system http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yoccoz.html (1 of 3) [2/16/2002 11:39:30 PM]

Yoccoz

it evolves according to Newton's laws but will it remain stable or, after many years, will one of the planets be ejected from the system? After Poincaré, the theory was developed by a large number of mathematicians such as Arnold, Fatou, Herman, Julia, Kolmogorov, Palis, Siegel, Smale and Yoccoz. Douady writes in [3]:... they have proved stability properties - dynamic stability, such as that sought for the solar system, or structural stability, meaning persistence under parameter changes of the global properties of the system. Yoccoz in his own address at the International Congress of Mathematicians in Zurich in 1994 said:The dynamical features that we are able to understand fall into two classes, hyperbolic dynamics and quasiperiodic dynamics; it my well happen, especially in the conservative case, that a system exhibits both hyperbolic and quasiperiodic features. ... we seek to extend these concepts, keeping a reasonable understanding of the dynamics, in order to account for as many systems as we can. The big question is then: Are these concepts sufficient to understand most systems? Douady describes Yoccoz's contributions in general terms in [2] but in more detail in [3] which was the article describing the mathematics leading to the award of the Fields Medal. We quote [2] and advise those looking for more technicalities to consult [3]:He combines an extremely acute geometric intuition, an impressive command of analysis, and a penetrating combinatorial sense to play the chess game at which he excels. He occasionally spends half a day on mathematical "experiments", by hand or by computer. "When I make such an experiment", he says, "it is not just the results that interest me, but the manner in which it unfolds, which sheds light on what is really going on." Yoccoz has developed a method of combinatorial study of Julia sets and Mandelbrot sets - called "Yoccoz puzzles" - which permit deep insight. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

Chronology: 1990 to 2000

Honours awarded to Jean-Christophe Yoccoz (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1994

Other Web sites

Encyclopaedia Britannica

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yoccoz.html (2 of 3) [2/16/2002 11:39:30 PM]

Yoccoz

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Yoccoz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yoccoz.html (3 of 3) [2/16/2002 11:39:30 PM]

Youden

William John Youden Born: 12 April 1900 in Townsville, Australia Died: 31 March 1971 in Washington, D.C., USA Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Youden's family were originally from Britain. His father was also named William John Youden and he was an engineer who had been born in Diver. Youden's mother, Margaret Hamilton, came from Carluke in Scotland. Although Youden was born in Australia, his family returned to England he was two years of age. For five years they lived in Dover, then when Youden was seven they emigrated to the United States. Arriving in Connecticut in 1907, Youden's family soon moved to the Niagara Falls. There he attended school but after a few years the family were on the move again, this time taking up residence in Rochester. Youden was sixteen years old when he began to live in Rochester and he was nearing the age to begin his university education. Indeed, he entered the University of Rochester in the following year and studied chemical engineering. When he entered university in 1917 World War I was in progress, and Youden had a break from university for three months in 1918 while he served in the Army. He returned to complete his studies for a B.S. which was awarded in 1921. In September 1922 Youden entered Columbia University to study, first for his Master's Degree which was awarded in 1923, then for his doctorate which was awarded in 1924. Both these postgraduate degrees were in chemistry. So far we have seen nothing which might indicate why Youden would be the right person to include in this history of mathematics archive. In fact up to this time Youden had shown no interest in statistics, the area to which he would eventually make a major contribution. After obtaining his doctorate in 1923 he obtained a post with the Boyce Thompson Institute for Plant research as a physical chemist. Youden did not become interested in statistics for a few years. In fact over the first few years that he worked at the Institute Youden became more and more disillusioned with the way that measurements were made in biology. At one point he was so upset by the methods used that he considered resigning his position. However, in 1928 he obtained a copy of Fisher's Statistical Methods which had been published three years earlier. This book opened up a new world for Youden who now saw that he had the perfect opportunity to carry out agricultural experiments which could be set up using the new experiment designs put forward by Fisher. In 1931 Youden published his first paper on statistical methods. It marks [1]:... the beginning of Youden's "missionary" efforts to acquaint research workers with statistical methods of value in their work. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Youden.html (1 of 3) [2/16/2002 11:39:31 PM]

Youden

In his work during the 1930s on the statistical design of experiments, Youden introduced new methods based on balanced incomplete block designs. Youden made Fisher aware of these methods in 1936 and Fisher called the new experiment designs 'Youden squares' in his work on Statistical tables published jointly with Yates in 1938. Youden published a paper in 1937 Use of incomplete block replications in estimating tobacco mosaic virus which used his new rectangular experimental arrangements. He introduced further designs in a paper which appeared in 1940. In [5] Preece surveys Youden squares. His own summary of what is contained in this papers follows:Youden squares were introduced in the 1930s as designs for experimentation on plants. We discuss their mathematical properties and their enumeration. Cyclic and partly cyclic methods of generating Youden squares are given particular attention. Some balanced Graeco-Latin designs that incorporate Youden squares are discussed; these have statistical interest because of their potential for use as designs for orchard experiments, and some have mathematical interest because of their role in the construction of Graeco-Latin squares. Youden squares played an important role in World War II being used for experimental trials in engineering and other scientific projects. During World War II Youden worked for the military as a operations analyst. He spent some time in Britain undertaking war work, mainly investigating the factors which control the accuracy of bombing. He also spent time in India, China, and the Marianas carrying out similar work. In 1948 Youden, by now recognised as a leading statistician as well as a chemist, joined the National Bureau of Standards. In that position he again was deeply involved in devising experimental arrangements for a wide range of different tasks from spectral analysis to thermometer and other instrument calibration. In 1974, three years after his death, Youden's final book was published. He had completed the manuscript of Risk, choice and prediction shortly before his death and this text was aimed at school children. Youden wrote in the introduction that the book was aimed at anyone:... who wants to learn in a relatively painless way how the concept and techniques of statistics can help us better understand today's complex world. Article by: J J O'Connor and E F Robertson List of References (5 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines

Birthplace maps Chronology

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Youden.html (2 of 3) [2/16/2002 11:39:31 PM]

Search Form

Youden

Glossary index

Quotations index

Mathematicians of the day JOC/EFR May 2000

Poster index Anniversaries for the year

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Youden.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Youden.html (3 of 3) [2/16/2002 11:39:31 PM]

Young

William Henry Young Born: 20 Oct 1863 in London, England Died: 7 July 1942 in Lausanne, Switzerland

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

William Young's father was Henry Young, a grocer, and his mother was Hephzibah Jeal. William was his parents' eldest son and he was brought up by his parents as a Baptist. He attended the City of London School where the headmaster was particularly fascinated by mathematics. This headmaster was Edwin A Abott, who was the author of the famous popular mathematical work Flatland. He immediately saw the potential that Young had for mathematics and he encouraged him in that direction. In 1881 Young entered Peterhouse, Cambridge to begin his undergraduate studies of mathematics. At Cambridge Young was an outstanding student showing far more mathematical ability than any of the other students in his year. However to achieve the position of First Wrangler (the top position in the list of First Class graduates) in the Mathematical Tripos required enormous dedication and training in the type of examination questions set in the Tripos. It would be fair to say that the First Wrangler was the most skilled at answering Tripos questions rather than the best mathematician and many of the great mathematicians who attended Cambridge failed to gain this distinction. Young was one such student for he made a very conscience decision that becoming First Wrangler was less important to him than having varied interests, both academic and sporting, at university. He was fourth wrangler in 1884. While at Cambridge he put aside the Baptist religion of his family and as baptised into the Church of England. Although many famous mathematicians who attended Cambridge failed to become First Wrangler, many of those who failed did become Smith's Prizemen. However Young did not even submit an essay for this prize but submitted an essay for a theology prize instead. He won the theology prize and he decided to remain at Cambridge earning money by privately coaching students for the mathematical tripos. He did not undertake any mathematical research although he was a Fellow of Peterhouse between 1886 and

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Young.html (1 of 3) [2/16/2002 11:39:33 PM]

Young

1892. One of the students Young tutored was Grace Emily Chisholm, who studied mathematics at Girton College. She then went to Göttingen where she was supervised for her doctorate by Klein, returning in 1885 after the award of the doctorate. It is extremely unlikely that Young would have ever become interested in research had it not been that he married Grace Chisholm (who, of course, then became Grace Chisholm Young) in 1896, and that Klein visited Cambridge to receive an honorary degree in 1897. Discussions with Klein seems to inspire Young. Grace Chisholm Young later wrote that after Klein's visit:... [William] proposed, and I eagerly agreed, to throw up lucre, go abroad, and devote ourselves to research. Together the Youngs, who formed a mathematical married partnership of real significance, left Cambridge and went to Göttingen. After a few months they went to Italy, together with the first of their six children (three sons and three daughters), where they lived for over a year. In September of 1899 they returned to Göttingen which was then home for them until 1908 when they moved to Geneva. However Young returned to Cambridge during term time where he both taught and examined. In 1913 he accepted two part-time chairs, one being the Hardinge Professorship of Pure Mathematics in Calcutta University which he held from 1913 to 1917, the other being at the University of Liverpool which he held from 1913 to 1919. He was the first to hold the Hardinge Professorship. In 1915, while holding his two part-time chairs, the Youngs moved their permanent home from Geneva to Lausanne. This remained the family's permanent home even after 1919 when Young was appointed to the chair of mathematics at the University College of Wales in Aberystwyth in Wales. He held this post until 1923. Burkill wrote in [2]:He did not meet the recognition he deserved. This was due in part to his late start, and in part to a certain conservative hostility to the modern theory of real functions - a theory which few Englishmen in the early years of this century understood. Even when his profundity and originality were better appreciated, he was passed over in elections to chairs in favour of men who might be expected to be less exacting colleagues. Young discovered a form of Lebesgue integration, independently but two years after Lebesgue. His definitions of measure and integration were quite different from those which Lebesgue had given but were shown to be essentially equivalent. He studied Fourier series and orthogonal series in general, the ideas which he put forward being further developed by Littlewood and Hardy. His results in this area are described by Burkill [1] as:... of striking simplicity and beauty ... Perhaps his most important contribution was to the calculus of several variables. He set out this theory beautifully in his treatise The fundamental theorems of the differential calculus (1910). All advanced calculus books now use his approach to functions of several complex variables. This 1910 book was one of three which Young wrote. The other two were written jointly with his wife: The first book of geometry (1905) was an elementary work clearly written by the Youngs with teaching mathematics to their own children in their minds, and The theory of sets of points (1906). Young was trapped in Lausanne when France fell in 1940. He was forced to spend the last two years of his life there very unhappy at being separated from his family. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Young.html (2 of 3) [2/16/2002 11:39:33 PM]

Young

He received many honours for his mathematical achievements despite his lack of success in obtaining prestigious chairs. The was elected a Fellow of the Royal Society on 2 May 1907, receiving the Sylvester medal from that Society in 1928. He was president of the London Mathematical Society from 1922 to 1924 and president of the International Union of Mathematicians from 1929 to 1936. He received honorary degrees from the universities of Calcutta, Geneva, and Strasbourg. Burkill paints this picture of Young in [2]:The immediate and abiding impression which Young gave was one of restless vitality; it was shown in his gait, his gestures, and his words. His appearance was striking; in early married life he grew a beard, red in contrast with his dark hair, and he wore it very long in later years. Many stories were current about him, all turning on his energy, mental and physical. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (7 books/articles)

A Quotation

A Poster of William Young

Mathematicians born in the same country

Honours awarded to William Young (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1907

Royal Society Sylvester Medal

Awarded 1928

London Maths Society President

1922 - 1924

LMS De Morgan Medal

Awarded 1917

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Young.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Young.html (3 of 3) [2/16/2002 11:39:33 PM]

Young_Alfred

Alfred Young Born: 16 April 1873 in Widnes, Lancashire, England Died: 15 Dec 1940 in Birdbrook, Essex, England Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Alfred Young's father was Edward Young, described by Turnbull in [4] as:... a prosperous Liverpool merchant and a Justice of the Peace for the county. Alfred was the first son of his father's second marriage. In 1879, when Alfred was six years old, the family moved from Lancashire to Bournemouth. After being educated at home with a private tutor, who also taught the sons of Edward Young's first marriage, Alfred went to Monkton Combe school near Bath. It was at this school that the extent of Alfred's talents in mathematics were recognised. Encouraged by his teachers, he sat the Cambridge Scholarship examinations, winning a scholarship to Clare College. Entering Clare College in 1892, Young combined sporting and academic interests. He was an excellent oarsman and as such he made his mark rowing during his first two years. At University he was described by one of his fellow students as:... a shy, clever lad with great humility of spirit which so marked him in his youth ... By his third year Young had begun to undertake research in mathematics and although this proved a good start to his mathematical career, it was not the best way to prepare for the Mathematical Tripos examinations. He was placed tenth Wrangler (tenth in the First Class) in 1895, the Senior Wrangler that year being Bromwich with Whittaker Second Wrangler. Again a description of Young at this stage is interesting for he was said to be:... the most original man of his year [who] would have occupied a higher place in the list had he directed his attention to the examination schedule ... In 1896 Young was placed in the Second Class of Part II of the Mathematical Tripos. He published his first paper The irreducible concomitants of any number of binary quartics early in 1899 in the Proceedings of the London Mathematical Society. His second paper The invariant syzygies of lowest degree of any number of quartics was published in the following year. Young was appointed as a lecturer in Selwyn College, Cambridge in 1901. He remained in that post until 1905 when he was elected to a Fellowship at Clare College where he became Bursar. He married Edith Clara in 1907. They did not have any children. In 1908 Young was ordained and became a Curate at Christ Church, Hastings. In the same year he was awarded a Sc.D. from Cambridge for his outstanding contributions to mathematics. He was to become Parish Priest at Birdbrook, Essex in 1910 and lived for the rest of his life in this village 25 miles east of Cambridge. Together with his wife he lived [4]:... in a typical country rectory, set in an old world garden full of colour and of great charm, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Young_Alfred.html (1 of 4) [2/16/2002 11:39:35 PM]

Young_Alfred

where a warm welcome awaited a visitor from Cambridge or elsewhere, young or old, who sought out in this secluded corner of Essex a master of abstract algebra, and found more than a mathematician, a friend. In 1926 Young began to lecture again at Cambridge. W L Edge attended Young's lecture course which began in January of that year. Edge wrote:I remember (who could forget?) very well my experiences of attending his first lectures. This was only a course of one lecture a week for one term; you can see for yourself how much he got through ... Doubtless it is all standard work to you, but it will be interesting to see how the old warrior entered the lists again and what he considered should be given to his first hearers. I went along on 19 January 1926, in my third year, just two terms before my Tripos, to Clare. ... there were eleven of us and I was the only undergraduate who ventured. Others in the class were, I think, Cooper, now at Belfast; Broadbent, now at Greenwich; L H Thomas, who got a Smith's Prize and a Trinity Fellowship and went to America; Dirac certainly ... I remember the tall clerical figure entering the room, and his surprise at so large an audience ... And so to linear transformations and Aronhold's symbolic notation.... At the end of the last lecture in March, Young said that he was so pleased that people had turned up that he would lecture again in the following term. And he was surprised, and I very embarrassed, when no other members of the class but myself showed up in April. It was my Tripos term but I was not going to miss his lectures... Young's work had, and continues to have, a major impact on the theory of groups, in particular on group representations. Turnbull writes [4]:From the outset Young's rapid and skilful handling of symbolic algebra bore all the signs of genius. In 1900 he introduced 'Young tableau', the method for which he is best remembered. He wrote a series of papers On quantitative substitutional analysis which arose out of the classical theory of invariants and contained his results in this area. I [EFR] have just attended the conference Groups-St Andrews 2001 in Oxford where one of the main speakers was showing how he was using Young tableaux in his latest research. Burnside, Frobenius and Weyl saw the power of Young's methods. Burnside, as referee of Young's papers, suggested how the papers could be written to emphasise their impact on group theory and he pointed Young towards the papers of Frobenius and Schur. Young did not read German easily and it was some years before he fully understood the work of Frobenius. This resulted in a delay in Young obtaining results on the representation theory of the symmetric group. Frobenius used Young tableaux for the first time in 1903 when he investigated representations of the symmetric group but it was not until 1906 that Young learnt of Frobenius's applications of his methods. In 1927 Young published further work now extending what Frobenius had done and relating it to his own work. He introduced what he called 'standard forms' which improved the efficiency of his methods. He was very pleased to find Frobenius using his ideas. He wrote:I am delighted to find someone else really interested in the matter. The worst of modern mathematics is that it is now so extensive that one finds there is only about one person in the universe really interested in what you are... Weyl also began to make use of Young's ideas and Young tableau appear in his famous book Theory of

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Young_Alfred.html (2 of 4) [2/16/2002 11:39:35 PM]

Young_Alfred

groups and quantum mechanics. In 1934 Young realised the significance of the sequence in which the standard tableau can be written and the following year he again related his work to that of Frobenius and Schur. His ninth (and final) paper On quantitative substitutional analysis was published in 1952, eleven years after his death. G de B Robinson, who edited this paper, writes in the introduction on Young's methods of working [8]:Young's habits of working were systematic and consistent throughout his life. Unable to devote his whole time to mathematics, he could lay down a piece of work and pick it up again after a lapse of time with little apparent loss of continuity. He worked on different aspects of what he called 'Substitutional analysis' and filled numerous folders designated A Z, A2, AB, ..., [Aux A], B2, and BC. As a subject developed he would write a paper on it including material, it may be, from different folders, but destroying the final draft and typescript after the paper appeared in print. It appears, however, that he kept his folders intact throughout the greater part of 50 years, though he only dated work done in the last few years of his life in any consistent manner. The three folders [Aux A], B2 and BC contain material he worked on during the last year of his life. The final folder BC contains the beginnings of a project started two weeks before his death aimed at making a systematic study of the representation theory of subgroups of the symmetric group. We should not give the impression that Young only contributed to group representations. He studied other mathematical topics and the breadth of his interests are illustrated by the fact that he was also an inventor. He invented an electric motor to pump water. In 1918 he patented a generator which converted mechanical energy into high frequency electric currents which could be used for wireless telegraphy. He patented another generator in 1919 but these were never developed commercially. Among the honours that Young received for his mathematical contributions we should mention that in 1931 he was awarded an honorary degree from the University of St Andrews and in 1934 he was elected a Fellow of the Royal Society of London. Young's wife Edith gave details of his death. She wrote [1]:... he was out with me on the Wednesday afternoon visiting in the Parish. He seemed as usual when suddenly after tea he was taken with a pain in the side. The doctor had him rushed off to hospital, operated on him but he never really came round and passed away on Sunday morning. I know it was what he would have wished - to die at his post. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Honours awarded to Alfred Young (Click a link below for the full list of mathematicians honoured in this way)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Young_Alfred.html (3 of 4) [2/16/2002 11:39:35 PM]

Young_Alfred

Fellow of the Royal Society

1934

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR September 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Young_Alfred.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Young_Alfred.html (4 of 4) [2/16/2002 11:39:35 PM]

Yule

George Udny Yule Born: 18 Feb 1871 in Morham (near Haddington), Scotland Died: 26 June 1951 in Cambridge, Cambridgeshire, England

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

George Udny Yule's father was also called George Udny Yule. George Udny Yule senior was involved in administration in India and he was knighted for his services. Yule came from a family with a strong reputation for scholarship, his grandfather, William Yule, being a renowned scholar in Persian and Arabic. George was born at Beech Hill, a house in Morham near Haddington in Scotland. He attended Winchester College, one of the oldest of the great independent schools of England situated in Winchester, Hampshire, until he was sixteen years of age when, in 1887, he entered University College London to read for an engineering degree. In 1890 Yule graduated with a degree in engineering and then for two years he was involved in the practical side of the subject, working in engineering workshops. It was an experience which made him decide that engineering was not the subject for him so, in 1892, he began to undertake research in physics. Yule spent a year in Bonn undertaking research in experimental physics under Hertz. This was a successful year in that he published four papers based on the research on electric waves which he undertook in Bonn, yet again Yule seems to have not found the topic one to excite him enough for him to want to work in that area for the rest of his life. In fact the influence of his work in engineering and experimental physics was less than one would expect for, as Maurice Kendall writes in [4]:It does not appear, in fact, that this early training left a permanent imprint on his habits of thought. One would not suspect an engineering background behind his mature work; the only point at which it exerted some influence was in his careful and expert draughtsmanship and his preference for diagrammatic representation. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yule.html (1 of 4) [2/16/2002 11:39:37 PM]

Yule

Yule returned from Germany to London in the summer of 1893 and was offered a post as a demonstrator in University College London by Karl Pearson. In fact Pearson had known Yule when he had studied at University College as an undergraduate so he knew that he was appointing someone with great potential. For the first time, Yule was inspired by the work which he undertook with Pearson and his first paper on statistics appeared in 1895 On the correlation of total pauperism with proportion of outrelief. This work [3]:... introduced correlation coefficients in studying two-way tables in the earlier volumes of the monumental work of Booth [Life and labour of the people of London (1889-1893)]. In 1895 Yule was elected to the Royal Statistical Society and over the next few years, inspired by Pearson, he produced a series of important articles on the statistics of regression and correlation. Yule's work entitled On the Theory of Correlation was first published in 1897. He developed his approach to correlation via regression over the next few years with a conceptually new use of least squares and by the 1920's his approach predominated in applications in the social sciences. He progressed from his appointment as a demonstrator to that of Assistant Professor at University College in 1896, but as he was paid scarcely enough to live on, he left his post at University College in 1899 to take up the the better paid position of secretary to the examination board of the City and Guilds of London Institute. This change did not lessen his research output in statistics, nor did it end his association with University College London, for over the next few years he gave the annual Newmarch lectures in Statistics. These lectures became the basis for Yule's famous text Introduction to the Theory of Statistics which he first published in 1911. The text was intended for those who possessed only a limited knowledge of mathematics and proved a great success. It was a book clearly reflecting Pearson's approach to statistics, but containing many of the notable contributions made by Yule. It ran to fourteen editions but, perhaps surprisingly, later editions sold very much better than the early ones. In the same year of 1911 Yule was awarded the Guy Medal in Gold of the Royal Statistical Society, their highest award. While commenting on his association with the Royal Statistical Society it is worth noting that Yule was secretary to the Society from 1907 to 1919 and President from 1924 to 1926. In 1912 he accepted a Lectureship in Statistics at Cambridge, taking a drop in salary but never regretting the move. He became a member of St John's College in 1913 and lived in the College for most of the rest of his life. He was made a Fellow of St John's College in 1922, which is the same year in which he was elected a Fellow of the Royal Society. During World War I Yule worked as a statistician in the army in the Contracts Department of the War Office, then at the Ministry of Food where he was Director of Requirements. After the war he was awarded a C.B.E. for this work. The years from 1920 to 1930 were the most productive ones for Yule. He wrote papers on time-correlation in which he introduced the correlogram and he did fundamental work on the theory of autoregressive series. In 1930 he retired from his post, by now a readership, in Cambridge. Although he was still active in research, and would be for many years to come, he had begun to regret that statistics had expanded into such a broad topic that he would never be able to keep up to date. When Karl Pearson died in 1936, Yule was deeply affected. Let us relate a story about Yule which tells us quite a bit about his character. He became interested in driving a car in the 1920s and would, it was reported, drive at reckless speeds. This desire for speed made him want to fly which he decided he would do when he had retired. However, after he retired he

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yule.html (2 of 4) [2/16/2002 11:39:37 PM]

Yule

discovered that he was too old to qualify for insurance and no company would teach him to fly. He was not to be stopped by such problems, however, and he purchased his own plane and qualified for his pilot's licence in 1931. He could beat the insurance companies but not his health, for sadly he suffered a heart problem in 1931 which prevented him for flying and made him a partial invalid for the rest of his life. In 1937 Yule produced a thorough revision of the text of Introduction to the Theory of Statistics for the eleventh edition published in that year. Maurice Kendall writes in [4]:The increasing popularity of the book did a great deal to counteract Yule's feeling of being left behind by modern developments. He professed to be astonished that the work fulfilled his earlier hope that it would be useful to new generations of students, but he was undoubtedly greatly pleased and comforted. The fourteenth and last edition of Introduction to the Theory of Statistics was written jointly with Maurice Kendall and published in 1950, shortly before Yule's death. The first half of the book deals with descriptive statistics: the theory of attributes, frequency distributions and their characteristics, correlation and regression, and curve fitting). The second half of the book deals with sampling theory: large and small samples, chi-square, analysis of variance. The last chapters discuss interpolation and graduation, index numbers, and time series. In his later years he applied statistics to literary style and published a book The statistical study of literary vocabulary in 1944. His paper Cumulative sampling: a speculation as to what happens in copying manuscripts (1946) is described in a review by Feller as follows:Variations in old manuscripts are to a great extent due to copying errors and these are in turn frequently related to "danger spots" in the outward appearance. Since the error removes the danger spot, variations due to copying errors will in general be more stable than the original version. The author uses an admittedly greatly oversimplified model of a random game to study the probable development within so-called families of texts. The mathematics is elementary and the interest of the paper lies in conclusions which apparently differ greatly from commonly accepted views. It is stated that the criterion has been applied to a particular case with results contradicting the philologists' conclusions. Yule did not develop any completely new branches of statistical theory but he took the first steps in many areas which proved important in their further development by later statisticians. Maurice Kendall's comment in [4] as to Yule's contribution is, however, very appropriate:A great deal of Yule's contributions to the advancement of statistics cannot come to light; they reside in the stimulus he gave to his students, the discussions he held with his colleagues on a host of subjects, notably agriculture and demography, and the advice he freely tendered to all who consulted him, for he was always a most approachable man. The story about Yule learning to fly tells us something of his character. In addition Maurice Kendall tells us in [4] that Yule was:... kindly, gentle and genial. His wide knowledge of many subjects and his love of an apposite story made him the best of companions. Article by: J J O'Connor and E F Robertson

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yule.html (3 of 4) [2/16/2002 11:39:37 PM]

Yule

List of References (6 books/articles)

Some Quotations (2)

A Poster of George Udny Yule

Mathematicians born in the same country

Honours awarded to George Udny Yule (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1921

Other Web sites

University of Minnesota

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR July 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Yule.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yule.html (4 of 4) [2/16/2002 11:39:37 PM]

Yunus

Abu'l-Hasan Ali ibn Abd al-Rahman ibn Yunus Born: 950 in Egypt Died: 1009 in Fustat, Egypt Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ibn Yunus's full name is Abu'l-Hasan Ali ibn Abd al-Rahman ibn Ahmad ibn Yunus al-Sadafi. As the name indicates, his great-grandfather was called Yunus, his grandfather was Ahmad, and his father Abd al-Rahman. It was a family of scholars, his father Abd al-Rahman being a noted historian. We know little of ibn Yunus's childhood but we do know that he grew up in a period of military conquest in Egypt. The Fatimid political and religious dynasty took its name from Fatimah, the daughter of the Prophet Muhammad. The Fatimids headed a religious movement dedicated to taking over the whole of the political and religious world of Islam. As a consequence they refused to recognise the 'Abbasid caliphs. The Fatimid caliphs ruled North Africa and Sicily during the first half of the 10th century, but after a number of unsuccessful attempts to defeat Egypt, they began a major advance into that country in 969 conquering the Nile Valley. They founded the city of Cairo as the capital of their new empire. Ibn Yunus was closely connected with the Fatimids and two Caliphs supported his scientific work. The first of these Caliphs was al-Aziz, who was the first of the Fatimid caliphs to begin his reign in Egypt. Al-Aziz became Caliph in 975 on the death of his father al-Mu'izz and, two years later, ibn Yunus began to make astronomical observations. Although there is uncertainty about the instruments that ibn Yunus used, it is claimed by early writers that al-Aziz provided ibn Yunus with at least some instruments. Famed for his astronomical observations, ibn Yunus was also an astrologer but he is most famous for his many trigonometrical and astronomical tables. Of course it was very reasonable for a Caliph to support the type of astronomical work that ibn Yunus was undertaking. The Muslim religion required considerable knowledge of the moon and the sun to determine the times of prayer during the year. The Muslin lunar calendar required that the new months be determined by actual visibility of the lunar crescent rather than duration of the lunar month, so it was necessary to know a number of different details such as how far the moon was from the sun to determine when it became visible. Perhaps al-Aziz would have given better support to ibn Yunus if he had not been so involved in military and political ventures in northern Syria trying to expand the Fatimid empire. For most of his 20 year reign he worked towards this aim while ibn Yunus toiled on his astronomical work. Al-Aziz died in 996 while organising an army to march against the Byzantines and al-Hakim, who was eleven years old at the time, became Caliph. We should note, however, that in [5] the author suggests that Yunus probably had the use of professional calculators in preparing his tables so perhaps al-Aziz or al-Hakim gave him more http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yunus.html (1 of 3) [2/16/2002 11:39:39 PM]

Yunus

support than we have suggested above. Certainly al-Hakim supported ibn Yunus in his astronomical work, although it is hard to determine the strength of that support. Perhaps al-Hakim's interest in astrology meant that he favoured ibn Yunus who is reported by his biographers to have devoted considerable amounts of time to making astrological predictions. Ibn Yunus and al-Hakim were both eccentrics, although al-Hakim's eccentricities were more damaging, while ibn Yunus's sound rather typical of someone totally absorbed in academic pursuits. Ibn Yunus was described by his biographer al-Musabbihi, who was a contemporary of ibn Yunus, as follows (see [1]):He was an eccentric, careless and absent-minded man who dressed shabbily and had a comic appearance. On the other hand al-Hakim ordered the sacking of the city al-Fustat, the city in which ibn Yunus sometimes observed from his great-grandfather's house. Al-Hakim ordered the killing of all dogs, since their barking annoyed him, and he banned certain vegetables and shellfish. However, probably because of his interest in astrology, al-Hakim kept some astronomical instruments in his house overlooking Cairo and we know that on at least one occasion ibn Yunus observed Venus from al-Hakim's house. Ibn Yunus's major work, an astronomical handbook, was al-Zij al-Hakimi al-kabir. 'Al-kabir' means 'large' which is apt and 'al-Hakimi' means that the work is dedicated to Caliph al-Hakim who certainly supported ibn Yunus. The book is certainly large, containing 81 chapters. There are lists of observations made by Yunus and also observations made by his predecessors. In fact it is a rather remarkable fact that singles the work out from all similar works of that period. Other authors never distinguished between their observations and those in their works which had been made by other scientists. He describes 40 planetary conjunctions accurately and 30 lunar eclipses which were used by Simon Newcomb in his lunar theory. To give an example of a planetary conjunction described in the Hakimi Zij we quote from [2] (see also [1]) having changed the dates given by ibn Yunus to those of a modern calendar:A conjunction of Venus and Mercury in Gemini, observed in the western sky: The two planets were in conjunction after sunset on the night [of Sunday 19 May 1000]. The time was approximately eight equinoctial hours after midday on Sunday ... . Mercury was north of Venus and their latitude difference was a third of a degree. We can confirm, using modern knowledge of the positions of the planets, that ibn Yunus was exactly right in his description and that the distance of one third of a degree that he gives is again exactly right. He also describes an eclipse of the moon [2]:This lunar eclipse was [on 22 April 981]. We gathered to observe this eclipse at al-Qarafa, in the Mosque of Ibn Nasr al Maghribi. We perceived first contact when the altitude of the moon was approximately 21 . About a quarter of the lunar diameter was eclipsed, and reemergence occurred about a quarter of an hour before sunrise. The first chapter of the Hakimi Zij gives calendar tables for Muslim, Coptic, Syrian and Persian calendars. Ibn Yunus gives tables to convert dates between these calendars. Tables to compute the date of Easter are also given. Trigonometric functions are given as arcs rather than angles. Spherical trigonometry reaches a high level of sophistication in this work. Many other tables have been attriributed to ibn Yunus. For example in [9] the author writes:http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yunus.html (2 of 3) [2/16/2002 11:39:39 PM]

Yunus

In this paper I describe a set of tables for finding the longitude of the moon, attributed to the tenth century Egyptian astonomer ibn Yunus. The underlying lunar theory is that of Ptolemy, but these tables are so devised that the user is spared the calculations which are associated with Ptolemy's lunar tables. Ibn Yunus's tables, if such they are, contain over 34000 entries ... . The tables are of interest as the earliest attempt by a medieval scholar to solve the computational problem of the determination of the lunar position according to the sophisticated Ptolemaic theory. In [5] ibn Yunus's Very useful tables for finding the time since sunrise, the hour angle and the solar azimuth from the solar altitude are described. The author notes that ibn Yunus used data for these tables that he had collected for the Hakimi Zij. The high degree of accuracy displayed by these tables suggests to D A King that ibn Yunus used systems of nonlinear interpolation. Perhaps it is worth mentioning that, contrary to claims which are often made, there is no evidence to suggest that ibn Yunus used a pendulum for time measurements. D A King in [8] shows that this myth was started in 1684 by the English historian Edward Bernard. Ibn Yunus predicted the date of his own death to be in seven days time when he was in good health. He tidied up his business affairs, locked himself in his house and recited the Qur'an until he died on the day he predicted. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

JOC/EFR November 1999 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Yunus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yunus.html (3 of 3) [2/16/2002 11:39:39 PM]

Yushkevich

Adolph Andrei Pavlovich Yushkevich Born: 15 July 1906 in Odessa, Ukraine Died: 17 July 1993 in Moscow, Russia Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Adolph Pavlovich Yushkevich was called Adolph Pavlovich throughout his life. Born into a Jewish family containing scholars in philosophy and literature, he attended the high school in St Petersburg, then continued his education in Odessa. Entering the Physics and Mathematics Faculty of the University of Moscow, Adolph Pavlovich was an undergraduate with Gelfond who became his friend for life. They were taught mathematics by Egorov, Luzin and other outstanding mathematicians. Adolph Pavlovich graduated in 1929 and then, from 1930, taught at the Moscow Higher Technical School. He was promoted to professor in 1940 and became Head of Mathematics there in 1941. In addition to his professorship of mathematics at the Moscow Higher Technical School, Adolph Pavlovich was appointed to the Institute of History of Natural Sciences and Technology in 1945. The cultural policy Zhdanovism of the Soviet Union was initiated by a 1946 resolution of the Central Committee of the Communist Party. There was an attempt to eliminate all traces of Westernism, or cosmopolitanism, from Soviet life. The policy continued until Stalin's death in 1953, becoming more anti-Semitic in its later stages. Yushkevich suffered under this policy being forced to leave his post at the Moscow Higher Technical School. He was able, however, to keep his post at the Institute of History of Natural Sciences and Technology. Yushkevich was one of the leading historians of mathematics in the world. His doctorate was on Russian mathematics during the 18th century and he began publishing in 1929 the first of over 300 works on the history of mathematics. He contributed 21 articles to the Dictionary of Scientific Biography which are referenced in this Archive. This Archive also references over 50 articles by Yushkevich about a wide range of mathematicians from the earliest to modern times. Yushkevich was arguably the leading world authority on Euler and he was one of the leading authorities on medieval mathematics. In [1] the quality of his work is described in these terms:Yushkevich's work was characterised by an exceptional skill in analysing historical sources, irreproachable logic, carefully considered assessments and historical judgements, and a

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yushkevich.html (1 of 2) [2/16/2002 11:39:40 PM]

Yushkevich

striking ability to illuminate specific problems by placing them in general historical setting. His interests outside the history of mathematics are described in [1] in these terms:Yushkevich was a man of considerable culture, a true representative of the Russian intelligentsia. He spoke many languages, including Latin, and was well acquainted with literature, particularly the works of Russian writers. He took pleasure in rereading Turgenev and Leskov, enjoyed poetry (his favourite poets were Goethe and Puskin), and even composed verses himself. He was keenly interested in painting, and held the French impressionists and Chagall in especially high esteem. He liked Ukrainian songs and Russian love songs of the 19th century, and enjoyed singing them at parties. And he also adored France, especially Paris, which he visited nearly every year during the last three decades of his life. Yushkevich received many honours for his scholarship in the history of mathematics. He was elected to academies in Germany, Spain, Czechoslovakia and other countries. He was awarded the Koyré Medal by the Académie internationale d'histoire des sciences in 1971, the Sarton Medal of the History of Science Society of the USA in 1978 and the May Prize of the International Commission on the History of Mathematics in 1989, the Prize of the Akademie der Wissenschaft der DDR in 1978 and again in 1983, and the Prize of the Académie des Sciences de France in 1982. Article by: J J O'Connor and E F Robertson A Reference (One book/article) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Yushkevich.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Yushkevich.html (2 of 2) [2/16/2002 11:39:40 PM]

Zarankiewicz

Kazimierz Zarankiewicz Born: 2 May 1902 in Czestochowa, Poland Died: 5 Sept 1959 in London, England

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Kazimierz Zarankiewicz was born and brought up in Czestochowa in south-central Poland. He attended secondary school in Bedzin, which is near Czestochowa, and for much of his time at school Poland was going through the difficult events of World War I. When we say that Zarankiewicz was brought up in Poland, this must be seen in relation to the political circumstances of the time. Poland had been partitioned in 1772 with the south, which was called Galicia, under Austrian control and Russia in control of much of the rest of the country. This situation, which was the position throughout the early years of Zarankiewicz's life, lasted until the outbreak of World War I in 1914. At this time Russia tried to win Polish support, particularly in Galicia, by promising the Poles autonomy. By the end of 1914 Russian forces controlled almost all of Galicia. However, the Central Powers (Germany and Austria- Hungary) recaptured Galicia and large parts of Congress Poland which had been under Russian control. A German governor general was installed in Warsaw and a new Kingdom of Poland was declared on 5 November 1916. The University of Warsaw, which had been a Russian language university for many years, became Polish again in November 1915 following the withdrawal of the Russian forces from Warsaw in August 1915. Zarankiewicz completed his secondary school education and entered the University of Warsaw in 1919. From the time its reopening the university had rapidly become a leading world centre for topology. Janiszewski and Mazurkiewicz were conducting a topology seminar there from 1917 onwards, Sierpinski arrived in 1918, and in 1919, the year Zarankiewicz arrived, Kuratowski had just graduated and was beginning his doctoral studies. Saks was also studying for his doctorate at this time.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zarankiewicz.html (1 of 3) [2/16/2002 11:39:42 PM]

Zarankiewicz

With such a concentration on topology, and the excitement of those studying this new discipline in their newly freed country, it is not surprising that this was the area which attracted Zarankiewicz. He wrote his doctoral dissertation on cut points in connected sets and, in 1923, he was awarded a Ph.D. Although his thesis was published, this did not happen until 1927. Zarankiewicz was appointed to Warsaw University as an assistant in 1924 and continued with his research on topological properties of the plane for his habilitation thesis. On acceptance of this thesis in 1929 Zarankiewicz was appointed as a dozent at the University of Warsaw. During the year 1930-31 he visited Vienna where he worked with Menger, and he also visited Berlin where he worked with von Mises, Bergman and others. On his return to Warsaw, Zarankiewicz taught both at the Polytechnic and at the Agricultural College. There was no position for him at the university at this time so he was not able to teach his specialist research topics, but rather he had to teach mechanics, and statistics. However, he taught a course on conformal mappings, one of his current research interests, for a semester at Tomsk in 1936. After he returned to Warsaw from Tomsk he substituted for the professor at Warsaw Polytechnic in 1937. He was put forward for a professorship himself in 1939 but the events of that year, namely the start of World War II, brought normal life to an end and his professorship would have to be put on hold to be considered again only at the end of the war. Zarankiewicz risked his life during the war teaching in the underground university which had been set up by the Poles in German occupied Warsaw to try to keep the intellectual life going. Kuratowski writes in [2]:Almost all our professors of mathematics lectured at these clandestine universities, and quite a few of the students then are now professors or docents themselves. Due to that underground organisation, and in spite of extremely difficult conditions, scientific work and teaching continued, though on a considerably smaller scale of course. The importance of clandestine education consisted among others in keeping up the spirit of resistance, as well as optimism and confidence in the future, which was so necessary in the conditions of occupation. The conditions of a scientist's life at that time were truly tragic. Most painful were the human losses. Zarankiewicz paid dearly for teaching in the underground university for in 1944 he was sent to a labour camp in Germany. He survived this experience and returned to Warsaw at the end of the war in 1945. He resumed his teaching duties in 1946 at Warsaw Polytechnic. The long delayed decision to promote him to professor at Warsaw Polytechnic eventually took effect and, in 1948, he became a full professor. Also in 1948 he went to the United States for several months and taught at a number of universities including Harvard. Zarankiewicz did important work in topology and graph theory. He also wrote on complex functions and number theory. His work on triangular numbers inspired Sierpinski to further work on this topic while Zarankiewicz also worked jointly with Kuratowski on topology. Another of his favourite research topics was complex function theory and, in this topic, he proved results which [1]:... played an important role in the development of the theory of the kernel and its generalisations to several variables, notably to pseudo-conformal transformations in space of more than three dimensions.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zarankiewicz.html (2 of 3) [2/16/2002 11:39:42 PM]

Zarankiewicz

Zarankiewicz made several other important contributions to mathematics. From 1949 until 1957 he coached the Polish Olympiad team of school pupils. He served as president of the Warsaw section of the Polish Mathematical Society from 1948 till 1951. His death in London in 1959 came during the Tenth Congress of the International Astronautical Federation of which he was the President. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zarankiewicz.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zarankiewicz.html (3 of 3) [2/16/2002 11:39:42 PM]

Zaremba

Stanislaw Zaremba Born: 3 Oct 1863 in Romanowka, Ukraine Died: 23 Nov 1942 in Kraków, Poland

Click the picture above to see four larger pictures

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Stanislaw Zaremba's father was an engineer. Zaremba attended secondary school in St Petersburg then, after graduating, he studied engineering at the Institute of Technology in that city. He was awarded his engineering diploma in 1886 and he then went to Paris where he studied mathematics for his doctorate at the Sorbonne. As a topic for his doctorate Zaremba looked to build on ideas introduced by Riemann in 1861. His doctoral thesis Sur un problème concernant l'état calorifique d'un corp homogène indéfini was presented in 1889. Zaremba made many contacts with mathematicians of the French school at this time which would provide him with international collaborators after returning to Poland. In particular he collaborated with Painlevé and Goursat. For eleven years he taught in schools in France, during which time he concentrated hard on his research. The fact that he published his results in French mathematical journals meant that his work became well known and highly respected by leading French mathematicians such as Poincaré and Hadamard. Zaremba returned to Poland in 1900 where he was appointed to a chair in the Jagiellonian University in Kraków. In the following years he achieved much in teaching, writing textbooks, and organising the progress of mathematics in Kraków. Stanislaw Golab, a differential geometer, wrote on the history of mathematics in Poland. He described Zaremba's teaching style (see [1]):[Zaremba's] teaching was characterised by absolute rigour and an insistence on an exposition of a subject's subtleties. His lecturing style employed long and convoluted sentences, whose logical progression became clear only after closer scrutiny. He enjoyed working on and solving difficult problems that bogged down other researchers. Always taking a philosophical view of a problem, Zaremba combined physical intuition with http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zaremba.html (1 of 3) [2/16/2002 11:39:44 PM]

Zaremba

enormous erudition, a method that enabled him to connect seemingly unrelated problems. When the Mathematical Society of Kraków was set up in 1919, Zaremba chaired the inaugural meeting and was elected as the first President of the Society. This Society went on to became the Polish Mathematical Society in 1920. For many years he served the Society as editor of the Annals of the Polish Mathematical Society. From very unpromising times up to World War I, with the recreation of the Polish nation at the end of that war, Polish mathematics entered a golden age. Zaremba played a crucial role in this transformation. Slebodzinski, one of the mathematicians to work for the re-establishment of Polish mathematics after the Nazi destruction of the World War II, stressed the importance of the role of Zaremba in the creation of the golden age between the wars (see for example [2]):With the appearance of these two scholars [Zaremba and Zorawski], Polish mathematics ceased to consume exclusively other people's thoughts and results and from that moment onwards began to participate actively in the development of its own science. The political circumstances of the period were such that, for a decade or more, Stanislaw Zaremba and Kazimierz Zorawski were the only representatives of Polish mathematics in contact with foreign countries. Much of Zaremba's research work was in partial differential equations and potential theory. He also made major contributions to mathematical physics and to crystallography. He made important contributions to the study of viscoelastic materials around 1905. He showed how to make tensorial definitions of stress rate that were invariant to spin and thus were suitable for use in relations between the stress history and the deformation history of a material. He studied elliptic equations and in particular contributed to the Dirichlet principle. In [3] his contribution is described as follows:In the work of the eminent Polish mathematician Stanislaw Zaremba (1863 - 1942), the problem of an axiomatic development of classical mechanics plays an important role, as is well known, this problem constitutes part of Hilbert's Sixth Problem. Starting with the works of G Hamel, this question has been studied by many specialists in mechanics, mathematics and logic. In [3] the authors describe Zaremba's axiomatic justification of the notion of time in classical mechanics which he worked on during the period from 1933 to 1940. I have spoken to someone who was a student at the Jagiellonian University in Kraków during this last period of Zaremba's life. By this time Zaremba was essentially retired from normal teaching duties but still came to give special lectures and was often seen by the students who held him in great respect and in some awe. He had a reputation as a hard examiner, someone who would expect a lot of his students and who thought up hard problems to spring on them in oral examinations. There was another side to him, however, for despite this reputation as a hard examiner, Zaremba showed great kindness and understanding to students who approached the oral examination in fear and trembling. Lebesgue, someone who seldom heaped praise on his colleagues, paid tribute to him in 1930 when Zaremba received an honorary degree from the Jagiellonian University in Kraków (see for example [1] or [2]):Zaremba's scientific activity influenced so many research areas that his name cannot be

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zaremba.html (2 of 3) [2/16/2002 11:39:44 PM]

Zaremba

unknown to anyone interested in mathematics. However, it seems that the power of the methods he created, and the originality of his imagination, can be appreciated best by those who work in the area of mathematical physics. There he showed his style and his name is imprinted forever. On the same occasion in 1930, Hadamard also described Zaremba's contributions (see for example [2]):One cannot help mentioning the ideas which he inspired in the domain of research pertaining to those fields to which French science of the present century has devoted the most effort. The profound generalisation due to him has recently transformed the foundations of potential theory and immediately became the starting point of research by young mathematicians of the French school. That generalisation, in a degree truly unexpected in that field, is marked by that simplicity and elegance which characterise ideas pertinently and profoundly grasping the nature of things. And as for my speciality, why, how could I forget the splendid results in the domain of mixed boundary problems and of harmonic functions, as well as of hyperbolic equations, research by means of which he opened a new path along which contemporary knowledge will proceed in the near future. Zaremba received many honours. In addition to the honorary degree from the Jagiellonian University mentioned above, he received honorary degrees from Caen and Poznan. He was elected to the Soviet Academy in 1925. Kuratowski writes in [2]:Stanislaw Zaremba is the pride of Polish science. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zaremba.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zaremba.html (3 of 3) [2/16/2002 11:39:44 PM]

Zariski

Oscar Zariski Born: 24 April 1899 in Kobrin, Belarus, Russian Empire Died: 4 July 1986 in Brookline, Massachusetts, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Oscar Zariski's father was Bezalel Zaritsky and his mother Hannah Zaritsky. Oscar, born into this Jewish family, was named Ascher Zaritsky by his parents. We will comment below on how his name came to be changed to the now familiar version of Oscar Zariski. Oscar's mother was the one who ensured that her young son had a good education. A tutor was provided for Oscar from the time he was seven years old and Oscar, under the guidance of the tutor, showed remarkable aptitude for the Russian language and for arithmetic. When the fighting associated with World War I reached Belarus, Oscar's family fled to Chernigov in the Ukraine. It was the first of several moves forced on him by political problems. Unable to enter the faculty of mathematics at the University of Kiev as all the places were full, he chose philosophy instead. He was a student of philosophy at Kiev from 1918 to 1920. However he was able to pursue his mathematical interests and studied algebra and number theory in addition to philosophy. However Zariski had carried out his studies through a period of turmoil in Kiev. In January 1918 Ukraine had become an independent state with Kiev as its capital. In the following month minor uprisings by workers in Kiev were suppressed but Red Army troops entered Kiev to give support to the workers. Kiev was then occupied by the Germans, but with the end of the war in November 1918, an independent Ukraine was declared again in Kiev. In November 1919 Kiev was briefly taken by the White armies, soon after to be replaced by the Red Army. There then followed the Russian-Polish War and, in May 1920, the Polish army captured Kiev but were forced out in a counterattack. Life for Zariski was just too difficult in this city so devastated by war, so he decided to go to Italy to continue his studies. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zariski.html (1 of 4) [2/16/2002 11:39:46 PM]

Zariski

In Rome Zariski came under the influence of the great algebraic geometers Castelnuovo, Enriques and Severi. He obtained a doctorate from Rome in 1924 for a doctoral thesis on a topic related to Galois theory which was proposed to him by Castelnuovo. Zariski married in 1924; having met his future wife Yole Cagli in Rome they returned to Zariski's home town of Kobin to marry. Returning to Rome he remained there as a fellow of the International Education Board until 1927. It was while Zariski was in Rome that Enriques suggested that Ascher Zaritsky, as he was then called, change his name to the Italian sounding Oscar Zariski. This was the name which he used on his first publication which was a joint paper with Enriques. Zariski wrote in [2] of how his mathematical interests differed from those of his supervisors Castelnuovo and Enriques:However, even during my Rome period, my algebraic tendencies were showing and were clearly perceived by Castelnuovo who once told me: "You are here with us but are not one of us." This was said not in reproach but good naturedly, for Castelnuovo himself told me time and time again that the methods of the Italian geometric school had done all they could do, had reached a dead end, and were inadequate for further progress in the field of algebraic geometry. Zariski had gone to Italy to escape the problems in Belarus and the Ukraine. However, the political situation in Italy began to deteriorate rapidly. In October 1922 Mussolini organized the Fascist "March on Rome" and he was asked to form a government. For 18 months he ran the country in reasonably democratic way but, during the years 1925 to 1927, he removed the right of free speech, and removed opposition parties and trade unions. The Fascist hatred of Jews made life for Zariski, because of his Jewish background, particularly difficult. Helped by Lefschetz, he escaped from the political problems of Italy in 1927 and went to the United States. There he taught at Johns Hopkins University, being a Johnston Scholar until 1929 when he joined the Faculty. He became a full professor at Johns Hopkins in 1937. Now Castelnuovo and Severi had encouraged Zariski to view Lefschetz's topological methods as being the road ahead for algebraic geometry, so between 1927 and 1937 Zariski frequently visited Lefschetz at Princeton. Zariski wrote [2]:I owe a great deal to [Lefschetz] for his inspiring guidance and encouragement. During this period Zariski wrote Algebraic Surfaces which was published in 1935. He explains in [2] how writing this monograph changed the direction of his work:At that time (1935) modern algebra had already come to life (through the work of Emmy Noether and the important treatise of B L van der Waerden), but while it was being applied to some aspects of the foundations of algebraic geometry by van der Waerden ... the deeper aspects of birational algebraic geometry ... were largely, or even entirely, virgin territory as far as algebraic exploration was concerned. In [Algebraic Surfaces] I tried my best to present the underlying ideas of the ingenious geometric methods and proofs with which the Italian geometers were handling these deeper aspects of the whole theory of surfaces ... I began to feel distinctly unhappy about the rigour of the original proofs (without losing in the least my admiration for the imaginative geometric spirit that permeated these proofs); I became convinced that the whole structure must be done over again by purely algebraic

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zariski.html (2 of 4) [2/16/2002 11:39:46 PM]

Zariski

methods. At Johns Hopkins University between 1939 and 1940 Zariski carried out his project of applying modern algebra to the foundations of algebraic geometry. He worked on the theory of normal varieties, local uniformisation and the reduction of singularities of algebraic varieties. An important year for Zariski was 1945 which he spent in Sao Paolo. There he gave a lecture course three days each week which was attended by André Weil and nobody else. Both Zariski and Weil learnt much in discussions, often arguments, about the material that Zariski was presenting. After spending the year 1946-47 at the University of Illinois, Zariski was appointed to a chair at Harvard where he was to remain until he retired in 1969. From the late 1970s he suffered from Alzheimer's disease and his last few years were difficult ones as his health failed. In 1981 Zariski was awarded the Steele Prize by the American Mathematical Society for the cumulative influence of his total mathematical research. The citation for the prize summarised Zariski's contributions to mathematics throughout his life [2]:After beginning his work in Italy in 1924 very much in the style of "Italian algebraic geometry," Zariski realised that the whole subject needed proper foundations. Thus in the period 1927 to 1937 he turned first to topological questions and then in 1937 he began to lay the commutative algebraic foundations of his subject. His topological work concentrated mainly on the fundamental group; many of the ideas he pioneered were innovations in topology as well as algebraic geometry and have developed independently in the two fields since then. In 1937 Zariski completely reoriented his research and began to introduce ideas from abstract algebra into algebraic geometry. Indeed, together with B L van der Waerden and André Weil, he completely reworked the foundations of the subject without the use of topological or analytic methods. His use of the notions of integral independence, valuation rings, and regular local rings, in algebraic geometry proved particularly fruitful and led him to such high points as the resolution of singularities for threefolds in characteristic 0 in 1944, the clarification of the notion of simple point in 1947, and the theory of holomorphic functions on algebraic varieties over arbitrary ground fields. The theory of equisingularity and saturation begun by Zariski in 1965 has also been of great influence and importance. All of Zariski's work has served as a basis for the present flowering of algebraic geometry and the current school uses his work and ideas in the modern development of the subject. Zariski's most famous book is Commutative Algebra, a two volume work written jointly with P Samuel. The first volume appeared in 1958, the second in 1960. The American Mathematical Society played a large role in Zariski's life and he contributed greatly to the Society over many years. He was vice president of the Society between 1960 and 1961 and president of the Society from 1969 to 1970. Zariski played an important role in mathematical publishing after his appointment as a full professor. He was an editor of the American Mathematical Journal from 1937 to 1941, served as a member of the editorial committee of the Transactions of the American Mathematical Society from 1941 to 1947, and also served on the editorial boards of the Annals of Mathematics and the American Journal of Mathematics. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zariski.html (3 of 4) [2/16/2002 11:39:46 PM]

Zariski

After going to the United States in 1927 Zariski spent considerable periods lecturing at other universities both in the United States and in other countries. We have already mentioned that he was a visiting professor at San Paulo in 1945 and a visiting professor at the University of Illinois in 1946-47. Before that, in 1936, he had lectured at the University of Moscow. Later, he lectured at Kyoto (1956), the Institut des Hautes Etudes Scientifique (1961 and again 1967), and the University of Cambridge (1972). He was awarded many honours for his work in addition to the Steele Prize described above. He was awarded the Cole Prize in Algebra from the American Mathematical Society in 1944 for four papers on algebraic varieties, two published in the American Journal of Mathematics in 1939 and 1940, and the other two in the Annals of Mathematics also one in 1939 and the second in 1940. He was awarded the National Medal of Science in 1965. Many academies and societies have honoured him by electing him to membership, including the U.S. National Academy of Sciences (1944), the American Academy of Arts and Sciences (1948), the American Philosophical Society (1951), the Brazilian Academy of Sciences (1958), and the Accademia Nazionale dei Lincei (1958). Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (5 books/articles) A Poster of Oscar Zariski

Mathematicians born in the same country

Honours awarded to Oscar Zariski (Click a link below for the full list of mathematicians honoured in this way) American Maths Society President

1969 - 1970

AMS Colloquium Lecturer

1947

AMS Cole Prize

Awarded 1944

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zariski.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zariski.html (4 of 4) [2/16/2002 11:39:46 PM]

Zassenhaus

Hans Zassenhaus Born: 28 May 1912 in Koblenz-Moselweiss, Germany Died: 21 Nov 1991 in Columbus, Ohio, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hans Zassenhaus's secondary education was at two schools in Hamburg, graduating from the Lichtwark Schule in 1930 and in the year entering the University of Hamburg. At first he studied mathematics and physics with the intention of specialising in atomic physics. However he had fine mathematics teachers in Artin and Hecke and, particularly Artin inspired him to undertake research in mathematics. Zassenhaus studied for his doctorate under Artin's supervision. During this time he proved Zassenhaus's lemma, a beautiful result on subgroups which can be used to give a simple proof of the Jordan-Hölder theorem. In his doctoral dissertation of 1934 he considered permutation groups whose elements are determined by the images of three points. These groups are called Zassenhaus groups today. In his dissertation Zassenhaus classified all 3-fold transitive Zassenhaus groups. These groups play an important role in the classification of finite simple groups coordinated by Gorenstein. From 1934 to 1936 Zassenhaus worked at the University of Rostock and wrote his famous group theory text Lehrbuch der Gruppentheorie (1937) based on Artin's lectures at Hamburg. He became Artin's assistant at Hamburg in 1936. His habilitation of 1938 studied Lie rings of prime characteristic. Zassenhaus found that a normal academic career was made impossible for him because of his intense dislike of the Nazi party. He worked on weather forecasting during World War II but, when offered the chair of mathematics at Bonn in 1943 he asked that he could postpone a decision until the end of the war.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zassenhaus.html (1 of 2) [2/16/2002 11:39:48 PM]

Zassenhaus

After the war he did not accept the Bonn post, preferring that it went to someone who had lost their chair under the Nazis. He continued to work at Hamburg, spending session 1948-49 at Glasgow in Scotland, then, in 1949, he accepted a chair of mathematics at Montreal. Although he visited many other universities during the next 10 years, he remained in the post at Montreal until he moved to the University of Notre Dame in 1959. Five years later he accepted the post of research professor at Ohio State University and held this position until he retired. We have mentioned his work in group theory and Lie rings above. The work on Lie rings extended to Lie algebras and he developed computational methods for studying them. In a long series of papers he applied Lie algebras to problems of theoretical physics. His work on computational algebraic number theory seems to have started when he visited Caltec in 1959 and collaborated with Taussky-Todd. He put forward a programme to develop methods for computational number theory which, given an algebraic number field, involved calculating its Galois group, an integral basis, the unit group and the class group. He contributed himself in a major way to all four of these tasks. Zassenhaus worked on a broad range of topics and, in addition to those mentioned above, he worked on nearfields, the theory of orders, representation theory, the geometry of numbers and the history of mathematics. He loved teaching and wrote several articles on the topic such as On the teaching of algebra at the pre-college level. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles) A Poster of Hans Zassenhaus

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR January 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zassenhaus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zassenhaus.html (2 of 2) [2/16/2002 11:39:48 PM]

Zeckendorf

Edouard Zeckendorf Born: 2 May 1901 in Liège, Belgium Died: 16 May 1983 in Liège, Belgium

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Eduourd Zeckendorf was an amateur mathematician whose name is given to the property that every positive integer can be represented uniquely as the sum of non-consecutive Fibonacci numbers, the sequence defined by F1 = F2 = 1 and Fn = Fn-1 + Fn-2 for n > 2. This is called Zeckendorf's theorem, and the subsequence of Fibonacci numbers which add up to a given integer is called its Zeckendorf representation. (Because F1 = F2, we need to exclude F1 from the representation to give uniqueness.) For example, 71 = 55 + 13 + 3, 1111 = 987 + 89 + 34 + 1. Zeckendorf qualified as a medical doctor, became an officer in the Belgian army in 1925 and subsequently also qualified as a dentist. Following the surrender of the Belgian army in May 1940 Zeckendorf was interned as a prisoner of war until 1945. He subsequently published several mathematical papers, nearly all of them in the Bulletin de la Société Royale des Sciences de Liège, mainly on elementary number theory. Article by: G M Phillips, St Andrews Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles)

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeckendorf.html (1 of 2) [2/16/2002 11:39:50 PM]

Zeckendorf

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR August 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zeckendorf.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeckendorf.html (2 of 2) [2/16/2002 11:39:50 PM]

Zeeman

Erik Christopher Zeeman Born: 4 Feb 1925

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Chris Zeeman's father Christian Zeeman was from Aarhus in eastern Jutland, Denmark. Zeeman was educated at Christ's Hospital School in Horsham, West Sussex, England. He served as a Flying Officer with the Royal Air Force from 1943 to 1947. Zeeman's university studies were at Christ's College Cambridge and he received his B.A. and Ph.D. from Cambridge. Zeeman was awarded a fellowship by Gonville and Caius College Cambridge in 1953. After being awarded a Commonwealth Scholarship he spent the year 1954-55 partly at the University of Chicago and partly at Princeton. Back at Cambridge, he was appointed a College Lecturer in 1955. During 1962-63 Zeeman was a member of the Institut des Hautes Etudes Scientifique. Then in 1964 he made the biggest move of his life when he went to the new University of Warwick in Coventry. There he lead the setting up of the Department of Mathematics and the Mathematics Research Centre. When the University took in its first undergraduates in October 1965, it seemed as though mathematics at Warwick was already established with an international reputation. This was largely due to Zeeman's remarkable leadership. Zeeman's style of leadership at Warwick was a very informal one. It produced an atmosphere in which mathematical research flourished. From 1964 Zeeman remained at Warwick until 1988, although he did spend 1966-67 as a visiting professor at the University of California at Berkeley. From 1976 till 1981 he held a senior SERC fellowship which enabled him to concentrate on research. He also held a visiting fellowship at Oxford during 1985-86. In 1988 Zeeman left Warwick, although he was made an honorary professor there on leaving. At this point he became Principal of Hertford College, Oxford and Gresham professor of geometry at Gresham College London. He retired from this post at Gresham College in 1994 and from his position of Principal of Hertford College in 1995. http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeeman.html (1 of 3) [2/16/2002 11:39:52 PM]

Zeeman

Zeeman has held important roles within UK mathematics. He served on the SERC Mathematics Committee from 1982 to 1985 and, in 1990, he chaired the committee which set up the Isaac Newton Institute in Cambridge. He continues to serve on the Steering Committee for the Isaac Newton Institute. Zeeman's research has been in a variety of areas such as topology, in particular PL topology, dynamical systems and mathematical applications to biology and the social sciences. His initial research was in topology and one of his theorems was the unknotting of spheres in five dimensions. Certainly his work in topology would make him one of the leading topologists of all time but he may be known principally for other work. Perhaps he is best known for his work on catastrophe theory for, although this theory was due initially to René Thom, it was Zeeman who brought it before the general public giving widespread publicity to applications of what was before that time thought of as pure mathematics. In particular Zeeman pioneered the applications of catastrophe theory in the biological and behavioural sciences, as well as the physical sciences. He invented the Zeeman Catastrophe machine which was a mechanical device to illustrate how a small perturbation can give rise to a discontinuous consequence. Among the books which Zeeman has published are the texts Catastrophe theory (1977), Geometry and perspective (1987) and Gyroscopes and boomerangs (1989). One of his many memorable quotes, from his Catastrophe theory text, says much about mathematical philosophy:Technical skill is mastery of complexity while creativity is mastery of simplicity. A shorter introduction to catastrophe theory than his 1977 book was given by Zeeman in his beautifully written survey article Bifurcation and catastrophe theory, Contemp. Math. (1981). The article introduces catastrophe theory in a unified way giving both elementary and non-elementary aspects. There is an elementary discussion of the cusp and the pitchfork and a statement of the classification theorem for elementary catastrophes. In 1978, Zeeman gave the Christmas Lectures at the Royal Institution, out of which grew the Mathematics Master classes for 13-year old children that now flourishes in forty centres in the United Kingdom. He was the 63rd President of the London Mathematical Society in 1986-88 and delivered the Presidential Address to the Society on 18 November 1988 On the classification of dynamical systems. He was awarded the Senior Whitehead Prize of the London Mathematical Society in 1982. During his period as president of the Society, he became the Society's first Forder lecturer in 1987. Elected to the Royal Society of London in 1975, he was awarded the Society's Faraday Medal in 1988. Zeeman was knighted in 1991 and he has received many honours in addition to those mentioned above. He has been awarded honorary degrees from many universities including Strasbourg (1974), Hull (1984), Warwick (1988), York (1988), Leeds (1990), Durham (1990) and Hartford (1992). I [EFR] first met Chris Zeeman in 1965 when I went to Warwick as a postgraduate student. He leapt out of his office to greet the new postgraduates with "Hi, I'm Chris". He is an exceptional lecturer with a remarkable ability to convince his audience that they understand the deep concepts that he is explaining, either in a research seminar of talking to non-mathematicians. The first year that Warwick opened for undergraduates, all the undergraduates and postgraduates could get into one lecture theatre. There was a course covering all aspects of study including arts, science and http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeeman.html (2 of 3) [2/16/2002 11:39:52 PM]

Zeeman

mathematics. Chris Zeeman gave the mathematics lectures and explained to an audience, most of whom had no more than a low level school mathematics qualification, knotting and unknotting spheres in high dimensions. The remarkable thing was that everyone said they understood what he was talking about! Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article)

A Quotation

Mathematicians born in the same country Honours awarded to Chris Zeeman (Click a link below for the full list of mathematicians honoured in this way) Fellow of the Royal Society

Elected 1975

London Maths Society President

1986 - 1988

Other Web sites

D Gokhman

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR June 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zeeman.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeeman.html (3 of 3) [2/16/2002 11:39:52 PM]

Zelmanov

Efim I Zelmanov Born: 7 Sept 1955 in USSR

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Efim Zelmanov attended Novosibirsk State University, obtaining his Master's degree in 1977. On being awarded this degree he was appointed to the staff at Novosibirsk State University and taught there while continuing with his own research. He received his Ph.D. from Novosibirsk State University in 1980 having had his research supervised by Shirshov and Bokut. The thesis he presented for his Ph.D. was on nonassociative algebra. In particular his work completely changed the whole of the subject of Jordan algebras by extending results from the classical theory of finite dimensional Jordan algebras to infinite dimensional Jordan algebras. Zelmanov described this work on Jordan algebras in his invited lecture to the International Congress of Mathematicians at Warsaw in 1983. In 1980 Zelmanov was appointed as a Junior Researcher at the Institute of Mathematics of the Academy of Sciences of the USSR at Novosibirsk. On the award of his doctorate (habilitation) in 1985, he was promoted to Senior Researcher. He was promoted again at the Institute of Mathematics of the Academy of Sciences in 1986, this time becoming a Leading Researcher. In 1987 Zelmanov solved one of the big open questions in the theory of Lie algebras. He proved that the Engel identity ad(y)n = 0 implies that the algebra is necessarily nilpotent. This was a classical result for finite dimensional Lie algebras but Zelmanov solved a big open problem when he proved that the result also held for infinite dimensional Lie algebras. In 1990 Zelmanov was appointed a professor at the University of Wisconsin-Madison in the United States. He held this appointment until 1994 when he was appointed to the University of Chicago. In 1995 http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zelmanov.html (1 of 4) [2/16/2002 11:39:54 PM]

Zelmanov

he spent the year at Yale University. The results mentioned above on Jordan algebras and Lie algebras would have guaranteed Zelmanov a place as one of the great algebraists of the 20th century. However, in 1991, Zelmanov went on to settle one of the most fundamental results in the theory of groups which had occupied group theorists throughout the 20th century. He solved the restricted Burnside problem. In 1994 Zelmanov was awarded a Fields Medal for this work at the International Congress of Mathematicians in Zurich in 1994. Let me explain the background to the restricted Burnside problem, the solution of which was the main reason for the award of the Medal, and also explain how Zelmanov, not a group theorist by training, came to solve one of the most fundamental questions in group theory. In 1902 Burnside first asked whether a finitely generated group in which every element has finite order, is finite. This problem is known as the General Burnside problem. The Burnside problem asks whether, for fixed d and n, the group B(d, n) having d generators and in which every element satisfies xn = 1, is finite. It is really easy to show the B(d, 2) is finite. Burnside himself showed that B(d, 3) is finite, Sanov showed B(d, 4) is finite and Marshall Hall showed B(d, 6) is finite. By the 1930s no real progress had been made on either of these problems and the Restricted Burnside problem was formulated (and so named by Magnus). It asks whether, for fixed d and n, there is a largest finite d generator group in which every element satisfies xn = 1. This is equivalent to saying that a positive solution to the Restricted Burnside problem would show that there are only finitely many finite factor groups of B(d, n). The General Burnside problem was shown to have a negative solution by Golod in 1964. In 1968 Novikov and Adian showed that the Burnside problem was false for large n. The greatest early contribution to the Restricted Burnside problem was by Hall and Higman in 1956 where they showed that, if the Schreier conjecture holds, then the Restricted Burnside problem has a positive solution if it could be proved for all prime powers n. The Schreier conjecture, that the outer automorphism groups of finite simple groups are soluble, was shown to be true as a consequence of the classification of finite simple groups. Magnus had reduced the case of the Restricted Burnside problem for n prime to a question about whether Lie algebras satisfying an Engel condition are locally nilpotent. Kostrikin, in 1959, proved that such Lie algebras were indeed locally nilpotent. However Kostrikin's proof is not entirely satisfactory and a corrected version only appeared much later. When Zelmanov began to work on the Restricted Burnside problem there were two major difficulties in pushing what had been achieved for n = p to n = pk. Firstly it there was no reduction of the problem to Lie algebras with the Engel condition, This Zelmanov achieved in 1989. Zelmanov next set about proving that a Lie algebra with an Engel condition was locally nilpotent. This he achieved in two papers, the first dealing with odd prime characteristic and the second dealing with n = pk which corresponds to Lie algebras of characteristic 2. Shalev writes in [3]:His stunning proof ... combines an amazing technical capability with highly original ideas from various disciplines. The proof uses a deep structure theory for (quadratic) Jordan http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zelmanov.html (2 of 4) [2/16/2002 11:39:54 PM]

Zelmanov

algebras, previously developed by McCrimmon and Zelmanov, as well as divided powers and other tools; it also relies on the joint work of Kostrikin and Zelmanov, which establishes the local nilpotency of the so-called sandwich algebras. While Lie algebras have long been considered a natural playground in the context of the Restricted Burnside problem, the appearance of Jordan algebras is unprecedented and quite surprising. At the Groups-St Andrews conference at Galway, Ireland in 1993, of which I [EFR] was a joint organiser, Zelmanov was one of the main speakers and he gave a series of five lectures on Nil rings methods in the theory of nilpotent groups. His lectures were beautifully constructed, models of clarity, showing what had been achieved and presenting many glimpses of possible directions for future research. Filled with humour, they were all delivered with Zelmanov's infectious twinkle in his eyes. In addition to the Fields Medal, Zelmanov has received other honours for his outstanding work. He received the College de France Medal in January 1992 and the Andre Aizenstadt Prize in May 1996. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (3 books/articles) Mathematicians born in the same country Other references in MacTutor

1. Chronology: 1980 to 1990 2. Chronology: 1990 to 2000

Honours awarded to Efim Zelmanov (Click a link below for the full list of mathematicians honoured in this way) Fields' Medal

Awarded 1994

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1998

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zelmanov.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zelmanov.html (3 of 4) [2/16/2002 11:39:54 PM]

Zelmanov

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zelmanov.html (4 of 4) [2/16/2002 11:39:54 PM]

Zeno_of_Elea

Zeno of Elea Born: about 490 BC in Elea, Lucania (now southern Italy) Died: about 425 BC in Elea, Lucania (now southern Italy)

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Very little is known of the life of Zeno of Elea. We certainly know that he was a philosopher, and he is said to have been the son of Teleutagoras. The main source of our knowledge of Zeno comes from the dialogue Parmenides written by Plato. Zeno was a pupil and friend of the philosopher Parmenides and studied with him in Elea. The Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, had been founded by Parmenides in Elea in southern Italy. His philosophy of monism claimed that the many things which appear to exist are merely a single eternal reality which he called Being. His principle was that "all is one" and that change or non-Being are impossible. Certainly Zeno was greatly influenced by the arguments of Parmenides and Plato tells us that the two philosophers visited Athens together in around 450 BC. Despite Plato's description of the visit of Zeno and Parmenides to Athens, it is far from universally accepted that the visit did indeed take place. However, Plato tell us that Socrates, who was then young, met Zeno and Parmenides on their visit to Athens and discussed philosophy with them. Given the best estimates of the dates of birth of these three philosophers, Socrates would be about 20, Zeno about 40, and Parmenides about 65 years of age at the time, so Plato's claim is certainly possible. Zeno had already written a work on philosophy before his visit to Athens and Plato reports that Zeno's book meant that he had achieved a certain fame in Athens before his visit there. Unfortunately no work by Zeno has survived, but there is very little evidence to suggest that he wrote more than one book. The book Zeno wrote before his visit to Athens was his famous work which, according to Proclus, contained forty paradoxes concerning the continuum. Four of the paradoxes, which we shall discuss in detail below, http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeno_of_Elea.html (1 of 5) [2/16/2002 11:39:56 PM]

Zeno_of_Elea

were to have a profound influence on the development of mathematics. Diogenes Laertius [10] gives further details of Zeno's life which are generally thought to be unreliable. Zeno returned to Elea after the visit to Athens and Diogenes Laertius claims that he met his death in a heroic attempt to remove a tyrant from the city of Elea. The stories of his heroic deeds and torture at the hands of the tyrant may well be pure inventions. Diogenes Laertius also writes about Zeno's cosmology and again there is no supporting evidence regarding this, but we shall give some indication below of the details. Zeno's book of forty paradoxes was, according to Plato [8]:... a youthful effort, and it was stolen by someone, so that the author had no opportunity of considering whether to publish it or not. Its object was to defend the system of Parmenides by attacking the common conceptions of things. Proclus also described the work and confirms that [1]:... Zeno elaborated forty different paradoxes following from the assumption of plurality and motion, all of them apparently based on the difficulties deriving from an analysis of the continuum. In his arguments against the idea that the world contains more than one thing, Zeno derived his paradoxes from the assumption that if a magnitude can be divided then it can be divided infinitely often. Zeno also assumes that a thing which has no magnitude cannot exist. Simplicius, the last head of Plato's Academy in Athens, preserved many fragments of earlier authors including Parmenides and Zeno. Writing in the first half of the sixth century he explained Zeno's argument why something without magnitude could not exist [1]:For if it is added to something else, it will not make it bigger, and if it is subtracted, it will not make it smaller. But if it does not make a thing bigger when added to it nor smaller when subtracted from it, then it appears obvious that what was added or subtracted was nothing. Although Zeno's argument is not totally convincing at least, as Makin writes in [25]:Zeno's challenge to simple pluralism is successful, in that he forces anti-Parmenideans to go beyond common sense. The paradoxes that Zeno gave regarding motion are more perplexing. Aristotle, in his work Physics, gives four of Zeno's arguments, The Dichotomy, The Achilles, The Arrow, and The Stadium. For the dichotomy, Aristotle describes Zeno's argument (in Heath's translation [8]):There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end. In order the traverse a line segment it is necessary to reach its midpoint. To do this one must reach the 1/4 point, to do this one must reach the 1/8 point and so on ad infinitum. Hence motion can never begin. The argument here is not answered by the well known infinite sum 1/

2

+ 1/4 + 1/8 + ... = 1

On the one hand Zeno can argue that the sum 1/2 + 1/4 + 1/8 + ... never actually reaches 1, but more perplexing to the human mind is the attempts to sum 1/2 + 1/4 + 1/8 + ... backwards. Before traversing a http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeno_of_Elea.html (2 of 5) [2/16/2002 11:39:56 PM]

Zeno_of_Elea

unit distance we must get to the middle, but before getting to the middle we must get 1/4 of the way, but before we get 1/4 of the way we must reach 1/8 of the way etc. This argument makes us realise that we can never get started since we are trying to build up this infinite sum from the "wrong" end. Indeed this is a clever argument which still puzzles the human mind today. Zeno bases both the dichotomy paradox and the attack on simple pluralism on the fact that once a thing is divisible, then it is infinitely divisible. One could counter his paradoxes by postulating an atomic theory in which matter was composed of many small indivisible elements. However other paradoxes given by Zeno cause problems precisely because in these cases he considers that seemingly continuous magnitudes are made up of indivisible elements. Such a paradox is 'The Arrow' and again we give Aristotle's description of Zeno's argument (in Heath's translation [8]):If, says Zeno, everything is either at rest or moving when it occupies a space equal to itself, while the object moved is in the instant, the moving arrow is unmoved. The argument rests on the fact that if in an indivisible instant of time the arrow moved, then indeed this instant of time would be divisible (for example in a smaller 'instant' of time the arrow would have moved half the distance). Aristotle argues against the paradox by claiming:... for time is not composed of indivisible 'nows', no more than is any other magnitude. However, this is considered by some to be irrelevant to Zeno's argument. Moreover to deny that 'now' exists as an instant which divides the past from the future seems also to go against intuition. Of course if the instant 'now' does not exist then the arrow never occupies any particular position and this does not seem right either. Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution. As Frankel writes in [20]:The human mind, when trying to give itself an accurate account of motion, finds itself confronted with two aspects of the phenomenon. Both are inevitable but at the same time they are mutually exclusive. Either we look at the continuous flow of motion; then it will be impossible for us to think of the object in any particular position. Or we think of the object as occupying any of the positions through which its course is leading it; and while fixing our thought on that particular position we cannot help fixing the object itself and putting it at rest for one short instant. Vlastos (see [32]) points out that if we use the standard mathematical formula for velocity we have v = s/t, where s is the distance travelled and t is the time taken. If we look at the velocity at an instant we obtain v = 0/0, which is meaningless. So it is fair to say that Zeno here is pointing out a mathematical difficulty which would not be tackled properly until limits and the differential calculus were studied and put on a proper footing. As can be seen from the above discussion, Zeno's paradoxes are important in the development of the notion of infinitesimals. In fact some authors claim that Zeno directed his paradoxes against those who were introducing infinitesimals. Anaxagoras and the followers of Pythagoras, with their development of incommensurables, are also thought by some to be the targets of Zeno's arguments (see for example [10]). Certainly it appears unlikely that the reason given by Plato, namely to defend Parmenides' philosophical position, is the whole explanation of why Zeno wrote his famous work on paradoxes. The most famous of Zeno's arguments is undoubtedly the Achilles. Heath's translation from Aristotle's

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeno_of_Elea.html (3 of 5) [2/16/2002 11:39:56 PM]

Zeno_of_Elea

Physics is:... the slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead. Most authors, starting with Aristotle, see this paradox to be essentially the same as the Dichotomy. For example Makin [25] writes:... as long as the Dichotomy can be resolved, the Achilles can be resolved. The resolutions will be parallel. As with most statements about Zeno's paradoxes, there is not complete agreement about any particular position. For example Toth [29] disputes the similarity of the two paradoxes, claiming that Aristotle's remarks leave much to be desired and suggests that the two arguments have entirely different structures. Both Plato and Aristotle did not fully appreciate the significance of Zeno's arguments. As Heath says [8]:Aristotle called them 'fallacies', without being able to refute them. Russell certainly did not underrate Zeno's significance when he wrote in [13]:In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance .... Here Russell is thinking of the work of Cantor, Frege and himself on the infinite and particularly of Weierstrass on the calculus. In [2] the relation of the paradoxes to mathematics is also discussed, and the author comes to a conclusion similar to Frankel in the above quote:Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets. In the end, however, the difficulties inherent in his arguments have always come back with a vengeance, for the human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable. It is difficult to tell precisely what effect the paradoxes of Zeno had on the development of Greek mathematics. B L van der Waerden (see [31]) argues that the mathematical theories which were developed in the second half of the fifth century BC suggest that Zeno's work had little influence. Heath however seems to detect a greater influence [8]:Mathematicians, however, ... realising that Zeno's arguments were fatal to infinitesimals, saw that they could only avoid the difficulties connected with them by once and for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or decreasing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please. We commented above that Diogenes Laertius in [10] describes a cosmology that he believes is due to Zeno. According to his description, Zeno proposed a universe consisting of several worlds, composed of http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeno_of_Elea.html (4 of 5) [2/16/2002 11:39:56 PM]

Zeno_of_Elea

"warm" and "cold, "dry" and "wet" but no void or empty space. Because this appears to have nothing in common with his paradoxes, it is usual to take the line that Diogenes Laertius is in error. However, there is some evidence that this type of belief was around in the fifth century BC, particularly associated with medical theory, and it could easily have been Zeno's version of a belief held by the Eleatic School. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (35 books/articles) A Poster of Zeno of Elea

Mathematicians born in the same country

Cross-references to History Topics

1. An overview of the history of mathematics 2. The rise of the calculus

Other references in MacTutor

Chronology: 500BC to 1AD

Other Web sites

1. Interactive Real Analysis 2. The Catholic Encyclopedia 3. Kevin Brown 4. Internet Encyclopedia of Philosophy 5. S M Cohen (Zeno's paradoxes) 6. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1999

Anniversaries for the year

School of Mathematics and Statistics University of St Andrews, Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/References/Zeno_of_Elea.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeno_of_Elea.html (5 of 5) [2/16/2002 11:39:56 PM]

Zeno_of_Sidon

Zeno of Sidon Born: about 150 BC in Sidon (now Saida in Lebanon) Died: about 70 BC in Athens, Greece

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Zeno of Sidon was born in the city of Sidon on the Mediterranean coast of what today is Lebanon. Sidon was one of the oldest Phoenician cities and, from its founding in the 3rd millennium BC, was ruled by many different peoples: Assyria, Babylonia, Persia, Alexander the Great, the Seleucids of Syria, the Ptolemys of Egypt, and the Romans. To understand the philosophy of Zeno we need to make some comments about the philosopher Epicurus who founded the Epicurean School to which Zeno later belonged. Epicurus, who lived from 341BC to 270 BC, founded his own School of philosophy based on his teachings. These teachings were designed to indicate a means of living ones life, and they aimed both to guarantee happiness and to provide a means to find it. Epicurus had no interest in science for its own sake and he was a severe critic of mathematics. On science he wrote:If we were not troubled by our suspicions of the phenomena of the sky and about death, and also by our failure to grasp the limits of pain and desires, we should have no need of natural science. His criticisms of mathematics were very superficial of little importance since he clearly had very little understanding of the subject. In 306 BC he founded his School in Athens in the garden of his house. Reasonably enough the School became known as The Garden. Apollodorus, the writer of more than 400 books, was a prominent follower of Epicurus who lived in the 2nd century BC. Zeno of Sidon was a student of Apollodorus and he studied, and later taught, in the Garden in Athens. Cicero heard him teaching there in 79 BC. Zeno was a man of great learning who wrote on a very wide range of topics. It is believed that, among

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeno_of_Sidon.html (1 of 3) [2/16/2002 11:39:59 PM]

Zeno_of_Sidon

the areas he studied, he contributed to logic, atomic theory, biology, ethics, literary style, oratory, poetry, the theory of knowledge, and to mathematics. Except for the last mentioned two topics, we know very little about the contributions which he made. Here we shall discuss the only two areas to which Zeno contributed where details of his contributions are quite well known to us, namely the theory of knowledge and to mathematics. Although Epicurus, the founder of the School to which Zeno belonged, had no real mathematical abilities and criticised the subject from a position of ignorance, this is far from true of Zeno who had a deep understanding of the subject. Zeno made deep criticisms of the axioms that Euclid set out in The Elements. For example he claimed that Euclid's first proposition assumes that two straight lines can intersect in at most one point but Euclid does not have this as an axiom, nor can it be deduced from the other axioms. Zeno also attacked Euclid's proof of the equality of right angles on the grounds that it presupposes the existence of a right angle. Proclus also says that an Epicurean (almost certainly Zeno but Proclus does not name him) claimed that Euclid assumes that every curve is infinitely divisible, but again this cannot be deduced from the axioms. Some modern authors have suggested that these claims give Zeno of Sidon some justification to be considered as having been the first person to consider the possibility of non-Euclidean geometry. This is a little far fetched particularly since Zeno's aim was certainly not this. Rather his aim was to give substantial arguments against mathematics supporting the anti-mathematical beliefs of Epicurus. Heath writes in [2] regarding comments by Proclus concerning Zeno:Zeno argued generally that, even if we admit the fundamental principles of geometry, the deductions from them cannot be proved without the admission of something else as well which has not been included in the said principles, and he intended by means of these criticisms to destroy the whole of geometry. Mathematicians of course, came to the defence of their subject, rather than to try to understand the deep and justified comments of Zeno. As von Fritz writes in [1]:Zeno's criticisms of Euclid are pertinent, however, and if any of the ancient philosophers and mathematicians who tried to refute them had been able to grasp their full implications, the development of mathematics might have taken a different turn. Many people gain an important position in history, or fail to gain such a position, as a result of luck. Had there been a mathematician following Zeno who could have continued to develop his ideas then we might know Zeno today as a major figure whose flash of mathematical genius changed the course of mathematics. This was not to be, however, and the brilliance of Zeno's ideas were not appreciated for many centuries. We know more of Zeno through one of his students Philodemus of Gadara. Philodemus studied under Zeno in Athens and then moved to Rome in 75 BC to work for the Roman aristocrat Lucius Calpurnius Piso. Philodemus then went to live in Lucius's villa at Herculaneum, near Naples, taking with him his considerable library of papyri. When Vesuvius erupted in 79 AD, Herculaneum together with Pompeii and Stabiae, was destroyed. Herculaneum was buried by a compact mass of material about 16 metres deep which preserved the city http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeno_of_Sidon.html (2 of 3) [2/16/2002 11:39:59 PM]

Zeno_of_Sidon

until excavations began in the 18th century. Special conditions of humidity of the ground conserved wood, cloth, food, and in particular Philodemus's papyri. The papyri contain remarkable information written by Philodemus describing the arguments of his teacher Zeno with the Stoics. Although Zeno's Epicurean philosophy of the desire for pleasure seems the direct opposite of the Stoic's ethic of duty, the consequences on how they lived their lives were quite similar. The arguments described by Philodemus concerned the foundations of knowledge. Von Fritz writes in [1]:In this dispute Zeno defended the old Epicurean doctrine that all human knowledge is derived exclusively from experience. What make it interesting, however, is that he bases his defence on a theory ... that is essentially an anticipation of John Stuart Mill's theory of induction. ... Zeno insisted that all knowledge is fundamentally derived by inference to all cases from a great many cases without observed counter-instance. Of course there are many examples in mathematics where Zeno's observations of many cases would actually suggest a false result. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (4 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zeno_of_Sidon.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeno_of_Sidon.html (3 of 3) [2/16/2002 11:39:59 PM]

Zenodorus

Zenodorus Born: about 200 BC in Athens, Greece Died: about 140 BC in Greece Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

We know little of Zenodorus's life but he is mentioned in the Arabic translation of Diocles' On burning mirrors where it is stated [3]:And when Zenodorus the astronomer came down to Arcadia and was introduced to us, he asked us how to find a mirror surface such that when it is placed facing the sun the rays reflected from it meet a point and thus cause burning. Toomer notes that his translation of 'when Zenodorus the astronomer came down to Arcadia and was introduced to us' could, perhaps, be translated 'when Zenodorus the astronomer came down to Arcadia and was appointed to a teaching position there'. Before the discovery of the Arabic text of Diocles' On burning mirrors, Zenodorus was known to us mainly because of references to his treatise On isometric figures which is lost. There is another interesting source of information however. When Vesuvius erupted in 79 AD, Herculaneum together with Pompeii and Stabiae, was destroyed. Herculaneum was buried by a compact mass of material about 16 metres deep which preserved the city until excavations began in the 18th century. Special conditions of humidity of the ground conserved wood, cloth, food, and in particular many papyri. The papyri contain remarkable information and in particular there is a biography of the philosopher Philonides. This biography speaks of Zenodorus as a friend of Philonides and, although complete certainty is impossible, we can be confident that this reference to Zenodorus is to the mathematician described in this article. Two visits by Zenodorus to Athens are described in the biography. Despite the loss of Zenodorus's treatise On isometric figures, we do know something of the results which it contained since Theon of Alexandria quotes a number of propositions from Zenodorus's work when he is giving his commentary on Ptolemy's Syntaxis. Pappus also made use of Zenodorus's On isometric figures in Book V of his own work and in fact a comparison with what Theon of Alexandria has presented shows that Pappus followed Zenodorus's presentation rather closely. In On isometric figures Zenodorus himself follows the style of Euclid and Archimedes quite closely and he refers to results of Archimedes from his treatise Measurement of a circle. Zenodorus studied the area of a figure with a fixed perimeter and the volume of a solid figure with fixed http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zenodorus.html (1 of 2) [2/16/2002 11:40:00 PM]

Zenodorus

surface. For example he showed that among polygons with equal perimeter and an equal number of sides, the regular polygon has the greatest area. He also showed that a circle is greater than any regular polygon of the same perimeter. To do this Zenodorus makes use of Archimedes result that the area of a circle is equal to that of a right-angled triangle of perpendicular side equal to the radius of the circle and base equal to the length of the circumference of the circle. The treatise contains three-dimensional geometry results as well as two-dimensional. In particular he proved that the sphere was the solid figure of greatest surface area for a given volume.

Article by: J J O'Connor and E F Robertson List of References (4 books/articles) Mathematicians born in the same country Cross-references to History Topics

How do we know about Greek mathematicians?

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR April 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zenodorus.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zenodorus.html (2 of 2) [2/16/2002 11:40:00 PM]

Zermelo

Ernst Friedrich Ferdinand Zermelo Born: 27 July 1871 in Berlin, Germany Died: 21 May 1953 in Freiburg im Breisgau, Germany

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Ernst Zermelo's parents were Ferdinand Zermelo and Maria Augusta Elisabeth Ziegler. His father was a college professor so Zermelo was brought up in a family where academic pursuits were encouraged. His secondary school education was at the Luisenstädtisches Gymnasium in Berlin and he graduated from the gymnasium in 1889. At this time it was the custom for students in Germany to study at a number of different universities and indeed that is precisely what Zermelo did. His studies were undertaken at three universities, namely Berlin, Halle and Freiburg, and the subjects he studied were quite wide ranging and included mathematics, physics and philosophy. At these universities he attended courses by Frobenius, Lazarus, Fuchs, Planck, Schmidt, Schwarz and Edmund Husserl. This was an impressive collection of inspiring lecturers and Zermelo began to undertake research in mathematics after completing his first degree. His doctorate was completed in 1894 when the University of Berlin awarded him the degree for a dissertation Untersuchungen zur Variationsrechnung which followed the Weierstrass approach to the calculus of variations. In this thesis he [1]:... extended Weierstrass's method for the extrema of integrals over a class of curves to the case of integrands depending on derivatives of arbitrarily high order, at the same time giving a careful definition of the notion of neighbourhood in the space of curves. After the award of his doctorate, Zermelo remained at the University of Berlin where he was appointed assistant to Planck who held the chair of theoretical physics there. At this stage Zermelo's work was

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zermelo.html (1 of 4) [2/16/2002 11:40:03 PM]

Zermelo

turning more towards areas of applied mathematics and, under Planck's guidance, he began to work for his habilitation thesis studying hydrodynamics. In 1897 Zermelo went to Göttingen, perhaps the leading centre for mathematical research in the world at that time, where he completed his habilitation, submitting his dissertation Hydrodynamische Untersuchungen über die Wirbelbewegungen in einer Kugelfläche in 1899. Immediately following the award of the degree he was appointed as a lecturer at Göttingen on the strength of his contributions to statistical mechanics as well as to the calculus of variations. The direction of Zermelo's research was soon to take a major change. Cantor had put forward the continuum hypothesis in 1878, conjecturing that every infinite subset of the continuum is either countable (i.e. can be put in 1-1 correspondence with the natural numbers) or has the cardinality of the continuum (i.e. can be put in 1-1 correspondence with the real numbers). The importance of this was seen by Hilbert who made the continuum hypothesis the first in the list of problems which he proposed in his Paris lecture of 1900. Hilbert saw this as one of the most fundamental questions which mathematicians should attack in the 1900s and he went further in proposing a method to attack the conjecture. He suggested that first one should try to prove another of Cantor's conjectures, namely that any set can be well ordered. Perhaps it would be helpful to give a definition of a well ordered set at this point. A set S is well ordered if it has a relation < defined on it which satisfies three properties: (i) for any elements a, b in S either a = b, a < b or b < a. (ii) for every a, b, c in S with a < b and b < c then a < c. (iii) every non-empty subset of S has a least element. The set of natural numbers with the usual ordering is therefore a well ordered set but the the set of integers is not well ordered with the usual ordering since the subset of negative integers has no least element. Zermelo began to work on the problems of set theory, in particular taking up Hilbert's idea to head towards a resolution of the problem of the continuum hypothesis. In 1902 Zermelo published his first work on set theory which was on the addition of transfinite cardinals. Two years later, in 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved that every set can be well ordered. This result brought fame to Zermelo and also earned him a quick promotion for, in December 1905, he was appointed as professor in Göttingen. The axiom of choice is the basis for Zermelo's proof that every set can be well ordered; in fact the axiom of choice is equivalent to the well ordering property so we now know that this axiom has to be used. His proof of the well ordering property used the axiom of choice to construct sets by transfinite induction. Although Zermelo certainly gained fame for his proof of the well ordering property, set theory at this time was in the rather unusual position that many mathematicians rejected the type of proofs that Zermelo had discovered. There were strong feelings as to whether such non-constructive parts of mathematics were legitimate areas for study and Zermelo's ideas were certainly not accepted by quite a number of mathematicians [1]:The proof stirred the mathematical world and produced a great deal of criticism - most of it http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zermelo.html (2 of 4) [2/16/2002 11:40:03 PM]

Zermelo

unjustified - which Zermelo answered elegantly in Neuer Beweis... As this quote indicates, Zermelo's reaction to these criticisms was to try to prove the well ordering property with a proof that would find more widespread acceptance, and this he succeeded in doing in the paper Neuer Beweis which he published in 1908. It was a paper which he specifically directed at critics of his work. On the one hand he emphasised the formal character of his new proof of the well ordering and on the other hand he argued that his critics, and other mathematicians, also used the axiom of choice when dealing with infinite sets. Zermelo made other fundamental contributions to axiomatic set theory which were partly a consequence of the criticism of his first major contribution to the subject and partly because set theory began to become an important research topic at Göttingen. The set theory paradoxes first appeared around 1903 with the publication of Russell's paradox. Zermelo had in fact discovered a similar set paradox himself but did not publish the result. Rather it prompted him to make the first attempt to axiomatise set theory and he began this task in 1905. Having produced an axiom system he wanted to prove that his axioms were consistent before publishing the work, but he failed to achieve this. In 1908 Zermelo published his axiomatic system despite his failure to prove consistency. He gave seven axioms : Axiom of extensionality, Axiom of elementary sets, Axiom of separation, Power set axiom, Union axiom, Axiom of choice and Axiom of infinity. Zermelo usually stated his axioms and theorems in words rather than symbols. In fact he did not often use the formal language for quantifiers such as or and binding variables that were then being used, instead, he used ordinary expressions such as "there exists" or "for all". It is worth commenting that Skolem and Fraenkel independently improved Zermelo's axiom system in around 1922. The resulting system, with ten axioms, is now the most commonly used one for axiomatic set theory. It enables the contradictions of set theory to be eliminated yet the the results of classical set theory excluding the paradoxes can be derived In 1910 Zermelo left Göttingen when he was appointed to the chair of mathematics at the Zurich University. His health was poor but his position was helped by the award of a prize of 5000 marks for his major contributions to set theory. The prize was awarded on the initiative of Hilbert and certainly it was an attempt to enable Zermelo rest and so to regain his health. When his health had not improved by 1916 Zermelo resigned his chair in Zurich and moved to the Black Forest in Germany where he lived for ten years. He was appointed to an honorary chair at Freiburg im Breisgau in 1926 but he renounced his chair in 1935 because of his disapproval of Hitler's regime. At the end of the World War II Zermelo requested that he be reinstated to his honorary position in Freiburg and indeed he was reinstated to the post in 1946.

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (6 books/articles)

A Quotation

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zermelo.html (3 of 4) [2/16/2002 11:40:03 PM]

Zermelo

Mathematicians born in the same country Cross-references to History Topics

The beginnings of set theory

Other references in MacTutor

Chronology: 1900 to 1910

Other Web sites

Paul Walker (A history of Game Theory)

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1999

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zermelo.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zermelo.html (4 of 4) [2/16/2002 11:40:03 PM]

Zeuthen

Hieronymous Georg Zeuthen Born: 15 Feb 1839 in Grimstrup, West Jutland, Denmark Died: 6 Jan 1920 in Copenhagen, Denmark

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Hieronymus Georg Zeuthen's father was the minister in Grimstrup where his son was born and began his education. However Zeuthen's father moved in 1849 from a church in Grimstrup to a church in Soro and at this time Zeuthen began his secondary education. After completing his schooling in Soro in 1857 he entered the University of Copenhagen to study mathematics. At Copenhagen Zeuthen studied a broad mathematics course attending courses on topics in both pure and applied mathematics. In 1862 he graduated with a Master's degree and was awarded a scholarship to enable him to study abroad. He decided to visit Paris and there he studied geometry with Chasles. This was extremely important for Zeuthen since his research areas of mathematics were firmly shaped by Chasles during this period. The first topic on which Zeuthen undertook research was enumerative geometry. In 1865 he submitted his doctoral dissertation on a new method to determine the characteristics of conic systems to the University of Copenhagen. Haas describes the thesis in [1]:In this work Zeuthen adhered closely to Chasles's theory of the characteristics of conic systems but also presented new points of view: for the elementary systems under consideration, he first ascertained the numbers for point or line conics in order to employ them to determine the characteristics. Up until 1875 Zeuthen worked almost exclusively on enumerative geometry. He was appointed as an extraordinary professor at the University of Copenhagen in 1871 and, in the same year, he became an editor of Matematisk Tidsskrift, a position he held for 18 years. He was the secretary of the Royal Danish

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeuthen.html (1 of 3) [2/16/2002 11:40:06 PM]

Zeuthen

Academy of Arts and Sciences for 39 years, during which time he was also a lecturer at the Polytechnic Institute. He continued teaching at the University of Copenhagen where he was promoted to ordinary professor in 1886. He was twice Rector of the University. After 1875 Zeuthen's contributions to mathematics became more varied. He began to write on mechanics and he also made significant contributions to algebraic geometry, particularly the theory of algebraic surfaces. As we mentioned above, he developed the enumerative calculus, proposed by Chasles, for counting the number of curves touching a given set of curves. The move towards rigour in geometry led to this theory being neglected for many years but recently some of the remarkable numerical results produced by it have been confirmed. He was also an expert on the history of medieval mathematics and produced important studies of Greek mathematics. He wrote 40 papers and books on the history of mathematics, some of which have become classics. Unlike many historians of science Zeuthen explained the ancient texts in the manner of a colleague of the ancient mathematicians. His historical studies covered many topics and several periods. In a major work in 1885 he looked in detail at the work of Apollonius on conic sections and showed that Apollonius used oblique coordinates. Caveing, in [3], looks at Zeuthen's ideas on the discovery of irrational numbers. Zeuthen argued that Pythagoras himself discovered that 2 was irrational when computing the diagonal of a square. The passage from Plato's Theaetetus where it states that Theodorus proved the irrationality of 3, 5, ... 17 was also carefully studied by Zeuthen. He suggested that the end of Theodorus's proof somehow involved the continued fractions for 17 and 19, a conjecture which is very much in line with modern ideas about Greek mathematics. Zeuthen's largest historical work was published in 1896. It looked in detail at the work of Descartes, Viète, Barrow, Newton and Leibniz as he traced the development of algebra, analytic geometry and analysis. In [7] Lützen and Purkert compare the historical approaches of Moritz Cantor and Zeuthen. They write:Moritz Cantor and Hieronymus Georg Zeuthen were probably the two most outstanding historians of mathematics at the end of the 19th century. However, their methods of work differed strikingly. Moritz Cantor was an encyclopaedist who ... followed the development of mathematics in a survey of an almost innumerable collection of original and secondary sources from antiquity to the end of the 18th century. ... Zeuthen's papers and books, on the other hand, present deep mathematical analyses of the methods found in classical works mostly from antiquity and from the 16th and 17th centuries in an attempt to capture their fundamental ideas. Kleiman gives an interesting biography of Zeuthen in [6]. This indicates that:[Zeuthen] was in many ways a leading light of the burgeoning intellectual life of his country, and with a selfless and generous disposition he seems happily to deserve Coolidge's epithet of the "ever kindly". Finally we quote Hass's comments in [1] on Zeuthen's style. He writes:Zeuthen saw things intuitively: he constantly strove to attain an overall conception that would embrace the details of the subject under investigation and afford a way of seizing http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeuthen.html (2 of 3) [2/16/2002 11:40:06 PM]

Zeuthen

their significance. This approach characterised his historical research equally with his work on enumerative methods in geometry. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (9 books/articles) Mathematicians born in the same country Other Web sites

Theseus

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zeuthen.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zeuthen.html (3 of 3) [2/16/2002 11:40:06 PM]

Zhukovsky

Nikolai Egorovich Zhukovsky Born: 17 Jan 1847 in Orekhovo, Vladimir gubernia, Russia Died: 17 March 1921 in Moscow, USSR

Click the picture above to see three larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Nikolai Egorovich Zhukovskii (or Zhukovsky or Joukowski) was the son of Egor Zhukovskii who was a communications engineer. Nikolai Egorovich attended the Fourth Gymnasium for Men in Moscow, completing his secondary education there in 1864. He then entered the Faculty of Physics and Mathematics at Moscow University where he studied applied mathematics. He graduated in 1868 and from 1870 he taught at the Second Gymnasium for Women in Moscow. After two years teaching at the Gymnasium, Zhukovskii received an invitation to teach mathematics at Moscow Technical School then, from 1874, he also taught theoretical mechanics there. While he was teaching these courses, Zhukovskii was also studying for his Master's Degree and in 1876 he was awarded this degree for a thesis on the kinematics of a liquid. It is worth pointing out that the Russian Master's Degree is essentially the equivalent of a British/American Ph.D. today while the Russian doctorate at this time was essentially the equivalent of the German Habilitation. After being awarded his Master's Degree, a special chair of mechanics was created for Zhukovskii at Moscow Technical School. Zhukovskii obtained a doctorate from Moscow University in 1882 for a dissertation on the stability of motion. He worked at the university, becoming the Head of the Department of Mechanics in 1886. By this time he had begun to receive awards for his outstanding work, having been awarded the N D Brashman prize for theoretical work in fluid dynamics in 1885. Over his career Zhukovskii had a remarkable publications record producing over 200 publications on mechanics. In 1886 he wrote a [13]:... memoir dealing with the motion of bodies filled with a homogeneous incompressible fluid.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zhukovsky.html (1 of 3) [2/16/2002 11:40:08 PM]

Zhukovsky

... the advantages of Zhukovskii's geometrical and sound mechanical approach to the problem, [means that] his memoir still remains quite up to date. Perhaps Zhukovskii is most famous, however, as the founder of the Russian schools of hydromechanics and aeromechanics. For his work in these areas he became known as the Father of Russian Aviation. Zhukovskii [1]:... became interested in the late 1880s in flight in heavier-than-air machines, a basic problem of which was lift. During 1890-91 he experimented with disks placed in currents of air and, in 1891, he began to study the dynamics of flight. In 1895 he visited Lilienthal in Berlin. Lilienthal was selling gliders produced in his factory in Berlin. Zhukovskii [2]:... observed several of Lilienthal's flights and was most impressed. After returning to Moscow, he spoke before the Society of Friends of the Natural Sciences: "The most important invention of recent years in the area of aviation is the flying machine of the German engineer Otto Lilienthal. Zhukovskii purchased one of the eight gliders which Lilienthal sold to members of the public. In 1906 Zhukovskii published two papers in which he gave a mathematical expression for the lift on an airfoil. Today it is known as the Kutta-Joukowski theorem, since Kutta pointed out that the equation also appears in his 1902 dissertation. In 1911 Zhukovskii wrote:The field of hydrodynamic phenomena which can be explored with exact analysis is more and more increasing. Zhukovskii was concerned both with theoretical and with experimental aspects of the subject. His theoretical work concentrated on lift, high-speed aerodynamics, vortex theory, longitudinal and cross stability but he complemented this work with appropriate experimental observations in every case. With this twin approach he became the Russian pioneer on both aspects of aviation. He went on to establish an aerodynamics laboratory and to teach courses on his theories of aerodynamics [1]:His lectures at the Moscow Technical School on the theoretical basis of aeronautics (1911-12) were the world's first systematic course in aviation theory and were based largely on his own theoretical research and on experiments conducted in laboratories that he had established. In mathematics today the conformal mapping of the complex plane z z + 1/z is called the Joukowski transformation. This gave Zhukovskii [2]:... a means of designing aerofoils using conformal mappings and the techniques of complex variables. Those Joukowski aerofoils were actually used on some aircraft, and today these techniques provide a mathematically rigorous reference solution to which modern approaches to aerofoil design can be compared for validation. Click HERE for more about the aerofoils. During World War I Zhukovskii taught a special course for pilots and he was the first person in Russia to study the theory of bombing from aeroplanes in 1915. In 1918 he organised the Central Aerohydrodynamic Institute and became its first head. The Institute was http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zhukovsky.html (2 of 3) [2/16/2002 11:40:08 PM]

Zhukovsky

renamed the N E Zhukovskii Academy of Military and Aeronautical Engineering in 1922 following Zhukovskii's death. Zhukovsky also worked on hydrodynamics and hydraulics, in particular shock waves in water pipes. In particular he solved problems concerning the bursting of pipes with his studies of hydraulic shock. Other problems he considered were the formation of river beds and the construction of dams, where again his expertise was invaluable in constructing power stations. Zhukovskii's works were published in 25 volumes from 1935 to 1950. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (13 books/articles) Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR May 2000

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zhukovsky.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zhukovsky.html (3 of 3) [2/16/2002 11:40:08 PM]

Zolotarev

Egor Ivanovich Zolotarev Born: 12 April 1847 in St Petersburg, Russia Died: 19 July 1878 in St Petersburg, Russia

Click the picture above to see a larger version

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Egor Ivanovich Zolotarev's father, Ivan Zolotarev, was a watchmaker. Egor Ivanovich attended the Gymnasium in St Petersburg completing his school education in 1863. It was a school career of great distinction and he received a silver medal when he graduated. After leaving school Zolotarev enrolled at the Faculty of Physics and Mathematics of St Petersburg University. There he attended lectures by Chebyshev and A N Korkin, graduating with his first degree in 1867. He then continued his studies at the Faculty of Physics and Mathematics investigating an indeterminate equation of degree three. For this work he was awarded a Master's degree in 1869. It is worth commenting that a Master's degree in Russia at this time was closer in standard to what might be expected of a doctoral dissertation in a British or American university in the present time. The doctoral dissertation that Zolotarev went on to write would be closer to the German habilitation. In 1874 he submitted this doctoral dissertation on algebraic integers and was awarded the degree. Two years later Zolotarev was appointed as Professor of Mathematics at St Petersburg Faculty of Physics and Mathematics. He also became an assistant in applied mathematics at the St Petersburg Academy of Sciences. Then he made two trips abroad, visiting Berlin where he attended lectures by Kummer, Weierstrass and Paris where he had many mathematical discussions with Hermite. We will indicate below some of Zolotarev's remarkable mathematical achievements some of which were consequences of problems which he discussed with Kummer and Hermite while on his trips. All his results were produced over a comparatively short time period for he died two years after being appointed a professor. His career ended

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zolotarev.html (1 of 3) [2/16/2002 11:40:10 PM]

Zolotarev

when he fell under a train and died shortly after of blood poisoning. In a short eleven year career Zolotarev produced fundamental work in approximation theory, quadratic forms, algebraic numbers and elliptic integrals. During this tragically short career he published 28 papers and books. Let us first comment on his work on algebraic numbers. He studied rings of integers in algebraic number fields, giving a divisibility theory for such rings developing ideas which had been introduced by Kummer. He studied local rings and semilocal rings and proved certain results on principal ideal domains. He also introduced ideas which were essentially what today are called valuations. The paper [7] looks at both the published work of Zolotarev, and also manuscripts preserved in libraries in Moscow and St Petersburg, relating to his work on elliptic functions. Nalbandjan was able to examine Zolotarev's notebooks, including lecture notes he made while a student. Zolotarev emphasised the relationship between elliptic functions and functions of a complex variable. In particular he applied his theory of complex integers to the integration of elliptic differentials. Abel had shown when certain elliptic differentials could be integrated in logarithms but his methods was of little practical use. Zolotarev was able to give a much more effective solution. Zolotarev worked with Korkin on quadratic forms. Hermite had posed the problem of finding the minimal values of quadratic forms in n variables whose coefficients were real. They were able to give complete solutions in the case of four variables and of five variables. Zolotarev posed four problems in approximation theory all of which he was able to solve. The first two of the problems he examined attempted to minimize max{ |p(x)| : -1 x 1} over polynomials p(x) whose coefficients satisfied a given condition. The third and fourth problems were concerned with optimal approximations of a rational function on a given interval subject to given restrictions elsewhere. He found the nth degree polynomial with two of its coefficients fixed which is closest to zero. In [10] the best approximation to 1/(1 + x) on [0, 1] by a quadratic polynomial is given as an example of how Zolotarev's methods can be used. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) A Poster of Egor Ivanovich Zolotarev

Mathematicians born in the same country

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zolotarev.html (2 of 3) [2/16/2002 11:40:10 PM]

Zolotarev

Mathematicians of the day JOC/EFR March 2001

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zolotarev.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zolotarev.html (3 of 3) [2/16/2002 11:40:10 PM]

Zorn

Max Zorn Born: 6 June 1906 in Germany Died: 9 March 1993

Click the picture above to see a larger version

Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Max Zorn received his doctorate from Hamburg in 1930, his doctoral work being supervised by Artin. He was appointed to Halle but, in 1933, he was forced to leave Germany because of the Nazi policies. Zorn emigrated to the USA and worked at Yale from 1934 to 1936 where he established 'Zorn's Lemma'. Zorn then moved to the University of California where he remained until 1946 when he became professor at Indiana. As well as his well known work in infinite set theory, Zorn worked on topology and algebra. One of his early results was to prove the uniqueness of the Cayley numbers, he showed that it was the only alternative, quadratic, real nonassociative algebra without zero divisors. Herstein was one of his students. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page A Reference (One book/article) A Poster of Max Zorn

Mathematicians born in the same country

Other references in MacTutor

Chronology: 1930 to 1940

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zorn.html (1 of 2) [2/16/2002 11:40:12 PM]

Zorn

History Topics Time lines Glossary index

Societies, honours, etc. Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR December 1996

Famous curves

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zorn.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zorn.html (2 of 2) [2/16/2002 11:40:12 PM]

Zuse

Konrad Zuse Born: 22 June 1910 in Berlin-Wilmersdorf, Germany Died: 18 Dec 1995 in Hünfeld (near Fulda), Germany

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Konrad Zuse was brought up in Braunsberg, East Prussia where he attended the Humanistisches Gymnasium. He entered the Technisches Hochschule of Berlin- Charlottenburg in 1927 where he took courses in civil engineering. It was these studies in engineering which led Zuse to become interested in developing a mechanical device for calculating around 1934. He found that he had to spend many hours [4]:... working through the long statics calculations which are so important in the training of civil engineers... After graduating Zuse joined the Henschel Aircraft Company where he worked on stress analysis. In particular he studied the stresses caused by vibrations of an aircraft's wing. His work again involved a great deal of calculation and so, to help him perform these calculations, Zuse built his Z1 computer in his parents living room. He wrote:I started in 1934, working independently and without knowledge of other developments going on around me. In fact, I hadn't even heard of Charles Babbage when I embarked on my work. Zuse completed the machine in 1938. It was entirely mechanical, with an arithmetic unit composed of large numbers of mechanical switches, and a memory consisting of layers of metal bars between layers of glass. One of its most innovative features was that it could be programmed by means of a punched tape. The main reason why Zuse succeeded in building his mechanical computer where Babbage had failed, was the fact that Zuse's Z1 was a binary machine with two position switches to represent 0 and 1. However, to say that Zuse succeeded with the Z1 is a bit of an exaggeration, for the machine did not

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zuse.html (1 of 4) [2/16/2002 11:40:14 PM]

Zuse

work very well. The memory was a successful feature, the way that commands were transmitted through mechanical linkages was not successful. Zuse's plans to develop a bigger and better computer the Z2 involved keeping the same memory system but replacing the mechanical arithmetic unit by electromechanical relays. However the project was interrupted by World War II when Zuse was called up for military service. He was put in the German infantry but persuaded the army to allow him to return to building computers. The Third Reich's Aerodynamic Research Institute funded his work and he completed building the Z2 which was still an experimental computer. He then progressed to build the Z3 which was the first computer which Zuse built to be used rather than to test out his ideas. The Z2 and Z3 computers were electromechanical relay machines and the Z3, completed in 1941, had an electromechanical memory composed of relays as well as an electromechanical arithmetical unit. Of course to make it useful for computation, the Z3 required many relays and it indeed it contained about 2600. It was the first operational program-controlled calculating machine and was used by the German aircraft industry to solve systems of simultaneous equations and other mathematical systems which were produced by the problems of dealing with the vibration of airframes put under stress. However when Zuse proposed a computer based on electronic valves, the proposal was rejected on the grounds that the Germans were so close to winning the War that further research effort was not necessary. Some of Zuse's computers were destroyed in bombing raids near the end of the war although the Z3 was reconstructed in 1960 for display in a museum in Munich. Zuse began work on his Z4 computer in 1942, and it was almost complete when, due to continued air raids, it was moved from Berlin to Göttingen. After only a few weeks Göttingen was in danger of being captured by the advancing Russian troops and the Z4 was moved again, this time to the small village of Hinterstein in Bavaria. The Z4 was coded the Versuchsmodell 4, or V4, and hidden in the cellar of a house. Ashurst writes [4]:Because of the association of V4 with the V1 and V2 flying bombs and rockets, the British and American troops who eventually found it (it had not been found by the French who entered the village first) were most surprised when their multifarious precautions were unnecessary and the fearsome V4 was just a conglomeration of mechanical bits and pieces. Finally the Z4 computer was taken to Switzerland where it was installed in the ETH in Zurich in 1950. It remained operational there until 1955 when it was moved to a French aerodynamical research institute close to Basel where it remained in use until 1960. In fact Zuse designed several computers other than those of his Z series. His S1 and S2 computers were used for computing the precise measurements necessary for the production of aircraft. For the S2 the computer included measuring devices to make measurements of the planes in production and to feed these directly into the calculations. The L1 computer which Zuse designed was not for solving arithmetical problems, but rather it was designed to solve logical problems. Only an experimental version was produced, no further work being done on this innovative idea. These developments were entirely independent of those of John Eckert, Mauchly and Howard Aiken in the USA, and Turing in England. Zuse later wrote:After the War was finally over, news of the University of Pennsylvania ENIAC machine went

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zuse.html (2 of 4) [2/16/2002 11:40:14 PM]

Zuse

all round the world - "18,000 tubes!". We could only shake our heads. What on earth were all the tubes for? ... The English development known as COLOSSUS was unheard of outside the circle of those working on it. It was only much later that the wraps came off this very interesting project. Zuse set up his own computing company in 1950 and it was taken over by the Siemens electronics firm in 1967. In 1965 he had received the Harry M Goode Memorial Award, a medal and $2,000 awarded by the Computer Society:For his contributions to, and pioneering efforts in, automatic computing; for independently proposing the use of the binary system and floating-point arithmetic; and for designing the first program-controlled computer in Germany - one of the earliest in the world. By 1958 he had reached the Z22 computer, which was one of the first to be designed with transistors. Zuse continued to undertake research on computers and acted as a consultant to Siemens after the firm took over complete control of Zuse's computer company in 1969. As well as his hardware developments Zuse was also interested in software and he developed the first algorithmic programming language known as "Plankalkül" in 1945. He used the language to design a chess playing program. Although the language was not widely used it formed the basis for the next generation language ALGOL which, of course, became a widely used language world-wide. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (8 books/articles) Mathematicians born in the same country Other Web sites

1. Kalamazoo 2. Virginia Tech 3. Esslingen, Germany 4. Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day

Anniversaries for the year

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zuse.html (3 of 4) [2/16/2002 11:40:14 PM]

Zuse

JOC/EFR July 1999

School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zuse.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zuse.html (4 of 4) [2/16/2002 11:40:14 PM]

Zygmund

Antoni Zygmund Born: 25 Dec 1900 in Warsaw, Russian Empire (now Poland) Died: 30 May 1992 in Chicago, Illinois, USA

Click the picture above to see two larger pictures

Show birthplace location Previous (Chronologically) Next Biographies Index Previous

(Alphabetically)

Next

Main index

Zygmund worked in analysis, in particular in harmonic analysis. He created one of the strongest analysis schools of the 20 Century. Zygmund obtained his Ph.D. from the University of Warsaw in 1923 for a dissertation written under Aleksander Rajchman's supervision. From 1922 to 1929 he taught at the Polytechnic School of Warsaw. After a year in England he took up a post at the university of University of Stefan Batory in Wilno, Poland (Vilnius, Lithuania as it is now). He held this post until he was drafted into the Polish army at the start of the Second World War. In 1940 Zygmund escaped with his wife and son from German-controlled Poland to the USA. After a number of posts he was appointed to the University of Chicago in 1947 and remained there until he retired in 1980. Zygmund was to create a major analysis research centre at Chicago. In 1986 he received the National Medal for Science for building this research school. He supervised over 80 research students in his years at Chicago. Zygmund's book Trigonometric Series (1935) is a classic that, together with later editions, is still the definitive work on the subject. Other major works include Analytic functions in 1938 and Measure and integral in 1977. His work in harmonic analysis has application in the theory of waves and vibrations. He also did major work in Fourier analysis and its application to partial differential equations.

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zygmund.html (1 of 2) [2/16/2002 11:40:16 PM]

Zygmund

Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References (10 books/articles) A Poster of Antoni Zygmund

Mathematicians born in the same country

Honours awarded to Antoni Zygmund (Click a link below for the full list of mathematicians honoured in this way) AMS Colloquium Lecturer

1953

Other Web sites

Encyclopaedia Britannica

Previous

(Chronologically)

Next

Biographies Index

Previous

(Alphabetically)

Next

Main index

Societies, honours, etc.

Famous curves

History Topics Time lines Glossary index

Birthplace maps Chronology

Search Form

Quotations index

Poster index

Mathematicians of the day JOC/EFR February 1997

Anniversaries for the year School_of_Mathematics_and_Statistics University_of_St_Andrews,_Scotland

The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zygmund.html

http://turnbull.dcs.st-and.ac.uk/~history/Mathematicians/Zygmund.html (2 of 2) [2/16/2002 11:40:16 PM]